## Intensional Semantic Calculus

**I. Introduction**

Semantics is a discipline which, speaking loosely, *deals with certain relations between expressions of a language and the objects* (or ‘states of affairs’) *‘referred to’* by those expressions. As typical examples of semantic concepts we may mention the concepts of *designation*, *satisfaction*, and *definition*…and the problem of defining truth proves to be closely related to the more general problem of setting up the foundations of theoretical semantics.[1] ~Alfred Tarski

The work that is here presented is concerned with the foundations of all language, namely the necessary criteria for its comprehension, use, and disambiguation. In accordance with these goals, this work formulates a novel theory of philosophy of language, as well as an innovative re-structuring of the semantics for first order modal logic and the creation of a new logic, Intensional Semantic Axiomatic Set Theory. A second logic, an epistemological logic that integrates probability calculus with first order logic, can be derived from the first, the outline of which is included in the appendix, as the purpose of this logic extends beyond the investigations of language. Concerning the foundations of theoretical semantics, the theory here prescribed is known as Logical Semanticism, so called to emphasize the importance of linguistic quantification, equivalency, and class relations regarding semantic investigations. Primarily, this theory is intended to solve problems concerning the referential paradoxes of language, both internal problems of scope and substitution, as well as external problems of translation; to explicate the possibility of quantifying and analyzing all semantic systems in an equivalent manner. Similarly, new questions and methods by which to accomplish these tasks are offered, as old debates are re-oriented by challenging certain traditional approaches in the philosophy of logic and language.

A necessary argument of this work is that the foundations of theoretical semantics is taken most fundamentally on the level of quantification, as constituting a complex ordering of semantic or intensional classes that conjoin necessary and possible semantic information. This requires a form of equivalency relation many contemporaries, charmed by the rhetorical brilliance and philosophical shallowness of “Two Dogmas of Empiricism”, deny as possible. This paper argues, however, that Rudolf Carnap and C. I. Lewis well understood—as did Kant—the nature of the analytic-synthetic, a priori-a posteriori, and necessity-possibility dichotomies, whereas Quine equivocated on all three. Quine’s obsession with transforming the quantification of semantics into metaphysical issues, a mistake which Alfred Tarski long argued against, further hurt and crippled Quine’s (and anyone that mistakenly followed him) into making errors in the philosophy of logic. The result has been that much of the last fifty years has been a needless detour in the philosophy of language, and that modal and epistemic logic have been led astray in the works of Kripke and Hintikka respectively. The extent of their corruption is implicit, a result of fallacious postulations in the semantics of how to interpret the operators their logics propose. Conceptually such problems lead to Kripkean rigid designation of transworld individuals, epistemological paradoxes of Plantinga’s alethic operators, David Lewis’ modal realism, and other truly stupid nonsense. Logical Semanticism requires New Semantics for both modal and epistemic logic, abandoning the notion of alethic modality as metaphysical, a mistake that has been with philosophy since its introduction in the Medieval era. Indeed, for all of Quine’s discussion in *Word and Object* concerning the “savage theology” of those that prescribe to ontological realism, it is peculiar that the sage of Boston never allowed for the possibility that a metaphysical interpretation of alethic operators was just as outdated a notion as the heavens and gods he so venomously fought against—that quantified modal logic was indeed free of this metaphysical burden, as free as the pure mathematics he so devotedly idolized.

The metaphysics of meaning and modality are eliminated, the ontological pursuits of Quine, the metaphysical semantics of Kripke, abandoned. No free variables ought to be allowed, as all semantic information that is modeled conforms to some semantically regulated modal operator. In relation to definitions, the upshot of this procedure of quantification is the successful overcoming of venerable paradoxes which plague both the philosophy of language and first order logic. Despite conceptual variations of Logical Semanticism compared to past ideas, the inspiration for this theory is most indebted to three philosophers in particular, the three greatest logicians of the 20^{th} century: Clarence Irving Lewis, Ruth Barcan Marcus, and Bertrand Russell. C. I. Lewis reformulated the philosophy of logic, introduced the possibility for first order modal logic with a semantics superior to Quine, Kripke, and Hintikka. Ruth C. Barcan (Marcus) quantified S2 into a successful calculus and greatly advanced the proper theory of reference. Russell, seeing the true nature of logic and language, greatly succeeded in conjoining the two fields into one. Despite the errors in Russellian quantification and blunders in the philosophy of logic, Russell was nevertheless inspirational, a pioneer of two developing fields.

Since the Analytic Turn of the 1950’s, when analytic philosophy became less rigorous and disturbingly attached to hard sciences, social sciences, and psychological conjectures, the projects of Montague, Tarski, and others have not been given their due (except by those such as Quine and Davidson which misinterpreted for their own deflationist pursuits). Logical Semanticism follows an older tradition, one that is not often or easily moved by the trends of the latter 20^{th} century. The place of the normative is re-instated, the distinction between the methodological analysis of philosophy and the content-filled knowledge of the other disciplines respected. An important part of this paper is retracing the history of philosophy, particularly the relationship of 20^{th} century philosophy to Kantianism and the modern era. Upon historical analysis, many of the contemporary problems in the philosophy of language manifest as much older than they are generally taken, and associated with philosophical problems many overlook, including quite fundamental philosophical concepts, such as analyticity, identity, and necessity. These and related topics are explored in relative detail.

**(A) Logical Semanticism: Its Fundamental Question and Philosophical Task**

The foundations of theoretical semantics entail the study of the necessary conditions for constructing formalized and “natural” languages—or better, extensionally intensional syntactic languages and intensional semantic languages. Theoretical semantics includes the necessary rules and postulations that must be adhered, and further requires some quantificational unit by which meanings are analyzable. Logical Semanticism follows the tenets of set theory as the basis by which all semantic concepts (including definitions, designators, etc.) can be classified, namely various kinds of intensional classes. All terms from definite singular terms to complex absolute general terms are orderly submitted to the set-theoretic systematization, a system of which is formulated with variables but the rules for applying the system incorporate the constant values inherent to intensional semantics. Only a system of variables, one lacking in constants (save operators), can be defined as a pure logic or mathematics. All other systems, ones with constants, are always applied logics or mathematical systems. The difference between science and mathematics is simply that science is the manipulation of constant values that form systems associated with empirical observation, for science is applied mathematics. And just as science is to mathematics, language is to logic, for language is the applied logic of the pure manipulation of logical symbols. The analogy holds on all levels.

Ultimately, all philosophical investigation is fundamentally grounded in its definitions. In recognizing this essential characteristic of philosophy, this work emphasizes the role of its terminological commitments, as well as the presuppositions thereby inherently entailed. Logical Semanticism is a theory by which can be summarized in a single sentence as a theory by which: *inference methodologically depends on reference, reference upon meaning, meaning upon intensional classes, and intensional classes upon formal set-theoretic systematization*. Formal set-theoretic systematization—Intensional Semantic Axiomatic Set Theory—is the ultimate foundations of theoretical semantics, that is, for the possibility of any language. The remainder of this paper is providing explication and justification of these specific philosophical claims and definitions, including the problems derived from adopting the postulates of Logical Semanticism.

Methodologically, the analysis of such fundamental matters as establishing a proper foundation of theoretical semantics generates numerous problems and potential investigations. However, the two most general categories by which the philosophical investigation into language is analyzable are: (1) the nature and structure of language in terms of theories of reference, meaning, and the relationship of syntax, semantics, and pragmatics; and (2) an outline of a theory of knowledge that conjoins the nature and structure of language with the philosophical theories of concept acquisition (i.e. concept and object learning through language education), epistemic judgment (e.g. object recognition and naming), justification (e.g. questions of causation and evidence of states of affairs), decision-making (e.g. theory of rationality), discursiveness (e.g. pragmatic/social themes), and valuation. In relation to this investigation, only the former category, analyzing and explicating the nature and structure of language proper, is taken as thematic; always preparatory for the possibility of investigating the linguistic questions proposed by the latter themes. The methodological priority of questions proposed by philosophy of language is not to discount or mitigate the importance of the latter category, but to suggest simply that formulating theories of the latter kind always presupposes—implicitly or explicitly—taking positions in terms of investigating the former category. To re-iterate, this methodological priority is true only in terms of investigating language.

However, this division is often never drawn, for the utility or correctness of the methodological order is controversial. The debate revolves upon a disagreement over the degree of inherent interrelation regarding experience, intelligence, intentionality, and semantics. For example, numerous philosophical approaches to the study of theories of meaning, reference, and the relationship of semantics and pragmatics emphasize some component of social or psychological affairs that are claimed to determine some aspect of language. For instance, it is often argued that if pragmatics always or generally determines semantics, then the analysis of the criteria constituting social-contextual use, habituation, anthropological status, etc., methodologically take primacy. Questions of semantic content are relegated to a methodologically dependency upon pragmatic criteria. Secondly, this tendency can particularly be observed in those philosophers incorporating themes present in either philosophy of mind or theoretical linguistics. Both fields offer themselves as potential aides to solving venerable paradoxes in the philosophy of language, emphasizing the descriptive, generally at the expense of the normative in terms of their purported solutions.

However, whether this descriptive concentration is detrimental or beneficial depends upon the purpose of the investigation, namely the problems a theory is intended to solve. Furthermore, the controversy surrounding the methodological which investigations are fundamental is contingent upon the questions proposed. For the purposes of this investigation, the primary task is re-orienting the fundamental question of philosophy of language, postulating presuppositions, problems, and solutions methodologically disparate from traditional ways of formulating such debates. In this context, a normative, as well as logical focus constitute the legitimate procedure, accounting for valid referential and semantic models for expressing the significance and intention of information. The fundamental question of the philosophy of language is: What is the necessary criterion for a quantificational procedure of semantic modeling capable of disambiguating referential paradoxes in linguistic expressions, and thereby establishing the possibility for referential transparency through successful commensurability, regarding both interchangeability within a particular semantic system and translation between two semantic systems? Referential disambiguation is possible through an explication of the criteria of semantic quantification, the problems of which form the basis of the paradoxes of philosophy of language. Thus, there is a more succinct means of formulating the fundamental question of the philosophy of language.

*What is the necessary criterion for the quantification of semantic information establishing the possibility for successful commensurability and referential disambiguation concerning any linguistic expression? *This is the fundamental question of the foundations of theoretical semantics, whereby the primary task of Logical Semanticism is providing a valid solution, namely through the nature and structure of intensional classes and their relations.

The nature of the question, being as fundamental as it is, eliminates the objections between the relationship of semantics and pragmatics, for the nature of this question is not concerned with any particular person, culture, history, kind of sapience, but with the possibility of any language through what is theoretically required of the quantificational rules, units, relations, and equalities of intensional semantic systems of any kind. The debates concerning much of 20^{th} century philosophy of language generally operate on a different level of interest and universality. Quine, for instance, makes the baffling remark in *Word and Object *that the majority of the book only applies to the English speaking world, as if the logical quantification of language could possibly be impeded, the linguistic necessity of definite singular and absolute general terms avoided in a robust intensional semantic language. Such an ignorant comment is but another testament to the infancy of philosophy of language. Descriptive examples of languages to the contrary are but instances of primitively incomplete systems. Creating and speaking a language are no guarantee of avoiding referential or semantic paradox. That a community speaks a particular language is no reason to philosophically take it seriously. Foundations of theoretical semantics is taking as thematic the necessary conditions for the possibility of successful commensurability of all its linguistic expressions, the procedure by which to separate sense of names from sense of objects, of etymology from direct reference. There is no reason to presuppose that all “natural” Earthbound languages either can or cannot do this. Some languages could potentially be so internally flawed that they ought to be abandoned.

Therefore, by implication, the themes discoverable in the philosophy of mind and theoretical linguistics are of no utility to the investigation of the foundations of theoretical semantics, no more than they could be to the foundations of mathematics. The major function of Logical Semanticism is the normative establishment of a pure logical system capable of quantifying and modeling all applied logical-languages. Just as mathematics could learn nothing from science, neither can Logical Semanticism be assisted by anthropology, psychology, neuroscience, etc. Philosophers of language have often turned to these fields for guidance. Unfortunately, this has been one of the greatest mistakes of the 20^{th} century, a methodological Wittgensteinian-inspired blunder derived from failing to appreciate the true nature of logic, mathematics, language, and their inherent relationships.

This is perhaps the first comprehensive attempt to treat the subject of language in this manner, the logic of semantics as a pure normative science.

**(B) Investigations into Logical Semanticism: Expositing the Theory **

The three major areas that Logical Semanticism explores, in conformity with investigation the nature and structure of language, are questions of reference, meaning, and the equivalency relation. Firstly, in relation to Logical Semanticism, reference is taken as the minimalist criteria of indicating some syntactic operator, concept, or complex relation of concepts. By implication, the criterion of reference as necessarily denoting empirical objects or states of affairs is taken as too stringent. Furthermore, direct reference, but not any causal-historical explanation, is taken as an important component of a comprehensive theory of reference. Secondly, the theory of meaning is a mixture of the referential theory of meaning, proposition theory, and truth-conditional intensional semantics. The truth-conditional intensional semantics abandons “possible worlds” in favor of logical-semantic systems. This maintains consistency with the theory of reference and the changes in the philosophy of logic, particularly the equivalency relation. Thirdly, material equivalence of extensional logic is exchanged for semantic analyticity of intensional logic, and the role of Lewis’ and Marcus’ strict implication is adopted. Of further requisite is an extensive explication of the role of necessity, principle of identity, and interchangeability. The development of these philosophical concepts directly diverges from Quinian accounts. Salmon, Davidson, and many others allow only for the analyticity of syntactic tautology, the manipulation of symbols. The basis of Logical Semanticism presupposes synonymous analyticity as not only valid but essential.

In relation to these three investigations, there are four major points that Logical Semanticism must successfully make in order for the theory to be adopted. The remaining portions of this paper are investigations into these four major points in terms of the antecedently noted themes involving reference, meaning, and the equivalency relation.

Firstly, like any contemporary philosophical theory of language, Logical Semanticism offers various solutions to the four fundamental problems derivative of early 20^{th} century philosophy: reference to non-existent objects, assertion of negative existentials, co-reference of two proper names, and substitutivity of identicals through interchangeability of co-referring definite singular terms with their associated descriptive properties. These quandaries are taken as entirely philosophical problems. Their solution derives from the philosophy of logic, whereby this paper shifts the nature of the debate surrounding these four problems to a set of different, and more enlightening questions.

Secondly, the solutions to these problems necessitate drastic modification of positions held my some contemporary groups of philosophers, namely those that dismiss a robust analyticity, deny quantified modal logic, claim to reject meaning as metaphysical essentialism, narrowly define reference as pertaining to empirical objects or states of affairs, maintain unfounded commitments to “the radical other” and unsolvable “problems of translation”, believe that referential opacity is both far-reaching and unsolvable, and take the essence of a theory of meaning to merely concern the inflationary and deflationary debate, or some epistemological game of choosing correspondence, coherence, or pragmatism. In effect, the philosophy of Quine needs to be comprehensively replaced with something tenable.

Thirdly, Logical Semanticism takes a number of positions in the philosophy of logic, including the prohibition of the Kripkean semantics of modal logic, a comprehensive refutation of the use of “all possible worlds” quantifiers. Logical Semanticism generates New Semantics for modal logic, by which “all possible worlds” is replaced with *logical-semantic systems*. A logical-semantic system presupposes a form of intensional equivalency relation—namely strict implication—requires a non-metaphysical reading of the concept of necessity, adopts Marcus’ direct theory of reference for definite singular terms, and further requires a semantic form of analyticity conforming to the intensional equivalency relation. All of these necessary conditions, when taken together, are sufficient to replacing the entire philosophical understanding of modal logic, of altogether abandoning the metaphysics of necessity and possibility, and avoiding the traditional 20^{th} century skeptical problems associated with these elements. This rendering of the modal alethic operators reformulates their semantic significance in a manner not challenged since Kant attempted in relation to the medieval era. In conformity with paradoxes of epistemology and language, neither such understanding proves sufficiently accurate.

Fourthly and finally, the positive implications of the theory establish a series of formation rules, semantic rules, and formal axioms to govern the basic quantificational unit of all language: intensional classes. These intensional classes presuppose all the positions in the philosophy of logic of Logical Semanticism and upon which generate Intensional Semantic Axiomatic Set Theory, the set-theoretic system that Logical Semanticism relies upon for the intensional classes that make the philosophical theory of language possible. The entirety of this axiomatic system is given in this paper. Furthermore, this axiomatic system has been extended elsewhere to cover epistemological theory, namely the extension of the system to logically recognize probability calculus. But these investigations are not developed in this paper, though further remarks and indications of where Logical Semanticism can be implemented elsewhere are topics developed in the conclusion.

The remainder of this paper is explicating and proving these points, with some extended comments related topics. Particular emphasis is placed on the presuppositions of the theory, the historical context of traditional philosophical problems, and the reasons behind the topical shifts introduced by Logical Semanticism. Further emphasis is placed on philosophy of logic topics and their under-valued relationship to philosophy of language.

**II. Fundamental Concepts and Categories in the History of Philosophy**

The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. ~Bertrand Russell

This section attempts to answer the fundamental question: What is the nature and structure of semantic definition? This question presupposes the philosophical concepts of necessity, the a priori, identity, and analyticity. Answering this question requires a historical and conceptual philosophical analysis of these concepts, the application of which promises to contribute to the solutions to the problems of the foundations of theoretical semantics.

The greatest European philosopher of the 20^{th} century is indisputably Martin Heidegger. One of the most remarkable qualities of this complex man was his peculiar aptitude for throwing into question philosophical concepts—being, identity, necessity, sufficient reason, the philosophical status of the proposition—of which most philosophers do not bother to analyze to the depths and multifarious dimensions of Heideggerian inquiries, presupposing their security through their venerability. That the established intellectual foundations of the history of philosophy could be re-opened to the extent and manner of *Identity and Difference*, *The Principle of Sufficient Reason*, or *The Fundamental Concepts of Metaphysics* ought to give pause to those philosophical investigations of which do not seriously reflect upon their fundamental methodologically necessary postulations. In this regard, Heidegger surprisingly serves as a philosophical model for the proper method of investigating the complexities of fundamental matters—despite the outpour of stylistic critiques. Without Heidegger’s influence this investigation would never have been possible.

Since the time of Locke, Leibniz, and Hume, the concepts of necessity, identity, analyticity, and the a priori have taken philosophical root. Identity and necessity, incepting in the origins of philosophy, however, their modern conceptions are dependent on more recent times. Under Kantian critical idealism many of these notions reached philosophical maturity. Nonetheless, in the period since the late modern era, despite the established longevity of such terms, the definitions have been both modestly and drastically ameliorated by philosophers in the 20^{th} century. As verificationism rushed to obliterate a priorism, as C. I. Lewis attempted to save a priorism through pragmatizing the concept, as Quine attempted to end four hundred years of analyticity, and as Paul Boghossian, Hartry Field, and other misguided thinkers’ attempts to consolidate these Quinian blunders, the 20^{th} century has seen a surprising amount of conceptual re-evaluation. Unfortunately, the majority of these shifts have met with false conclusions, many of which are currently in vogue. This section is dedicated to re-tracing the historical nature of these fundamental concepts and categories, the problems and debates surrounding them, and offering definitions of which Logical Semanticism adopts.

**(A) The Concepts of Necessity and the A Priori: Beyond Metaphysics**

Necessity is a concept as old as philosophy itself, possessing a privileged role in the dialectic investigations of the one and the many. For the one is always the aggregate of things necessary to be the one, of the necessary conditions Y_{1}…Y_{n} for the oneness of X. If only the Greeks did not take this definition metaphysically, then millennia of philosophical category mistakes could have been avoided. This section develops the dichotomy between metaphysical and semantic necessity, arguing that the framework of philosophical debate has been flawed since the origins of Western philosophy and continues to the present. Simply rejecting metaphysical necessity is not philosophically sufficient, which has been the response of many philosophers of the 20^{th} century, such as in regards to Kripkean semantics of quantified modal logic. Philosophers must go further in a way that none have thus far investigated.

Even in ancient times, the general concept of necessity was understood, however limited it was to either a metaphysical or syllogistically inferential character. This was not only true of antiquity, for the Christianized “philosophy” of the Medieval era originally secluded necessity to the spheres of metaphysics and deductive inferences. Indeed, this tendency was prominent well into modern philosophy, detectable in the writings of Descartes, particularly *Rules for the Direction of the Mind* and beyond to the religious rationalism of Leibniz. This interpretation of the metaphysical nature of necessity ultimately led to Humean skepticism of both Cartesian-Leibnizian rationalism and empiricism, insofar as necessary connections are taken as states of affairs metaphysically necessary, a problem that threatened to consume inductive inference altogether. For how could it be said that inherent to the operations of the world persists an unchanging necessity native to the things-themselves and their external relations?

The Kantian response, for this very reason, was the terrifyingly wrong conjoining of a priorism with epistemological necessity. This is not to say that the Kantian formulation of the a priori did not have its advantages. Indeed, compared to the rationalism preceding critical idealism, Kant successfully shifted philosophy from the conjoining of metaphysical necessity with empirical analysis of causation, while simultaneously putting Hume on the defensive. However, the price of Kant’s anti-skeptical critiques against Humean aversions to necessary causal connections is the historical association of a priorism and necessity that persists to this day—of either an epistemological or metaphysical manifestation. In the *Critique of Pure Reason*, Kant establishes the three analogies of experience which philosophically serve to account a priori for the relations of states of affairs by *imposing* laws onto nature and experience, particularly causal connection. More broadly, all the architectonic categories of the Table of Logical Judgments, as well as the alleged a priori forms of sensible intuitions impose conditional rules on reality. In this manner, the a priori takes on the philosophical status of limitation, conceptual rules that directly not only order but limit what can be experienced as real antecedently. C. I. Lewis protests against this Kantian conception of the theory of the a priori. The categories, as Lewis rightfully asserts, “do not compel the mind’s acceptance” for “the necessity of the a priori is its character as a legislative act. It represents a constraint imposed by the mind, not a constraint imposed upon mind by something else.”[2] This is because the a priori is independent of experience, and not “because it prescribes a form which experience must fit or anticipates some pre-established harmony of the given with the categories of the mind, but precisely because it prescribes nothing to the content of experience.”[3] The ultimate refutation that Lewis is after is the a priori modes of the forms of sensible intuition, of re-conceiving of the a priori as not a form of epistemological necessity which limits objects to spatial-temporal understanding but imposes no a priori limits upon experience the real.

That Kant is committed to this form of necessity is all the more obvious from the general postulates of empirical thinking, the modal definitions Kant assigns of the categories of necessity, actuality, and possibility. These definitions are comprehensively epistemological, having to deal with the formal (possible), material (actual), and universal (necessary) conditions Kant postulates of experience. The stipulated conditions concern the role of concepts and intuitions that allegedly govern reality. Kant formulates an empirical opposed to transcendental use of alethic modality. Thus, Kant writes that “the postulate of the possibility of things requires also, that the conception of things agree with the formal conditions of experience in general.”[4] But Kant has not strayed as far from the tradition as one might think. For Kant proposes that one is entitled to ask concerning the empirical status of an “object”, namely whether or not it is real. Since epistemological (contingent) necessity ultimately concerns criterion for actuality and the conceptualization of reality, the metaphysical nature of even the Kantian semantics of alethic operators, including necessity, is unavoidable. Kant is still engaged in the framework of a metaphysical debate, arguing transcendental versus empirical conditions. No matter how formal these conditions, the results, as Lewis notes, are still the same.

Therefore, it is not surprising that logical positivism, taking the origins of the synthetic a priori metaphysically, began the 20^{th} century assault on both philosophical concepts, from the attacks on quantified modal logic to the epistemological siege on all forms of rationalism. The grounds for such critiques were derived from Kant. But logical positivists go too far, for neither necessity nor a priorism can be abandoned, though many conceptions of both ought to be disregarded. Metaphysical conceptions of a priorism and necessity are only the same under certain forms of rationalism, which if abandoned lead easily to non-metaphysical conceptions of both philosophical terms. Thus, the key is to alter the notion of necessity in an epistemological context radically, taking the first steps that Kant made and going considerably further.

But what is necessity? In the most general sense, necessity is a set of specifiable conditions of the nature and structure of a concept, object, or state of affairs, for its possibility. Though the basic definition is agreeable, there is often confusion concerning the multitude of extraordinarily disparate connotations of necessity, the equivocation of which generates grave category mistakes. This danger is not simply historical but is ever present. The following exhaustive distinctions concerning the definition of necessity are possible: (1) Logical-Mathematic; (2) Logical-Semantic; (3) Metaphysical; (4) Deontic; (5) Contingent; and (6) Modal. Temporal and causal necessity would be considered sub-categories of contingent necessity and modal necessity would ultimately collapse traditionally into metaphysical necessity, though other times it is taken as a form of epistemological (contingent) necessity.

Most problematic is the tendency to mix necessity with other fundamental philosophical concepts and categories, ones that ought to remain distinct. Quine, in particular among contemporaries, conflates modal and metaphysical necessity and further indistinguishably asserts that analytic statements entail metaphysical commitments vis-à-vis epistemologically learning the purported “essences” of empirical concepts. Since these essences are in fact contingent upon both the language and the context of their learning, the necessity of such conditions is alleged to be undermined, because Quine argues they have lost their a priori stature. Thus is the mass category confusion of “Two Dogmas of Empiricism”, of which is most prevalent in the first three sections of this very flawed article.

Fortunately, this confusion is avoidable. Indeed, upon reflection there are certain advantages concerning older formulations of analyticity, though as C. I. Lewis has rightfully pointed out, the rationalism of Leibniz, Descartes, and others erred in so approximately embedding a priorism and necessity. Historically, the two were so affiliated that many up to this day have trouble separating the two concepts. But a priori statements say nothing about the actual world, as they are not metaphysical at all. The kind of necessity that binds mathematics and definitions in general is of a logical sort, pertaining to the internal deductive rules of mathematics and logic or the postulations and classificatory rules of a language. That a bird is a parrot is necessary is to neither say that the existence of either birds or parrots is requisite, but simply that there is a relationship of class inclusion, one that in no way affects what exists in the world.

In this context, analyticity, necessity, identity, and the a priori are mutually interrelated in crucial methodological ways. In order to prove this, the six possible definitions of necessity ought to be defined and analyzed.

Deontic necessity is straightforward, having to do with obligatory action and the quantification of these obligations. Metaphysical necessity is what is generically thought of as necessity, that some object or state of affairs is somehow obligated to exist, either in the empirical world or in the logical space of reasons. Contingent necessity is the expansive category of epistemological necessity, concerning causal-epistemological and temporal necessity among its sub-categories. The notion of contingent necessity is that within the parameters of either actual minds or the actual rules of this particular physical reality, there is some set of conditions X that are necessary. Thus, causal-epistemological necessity, first developed by Quine’s student Fǿllesdal, is the metaphysical rendering of scientific realism in an epistemological framework. These definitions of necessity are not the thematic investigation of this paper but are of critical importance, even if for their present time the extent of that importance is preventing confusion with these definitions.

The relationship of causal-epistemological necessity to semantic necessity is both fundamental and universal. The preliminary basis of the extent of this relationship is developed in the appendix, the possibility for an epistemological logic that conjoins probability calculus, causality of states of affairs, and quantified modal logic.

In the past there was only logical necessity, which modally was construed as signifying “true in all possible worlds”, which represented the normative universality of logic and mathematic reasoning in relation to the descriptive possibilities of the empirical order. But there is abundant reason to divide this definition further, into the Logical-Mathematic and Logical-Semantic, the pure and the applied. Logical-Mathematic necessity is the necessity that follows from syntactic inference, the rules of syntactic operations and proof theory. The law of transitivity, the functions of arithmetic, and syllogistic reasoning are all extensional examples of Logical-Mathematic necessity. Logical-Semantic necessity is semantic analyticity, or what Quine refers to as cognitive synonymy. This form of necessity is the epitome of the Greek definition of the one, except the definition is entirely non-metaphysical. Whatever constitutes X requires conditions Y_{1}…Y_{n}. These conditions are taken in logical space, their absence or presence in the empirical world never altering the semantic relations of such necessary conceptual connections. If the definition of the term ‘chair’ is such that a chair is “any piece of furniture with one or more legs, one and only one back, one and only one sit, and is intended for the purpose of sitting”, then the predications of this definition are the intensionally necessary properties of the concept chair. This is what is meant by semantic necessity, whereby semantic possibility would simply be the set of augmented properties consistent with the semantically necessary properties, such as “leather” is a possible predication of chair, since leather is not inherent to the definition of chair, relative to the definition. The difference between Logical-Mathematic (syntactic) and Logical-Semantic necessity is that all semantically necessary concepts and relations are not quantifiable, analyzable, or even understandable in a purely syntactic context. The intricacies of this separation, the challenges raised against the possibility of logical-semantic necessity, these require the concept of analyticity.

** (B) The Concept of Analyticity: A Logical-Semantic Category**

Now the credibility of *some* sentence types appears to be *intrinsic*—at least in the limited sense that it is *not* derived from other sentences, type, or token. This is, or seems to be, the case with certain sentences used to make analytic statements. The credibility of *some* sentence types accrues to them by virtue of their logical relations to other sentence types, thus by virtue of the fact that they are logical consequences of more basic sentences.[5] ~Wilfrid Sellars

The analytic-synthetic distinction originates in the early modern era, in the works of such figures as Locke, Leibniz, and Hume. With Locke the distinction is generally left implicit, but Leibniz generates the crude distinction between truths of reason (analyticity) and truths of fact (the synthetic). What is peculiar, as will be discussed later, is that Quine’s definition follows Leibniz instead of Kant, for Quine defines the synthetic as empirical facts, which is not even close to being the proper definition. The analytic-synthetic distinction has nothing to do with either epistemological or metaphysical categories, as the notion of empirical is foreign to the analytic-synthetic distinction. For this reason, Hume’s definition is similarly suspect; with the distinction between Relations of Ideas (analyticity) and Matters-of-Fact (the synthetic). Hume’s definition of analyticity is superior to Leibniz, but Hume makes the same error of Quine in defining the synthetic on the basis of empirical fact. The problem is that this conflates the distinction between necessity-contingency (metaphysical category), a priori-a posteriori (epistemological category), and the analytic-synthetic (logical category), as shall be seen.

Kant defines the analytic-synthetic distinction in paragraph 2 section (a)-(c) of the *Prolegomena to Any Future Metaphysics* and section IV (A7-10/B11-14) of the *Critique of Pure Reason*. There are numerous reasons why Kant’s definition is superior and why Quine’s quick dismissal undermines “Two Dogmas of Empiricism”. A caveat should be offered, however, as a careful reading of Kant garners not one but three definitions of analyticity in Kant’s work. Fortunately, these definitions can all be made equivalent, however, it is important to realize *how*, for in many alternative philosophical conceptions they are irreducible. For instance, if one takes a Salmonian reductivist account of the a priori as dealing with logical form (i.e. syntax); instead of the intensionality of the concept as Kant, there are quite fundamental differences.[6] Therefore, it is particularly important to understand Kant’s definitions. Analyticity has been defined as true by virtue of its meaning (e.g. C. I. Lewis), self-contradiction (e.g. Frege), substitutivity of equivalent terms (e.g. Russell and Frege), implicit definition (e.g. Boghossian), and logical form (e.g. Quine, Salmon). There is good reason to maintain all these definitions of analyticity are not reducible to the same definition but that issue will be explored later.[7]

Kant’s most recognized definition defines analyticity as explicative and the synthetic as ampliative. Here the analytic is defined as a declarative sentence of subject, copula, and proposition, whereby any declarative sentence that possesses a predicate contained in its subject is analytic. The thinking Kant has in mind is that analysis is the breaking up of constituent concepts *implicitly* and *intensionally* contained in the very *definition* of the subject. Also of importance to note is that analyticity is epistemologically trivial, for it asserts no novel information not already known in formulating the relations of some object to its own constituent property. A synthetic statement is simply any declarative sentence where this condition is not present. This is an extraordinarily fundamental point. Persons of no less stature as Quine, as will soon be argued, misunderstand the analytic-synthetic distinction—the greatest reason being lack of reflection on the definitions here expressed (namely, Quine misinterprets synthetic as *empirical* fact, which conflates epistemological and logical questions, discussed later).

The second definition is associated (by Kant) as being identical with the definition of analyticity as the containment of the predicate in the subject. The notion of self-contradiction, necessary of all analytic statements, is taken as fundamental and complimentary, whereby all synthetic statements are judgments of experience. Kant writes, “judgments of experience, as such, are one and all synthetic”, insofar as analyticity requires “no need to appeal to the testimony of experience in its support” because all the concepts are contained in the subject already *a priori*.[8] The crucial word is *judgment*, opposed to the notion of *concept acquisition*. As explication of this distinction, consider that the analytic a priori character is not diminished even if the concepts are attained empirically, for there is a distinction to be strictly adhered between the learning of concepts and attaining the knowledge of concept formation empirically versus the act of judging the internal logical-semantic nature of a concept into its constituent parts, which can be done logically without need of causal considerations. Thus, “‘Gold is a yellow metal’ requires the subject to learn this definition, to learn what the term gold signifies, but “no experience beyond the concept of gold” is needed in order to judge this is in the future insofar as yellow and metal are now contained in the thought of gold.[9]

The third definition is occasionally employed as anything that is known through the practice of reason. Reason and practice as explanatory terms are ambiguous in relation to a definition of analyticity. This will not be pursued further but for the reasons of hermeneutic charity will give Kant the benefit of the doubt. But keep in mind that the way in which these definitions can be assimilated is *if and only if* they are taken *intensionally*.

In any case all three of these definitions, whether they are slightly different, or indeed ultimately equivalent, encounter a seemingly unanticipated set of difficulties in light of the Kantian explication of the analytic-synthetic distinction. For despite Kant’s correct formulation of the analytic-synthetic distinction incorporating semantics, there are problems associated with Kant’s version. Quine attempts to show the arbitrariness of the distinction in light of alleged difficulties with accounting for cognitive synonymy of analytic definitions based on intensional logic. But the analytic-synthetic distinction is not *arbitrary*, for his arguments fail in this regard. However, Quine was correct to the extent that Kant’s formulation is similarly faulty, for the analytic-synthetic distinction turns out to be surprisingly *relative*. In other words, Quine had some interesting insights about the nature of analyticity but ultimately proposed an untenable set of radical inferences.[10] Therefore, the goal of this section is to defend the distinction as non-arbitrary by addressing Quine’s arguments, which in turn provides the ground later for the explication of the relativity of the distinction.

An arbitrary distinction is to do away completely with the meaning of analyticity. However, this is a misleading understanding of Quine’s “Two Dogmas of Empiricism” to begin with, insofar as Quine does not reject the analyticity of logical forms because these propositions follow an *extensional *logic of a syntactic logistic form (e.g. “An unmarried man is unmarried” whereby this is true in virtue of the tilde sign representative of “un”; no knowledge of the meanings are necessary). In other words, Quine is attacking analyticity as defined as “truth in virtue of its meaning”, “intensional synonymy”, and any equivalency relation pertaining to a *logical-semantic* account. Since Kant’s definition of analyticity involves the definition of the concept of the subject, this falls directly into Quine’s critique. It is in this specified sense that the Kantian understanding of analyticity would become arbitrary, if Quine is correct.

In Appendix F of *Mind and the World Order*, C. I. Lewis explicates in more detail a Kantian-inspired analytic-synthetic distinction. Lewis remarks that “all a priori truths are definitive” and that they “explicate criteria of classification” whereby all criteria is “a priori and independent of experience because they concern the classification or interpretation of the given content.”[11] Lewis further states the position that a priori truths are true in intension and synthetic a posteriori truths are true in extension. Lewis sees these as interchangeable terms. What is important, for the moment, is that Quine follows Lewis on this point.

Lewis gives a number of examples of what he takes to be the difference. An example of an intensional universal proposition (i.e. AAA) would be “Parrots are birds.”[12] An example of an extensional universal proposition would be “Parrots have a raucous cry.”[13] The idea is that parrots are birds is true by definition because it is a statement of classification, whereby parrots do not have to exist for it to be true, for even if there were no birds the statement holds true. Furthermore, classificatory definitions do not affect real particular objects, for whether existent things fall into analytic definitional categories is an empirical generalization, which is a different philosophical procedure. With “Parrots have a raucous cry” this is an empirical generalization and hence synthetic, for Lewis. Lewis sees such synthetic statements as probable only because their properties hold contingency. This supposes that the difference between necessary and contingent predicates pertains to their association to the singular term, for in this given language, the raucous cry of a parrot is taken as probable only. For instance, all swans are white is inductive, for the definition of color is not strictly implied by the accepted definition of a swan, and so the property of color is a contingent truth that must be observed.

The problem Quine has is precisely with the Lewisian account here outlined. As Quine states: “interchangeability *salva veritate*, if construed in relation to an extensional language, is not a sufficient condition for cognitive synonymy” and that intensional languages use the word necessary, but that “such a language is intelligible only in so far as the notion of analyticity is already understood in advance.”[14] This last quote refers to Quine’s “circle of terms argument”, whereby analyticity cannot be employed unless it is understood on its own without naming it synonymy, equivalency, etc. The “Two Dogmas of Empiricism”* *is really an attack on *metaphysical necessity*, construed as manifest in intensional languages involving the “dreaded” notion of meaning. This can be proved further insofar as the third to last paragraph of section II (Definitions) of “Two Dogmas of Empiricism” presupposes a theory of nominalist formalism, proven in Quine’s application of logical economies in regards to the fourth category of definitions, involving mathematical and logical rules of transference. The economy of grammar and vocabulary betray Quine’s assumptions in regards to an extensional logic and formalistic nominalism, from which point Quine’s arguments are not so inevitable but require a controversial set of metaphysical and epistemological principles to antecedently grant Quine.

That which destroys Quine’s arguments against analyticity are two basic problems. There is the “circle of terms argument” that Quine generates against the definition of analyticity and the problem of Quine’s definition of synthetic. The first of these problems leads Quine to demand something that cannot be given and then make unfounded conclusions. The second problem is that Quine’s definition of the synthetic conflates the distinction between epistemology and logic inappropriately.

The circle of terms argument is Quine’s dialectic move that claims that the notion of analyticity is unintelligible, insofar as its reliance on other technical philosophical terms leaves the notion of analyticity “dubious” and lacking in “explanatory value”, as Quine claims in the introductory paragraphs. The circle of terms is apparent early on when Quine states: “’synonymy’ [is] in no less need of clarification than analyticity itself.”[15] The solution to Quine, in this regard, is to be credited to Strawson and Grice in their joint article “In Defense of a Dogma”, whereby they make the correct observation that Quine introduces an intelligible form of synonymy in “Two Dogmas of Empiricism”, which Quine himself admits. As Quine writes: “Here we have a really transparent case of synonymy created by definition; would that all species of synonymy were as intelligible.”[16] But as Bonjour inquires, why is it the case that once Quine has established the “transparent meaning” of synonymy, a definition that is even asserted independently of analyticity, that this understanding cannot serve as the basis for synonymy in general now that the definition is established? Quine’s argument that the term analyticity and synonymy are unintelligible is undermined by Quine’s own remarks in section 2 of his article. It is not that these terms are unintelligible after all but that they are mutually dependent. But this seems no great philosophical danger. If this is the case, much of the dialectic force of the article is deflated.

The more substantive critique is that Quine equivocates on the a priori-a posteriori and the analytic-synthetic distinction. Quine’s conflation originates early in the article where in a single sentence refers to what he presupposes as the definition of the synthetic as “grounded in fact” whereby its contrary is those judgments “grounded in meaning”.[17] Indeed, this is not what Kant is after at all, nor is it what the synthetic is in general, whether that be Kant or not. Indeed, this mistake is even more apparent in the first paragraph of section II, where Quine argues that empirically acquired definitions (i.e. from the lexicographer) cannot be the basis of analyticity. But this is simply wrong. It is precisely that factually attained meanings can be judged in relation to other meanings a priori, even if they are empirical concepts.

Quine is very confused. What is even worse is that there is further reason to suppose that Quine also conflates the necessity-contingency distinction with the other two distinctions, for this is apparent in section III, entitled “Interchangeability”. Quine adopts an extensional language and the opposition is an intensional language, which Quine erroneously prescribes as pertaining to metaphysical necessity. The result is mass category confusion at the very basis of the structure of the arguments of “Two Dogmas”. The implication is that Quine is unable to prove what he is after, for analyticity and synonymy are indeed intelligible concepts and do not require the standards Quine requires of them. Furthermore, Quine’s article is directed at the refutation of necessity and the a priori, which he confuses with the notion of analyticity. The result is that his radical empiricism crumbles, in this regard, when it is admitted that empirical concepts can logically function a priori and admit of their empirical originations. If Quine proved anything with his article, it is that extensional language, by itself, fails to account for meaning, synonymy, or intensional semantics in general. What Quine proved to the world is that an extensional language is not sufficient. This is because an extensional account of semantics leads to unsolvable referentially opaque antimonies. As textual support, it is worth noting that Quine mentions in passing that an intensional language does *not *fall victim to his critiques of synonymy (and hence analyticity) but only an extensional language (which is the sort of language Quine presupposes in the article because it is what *he *wants):

“For most purposes extensional agreement is the nearest approximation to synonymy we need

care about…extensional agreement falls short of cognitive synonymy…if a language contains an

intensional adverb ‘necessarily’ in the sense lately noted, or particles to the same effect, then

interchangeability salve veritate in such a language does afford a sufficient condition of cognitive

synonymy; but such a language is intelligible only in so far as the notion of analyticity is already

understood in advance.”[18]

In light of this quote the case has been made. Quine has undermined himself, since analyticity is intelligible and since an intensional language does accurately account for synonymy (interchangeability). This refutes Quine’s article, sections 1-4 but not 5-6.

Quine was motivated against the analytic-synthetic distinction for the wrong reasons and with too radical a conclusion. The result is the more modest and philosophically less problematic *relativization* of the analytic-synthetic logical distinction, namely the relativization of terms to their postulated semantic content. This maintains the Kantian understanding of the analytic while refuting Quine’s attack, but also recognizing the slight element of truth in Quine’s philosophy, the non-arbitrary but relative character of conceptual identity.

The difference between first order and second order considerations is relative to the parameters of the investigation established by the logician. Take, for instance, the distinction between an object language and meta-language. As Tarski elaborates in detail, the distinction is dependent upon which linguistic objects and relations are under analysis, insofar as “a meta-language becomes automatically our object language” if the notion of truth applies directly to sentences.[19] This does not make the distinction between first and second order arbitrary but demands an awareness of the *parameters* and *rules* of the investigation in a more “holistic” manner. This is similarly true for the distinction between meta-mathematical and mathematical inquiry and even between philosophy and meta-philosophy. The claim is that the analytic-synthetic distinction, while not a distinction of the type of that between first and second order considerations, operates in a relative non-arbitrary capacity equivalent to these distinctions.

That which can be taken as analytic can be taken as synthetic in other circumstances, but this is only possible by examining the “background” presuppositions that are being inputted to the judgment of whether or not something is analytic or synthetic. These conditions are essentially necessary to know in order to make the determination. This is true even of the propositional theory of meaning in which propositions are seen as universally and eternally true or false (e.g. Moore and Russell).

An analytic statement is a definition, a definition of the *essential* properties of a term. These properties, as Russell has shown, are equivalent to what in ordinary language takes on the name adjective. These properties, or adjectives, are empirically acquired. However, since definitions are the relation of essential properties, the judgment of which can be composed independent of experience, is not requisite for the assimilation of properties that create concepts (whereby a concept is a cluster of essential properties) to exist empirically, even when those concepts are in relation to empirical ideas. What is interesting is that there is an infinite series of terms corresponding to various intensional equation combinations of essential and non-essential properties. Since a concept is a cluster of essential properties, the name for the concept is arbitrary. Similarly, the term that designates a concept is merely a stand in marker of the intensional content. This is, of course, why an analytic statement a priori is said to be trivial information, as its logical-semantic content is a redundancy.

Take the declarative sentence, “A parrot is a bird”. This declaration can be taken in one of two ways: epistemically informative or epistemically redundant, the determination of which is relative to the extent of the subject’s knowledge of the concepts involved. This is *not* the category of relativity that this paper is prescribing. For if the statement is epistemically informative this indicates that the subject is *learning* and therefore *acquiring* the meanings of the terms involved. Once this procedure is “finalized”, such statements collapse into the epistemically redundant category.

Thus, the relativization of the distinction is not due to the analytic judgment but surprising to that of the synthetic. Why is “A parrot has a raucous cry” synthetic? The answer was because it is taken as contingent, a contingency of the empirical world. But this is precisely the sort of category confusion brought on by Lewis and Quine that must be avoided. Therefore, surprisingly, “A parrot has a raucous cry” cannot be synthetic because the statement is simply epistemically informative, for this can be a necessary but not sufficient condition if there is to be not only a genuine but unalterable (i.e. non-relative) distinction between the analytic and synthetic judgment. If it is the case that the analytic-synthetic distinction is a logical-semantic distinction and not an epistemological (or psychological) distinction, then it necessarily follows that the defining of concepts logically bars no types of properties in logical-semantic space from acquiring the role as essential or non-essential. The rules for doing so are language-dependent and are always alterable, and therefore relative to the system of intension in practice. For instance, if the definition of parrot, when fully described is “a usually brightly colored zygodactyl tropical birds with short hooked beaks and the ability to mimic sounds”, then the analytic statement reads “a usually brightly colored zygodactyl tropical birds with short hooked beaks and the ability to mimic sounds is zygodactyl.” And, to re-iterate, the statement is epistemically trivial in its semantic redundancy. This is important, for Quine wrongly argues that semantic based analyticity is not trivial but epistemically informative, which helps lead to Quine’s confusion.

But the important thing to bear in mind is that the only thing that separates the property of being a bird and the property of possessing a raucous cry is that one is essential and one is non-essential to the definition of parrot. This is quite relative for there is nothing logically necessary in this determination.

The following five definitions of the term parrot are materially co-extensive of the empirical world, but intensionally different in logical space, and are therefore not cognitively synonymous, (to use Quine’s phrase):

- Parrot A = Zygodactyl tropical birds with short hooked beaks and the ability to

mimic sounds.

- Parrot B = Brightly colored zygodactyl tropical birds with short hooked beaks and

the ability to mimic sounds.

- Parrot C = Brightly colored zygodactyl birds with short hooked beaks and the ability to mimic sounds.
- Parrot D = Zygodactyl tropical birds with short hooked beaks.
- Parrot E = Bird with the ability to mimic sounds.

How can one tell which is metaphysically necessarily only a matter of empirical probability due to its inherent contingency? Say that red is postulated as part of the definition of apple. If an apple is red, then formulating this information as a simple problem is analytic *if and only if* *one prescribes the rule* that the property of redness is essential to the definition. In the last several months a pig in Japan was genetically altered to have light green skin. Depending on how one defines a pig, perhaps this is not a pig at all? This does not only apply to color but all considerations of properties, both intrinsic and extrinsic. In regards to definitions, all properties in relation to concepts are relatively assigned; the way in which they “correspond” with particular objects in the world is a matter of *empirical generalizations*. *Whether or not particular objects have their properties essentially or non-essentially is an entirely other question to that of which the conceptual definitions that are to conform in relation to these particular objects are essential or non-essential.* The former question is metaphysical the latter question is logical. Quine attacked the metaphysical nature without seeing that the logical question could be “detached” without philosophical controversy. This conflation can be seen easily enough in Quine’s article “Reference and Modality”.

But there is more to say on this matter. If the five “parrots” above have different essential properties, then the *term* for parrot may be the same but these are essentially five different concepts. Furthermore, even if in the material world it just happens to be that all five of these definitions are co-extensive with all the exact same members, this would change nothing. Whether or not these definitions constitute analytic or synthetic judgments *contingently depends on the rules of the properties themselves*, the way in which the language or the logic intends to *classify and order* them. It is a *normative* choice whether or not a raucous cry is essential or non-essential. *But similarly this does not make practices, discursive commitments, or normative choice in general arbitrary, but simply admits of its relativity*.

From the standpoint of Logical Semanticism, the basis of the project requires a robust semantic analyticity, one capable of using semantic necessity in an analytic context to relate the syntax of terms to the meanings thereby entailed. The arguments here presented are concentrated on attacking ulterior conceptions of the analytic, namely Quine. More work is necessary to provide positive application of semantic analyticity.

**(C) The Principle of Identity and the Problem of Referential Opacity**

In the fields of mathematics and philosophy, no principle is more fundamental than that of identity. Despite the fact that it is universally agreed upon that A = A, remarkably there is irreconcilable disparity between different philosophical interpretations of precisely what identity entails. The nature of this disparity is most acutely felt in the philosophy of logic and language, more so than any other, even mathematics. The heart of the problem is the relationship of syntax, meaning, and identity. Disambiguating these relationships, with the assistance of a thorough investigation into the principle of identity, results in solutions to referential opaque contexts. This section provides arguments in the philosophy of logic that result in possible solutions to the philosophy of language, a claim critical for the possibility of Logical Semanticism.

Referential opacity, always a semantic problem, is present for any instance that a term procures its semantic content from multiple sources, namely an admixture of the sense of the name and the sense of the denoted object. Such fickle augmentation manifests itself as a most severe impediment to quantificational logic, as the separation of the two senses often proves impossible; namely when the scope of a description is indifferent to its truth value in a propositional function or when synonymy is *only* taken as a matter of material equivalence of the members of one set to that of another.[20] In relation to quantification, the principal quandary entails substitutivity of identity claims, whereby many logical terms intended to designate identical properties, classes, and relations cannot be interchanged for seemingly equivalent terms. Consequently, without the determinate means of substitutivity, one of the primary functions of quantification is compromised.[21] There are a plethora of examples pertaining to substitution failure in both first order predicate and modal logic: (1) the substitutivity of proper names in predicate logic; (2) the quantification of terms that are bound to necessity operators in modal logic; and (3) intensional semantics in general, such as the relation of terms to definite descriptions. Logical Semanticism is the general method to overcome all three, albeit in different ways.

The extensional logic of Russell, as it applies to language, leads to an antinomy, namely the insufficiency of accounting for semantic equivalence in most contexts. Pure logic, like pure mathematics, deals with variables alone. The realm of application requires set values. To give an analogy, science is to mathematics as language is to logic, for the relationship of fields of application to those of pure a priority are the same. But applied logic, and the quandaries of logical constants that are thereby entailed, suffer from numerous quantificational difficulties not shared with pure logic. These difficulties dramatically increase when specific semantic constants are reduced to a purely logical formula, such as a propositional function (e.g. the logical elimination of singular terms in linguistic sentences). Frege, Russell, and even Peirce were aware of some of these substitutivity difficulties but it was Quine that discovered the depth of these antinomies when he derived the extensional classes of his logic from his adherence to the material equivalency relation as they apply linguistically to definition and analyticity. With the application of these classes to the principle of the indiscernibility of identicals, Quine realized that substitutivity becomes a haphazard enterprise. Furthermore, since analyticity is a stronger claim than even synonymy (at least for Quine), the notion of logical necessity applies only to syntactic logical form and not semantics at all, being the source of Quine’s disdain for modal logic.

These conclusions, however, are less a result of logic than an epistemological and metaphysical prejudice, for ultimately, Quine had two choices. It was possible for Quine either to admit a more intensional based system of logic to account for the insufficiency of the material condition that led antecedently to the antinomy in the first place or Quine could deny meaning altogether and establish a purely material referential apparatus for quantification. Quine chose the latter and dismissed meaning as a variant of Aristotelian essentialism, which antagonized his metaphysical nominalism. Furthermore, Quine believed that epistemologically there is a confusing of the empirical acquisition of concepts vis-à-vis their rudimentary behavioral and societal linguistic role, with that of presupposing a relationship of meanings and referential objects as necessarily related in ways not conforming to this epistemological view, by which Quine believed is the case of semantic based analyticity.

Quine develops these issues in many of his writings, the most famous being “Two Dogmas of Empiricism”. The careful reader will notice, however, that Quine’s critique against cognitive synonymy presuppose a strong extensional language, a point which Quine confesses in section 3, the section on interchangeability. In fact, as Quine himself stipulates, the only condition by which an intensional language suffers philosophical problems is in relation to the definition of analyticity, a notion that must be take for granted as unproblematic for the intensional account to succeed. Consider Quine’s own words:

“For most purposes extensional agreement is the nearest approximation to synonymy we need

care about…extensional agreement falls short of cognitive synonymy…if a language contains an

intensional adverb ‘necessarily’ in the sense lately noted, or particles to the same effect, then

interchangeability salve veritate in such a language does afford a sufficient condition of cognitive

synonymy; but such a language is intelligible only in so far as the notion of analyticity is already

understood in advance.”[22]

Quine is attacking modal quantification by arguing that the intensional adverb ‘necessarily’ cannot be used, insofar as the notion of analyticity is unintelligible. The result is that cognitive synonymy of terms that would express semantic equivalence is known not through logic a priori but through the empirical acquisition of the concepts and terms determined a posteriori and synthetically in the realm of behavior, use, and learning. This is the point Quine makes against analyticity in the first paragraph of section 2, the section on definition. It is this section that betrays Quine’s erroneous conflation of epistemological and logical philosophical categories.

Recall that the analytic-synthetic distinction is a purely logical category, whereas the a priori-a posteriori dichotomy is an epistemological one. That a predicate is contained in a subject says nothing of how that information is acquired. As Kant argues, even empirical concepts are analytic, insofar as the semantic content is learned, but this in no way modifies the logical-semantic character of the relationship of the terms involved. Quine equivocates this distinction by arguing as early as the second sentence of “Two Dogmas of Empiricism” that by synthetic Quine means “fact”. And by fact, Quine means empirical fact. This confusion is disastrous, for it undermines the argumentative force against analyticity as definitional. The synthetic does not mean fact at all; it means any logical relation where the predicate is not contained in the subject, which in no way corresponds to a posteriori claims. No one is arguing that a bachelor is an unmarried male is not a fact that is empirically acquired, by analyticity all that is meant is that the logical-semantic conditions of the terms are necessarily equivalent when defined in the way that they are. Indeed, this makes the relationship trivial, but this is precisely what is meant by analyticity. Modal quantification is not after the novelty of non-trivial information, it is more modestly after the ability to substitute equivalent terms that are antecedently agreeable in the universe of discourse. No one is arguing that this universe of discourse can be significantly altered. This merely changes the values of the constants in the language, for this would in no way affect the underlying calculus formulas of quantified modal logic or predicate logic, which is the more general point.

But what is even more problematic for Quine’s account is that his attack against the unintelligibility of analyticity is grounded upon the flawed argument of the “circularity of philosophical terms.” Recall that if analyticity is an intelligible concept, then there is nothing that hinders the employment of an intensional logic, as Quine himself states. The intelligibility of a philosophical concept is not a hard standard to meet, what makes it difficult is making the claim that a philosophical concept must have all its meaning without support of other terms. In other words, analyticity cannot be defined in terms of self-contradiction or cognitive synonymy insofar as these philosophical notions are equally “mysterious” and “unintelligible”. But Quine contradicts himself on this point in the very heart of this argument:

“Here the definiendum becomes synonymous with the definiens simply because it has been

created expressly for the purpose of being synonymous with the definiens. Here we have a really

transparent case of synonymy created by definition; would that all species of synonymy were as

intelligible.”[23]

The outstanding question is: why could not analyticity be of the version of synonymy grounded upon the conveyance of a specific meaning to other novel notations that are stipulated as such? This form of synonymy indicates triviality of information, which would be of the precise nature of analyticity. What is unintelligible is not the concept of analyticity or synonymy but how these concepts are accounted for in Quine’s version of extensional logic. Indeed, analyticity and synonymy do not look unintelligible at all, which has been demonstrated even within the confines of Quine’s own assumptions.

Quine’s shortcoming on this matter merely indicates a deeper problem in twentieth century logic. Russell made the noble effort to quantify a theory of definite descriptions and a descriptive theory of proper names. These two entirely different theories have played an integral role in the applied logic of propositional calculus in relation to language. Unfortunately, there are quandaries, the most problematic of which being not even referential opacity as such, but the underlying cause of how referential opacity is inevitably derivative of Russell’s logic. The argument to be developed will demonstrate that *referential opacity is an unavoidable outcome of the Russellian method of quantifying, making the problem of substitutivity inherent to the accepted employment of extensional logic in general, from Russell to Quine and beyond*.

The problem concerns the Russellian approach to the denotation of terms in predicate logic. Russell’s three premises for the descriptive theory of proper names is that a proper name, or more generally a singular term, logically quantifies into three necessary conditions. For example, take the proposition: “William Howard Taft is really fat.” Russell would say that this statement logically reduces to the tripartite assertion: At least one person is William Howard Taft, and at most one person is William Howard Taft, and whoever is William Howard Taft is really fat. The terms and the intension can be abstracted to a generalized propositional function. This formalization would appear extensionally as:

(1) ( x)(Tx & ((y)(Ty y ≡ x) & Fx))

However, the argument to be developed is that this approach is fundamentally insufficient, insofar as there are other equally necessary conditions implicit in the quantification of singular terms. The optimal way to demonstrate further necessary conditions manifests in the problems of identity and of substitutivity in relation to the formation of propositional functions, such as the example given above.

Take the quantification of proper names. One of the fundamental problems with this relationship is the quantification of the proper name as it relates to the definite description is oversimplified. Opacity exists in the variables themselves since extensionally denoting the object as a single variable leads to confusion over whether the relationship of the definite description is attached to the reference the name represents, the meaning the name represents, or both. Thus, from the above example, referential opacity is a foregone conclusion from a quantificational approach to proper names, or more generally singular terms, insofar as there is an underlying failure to differentiate the variable “x” and the existential quantification of “William Howard Taft” into its constituent intensional parts.

What is it to say that William Howard Taft is really fat? Is the property of fatness a possession of the meaning of the name ‘William Howard Taft’ or the designated object of reference William Howard Taft? To re-iterate, one could never tell from Russell’s system or any other logic that takes this approach for granted. *But to ask for this differentiation is not to undermine the character of a propositional function*. The variable can still stand for the input of any value, which maintains the necessary level of abstraction that a variable and propositional function in pure logic must conform; but what needs to change is which values constitute valid truth affirming inputs in the applied logic representative of intensional language. This debate seems to be novel, and since this paper is apparently the first to introduce this new logical controversy, the traditional approach will be referred to as the quantifying of “simple or irreducible constants”, whereas this paper is calling for interpreting applied logic as requiring “complex or reducible constants” in order for quantification to be sufficiently accurate and comprehensive. The result is that complex constants require a more robust model of quantification, whereby the payoff is an intensional system capable of overcoming many problems of referential opacity. Recall that this is the antithetical horn of the one Quine ultimately chose in order to deal with the extensional antinomies: the move into a logical system with the capacity to deal with intensional equivalency relations and their substitutivity conditions.

When the singular term is quantified in relation to a definite description, the term of the equation is a genuinely multifaceted variable. Overlooking this component leads to Russell’s problem. Consider the traditional approach to sentential quantification. Russell’s descriptive theory of proper names asserts that a term is reducible to its intensional description. For instance, the proper name Henry Kissinger could be reduced to the definite description: “Secretary of State under President Nixon”. Quine also capitalizes on this ability to eliminate terms into the representations of bound variables. The argument to be developed here is that the quantificational role the singular term signifies is not categorially equivalent among the three necessary conditions that a declarative sentence between a subject and predicate is alleged by Russell to conform.

Recall that if given the statement, “Henry Kissinger is the Secretary of State under President Richard Nixon” that this statement is supposed to logically signify that at least one object denotes Henry Kissinger, that at most one object denotes Henry Kissinger, and that whatever is Henry Kissinger is the Secretary of State under President Richard Nixon. Notice, however, that when singular terms are quantified as bound variables of existential quantifiers that the first two conditions are internal representations of what the singular term, in this case a proper name, logically signifies. The third condition, that is, the relationship of the term to the intensional content predicated of the declarative sentence, is external to the term and stands as a relationship of equivalence. But the way in which Russell quantifies the sentence into a single propositional function does not reflect this distinction, as it is partially lost in quantificational translation.

Russell’s quantification of singular terms is inherently indeterminate. The first example that will be offered is: “Rachel is the wife of Scott.” Etymologically, Rachel means “ewe”. Therefore, Rachel is a female sheep. Is it the case that: “A female sheep is the wife of Scott”? If the descriptive theory of proper names allows for the eliminability of terms in relation to its expressed definite description, then the syntactic term ‘Rachel’ signifying a female sheep with the syntactic term Rachel that signifies a unique object of reference cannot be separated in terms of Russell’s logic. For the two terms syntactically appear the same and the substitutivity of one for another fails. What an extensional account of singular terms under the interpretation of simple constants overlooks is the ability to categorize meanings of names as legitimate without crossing into a category mistake when attempting to substitute two entirely different kinds of intensions relating to identical syntactic notation. The problem is that the fact that Rachel etymologically signifies “ewe” says nothing of an actual object of reference. However, this is precisely the problem of substitutivity of terms for relating two names to a definite description that do relate to the referent.

All instances of referential opacity in relation to predicate logic exploit this conflation. For instance, take Quine’s most famous example from his article “Reference and Modality”, whereby Quine begins with the alleged identity of two terms that denote the same object of reference: “Giorgione = Barbarelli”. For the indiscernibility of identicals to succeed, interchangeability of both terms must be validated for each instance of substitution. Failure of substitution is possible for any declarative sentence that relates the meaning to the name and not the object, whereby the introduction of another term will therefore not succeed. Quine gives an example of this, he believes, by postulating to the reader the declarative sentence: “Giorgione was so called because of his size.” Quine then proclaims that the reason why Barbarelli cannot be substituted is because the meaning in relation to Giorgione applies to both the name Giorgione and the reference Giorgione, and substituting Barbarelli for both generates a falsehood because it presupposes that Giorgione is only one variable, the single Russellian variable “x”. This is, of course, what is meant by referential opacity. Quine’s assumption, however, is the employment of the quantification of Russellian extensional logic, as the terms involved are all representative of the same variable. This is incorrect. Quine is only correct to the extent that there occurs a quandary in relation to substitutivity. *But* *the fundamental problem is not substitutivity, the problem is faulty quantification. *

There is an alternative. Ruth Barcan Marcus introduced a direct theory of reference, whereby a proper name is what she referred to as a “tag”. Tags are logical markers in a quantificational formula that indicate that a single object is denoted and nothing else. In other words, a logical tag is not necessarily associated with any intensional properties other than the extensional definition of its unique denotation. Marcus developed this theory as an outgrowth of QML (quantificational modal logic), which was her pioneering contribution, as embodied in her article “A Functional Calculus of First Order Based on Strict Implication” and expanded in “The Deduction Theorem in a Functional Calculus of First Order Based on Strict Implication.” This direct theory of reference is antithetical to the quantificational eliminability of logical terms into the vocabulary of definite descriptions, as proposed by Russell. Marcus’ great insight was specifically concentrated on answering certain problems that Quine and others raised in regards to QML. But the argument of this paper is that the notion of tags can be applied to predicate logic, using Marcus’ insights, and be formulated in such a way to overcome the critiques leveled against her theory by Quine, Kripke, William G. Lycan, and others.[24]

Consider the general four step procedure of all of Quine’s counter-arguments to the substitutivity of identical terms:

1. Proper Name A = Proper Name B (Postulation of the Identity of Terms)

2. Proper Name A = Definite Description X (Intensional Equivalence of Terms)

3. Proper Name B ≠ Definite Description X (Substitutivity of Identical Terms)

4. Proper Name A ≠ Proper Name B (Conclusion)

What is most interesting is that the problematic component of Quine’s formulation is not step 3 but step 1, the postulation of the identity of terms. The problem is that the notion of a term has two values that are augmented into one. Recall Quine’s famous example, that “Giorgione was so called because of his size.” This statement can be expanded more properly to the propositional function “x was named x because of the size of x.” What ought to become immediately apparent, however, is that the substitution of a term in the place of variable x is not the same for all instances of x in the propositional function. The mistake that the traditional account is guilty is that all stated variables are not interchangeable. The first instance of Giorgione just denotes the one and only one denoted object of reference, it is a purely referential component of the function. The term is reducible to the notion of a logical tag. The second instance of Giorgione denotes nothing at all, not even a tag, as the second variable is representative simply of the sequentially ordered syntactic notation ‘Giorgione’. There is no semantic or referential component involved. The third instance, in this case, first appears ambiguous. What gives the appearance of referential opacity is that the third variable could refer either to the sense of the denoted object or the sense of the name. But what makes Quine’s example solvable is that the third variable truly refers to both simultaneously, for the meaning of the sequentially ordered syntactic notation “Giorgione” intensionally signifies the property of size which is precisely what the denoted unique object that the referential term Giorgione itself possesses. The opacity is not contained in what is semantically comprehensible but the way in which this knowledge is lost in Russellian quantification.

Properly reformulated, Quine’s declarative sentence is transformed into: “The class of terms that denote the one and only one object X includes by implication the sequentially ordered syntactic notation Y, whereby the intensional property Z is biconditionally equivalent to X and Y.” Substitutivity succeeds for any term that denotes the object X, insofar as this knowledge must be known antecedently. This is what separates a purely logical quantificational procedure from the epistemologically sensitive doxastic conditions of propositional attitudes. Of course identity of terms does not succeed if the subject is not aware of the relationship! How is this possibly problematic? The genuine logical issue concerns postulating from the inception which syntactic strings of symbols operate as members of the class of terms that denote the same object. This is the only way in which identity can succeed. If Giorgione = Barbarelli, then by definition, what this signifies is that two terms equal the same denoted object. A proper name by itself, serving the role as a direct referential tag, implicitly represents the class of one term that denotes the singular object X. However, when knowledge of Giorgione = Barbarelli is given, then this class is expanded to include two terms. This is what allows for substitutivity because the interchangeability of terms affects only the directly referential variable that only contains the notion of reference to an object and stipulates nothing concerning the sense of the name. Since this has been separated, referential opacity, specifically in relation to the problem of substituting identical proper names, will always be made transparent under the procedure of this method of quantifying.

The solution to the problem is re-orienting the indiscernibility of identicals to qualified equivalency relations, whereby the transitivity of notation of terms, semantics, and referential objects are all interrelated in multifarious ways.[25] What fails in the traditional account is that syntax, meaning, and reference are transitively related as two biconditionals. But this is not accurate. When a declarative sentence is made it relates the meaning of the predication to either the name qua syntax or qua object as such, whereas the condition of indiscernibility takes it for granted that it must apply to both. The indiscernibility of identicals does not account for this differentiation. What is necessary is a more qualified means of accountability.

The development of the precise quantification of this intensional approach and the necessary equivalency relations is made possible by Logical Semanticim’s Intensional Semantic Axiomatic Set Theory. The goal of this section has been to underlie the untenable problems of a strong extensional logic based in material equivalence as it specifically relates to quantifying language. The suggestion of this section is to re-assess the nature of the logic that led logicians into these paradoxes and to re-organize their efforts under the auspice of a system of logic with the means of overcoming the quandaries here outlined. To this end, solutions have been given in order to demonstrate the values of a new approach and to invite the philosophical community to adjudicate these demonstrations as justification to employ the methodology here prescribed. The next goal is to give the full system envisioned by the consequences in the philosophy of logic, namely the change in the equivalency relation, here developed. From here it is necessary to develop the theory of reference and meaning, which provides the basis for Intensional Axiomatic Set Theory.

**III. Logical Semanticism and as a Philosophical Theory of Langauge**

Now that the methodologically necessary philosophical concepts have been articulated, Logical Semanticism can be directly developed. The optimal place to incept is with the referential theory of meaning, as it is both the oldest and most commonsensical of the theories of meaning. From a historical standpoint, the theory often is overlooked. However, this section re-examines such hasty tendencies in order to inquire into the nature of the difficulties the theory is purportedly jeopardized. From there, the notion of truth conditional theories is considered in turn, the mistakes brought into prominence. Logical Semanticism is then offered as a theory that takes reference ultimately to intensional classes which are always truth-conditioned. But these truth-conditions are non-metaphysical, lacking in both Davidson’s and Kripke’s formulations. Instead of “T-sentences” or “all possible world semantics”, Logical Semanticism develops logical-semantic systems. The nature and structure of these logical-semantic systems, their fundamental role in language; these topics are pursued in this section.

**A. The Referential Theory of Meaning **

Traditionally, the referential theory of meaning is taken naïvely, receiving little serious philosophical attention. The basic idea is well enough understood, insofar as terms in a given language are representative of some specifiable denotation, symbolizing particular things. Simply put, words directly stand for something exacting and commensurable. Tarski’s student Montague, one of the few to have deemed the referential theory of meaning as worth pursuing, extended reference to cover quite abstract terms, such as grammatical quantifiers. In Montague’s work, one could thus “reify” definite singular terms, absolute general terms, and even words or phrases such as “such that”, “the”, “and”, “for”, and other similar quantifier terms. Now there are a number of problems raised against the referential theory of meaning, and these are indeed serious concerns. However, there is an important question worth asking about referential theories of meaning. Are all referential theories of meaning inherently naïve or is it the case that there are particular definitions of reference that are more susceptible to generating philosophical quandary? In general, philosophers do not ask this question. For this reason, there is justification for re-assessing the theory, namely through evaluating the concept of reference as such.

Perhaps one of the reasons that the referential theory of meaning procures so many difficulties is the limited scope given to the philosophical concept of reference. A major argument against the referential theory is that there are many terms that do not denote an empirical object or state of affairs, naming and purporting some sort of information which refers to nothing actual. What is here presupposed is that reference must entail something empirical. The question is never proposed as to whether or not reference ought to be defined so narrowly. Transforming reference into the metaphysics of actuality severely limits the utility of reference as a philosophical concept. Hardly surprising is the number of problems with abstract terms and definitions, or terms denoting syntactic connectors. No reflection is necessary to comprehend the difficulties in ostensive location of pronouns or terms such as “or”, “is”, “as”, or “dint”.

So, then, what is the appropriate definition of reference? Are Russell, Quine, and so many philosophers correct in defining reference narrowly to the empirical objects and states of affairs? Certainly the problem of the apparent reference to non-existents is an inherent outcome of a narrow definition of reference. But what happens if the definition of reference is expanded? What is the alternative? What are the detriments and problems of such definitional expansion?

For the purposes of this paper, the definition of reference is taken at bare minimum as the aggregate of the necessary conditions for understanding the denotation constituted by some particular string of syntax. Denotation is the *identifiable membership* associated with what is linguistically expressed. Identifying membership of what language refers does not always require empirical conformation, or disconfirmation in the case of non-existents. Such denotation through the succeeding of reference requires commensurability by comprehending what is intensionally signified through the employment of language. Reference depends on sense, insofar as denotation depends on connotation, extension on intension. Therefore, reference is answerable to the intensionality, for denoting the membership antecedently requires knowledge of what is signified. Enumerating all the parrots in the world requires referring to all the objects in the world that are parrots. Referring to parrots requires knowing what is signified by the abstract general term “parrot”. Similarly, that Russell can say that “the” or any proper name logically quantifies into “one and only one object such that…” is purely denotative, presupposing the semantic rules for prescribing what terms constitute proper names and that “the” has the pure referential properties that it does.

As for Quine’s blind contempt for Pegasus, one can only refrain from emulating his eccentric war against harmless abstract terms[26]. Quine’s shallow theory of meaning, derived from his *onto*-logical interpretation of meaning and consequent rejection of Aristotelian essentialism, leaves fiction, poetry, and many forms of linguistic expression in theoretical limbo. When the professor of classical literature refers to Prince Andrew Bolkonski’s distinguished military career or Natasha Rostov’s romantic entanglements is it the valid response to proclaim reference does not succeed because neither of these definite singular terms denote empirically factual objects? No one would want to say that fictional literature, such as Tolstoy’s *War and Peace* is meaningless or that two minds cannot both understand one another on such matters. The reader employs language to refer to the intelligible fictional characters and events, to refer to this identifiable context. This can be accomplished without logically requiring ontological qualification. Reference need not be ostensive, in practice or theory. But, then, what is the purpose of a narrow definition of reference when there is a tremendously large class of empirically non-existent objects and states of affairs that are intelligible, commensurable, and the understanding of which can be denoted in accessible terms? If reference is not equipped to serve this large portion of language, then reference is not a robust enough philosophical concept, thus necessitating either expansion as a concept or theoretical replacement to compensate its massive shortcomings.

The fields of mathematics and logic ought to be included in this referential predicament. Arithmetical, geometrical, and higher forms of mathematical computation are not ostensive or empirical, as anyone that takes mathematical philosophy seriously understands. Two is a set of all couples, and such an abstract intension is not referential to anything in the world. This is equally true of Sheffer’s equation for polynomial combinatorial functions, an equation of referential status equal to that of Pegasus.

The alleged danger often associated with expanding the definition of reference is that the Fregeian distinction between sense and reference collapses. The fear is Meinongism, an overabundant metaphysical ontology. Quine worries that reference as dependent on sense generates three “entities” for Venus is the morning and evening star, whereby morning star, evening star, and Venus would thus denote three disparate referents, where Quine contends only one is justified. Since in reality each sense only denotes one and the same empirical object, three referents to one object appears counter-intuitive. But intuitive and counter-intuitive are relative concepts, contingent upon how one analyzes, for in another context the situation is the reverse. The debate hinges upon two methods of appropriating sense and reference to concepts. In accordance with the Venus example, one can either refer to such and such object constituting three different senses or one can refer to three different senses and consequently judge those senses to be materially equivalent, sharing the same members. The only reason why morning star refers to the same object as Venus is simply that the definition of morning star is any astronomical planet rising after midnight, in truth referring to both Venus and Jupiter at different times. The example is a perfect reason why one ought to draw the opposite conclusion of the persons’ that employ it.

Of course, the even more fundamental problem is not respecting a sharp distinction between semantics and metaphysics. Allowing for the unique reference to a particular sense, one that does not empirically obtain only increases the universe of discourse, not the metaphysics or belief in non-existent objects. Purported objects of an empirical nature that do not exist in reality are not objects at all but merely concepts. Allowing for the linguistic use and commensurability of such concepts does not increase the ontology. Meinongism never gets off the ground, nor does the sense and reference distinction collapse, insofar as sense is never understood metaphysically. This mistake has been repeated throughout a wide range of philosophical literature, from language to modal logic. This point will be returned to in later sections, where it shall be expanded.

The problem at hand is the ambiguity of reference as a philosophical concept. Reference generally serves two functions that are not well distinguished. Firstly, reference is said to succeed when two minds have procured the same object, concept, or state of affairs and commensurability ensues. Secondly, reference is denoting the member or members that a term designates. Thus, in relation to the first aspect of reference, Pegasus refers to the properties of winged horse and conforming to the second aspect of reference has an extensional membership of zero. Reference succeeds to the properties allowing for commensurability, but reference does not succeed to anything material to correspond. Non-existent objects are said to refer to nothing real but by doing so, a narrow definition of reference, one that does not differentiate between the two aspects of what reference can philosophically apply, generates the problem of categorizing fictional, mathematical, and logical information. Emphasizing the first aspect of reference overcomes these problems but makes many philosophers nervous about the relation to sustainability of the sense and reference distinction.

The sense and reference distinction reduces to concerns with intension and extension. What can be agreed upon is that reference involves denotation. However, extensional definition is too limited to encompass the full application of expressively denoting particular things. Instead, words ought to be taken as referring to intensions, namely intensional sets or classes. Extension, as indicated earlier, always derives from intension. But in any case, identifying reference simply with positive extension and sense with intensional semantics is an oversimplified dichotomy. Reference functions as the denotation of sense, providing the means of commensurability through shared context. Material correspondence is simultaneously incorporated into this process, establishing the extension of what is intensionally referenced. The intensional classes of gold mountain, virtuous horse, and liche produce the necessary predications constituting each respective term. Minimally, what is required of reference is the explication of the necessary properties semantically constitutive of any term. A gold mountain is a topographically prominent landmass colored gold. A virtuous horse is a four legged hoofed mammal with snout, hair, tail, etc., ethically disposed towards praiseworthy action and intention. A liche is an undead sorcerer, capable of escaping death through the use of magic; isolationist by nature, a liche spends most of its time in contemplation. None of these things are likely to exist. All reference requires is commensurability, the standard is one of understanding the context and predications of terminology. Nothing about commensurability requires the criterion of existence, otherwise fictional literature, poetry, and a very great proportion of language would be taken as nonsensical. Reference succeeds to the extent that mutual understanding obtains.

Therefore, the sense and reference distinction does not collapse, for reference is both aspects of establishing extension of anything denoted and providing the criteria for commensurability. But the former depends on the latter and is consequently the more fundamental. As to Quine’s problem of an overabundant ontology, it is wrong from the inception to assume Venus, morning star, and evening star are indeed the same object. One can refer to one concept without entailing one of the others. Indeed, the morning star and evening star are not intensionally identical. In fact, Venus is not the only morning star or evening star, so they are not even extensionally equivalent. But the only way one can be cognizant of this state of affairs is by knowing the intensional semantics of the terms and only then epistemologically judging what objects such terms denote in the empirical world.

But surely the problem of apparent reference to non-existents generally enters not on the subsentential but the sentential level. The problem surrounds such statements as “Gold mountains exist”, “I saw a virtuous horse”, or “The liche was quite rude”, instead of “gold mountain”, “virtuous horse”, or “liche”. Similarly, any of these propositions could be true if under the right qualification they are contextually limited to a situation where truth could obtain, such as if a character in a fictional novel, such as C. S. Lewis’ *The Horse and His Boy* says that they have seen a virtuous horse. However, this is still referential, in even these contextual cases, since one is referring to the valid truth bearing instances for such propositions.

Such propositions cannot be said to be meaningless. There is great difference in “square circle” with that of positing the existence of Pegasus, for the concept of non-existing empirical objects is quite well understood when existence is predicated. Such propositions always depend on context, propositional attitudes and doxastic quantifiers, and metaphysical qualification for both their meaning and truth-value. Propositions are complex judgments, the reference of which demands contextual relation. In correspondence with Russell, the truth conditional nature of propositions demand that “Gold mountain exist” be taken as false, insofar as “exist” signifies empirical existence and not semantic “existence” in a universe of discourse—Meinongism changes the connotation of “exist” to something quite different, of which Quine is correct. The ambiguity of hundreds of years of debate on the subject derives from the ambiguity of “existence” as either *object*ive existence or *concept*ual existence. The former adherents make the point that the narrow definition of existence is preferable, retaining the meaning of the word and adhering to the metaphysics of actuality, whereas the latter adherents make the valid proposal of giving attention to the many possibilities of creative concept formation and attesting to the role gods, demons, ghosts, otherworldly alien abduction, and other non-existent phenomena have upon the real.

But even more to the point is recognizing the difference in reference, between referring to existing objects and referring to anything that is meaningful and relatable. The limited definition of existence conjoined with an expanded definition of reference is the valid procedure and here adopted.

The paradox of apparent reference to non-existents has proven to be a pseudo-problem derived from a narrow and inadequate interpretation of the philosophical concept of reference, as well as ambiguous debates concerning the truth or falsity of propositions containing non-existent subjects. The valid theory of reference incorporates the fundamental notion of words representing intensional classes. But the nature, structure, and philosophical commitments of such intensional classes have not yet been developed.

As for the referential theory of meaning, it is significant precisely because it is embedded in the understanding of an intensional truth-conditional theory of meaning, whereby reference is seen as performing the function of denoting an intensional class, some sort of general absolute term (i.e. universal, if a universal is here understood without necessarily implying any ontological commitments). So, in truth, the referential theory of meaning is saved by ameliorating the definition of reference. Of course, if reference is understood in this way, the referential theory of meaning cannot perform the function of accounting for meaning as such, since this falls within the domain of intensional classes, the theoretical commitments of Logical Semanticism. Reference becomes, as Tarski notes, the designation of semantic content. Thus, the truth-conditional theories of meaning, which reference in some way always refers, is the proper context to move forward in solving the problems of the philosophy of language.

**(B) Critiquing Traditional Truth-Conditional Theories of Meaning**

The two general 20^{th} century approaches to truth-conditional theories of meanings are Davidson’s “T-theory” and Kripkean intensional semantic “all possible worlds” theory. Traditionally opposed to truth-conditional theories is the inferential or “use” theory of meaning developed in Wittgenstein, Sellars, Brandom, and others. This section attempts to critique the traditional approach to truth-conditional theories, but provides reasons other than the problems given by proponents of inferential theories of meaning, or inferential semantics. As for “use” theories, recall that the introduction of this paper stated that methodologically the issues of pragmatism proceed after considerations on the level of how semantic information is referenced and quantified. All inferential theories involve social context, reasons for employing or uttering statements to fit specific intentions of the speaker, and entail issues related to the social sciences. Logical Semanticism does not deny the need for components of the inferential theory but maintains the need for a robust conception of reference, the traditional weakness of use theories, even Brandom’s. That a chair is for sitting, that a house is for living, or that a car signal is for indicating a change in direction is but one component of an object or state of affairs, and does not exhaust the necessary content of empirical or other kinds of definitions. Indeed, the greatest benefit of pragmatic oriented conceptions of meaning is not useful for the foundations of theoretical semantics but is often a replacement or amelioration of epistemological issues. Thus, there is no coincidence that those following Wittgenstein, Rorty, and other such figures take language as replacing traditional epistemology—indeed, this follows from the pragmatic nature of inferential theories. Since such questions can only manifest after the development of the methodological pursuit of the necessary conditions of the foundations of theoretical semantics, such inferential and epistemological investigations must be left for another work.

This is not similarly true of truth-conditional theories of meaning. Davidson, Kripke, Russell, and others are interested in the assignment of truth and falsity to all propositions, which is certainly fundamental to logic. Doing so does not require representational epistemology. However, there are still disadvantages to both traditional approaches of truth-conditional theories of meaning.

But recall that the referential theory of meaning is significant precisely because it is embedded in the understanding of a truth-conditional theory, whereby reference is taken as performing the function of designating an intensional class, some absolute general term (i.e. a “universal”, if universal is not taken metaphysically). So, in truth, the referential theory of meaning is modified in this manner from the previous section, and reference to intensional classes can always be judged as either being true or false.

Now the greatest problem with Davidson’s theory is that it is barely capable of working with intensional semantics, as Davidson formulates not only a deflationist theory of truth but follows Quine in a deflationist theory of meaning. But the second problem is that the Davidson, like inferential theories of meaning, takes the sentence as the smallest unit of linguistic analysis—the “T-sentence”. The sentence is not the most primitive form of meaning, for the individual objects themselves, as stipulated by intensional classes, are genuinely complex strings of information, the meanings of which are implicit.

But the implicit character of these meanings cannot be quantified or worked with under Davidson’s approach, which leads to Davidson’s second problem. This is the same difficulty that Quine suffered concerning analyticity, where Davidson is unable to use cognitive synonymy and thus offer a method for explaining how the substitutivity of identicals, on an intensional level, is successful. Like Quine, Davidson says that a sentence is interchangeable if and only if two sentences share all the same truth conditions. But this co-extensive definition of synonymy, and hence interchangeability, suffers from the traditional problems of extensional identity (e.g. “creature with a kidney” and “creature with a heart”), for there is no mechanism in Davidson’s deflated theory of meaning to deal with intensional semantics on the necessarily robust enough level. Furthermore, if Davidson’s theory did change this point, to allow cognitive synonymy through intensional logic of strict implication, Davidson’s theory would have to be abandoned altogether.

The third problem with Davidson’s theory is simply its lack of universality. Like Quine’s *Word and Object*, Davidson is postulating a method for intended use of the natural language of English. As mentioned in the introduction, this is entirely unsatisfactory. The limitation of Davidson’s system becomes specialized to certain eccentricities of English, losing sight of what is fundamental to language in general, to the foundations of theoretical semantics, that which holds for all reference to meanings.

For these reasons, Davidson does not offer a tenable procedure for solving problems in the philosophy of language. By definition, Davidson can never solve the problem of the substitutivity of identicals and hence can never overcome referential opaque contexts or ambiguities in scope of certain propositions, whatever language that might be.

The more promising alternative is Kripke’s, for the possible worlds’ semantics approach has some critical advantages. For one, this theory of meaning is a genuine truth conditional theory, one that is inflationary and one that allows for the exchange of intensionally synonymous terms, able to go beyond extensional agreement for identity relations. Secondly, the theory is universalizable, able to be applied not only to English or the romantic languages but to any language. Thirdly, since the theory does correctly separate the nature of intension from extension, certain epistemological problems are easily avoided, a point developed in the section dedicated to analyticity and C. I. Lewis. And finally, it is much easier to discuss hypothetical situations with Kripke’s possible world semantics than Davidson’s rigid system.

In spite of these benefits, the Kripkean system is utterly flawed, to the point of being almost unsalvageable. But the fundamental question to ask of this theory is whether or not it is equipped to fully develop reference to intensional classes. One of the problems has been the causal-historical theory of reference is conjoined with the Kripkean theory of meaning. But rigid designation leads to a number of philosophical problems, ones that adversely affect the possible world semantics.

As stated many times before, the metaphysical conception of modality is a mistake, the possibility and necessity operators ought not to be concerned with debates between nominalism and realism. The problem with metaphysical conceptions is the epistemological paradoxes that inevitably ensue, a point that Kant discovered so many centuries ago. Apparently, Kripke and Plantinga have decided that if they simply ignore Kantian epistemology the problem with just magically go away if they ignore the conceptual developments in the history of philosophy.

To see the extent of these epistemological paradoxes, simply apply the venerable antinomies of the Transcendental Dialectic of the *Critique of Pure Reason*; say the antinomy of the cosmological origin, to the Kripkean possible world semantics, metaphysically conceived of the alethic modal operators:

1. It is logically possible that the cosmological origin was a matter of determinable causal processes, terminating in a scientifically explainable first cause.

2. It is logically possible that the cosmological origin was a matter of extra-scientific phenomena, the explanation by which is neither accountable by observation nor empirical reason.

3. It is logically possible that the notion of a cosmological origin was not true of this reality; that time has no beginning and the cosmos has always perpetuated in some manner.

Now the problem with “actualistic” modal logics, such as Kripke’s and Plantinga’s, is precisely the metaphysical indeterminacy of states of affairs of this sort, of the Kantian antinomies. The problem entails not knowing W from W’, of being unable to discern which world is actual. This makes the heuristic value of modal logic rather worthless.

But the problem does not end here. Just as damaging to metaphysical interpretations of de re and de dicto operators is the definition of possibility. Recall that Plantinga and others stipulate that for something to be modally possible it must be an object or state of affairs that is “actually possible”, that is, some X within the framework of the laws and empirical order of the actual reality W. Plantinga states that all possible worlds, including actual world W, are governed by two criterion: (1) they are “maximal”, meaning that all possibilities inherent to that world are entailed, making the world W’ “complete”; and (2) all such possibilities are “actually” possible internal to the rules governing that world. How this is epistemologically determinable is, however, impossible, as Kant’s antinomies proved so long ago. For how could one ever determine if all the logical possibilities of the cosmological origin are “actually” possible? Perhaps by some currently unknown law the steady-state theory, the theory that says there was no cosmological origin, is false and can be proved as such? Or, even worse, if Plantinga is correct concerning the actual world W must obey possibilities that are actually possible, then the other two logical possibilities of the cosmological origin are not genuinely possible at all. In other words, if one accepts the criterion the modal logician’s prescribe, then most possibilities concerning metaphysical antinomies (free will, souls, cosmological origin, God, etc.) cannot be said to be possible in W. But, epistemologically, there is no way of making such metaphysical judgments in accordance with W. Thus, explanatory value of such logic quickly crumbles, for these problems are not only confined to the traditional metaphysical antinomies but extend to problems of all causal necessary relations concerning the possibility of laws of nature and their metaphysical status. The metaphysical nature of the modal operators is philosophically powerless to make any epistemological decision, which leads modal logic into nothing other than pure speculation, a speculation worse even than philosophy of mind.

As for David Lewis, the idea of modal realism, the equal metaphysical existence of all possible worlds, is too outlandish of an idea to take serious or even mention, except to say that the only intellectually mature thing to do is ignore it and hope that in the future doctoral degrees in philosophy and seats in graduate schools are not given out so liberally to the undeserving, perhaps people that take philosophy as a profession serious and not a forum to play childish games that no grown adult could possibly believe.

If one accepts the metaphysical of modal logic, one must either be an ontological realist (Plantiga) or nominalist (David Lewis). Neither is a satisfying position for the simple reason that Kripke’s semantics that led to the metaphysical understanding of necessity and possibility, either in de re or de dicto, fails on all levels. Fǿllesdal’s quantification of necessity as a form of epistemological-causal necessity, similar to Kantian critical idealism, was more on track. Logical Semanticism, however, goes even further, changing the meaning of such operators to deal exclusively with semantic relations of syntax to semantics, and reference to semantics.

**(C) Revolutionizing QML: The Methodology of Logical Semantic Systems**

There are a number of differences concerning logical semantic systems and the Kripkean development of possible worlds’ truth conditional theories of meaning. There are a number of differences, for one, part of the theory of reference the logical semanticist prescribes entails a direct referential theory to proper names, whereby names are taken in Millian terms, they are the intensional null class. Secondly, the notion of rigid and flaccid designators, as in Kripke and Putnam, is done away with. Thirdly, the causal-historical theory is not adopted, in favor of Searle’s cluster theory of descriptions (save for the way in which the direct referential theory is used for definite singular terms for quantificational purposes, whereby all clusters of descriptions are taken as empirical generalizations of semantic possible kind—the epistemological issues of learning the descriptions and their relations to names not being part of the direct referential theory). Fourthly, and the issue of this post, the Kripkean notion of possible worlds is jettisoned as unnecessary metaphysical baggage, at least, for a theory of meaning.

The traditional Quinian critiques, and the others like it in the 20^{th} century, are attacks upon the Kripkean modal semantics that leads to Plantiga’s and related systems. Quine’s greatest problem with all quantified modal logic, even Ruth Barcan Marcus’ and C. I. Lewis’, is that they lead to referential opaque contexts of the substitutivity of terms that are bound variables to quantifiers of necessity. These critiques cannot apply to Logical Semanticism, namely the logical semantic systems the theory adopts.

It never occurred to Quine that the only reason for this opacity and inability to substitute terms was not because they were bound to necessity but because they were taken to be bound to a type of metaphysics. For Quine, this was taken as the same thing without reflection. This is easily proven, for as Quine writes in *Word and Object*:

“But in connection with the modalities it yields something baffling—more so

even than the modalities themselves; viz., talk of difference between necessary

and contingent attributes. Perhaps I can evoke the appropriate sense of

bewilderment as follows. Mathematicians may conceivably be said to necessarily

be rational and not necessarily two-legged; and cyclists necessarily two-legged

and not necessarily rational. But what of an individual that counts among his

eccentricities both mathematics and cycling? Is this concrete individual

necessarily rational and contingently two-legged or vice versa? Just insofar as we

are talking referentially of the object, with no special bias toward grouping of

mathematicians as cyclists or vice versa, there is no semblance of sense in treating

some attributes as necessary and others as contingent. Some of his attributes

count as important and others as unimportant, yes; some as enduring and others as

fleeting; but none as necessary or contingent.”[27]

Actually, of course, what Quine means by this very last statement is that all attributes are contingent, conforming to the sort of semantic holistic empirical cluster theory that defines his web of beliefs, the inessential character of all things, from the law of non-contradiction to the norms of society and everything between and beyond. If Quine (or Rorty) was correct, it would truly be a miracle that commensurability takes place, that people can extensionally point to examples that miraculously attach to the same intensional concept, from “love” to “justice” to “itchy”.

What makes a word meaningful are those necessary attributes composed of the term. Establishing such relative relations of meaning to syntax is always contingent, but the nature of the relation itself, once postulated and accepted, either individually or communally, becomes semantically necessary. Where “all possible worlds” semantics and logical semantic systems part company is precisely over this issue of the purely intensional nature of syntactic-semantic-referential relations.

As for the Quinian excerpt, the beret-wielding thinker is confusing words and objects, or, more appropriately, concepts and objects. There exists no contradiction in what Quine has to say. A mathematician, a cyclist, a philosopher, an economist, a radar gun, a parrot, a grapefruit, a color, a pulsar, an action of any kind, an emotion, all of these things contain attributes which define these indefinite singular terms (all of which are reducible to absolute general terms—i.e. universals). If these attributes to not empirically exist, then simply the object or states of affairs that are necessary of a particular term also becomes impossible. A mathematician does necessarily have to be rational, an economist does necessarily have to know economics, and a parrot necessarily must be a kind of bird. These classificatory structure of intensional classes is dynamic. Thus, it is never the case that a concrete particular, say ‘Willard Van Orman Quine’ must necessarily be a philosopher. This is not the kind of necessity that is appropriate, which has to do with the absurd rigid designation of “transworld individuals” of Kripke.

Logical semantic systems are more modest. They do not concern themselves with anything other than semantics. This is why concrete particulars, such as proper names must philosophically be taken as empty of definite descriptions; for there is nothing necessary about a proper name other than that it refers to a single bearer, a referent. ‘Quine’ is not necessarily a philosopher, nor is ‘Lincoln’ necessarily the 16^{th} president of United States. Assigning descriptions to names is always an empirical generalization, which makes the denoting of any attribute to concrete particulars a semantic possibility. The only kind of attributes that can be said to be necessary to a proper name are ones that have to do not with reference, but with the sense of a name. Take etymology for instance. The name ‘Dasha’ means gift of God, and thus is a necessary component of the name in the logical semantic language of Ukrainian and Russian. Even if no one was named Dasha in the entire universe, and the name referred to no empirical object, if the semantic system dictates the etymological relationship of one term to a set of attributes, those attributes are necessary, as there is no relationship to reality but only logical space. Thus, the etymology of Dasha in logical semantic system X can be expressed as the relation of syntax to its postulated necessary attributes: φ(‘Dasha’) = □φ(gift of god).

There are also numerous explicit advantages over Kripke and other related systems. What is a possible world? A possible world, as Plantinga defines it: “the way the universe could have been–the total way”.[28] Possible worlds are like alternate realities, the way the actual universe could have been otherwise. So one can apply truth conditions to declarative statements and hypothetically calculate the range of possible worlds they are true or false in. There is no procedure by which this calculation can be made without making metaphysical judgments, which has already been shown. Thus modal logic, as it is practiced now, gets bogged down in conceptions like “transworld individuals”, “Twin Earth water molecules”, and other unsolvable debates. The biggest problem in modal logic is what to do with individuals. This seems to be the least consequential thing to worry about and yet it is the most pertinent to Plantinga, Kripke, and David Lewis. But Logical Semanticism circumvents all these problems altogether.

QML ought to be replaced with a new semantics, a new way of appropriating the meanings of modal operators, in either de re or de dicto form. The idea is so simple and so far-reaching and so very different from what modal logicians do that makes Logical Semanticism so revolutionary. There are two ways of understanding QML, either in the notion of “all possible worlds” or “a given logical semantic system”. What is a logical semantic system? Simply put, it is a non-extensional, purely intensional system of classifying intensional classes in a logic of relations, from simplest concepts, to more complex augmentations of concepts, that are defined in terms of the necessary intensional components that govern the classes. The necessity operator comes to represent a property that is, either true or false, a constitutive component of that class. The possible operator comes to be taken as any intensional property, true or false, that is not necessary in that functional relation, that proposition. All problems referring to individuals are eliminated because proper names are taken as the empty sets. Transworld individuals, rigid designators, and metaphysical context are finally eliminated. Similarly, implicitly, this is an adoption of Searle and Russell’s point, insofar as all properties are taken not as rigid designators.

All metaphysical indeterminacies are eliminated. Metaphysical possible worlds are replaced with intensional classes, which require no metaphysical commitments. States of affairs are quantified intensionally as “aRb” relations, but the relations don’t require a metaphysical commitment to what is necessary to impose on the actual universe. Take, for example, the definition of “causing the cosmological origin”, where the Kantian antinomy is now quantified entirely semantically. The three modal possibilities can be merely intensionally: (scientific explanation, religious explanation, no cosmological origin). Taken metaphysically, one has the familiar problem of epistemological indeterminacy, for the actual cosmological origin is unknown. However, taken semantically, the possibilities truly are mere possibilities, different possible states of affairs that each have an equal stake in the definition of a cosmological origin, or the very definition of a cosmological origin entails three conceptual possibilities, and thus become the intensional class of the three. The entire epistemological problem is avoided, which is true of all such antinomies and related difficulties.

Also of great importance is that this system is dynamic, not static. The system can always be re-defined and altered. There is nothing in the logical system that makes re-legislating rules an impossibility. The only rules the system is self-governed by necessity are those of logical inference. This avoids the stupid questions of metaphysical logic like “Are the rules of logic true in all possible worlds?” This allows for the possibility of truly establishing all meaning in an axiomatic set theoretic system no less robust than mathematics. Modal logic is revolutionized, and so is philosophy of language. If this is true, and what has previously been stated in relation to the possibility of Logical Semanticism, philosophy of language is almost completely made referentially disambiguated, the foundations for determination (quantification) of all meaning, made possible through the robust calculus of the strict implication of quantification modal logic, made to conform to language.

The development of all of these ideas, replacing truth-conditional theories of meaning, adhering to the referential aspect of the referential theory of meaning (in this case, to denoting intensional classes), direct reference to definite singular terms (including proper names), disambiguating the difference between sense of names (e.g. Dasha, Rachel, and Scott examples) and sense of objects (e.g. the pure direct referential denotation of logical tags—that is, definite singular terms and concrete particulars), semantic analyticity, semantic necessity, and the role of necessary attributes, all lead to the Intensional Axiomatic Set Theory that quantifies, regulates, and makes the application of Logical Semanticism possible. This theory will now be turned to as the rules that make all logical semantic systems possible, whether they be “natural” languages, formal languages, the language of artificial intelligence, etc. All languages need definite singular terms (the null class), absolute general terms (the infinite set of intensional classes from one to infinite integers), and rules for the relations of terms to one another (of aRb relations, such as states of affairs and complex linguistic statements). The rules for each of these necessary components of language are developed in the formation rules and axioms of the following system. The axioms provide ways of overcoming problems in the philosophy of language, such as the substitutivity of identicals, which will be made explicit later on.

**IV. Intensional Semantic Axiomatic Set Theory**

This section lists all necessary details for the quantificational logic that grounds the foundations of theoretical semantics. Rules are given for all linguistic expressions, from proper names and other definite singular terms to the most general of absolute terms, including complex relations and states of affairs. The system covers semantic information from nothing to infinity and everything in between. The following five categories—Vocabulary, Formation Rules, Rules of Inference, Semantic Rules, and Axioms—are the minimal requirements for the operation of the system. Interpretation of the set-theoretic system follows in the last sub-section, answering questions concerning the relation of this theory to Russell’s theory of types, similarities to portions of Zermelo’s system of 1908, the adoption of the need for a finite axiomatized system (following John von Neumann), and other philosophy of mathematics and philosophy of logic issues.

**A. Vocabulary**

**Logical Operators and Meta-Symbols:** ( ) {parentheses}, (, ) {punctuation},

~ {negation}, & {conjunction}, V {disjunction}, {implication},

≡ {biconditional}, ( ) {universal quantifier}, ( ) {existential quantifier},

□ {necessity operator}, ◊ {possibility operator}, {set symbol}, ⊆{power set

symbol}, I {identity relation symbol}, R {relation symbol}, Ø {null class},

α {predicate letter}, ( ! ) {quantificational uniqueness}, Φ {Intensional symbol}.

**Intensional Classes/Types {well-formed formulae}:** Φ = [φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1},

π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…].

**Functional Variables: **Φ = [F, G, H, J,…F_{1}, G_{1}, H_{1}, J_{1},…F_{n}, G_{n}, H_{n}, J_{n},…].

**Intensional Variables {intensional properties}:** Φ = α_{1},…α_{n} = [P, Q, N, M,…P_{1}, Q_{1},

N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…].

**Logical Constants {denotation tags}:** Ø^{Φ} = [A, B, C, D…A_{1}, B_{1}, C_{1}, D_{1},…A_{n}, B_{n}, C_{n, }

_{ }D_{n},…].

**Intensional Variables {intensional properties}:** Φ = α_{1},…α_{n} = [x, y, z, t, u, v, w, x_{1}, y_{1},

z_{1}, t_{1}, u_{1}, v_{1}, w_{1},…x_{n}, y_{n}, z_{n}, t_{n}, u_{n}, v_{n}, w_{n}…].

**B. Formation Rules**

**I.** Validation of the vocabulary of the system is judged by the following rules:

1. Let t_{i} be any term with unspecified valence.

2. Let t_{i} be any constant, including primitive symbols, *if and only if* t_{i} is a term

with no free variables.

3. Let t_{i} be any variable *if and only if* t_{i} is a term whose only free variable is itself.

4. Let t_{i} be any expression of the function P(t_{1}…t_{n}) of valence n ≥ 1 be a term,

whereby valence is the n-adic determination of free variables assigned to t_{i}.

5. Closure clause: only terms meeting rules *I.** 1-4* are terms; nothing else is a term.

**II.** Let the infinite sequence [(A, B, C, D…A_{1}, B_{1}, C_{1}, D_{1},…A_{n}, B_{n}, C_{n, }D_{n},…)] denote functional constants and the infinite sequence [P, Q, N, M,…P_{1}, Q_{1}, N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…] denote functional variables equivalent to the infinite sequence [F, G, H, J,…F_{1}, G_{1}, H_{1}, J_{1},…F_{n}, G_{n}, H_{n}, J_{n},…] of functional variables. Functional variables are defined in terms of n-adic variables, whereby functional variables of degree 1, 2,…n…: monadic functional variables F_{1}, G_{1}, H_{1}, F …; dyadic functional variables F_{2}, G_{2}, H_{2}, F …; n-adic functional variables F_{n}, G_{n}, H_{n}, F …;….

**III.** A formula is defined as any finite sequence of primitive symbols. Write “wff” for well-formed formula and “wf” for well-formed. A wff is defined recursively under the following formation rules:

1. An intensional variable (property) of the infinite sequence [x, y, z, t, u, v, w, x_{1}, y_{1}, z_{1}, t_{1}, u_{1}, v_{1}, w_{1},…x_{n}, y_{n}, z_{n}, t_{n}, u_{n}, v_{n}, w_{n}] is wf.

2. Functional constants, functional intensional variables, and functional variables are already stipulated in formation rule II. A functional constant or variable is wff.

3. Intensional classes are wff. Intensional classes are complex functional variables constituted by variable relations.

4. If intensional classes are wff’s, then φ and ψ are wff in: ~ φ, (φ & ψ), !(φ), (φ), ◊φ.

5. Closure clause: The only wff are those derivable from the wff enumerated in rule *III. 4*. This includes but is not exclusive to: ( φ), φ ψ, φ ⊆ ψ, and □φ.

**IV.** **Rules of Class Construction:** Let the following rules determine the formation rules governing the infinite sequence [φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1}, π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…], which are wff. The rules governing wff are defined recursively as follows:

1. Let the infinite sequence [φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1}, π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…] denote the infinite series of intensional classes.

2. Let the infinite sequence [x, y, z, t, u, v, w, x_{1}, y_{1}, z_{1}, t_{1}, u_{1}, v_{1}, w_{1},…x_{n}, y_{n}, z_{n}, t_{n}, u_{n}, v_{n}, w_{n}…] denote intensional variables of intensional classes. Let the infinite sequence [P, Q, N, M,…P_{1}, Q_{1}, N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…] denote relations of intensional variables, whereby relations are a category of functional variables. Let any functional relation variable (relative general term) be transformable into an intensional variable (absolute general term). Let intensional variables and functional relation variables constitute α_{1},…α_{n}.

3. Let any intensional class denote: (φ)(A_{0}…A_{n} V α_{0},…α_{n}), whereby the disjunction is inclusive. Let A_{0}…A_{n} denote the range of singular terms. Let α_{0},…α_{n} denote the range of intensional variables. Let no intensional class be empty of both functional variables and intensional variables, for an entirely empty intensional class is semantically classified as *meaningless*. Let all intensional classes be governed by semantic necessity and/or semantic possibility such that all intensional classes denote the form: (φ) φ [□ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})].

4. Let the infinite series of intensional variables relate to intensional classes, whereby these variables are taken as predicates, whereby φ_{x} is a mono-predicate bound class, φ_{xy} is a dyadic-predicate bound class, φ_{xyz} a triadic-predicate bound class, and φ_{n} an (n-adic)-predicate bound class.

5. Let the infinite sequence of functional variables [(A, B, C, D…A_{1}, B_{1}, C_{1}, D_{1},…A_{n}, B_{n}, C_{n, }D_{n},…)] denote 0 valence. When φ is equal to 0 valence, written as φ = 0, whereby 0 indicates some functional variable A_{n}. This rule is in conformity with rules *IV. 1-4*.

6. Let [(A, B, C, D…A_{1}, B_{1}, C_{1}, D_{1},…A_{n}, B_{n}, C_{n, }D_{n},…)] form unique intensional classes of the form “[φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1}, π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…]” whereby [(A φ), (B ψ), (C ω), (D π),…(A_{n} φ_{n})]. A dyadic-predicate bound class takes the form (A, B φ_{AB}), a triadic-predicate bound class the form (A, B, C φ_{ABC}), and an (n-adic)-predicate bound class the form (A_{1}…A_{n} φ_{A1…An}).

7. Let all intensional classes of the form φ(A_{1}…A_{n} φ_{A1…An}) be quantified as [ (A_{1}…A_{n} φ_{A1…An}) = φ_{A1…An}].

8. Closure clause of *IV. 5-7*: Let no other sequence, finite or infinite, constitute 0 valence, and hence interchange in this regard, except under the specified conditions of *IV. 5-7*.

9. Let any term denoting any reflexive relation …R… be written with the infinite sequence of functional relation variables of the form [P, Q, N, M,…P_{1}, Q_{1}, N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…], whereby the infinite sequence of variables [x, y, z, t, u, v, w, x_{1}, y_{1}, z_{1}, t_{1}, u_{1}, v_{1}, w_{1},…x_{n}, y_{n}, z_{n}, t_{n}, u_{n}, v_{n}, w_{n}…] constitute the possible properties of any relation …R…. Let the possibilities of any relation …R… denote the possibilities of valid formation in first order predicate logic, such as: Px, x = y, or P = Q.

10. Let α_{1},…α_{n }signify the infinite sequence of variables of any valence, whereby α_{1},…α_{n}Rα_{1},…α_{n} signifies the possibilities of the infinite sequence of functional relation variables delineated in rule *IV. 10*: ARx, xRy, ARB, xRx, ARA…; A_{1}…A_{n}RA_{1}…A_{n}, A_{1}…A_{n}Rx_{1}…x_{n}, x_{1}…x_{n}Rx_{1}…x_{n}, x_{1}…x_{n}RA_{1}…A_{n}…;… equate to P(Ax), P(x,y), P(A,B), P(x), P(A)…; P(A_{1}…A_{n}, A_{1}…A_{n}), P(A_{1}…A_{n}, x_{1}…x_{n}), P(x_{1}…x_{n}, x_{1}…x_{n}), P(x_{1}…x_{n}, A_{1}…A_{n})…;….

11. Let Q(P) denote a dyadic relation, N(Q(P)) a triadic relation, M(N((Q(((P)))) a quadratic relation, and P_{1}…P_{n} an n-adic relation.

12. Let [P, Q, N, M,…P_{1}, Q_{1}, N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…] form intensional classes

[φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1}, π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…], whereby [(P φ), (Q ψ), (N ω), (M π),…(P_{n} φ_{n})]. A dyadic-predicate bound class takes the form (P, Q φ_{AQ}), a triadic-predicate bound class the form (P, Q, N φ_{PQN}), and an (n-adic)-predicate bound class the form (P_{1}…P_{n } φ_{P1…Pn}).

13. Let all rules governing [P, Q, N, M,…P_{1}, Q_{1}, N_{1}, M_{1}…P_{n}, Q_{n}, N_{n}, M_{n},…] apply to [φ, ψ, ω, π,…φ_{1}, ψ_{1}, ω_{1}, π_{1},…φ_{n}, ψ_{n}, ω_{n,} π_{n},…], whereby [(φ φ_{1}), (ψ ψ_{1}), (ω ω_{1}), (π π_{1}),…(φ_{n} φ_{n})]. A dyadic-predicate bound class takes the form (φ, ψ π_{φψ}), a triadic-predicate bound class the form (φ, ψ, ω π_{φψω}), and an (n-adic)-predicate bound class the form (φ_{1}…φ_{n } φ_{1…}φ_{n}).

14. Types and Classes: In accordance with Russell’s theory of types, a class of the equivalent type to a second class cannot be contained in the other, they can only be equivalent. For example, the null class contains the infinite series of definite singular terms, whereby one such term that forms one unique intensional class can be equivalent to another unique intensional class but cannot be contained in the other class. Class membership can only be formed with a higher type.

15. Closure Clause: Any terms t_{i} is a term of this system *if and only if* rules *IV. 1-14* are followed; nothing else is valid. Any formula f is a wff of this system if and only if rules *IV. 1-14* are followed; nothing else is valid.

**V.** Bound and Free Variables: An occurrence of an individual variable α in a wff φ is a bound occurrence if it is in a wf part of φ of the form ( α)φ. Otherwise it is a free occurrence.

**C. Rules of Inference**

Assume definitions, postulates, axiom schemata, and rules of inference of first order predicate logic and quantified modal logic (i.e. Ruth Barcan Marcus’ functional calculus of first order based on strict implication and subsequent deduction theorem). All rules of inference follow the standard rules of first order symbolic logic in relation to *functional variables.* Let the formation rules of class construction as delineated in section 2 rules *IV. 1-15 *superimpose upon first order predicate calculus, for any rule, axiom, definition, postulation, or axiom schemata inconsistent with this system.

Let φ_{x} and ψ_{x} be predicative functions of a single variable x. If φ_{x} be a function, then ├ is that which asserts φ_{x}.

From ├ φ and ├ φ ψ derive ψ.

If ├ φ, then ├ φ.

If ├ φ, then ├ □ φ.

From these rules derive all further rules of inference of this system.

**D. Semantic Rules**

**I.** The following rules govern the significance of □, ◊, , ≡, , I, and Φ. The semantic rules of □ and ◊ follow Belcher’s semantics for modal semantic logic, namely the rules of Logical Semanticism, such that:

1. Let “strictly implies” be denoted by . Follow C. I. Lewis and Langford.

2. Assert □F F = □Fx Fx = □ (Px_{1}…x_{n}) (Px_{1}…x_{n}) = □φ φ. Let any functional variable of modal logic in conformity with the modal logic axiom □F F be translatable into an intensional class.

3. Assert □A A = □∃!(A) ∃!(A) = □φ φ. Let any logical constant of modal logic in conformity with the modal logic axiom □A A be translatable into an intensional class.

4. **Prohibition on Kripke:** The semantics of modal operators and modal quantifiers do not signify “all possible worlds”, “books”, “worlds”, “models”, etc. of the Kripkean modal semantics. These significations are prohibited by the system.

5. Let x denote intensional property x, whereby x is a predicate. Let □ denote *semantic necessity*. Let ◊ denote *semantic possibility*. Let semantic necessity take an epistemological and semantic analytic sense, whereby anything that is semantically necessary is true in virtue of its meaning, related through cognitive synonymy. Let semantic possibility take an epistemological signification of “empirical generalization”, whereby a semantic possibility is a logically consistent, non-contradicting possible property or relation. Let there be no metaphysical commitments entailed in the semantics of the system.

6. Let □( x) denote: “It is semantically necessary all properties x”, whereby x can denote any number of properties. Let ~x denote the semantics of “no intensional property x”, derived from the formation rules. Let □( x) always correspond with some intensional class φ. Let the same rules apply for □( x) whereby “all” is replaced by “some”. Let all formation rules apply.

7. Let necessity for definite singular terms of the form □( x_{i})(A_{i}x_{i}) establish an etymological meaning, whereby no reference to an actual object is established, but simply the relationship of a property to the name of a term, of x_{i} to A_{i}.

8. Let x denote intensional property x, whereby x is a predicate. Let ◊ ( x) denote “It is semantically possible all properties x”, whereby x can denote any number of properties. Let ~x denote the semantics of “no intensional property x”, derived from the formation rules. Let ◊( x) always correspond with some intensional class φ. Let the same rules apply for ◊( x) whereby “all” is replaced by “some”. Let all formation rules apply.

9. Let □φ(x φ) denote: “The *intensional* class φ *strictly implies* all semantically necessary properties x.” Let ◊φ(x φ) denote: “The *intensional* class φ *strictly implies* all semantically possible properties x.”

10. Let Ø^{Φ} signify to the class of all definite singular terms, including proper names, which is equivalent to: [ ! (A, B, C,…S, A_{1}, B_{1}, C_{1},…S_{1}, A_{2}, B_{2}, C_{2}…)]. Let each term take the form of a unique class. Let each term be intensionally empty, denoting only direct reference to one and only one object. The rules of types and classes, formation rule *IV. 14,* is here applicable, whereby any empty *class* of the null set *type *can be equivalent but not contained in an empty class of the null set type.

11. Let ≡ be interchangeable with the identity relation I for any reflexive relation.

12. Let the I of the identity relation φIψ denote numerical, qualitative, or classificatory identity, contingent upon the values and subsequent relations of □ φ(A_{0}…A_{n} V α_{0},…α_{n}), □ ψ(B_{0}…B_{n} V β_{0},…β_{n}), and □ ω(C_{0}…C_{n} V γ_{0},…γ_{n}), whereby: (1) when φ and ψ and ω represents Ø^{Φ}, I is *numerical*, is representative of one object; (2) when φ and ψ and ω represent absolute general terms (concepts), I is *qualitative*, and is representative of one (semantically synonymous) intensional class; and (3) when φ represents Ø^{Φ} and ψ and ω represent a concept(s) or when ψ represents Ø^{Φ} and φ and ω represents a concept(s) or when φ and ψ represent Ø^{Φ} and ω represents a concept(s) or when φ and ψ represent a concept (s) and ω presents Ø^{Φ}, then I is *classificatory*, and represents an intensionally classed (identified) object.

13. Let φIψ denote a higher intensional set type ω, whereby φ and ψ are its members.

14. Let Φ^{n} denote any series of intensional predicates numbering one or higher α_{1},…α_{n.}

15. Let Φ^{∞} denote the infinite intensional type. Let the semantics of Φ^{∞} denote “meaning”, whereby “meaning” is the infinite type category. Let meaning be contained in the notion of meaning, ad infinitum. Let anything meaningful be meaningful.

**II. **All additional semantic rules must be in conformity with the rules of Logical Semanticism.

**III.** Derive the Axiom of Intension from the semantics, rules of inference, formation rules, and vocabulary here employed in the system.

**E. Axioms**

**First Axiom: Axiom of Intension**

├ [□ (x) □ (y) □(x P ≡ y P) ∃(φ)][29] ≡ □ (Pxy) φ_{xyz}

Summation of the Axiom: This axiom presupposes the New Semantics of Belcher’s modal logic and sections 1-4 of this minimalist intensional system. These necessary presuppositions for the deduction of this first axiom can be expressed as: ├ □Fx Fx = □ (Px_{1}…x_{n}) (Px_{1}…x_{n}) = □φ φ. Let □F F be the first axiom of modal logic, where F is a functional variable and never a logical constant (see the third axiom for the quantification of logical constants, where ~x & ~y). In accordance with the New Semantics, let a functional variable always be implicitly accompanied equivalently expressible as an intensional variable: P_{1}…P_{n} = x_{1}…x_{n}. Let these properties be equivalently expressible in the form of an intensional class. Let the formation rules of class construction serve as the translation rules from first order logic to the mathematical logic of intensional classes and vice versa. P becomes z in relation to φ in □ (Pxy) φ_{xyz}. The axiom asserts that when intensional terms *x* and *y* have identical properties/attributes/meanings, that x and y are semantically synonymous in relation to P and thereby interchangeable terms belonging to the same intensional class.[30] All variables are bound. For instance, if P = ‘bachelor’, x = ‘unmarried male’, and y = ‘single man’, then all terms strictly imply the other. This can be semantically exhibited in any case whereby any term of three terms can be P, which establishes the equivalency of strict implication.

**Second Axiom: Axiom of Intensional Empty Set**

├ □A A = □∃!(A) ∃!(A) = □φ φ = (∃φ) □(∀~Φ^{n})[(∃~x)(φ_{~x} (~y)(φ_{~y} ~y = ~x) φ ] ≡ Ø^{Φ}

Simplification of the Axiom: ├ (∃φ) ~(Φ^{n} φ) ≡ Ø^{Φ}

Summation of the Axiom: This internally redundant axiom asserts that there is set, a null class, whereby no intensional members exist. Any term is identical to the null set if and only if this criterion is met. This axiom in itself does not give a distinguishing criterion to separate one logical constant (definite singular term) from another. This requires the sixth axiom, the axiom of empty set regularity, which regulates this axiom.

**Third Axiom: Axiom of Pairs**

├ □∀(*x)*□∀(*y)*□∀(*z)*∃(*φ)*(*z *∈*φ* ≡ *z *= *x *V *z *= *y*) = □∀φ(x, y)

Summation of the Axiom: The axiom asserts that given any terms x and y there exists a pair set □∀φ(x, y), or just (x, y), whereby (x, y) contains all intensional properties of x and y together. The intensional variables x and y are replacable with φ and ψ with the aide of the fourth axiom, the axiom of the intensional power set.

**Fourth Axiom: Axiom of Intensional Power Set**

├ φ ⊆ ψ = {∃(φ) φ [□∀(A_{0}…A_{n} V α_{0},…α_{n}) V ◊∀(B_{0}…B_{n} V β_{0},…β_{n})] & ∃(ψ) ψ[□∀(C_{0}…C_{n} V γ_{0},…γ_{n})] V ◊∀(D_{0}…D_{n} V μ_{0},…μ_{n})]} ≡ {((α_{i} V A_{i}) V (β_{i} V B_{i}) ∈ φ))((γ_{i} V C_{i}) V (D_{i} V μ_{i}) ∈ ψ)) ∃(ω) ω [□∀(E_{0}…E_{n} V ν_{0},… ν_{n}) V ◊∀(L_{0}…L_{n} V ξ_{0},…ξ_{n})](ω ∈ φ ≡ ω ∈ ψ)]}

Summation of the Axiom: This axiom establishes the rules governing the formation of

any intensional class, namely union through intensional identity.[31] This axiom follows from the formation rules, axiom of intension, axiom of pairs, and axiom of intensional null class. This axiom explicitly allows for the conjoining of classes of infinitely different types. Unlike the axiom of intension, the axiom of intensional power set conjoins logical constants and variables, allows for the identity relation of an infinite level of complex classes. The semantics of this axiom is the basis of the entire system. The axiom of intensional power set also allows for the class operation of union, an explicit axiom for which would be redundant. This axiom is regulated by the seventh axiom, the axiom of intensional type regularity.

**Fifth Axiom: Axiom of Intensional Infinity**

├ (∃*φ)*[Ø^{Φ} ∈ *φ* & □∀(Φ^{n} )(Φ^{n} ∈*φ* )] ≡ Φ^{∞}

Summation of the Axiom: This axiom establishes the intensional class of infinity. This axiom is derived from the axiom of intensional power set. This allows for the creation of both infinite classes and infinite types of those classes.

**Sixth Axiom: Axiom of Intensional Empty Set Regularity**

├ (∃φ) ~(Φ^{n} φ) (∃!A)(A ∈φ & (∃!*B)*(*B *∈φ ~(*B *∈*A*)))

Summation of the Axiom: Avoids the collapse of all definite singular terms (logical constants) as interchangeable while still accounting for their complete lack of intensional properties, allowing for the uniqueness of eact term. This allows for the identity of terms through that axiom of intensional power set, but this axiom provides for the possibility of the non-identity of specific classes of logical constants.

** **

**Seventh Axiom: Axiom of Intensional Type Regularity **

├ (∃*φ)*[*φ* ≠ Ø^{Φ} *(*∃*ψ)*(*ψ *∈ *φ & (*∃*ω)*(*ω *∈*φ* ~(*ω *∈*ψ*)))]

Summation of the Axiom: This axiom avoids the circularity of intensional classes, whereby classes are ultimately differentiable and free of chains of set types equaling one another; also providing the possibility of class uniqueness. Specifically, this axiom applies to the fourth axiom, the axiom of intensional power set.

**F. Interpretation of the Axiomatic System**

§1. Expansion and Non-Completeness of the Finite System

The minimalist intensional system here outlined provides the basis by which any supplemental series of axioms, including an infinite axiom schemata, can be augmented if and only if the axioms to be added entail no contradiction of the rules governing the minimalist system. The system, therefore, is not complete, for other consistent axioms can be augmented. An axiom could be attached here expressing this point, namely the augmentation of any futher formulae in consistency with the system thus developed.

The minimalist system itself, however, is truly a finite axiomatic system, for it lacks axiom schemata in its present form. As John von Neumann’s set theory is also finite, this system has modeled itself in the spirit of Neumann’s work, opposed to Zermelo’s on this point. The greatest advantage of a finite system is not the elegance or aesthetic appeal, but the practicality of its application. The system can afford to be finite because the formation rules presuppose the necessary philosophical concepts.

§2. Derivability of Axioms

The axioms of the minimalist system are derivable from one another. The axiom of intensional regularity is derivable from the axiom of intensional power set. The axiom of intensional null class regularity is derivable from a combination of the axiom of power set and the axiom of intensional empty set. The axiom of intensional regularity and axiom of intensional empty set regularity could be combined to form one complex axiom, which would still be derivable from the already named axioms. The axiom of intensional infinity is derivable from the axiom of intensional power set. The axiom of intensional power set is derivable from the combination of the axiom of intensionality and axiom of pairs, whereby the axiom of intensional empty set is an applicable axiom to the function of the axiom of intensional power set. The axiom of pairs is derivable from the axiom of intension. The axiom of intensional empty set is derivable from the axiom of intension. The axiom of intensionality is derivable from the formation rules for class construction of the system and the semantics of the system, whereby the axiom of intensionality takes on the status as the postulation of what constitutes identity. Furthermore, the notion of class union is implicit in the axiom of intensional power set and thus requires no additional axiom. This completes the minimal requirements for the operations of classes. The system itself is thus not independent insofar as the axioms are interrelatedly derivable and not independent.

§3. Syntatic Affinity and Semantic Disparity to Zermelo

An abundance of similarities exist syntactically in relation to the work of Zermelo’s axiomatization of mathematics. The significance of what that syntax demonstrates is, however, entirely disparate. The axiom of intensionality is antithetical to the axiom of extensionality, for extensionality is a matter of material equivalence, whereby the semantics of intensional logic requires synonymy of meaning, and not the worker requirement of equivalent members. The axiom of pairs is identical to Zermelo in every practically every way. The axiom of intensional empty set, the axiom of intensional infinity, and the axiom of intensional class regularity are syntactically similar to Zermelo but represent disparate semantics to that of their mathematical extensional formulations, as the formation rules demonstrate. However, the axiom of intensional class regularity has been syntactically ameliorated in quite important and novel ways. The axiom of intensional power set is an entirely novel discovery, as is the axiom of intensional empty set regularity, which have no similarities to Zermelo or any axiomatic system. These two axioms are the major force of the system. Similarly, most of the axioms of the system are derivable from the axiom of power set. Furthermore, the intensional semantic axiomatic theory here outlined provides a means of relating the mathematization of classes, natural language, and quanitifed modal logic. The axioms and the semantics of a system generation intensional classes has been considerably reworked to the point where contribution is due to Zermelo but this is no slight modification of his 1908 system.

§4. The Russellian Theory of Types

The Russellian theory of types is adopted in the system. The types are categorized based upon their structure as intensional classes, in terms of definite singular terms and absolute general terms. Definite singular terms are taken as the lowest type. Increasing levels of types are constructed based upon their complexity as containing properties/attributes. The theory of types is thus implicit in the axioms.

§5. Philosophical Implications

This system provides a method of disambiguating quantificational contexts of otherwise referentially opaque descriptive scopes, particularly the relation of definite singular terms and abstract general terms. Such quantificational contexts are made transparent, allowing for the substitutivity of identicals, the proof of which can be developed from what has here been provided. Furthermore, the system replaces the semantics of modal logics with a non-metaphysical understanding, replacing all possible worlds with logical-linguistic systems. The philosophical relationship of the logical-linguistic system is epistemological, the elimination of the traditional philosophical problems of modality.

**V. Logical Semanticism and Its Application**

Applying intensional semantic axiomatic set theory to the quantification of definite singular terms, objects, and states of affairs is the intended purpose of the theory. This section is dedicated to outlining the technicalities of this process of quantification concerning the philosophically pertinent categories. The section ends by linking such quantification to epistemology and alluding the further possibilities of which are also indicated in the appendix.

In the broadest sense, an intensional class is denoted: (φ) φ [□ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})]. The disjunctions are inclusive, which conveniently allows the universality of intensional classes to this single formulation, fitting all various kinds of definite singular terms and absolute general terms. The upper case letters represent the set of all logical tags, that is, the syntactic names which equivalently correspond to the same object (and hence belong to the same intensional class). The lower case letters represent the attributes or properties that have a relation to the intensional class. The nature of this relation, as either necessary or possible, depends on the semantic modal operator that the attribute or property is bound. The universal quantifier denotes that these relations hold for all cases of the intensional class that is being postulated. The existential quantifier an association with the intensional class itself denotes the nature of the class as a postulation in logical space, that there is some information such that X. This signifies the syntactic quality of an intensional class, “some syntactic string of symbols exists such that…”. That which is quantified as universal is the “such that” of “some syntactic string of symbols”. This is because there is nothing metaphysically necessary about naming, for it is not the case that all “cardinals” must signify red, for some uses signify Catholic religious leaders, a passerine bird, and a special principle. A semantic translation matrix can be drawn to separate such syntactically identical but semantically and referentially disparate intensional classes from one another. But the full nature of these translation matrices cannot be developed and must be left to an epistemological investigation, where it is more useful.

Some intensional classes will have both necessary and possible attributes, whereas others will only have one kind of attribute; others still will have no attributes at all. But an intensional class has to have some information, or else the class is meaningless. An empty intensional class, having neither logical tokens to denote a reference to syntax or attributes to refer to the meanings of syntax makes the term intensionally (conceptually) empty. This is the very definition of meaninglessness. As shall be seen, the null class in intensional logic is not that which is meaningless, for this sort of situation is said to be outside the domain of the system, for it cannot be part of the formation rules. Instead, logical tags, ones that contain no attributes and only denote singular reference (they tag something as something) are all members of the intensional null class, for their semantic (non-metaphysical) “substratum” role. This will be investigated in the quantification of definite singular terms. The quantification of attributes will be discussed in the two sections on the quantification of objects and the quantification of states of affairs.

**(A) The Quantification of Definite Singular Terms**

Set theory requires a null class and in this regard definite singular terms, representing the bare notion of singular denotation to some object, are quantified as semantically void of necessary attributes.

The conceptual justification for this direct theory of reference of definite singular terms results in earlier arguments concerning analyticity, principle of identity, and the problem of substitutivity of identicals. The quantificational procedure by which to transform the conceptual arguments into a working logical theory require two particular axioms: one axiom governing the null class and one regulating the relation of subsets of the null class. Recall that the second axiom asserts: □A A = □∃!(A) ∃!(A) = □φ φ = (∃φ) □(∀~Φ^{n})[(∃~x)(φ_{~x} (~y)(φ_{~y} ~y = ~x) φ ] ≡ Ø^{Φ}. This states that the first rule of quantified modal logic (the necessitation rule) is equally expressible in this logical system, resulting in some specific class that is taken as empty if and only if that set is empty of attributes. Existentially (syntactically) unique classes, such as ∃!(A) are the intensional classes which denote definite singular terms of which are all unique sub-sets of the null class. But this axiom is only capable of asserting the logical existence of the null class in general and does not have the capacity to differentiate one definite singular term from another. In other words, if this were the sole axiom that governed the null class, there would be no logical or linguistic mechanism to separate proper names or other references to singular objects. In other words, all definite singular terms would be equal and always referred to whenever one was invoked. This would be a massive quantificational problem.

Fortunately, there is a second axiom which specifically clarifies the unique nature of each unique referent, strong enough to allow separation of each definite singular term from one another but not so strong to make it impossible that two or more definite singular terms could indeed refer to the same object. This is important, for if the axiom was too strong there could be no such thing as two names that refer to the same object, but if it too weak there would be no procedure by which to differentiate any name from any other. The quantificational tool that avoids all confusion is sixth axiom which asserts: (∃φ) ~(Φ^{n} φ) (∃!A)(A ∈φ & (∃!*B)*(*B *∈φ ~(*B *∈*A*))). This specifically details how logical tags are all materially equivalent (because they do not have any attributes) but they are intensionally differentiable (since they each denote a different object through their uniqueness).

Of course, this is on the level of quantification, and names can be made to refer to the same object. If one desires to specify that ‘Mark Twain’ and ‘Samuel Clemens’ or ‘Giorgione’ and ‘Barbarelli’ are co-referential, a propositional attitude expressing a doxastic assertion of such equality requires the knowledge be antecedently known, the situation must be tautological. These conceptual issues were previously addressed in section II. As for their quantification, one simply would stipulate that there exists some intensional class such that A strictly implies B, guaranteeing the co-referential status, and eliminating the problem of identifying definite singular terms. This also eliminates the problem of the substitutivity of identicals, at least, that of definite singular terms (for the time being) insofar as there is a quantificational procedure by which to interchange terms freely that are known to be co-referential. Otherwise, there is no justification for substitutivity, whether or not one adopts this system.

The quantification entailed in Logical Semanticism allows for all definite singular terms to be treated equally, belonging to the same Russellian type, that of intensional nullity. This quantification makes Marcus’ direct referential theory possible while avoiding the problems generally associated with her work, particularly because a metaphysical conception of the relationship of names to objects and names to meanings are universally avoided. Scope ambiguity and referentially opaque contexts are likewise eliminated, as demonstrated by the axioms governing definite singular terms, for the system treats logical tags neither too strong or too weak in their relationships.

**(B) The Quantification of Objects**

The basis of the theory is meant to quantify objects, to enumerate the monadic, dyadic, triadic, n-adic attributed predications constituting the meaning of some object. The quantification of objects entails the second side of the inclusive disjunction that regulates all intensional classes. Objects, as such, are the set of all intensional classes that do not contain definite singular terms. In effect, the quantification of objects represents the class of all absolute general terms, the universals. Thus, as opposed to logical tags that only concern reference, absolute general terms, objects, do not have a referential component. These terms, when quantified, are entirely semantic and only require the enumeration and relation of attributes, or properties, and nothing else. The number of these attributes, as made possible by the axiom of intensional infinity, ranges from one attribute to infinity. Conceptually, what could an infinitely predicated concept practically entail in a “natural” language? Perhaps the notion of God, being, or even meaning could be potential terms with an infinite number of necessary components. There are many more terms that could potentially have an infinite number of possible attributes, as one need only endlessly assign adjectives to a universal. Russell is correct in arguing that infinity is not practical but the practical is ultimately subservient to the nature of logical and mathematic truth.

There is great importance in allowing for intensional infinity, the possibility of infinitely complex concepts and objects. There should be no limiting procedure for this would be artificial. The genius of Russell’s theory of types is precisely the recognition that finite systems and finite classes is simply imposing human limits onto an infinite mathematical plane, whereby each type is simply one greater ordinal. Similarly, the ordinals are given unlimited possibility.

Practically, however, for the purposes of the overwhelming uses of everyday language, there will generally be little need to go beyond a low number of predicated attributes. Necessary attributes are those predications that, when taken together, define a term to give some syntax its meaning. ‘A chair’ is an indefinite singular term but as Quine shows easily reduces to an absolute general term (say ‘chairhood’). Thus, a chair has a necessary number of predicates, as previously shown, necessary for the possibility of the concept. This is true of all words, and whether or not their lexicographic definitions are chosen or more specialized definitions is not the issue. The issues is avoided precisely because each of these possibilities is its own logical semantic system, which ever one is ultimately chosen is a practical (and hence) epistemological choice based on justifications, social conventions, habits, etc., all of which are concerns not of the foundations of theoretical semantics. What is important is simply *how* to quantify *all* systems of definitions, *all *languages, and *all *procedures by which intelligent agents *do *make choices as to how to understand, classify, and differentiate the objects and concepts of the existential space of their being-in-the-world.

Thus, in this regard, the quantification of objects follows the earlier dictates granted of Lewis’ strict implication, of Carnap’s model of semantic analyticity. The process of quantification is simply enough. If “one or more legs” is taken as a necessary attribute of what constitutes ‘a chair’, then it becomes semantically necessary and if “leather” is taken as a contingent attribute, then it becomes semantically possible in relation to the intensional class ‘chairhood’. This procedure is universally true of all objects of all languages, and conforms to the basis of the intensional class.

**(C) The Quantification of States of Affairs**

Quantifying states of affairs is a more complex version of addressing objects. Firstly, the set of all possible intensional classes that have both definite singular and absolute general terms are states of affairs. Also, the set of all intensional classes that relate to other intensional classes, concerning objects, are states of affairs. Thus “Steven is a good husband” is an example of the first kind and “The billiard ball struck the table” the second. Quantifying these sorts of examples follows a more complex but similar procedure to that of definite singular terms and objects.

The quantification of states of affairs is straightforward enough, however, the conceptual details of the procedure unavoidably relates to epistemology and philosophy of science. The precise relationship of these subjects is critical to assessing the application and role of such quantification, which this paper is unfortunately limited in investigating. However, the appendix provides the outline of a method to relate the quantification of states of affairs to epistemology and philosophy of science, through the development of the relationship between probability calculus and the language and semantic modality of Logical Semanticism. The procedure here given is one that relates states of affairs to the possible range of correlation of coefficient values, establishing the subjectivized epistemic possibility and necessity of states of affairs based upon the rules of the epistemic logic. Thus, the full quantificational details of states of affairs cannot be given in this paper, for that requires epistemological and other issues, the scope of which far exceeds the capacity of this paper to demarcate. For the time being, Logical Semanticism suffices to cover the issues of definite singular terms, object naming, and quantifying precisely how objects are constructed and made commensurable in whatever logical semantic system is adopted.

**(D) The Relationship of Quantification to Epistemology**

States of affairs are fully quantified through epistemic logic, which is derived from the intensional semantic axiomatic set theory of Logical Semanticism. Like philosophy of language, requiring reformulation in epistemology is the analytic-synthetic distinction, the role of the a priori, the relationship of intuitions and concepts, the disambiguation of “empirical generalization” (“the class φ of empirical *objects* a, b, c…” versus “the temporal spread of *presentations* interpreted as inferring concept ψ, justified upon complete intrinsic intersecting properties (i.e. qualitative identity) of what concept ψ strictly implies to that of which is empirically observed and hence inductively generalized”), and ultimately the incorporation of a strict theoretical distinction between concept acquisition, concept recognition, and epistemic justification. The upshot is a thorough respect for the Sellarsian distinction between sense perception as the process of learning (concept acquisition) with that of sense perception as “the necessary conditions of empirical knowledge as providing the evidence for all other empirical propositions.”[32] However, what is lost in the Sellarsian dichotomy is the precise status of concept recognition, which is left ambiguous. This ambiguity is overcome when the role of learning and the role of justification are properly explicated, as there will be a crucial conjoining of particular theories historically taken as contrary. This can be seen in the surprising convergence of much of the Lewisian and Sellarsian traditions. These principles, how they are mutually related, and their consequences cannot be fully explicated or defended within the limited confines of this paper but provide an outline for a broader theory.

What might such modification entail? In regards to other aspects of Kantian transcendental idealism, the implication is a re-evaluation of certain basic tenets, whereby the a priori is taken as *definitive*; the application of pre-conceived linguistic categories to the classification of conceptually interpreted-sense-perception, constitutive of the subject’s world awareness. The explanation of such a procedure does not require strong foundationalism in the Cartesian sense, nor is it the free-floating Sellarsian/Davidsonian logical space of reasons, but involves an integration of both elements in a non-representational, non-copy theory of knowledge, taking all empirical statements, including that of naming particulars, as probable only. This is ultimately accomplished by appealing to some definitive a priori classificatory structure, a structure always open to re-interpretation with further experience and rules of germaneness, but a process that is not inflicted by skepticism. This is the outline of the basic structure of Lewis’ epistemology, including the fact Lewis is not susceptible to the myth of the given, for he is neither a phenomenalist nor a linear strong foundationalist that grounds all justification of empirical propositions in similar fashion. Lewis both recognizes that givenness is not knowledge of direct acquaintance of any version and that “givenness” is always conceptually interpreted. Indeed, most of Sellars’ critiques against phenomenalism, including Sellars’ examples are taken, without credit, from the pages of *Mind and the World Order*, particularly chapter V, “The Knowledge of Objects” (e.g. the “toothache” example, the “elliptical penny”, etc.).

The broader philosophical consequences entail that the classifications of objects can be constructed to be analytic for any sort of predicates, even those generally taken as empirically acquired synthetically structured generalizations only of a probable nature. That a “parrot has a raucous cry” is probable only can be altered within the language by re-assigning what constitutes the class of parrots as a priori, and therefore definitionally, requiring the predication of a raucous cry. Epistemologically, what becomes significant, therefore, is not the analytic-synthetic distinction but the general normative pragmatic procedure by which linguistic and discursive rules are applied to our definitions and the judgments we thereby make. When I give the ostensive report “This is a chair”, I am committing myself to an analytic statement. The concept chair is defined as (strictly implies) “a piece of furniture consisting of one seat, one or more legs, and one back for intended use by one entity.” This is strictly implied because my definition (concept) of chair a priori fits these necessary predications that when amalgamated is sufficient for my judgment that something in the world can be interpreted as that concept. Therefore, an empirical object is definable as a chair if and only if these four necessary and sufficient intrinsic properties are observable. Since it is the case that the one and only one object that ‘I’ *refer* to presently, at* time t*,* *does now *appear* to *me* to have all the necessary conditions of the concept chair, the particular object therefore meets the standard of justification to be classified and therefore named as a member of the intensional class of chairness. Indeed, “looks” talk in this case antecedes “is” talk in the sense of the empirical generalization of the conceptually interpreted recognition of the Heideggerian as-structure, something *as* something (i.e. a particular as a universal). But, on the other hand, Sellars is correct that “looks” talk comes after “is” talk in the sense of conceptualization as already being part of the mind, for the concepts and the holistic sense of their relations is *presupposed *and therefore there is no direct acquaintance. Psychological nominalism does indeed prevail. But Lewis is correct in emphasizing and outlining what Sellars mentions only in passing, which is the method of such conceptual acquisition is “foundationalistic”, that is, *non-epistemically* scientifically traceable to the social practices, language, and behavioral conditions by which the rudimentary concepts were *mind acquired*. But once these concepts manage to “make it into the mind” there is epistemological justification that can be and must be grounded on the a priori grounds and the discursive commitments that ground classificatory rules/definitions. What Sellars contributes is the need to take inference of recognizing particulars as possible only through antecedent conceptual acquisition, whereby Lewis makes clear the anti-skeptical element in all of this is the a priori nature of the logical space of reasons. To say that “a parrot is a bird” is a priori is to not influence the world at all, but only assign empirical objects into classes, not to confine what is experienced. And what allow for both traditions to come together are their mutual Kantian roots. In an age where such important American philosophers are fallaciously rendered as theoretically antithetical, a closer inspection reveals that in the broader pragmatic family, the similarities far exceed the differences.

Epistemologically, it is therefore important to conjoin the work in the foundations of theoretical semantics with investigations into theory of knowledge? Why? As Tarski rightfully understood, the theory of truth is a semantic concept and as such is intimately related to epistemological investigations into justified, true, beliefs. The major purpose of all epistemology is to outline not simply anti-skeptical critiques, but to given an outline for a theory of the relationship between concepts and objects, between first and second order truths, of justification and belief to that of empirical truth. Since truth is connected primordially to the sort of semantic inquiries of Logical Semanticism, there is little wonder that these issues are ultimately all connected.

**VI. Conclusion**

Logical Semanticism stipulates that the method for solving the basic linguistic problems of philosophy entails re-orienting the debate away from questioning the principle of identity or the principle of substitutivity, away from an ontological understanding of the problem of reference to non-existent objects. The nature of the philosophy of language debates in these matters is not well served by any of these investigations, for none of these four quandaries are genuine or difficult philosophical problems, despite their longevity as controversial. So, then, why has an adequate solution failed to attract philosophers? Solutions have been insufficient insofar as the wrong questions are being posited. The four fundamental problems derived from early 20^{th} century philosophy are quantificational anomalies of first order predicate logic and the philosophical presuppositions assumed of the unquestioned methodology of such quantification.

The problem is not the philosophy of language; the problem is the philosophy of logic. Therefore, the nature of the debate ought to be shifted to the level of quantification, to the question: Why do all these traditional quantificational logics generate linguistic referential and semantic paradoxes? Is there a method of quantification that circumvents these referential and semantic paradoxes? If there are such methods what would these new approaches to quantification necessary entail to be a possibility?

Therefore, it is more accurate to emphasize that Logical Semanticism is a philosophical theory of language through a particular set of presuppositions from the philosophy of logic. The primary quantificational unit of the theory entails representation of all linguistic expression into the logical terminology of definite or indefinite singular terms and relative or absolute general terms. This classificatory structure is taken as exhaustive. Any linguistic expression fitting this logical system of symbolic quantification is further transformable into denoting particular *types* of *intensional classes*. Both intensional classes as a general concept and moreover the philosophical properties assigned to the particular types of classes are necessary and fundamental to the theory.

Furthermore, the intensional classes and their respective types conform to a finite set of formation rules, semantic rules, and axioms of which govern the implementation, computation, and scope of both the quantificational and philosophical possibilities of Logical Semanticism. This formally axiomatized system is the quantificational component, given the name: Intensional Semantic Axiomatic Set Theory (for now on simply called: semantic set theory). The entirety of this novel set theory is included in the paper. Semantic set theory is a necessary component is the overall system, providing the proofs and logical consistency on the quantificational level of what Logical Semanticism thus ultimately depends.

Logical Semanticism presupposes the philosophical sustainability of the analytic-synthetic distinction, namely the employment of “cognitively synonymous” or “intensionally identical and thereby terminologically interchangeable” as valid instances of analyticity. Necessity is presupposed as both valid and *non-metaphysical*, thereby prohibiting Kripke’s or Hintikka’s semantics. Their work in modal and epistemic logic are replaced with a two-tiered semantics for the possibility and necessity operators, of which the epistemic logic is derivable from the intensional semantic logic, one set of operators from the other: semantic necessity (i.e. □), epistemological necessity (i.e. ■), semantic possibility, semantic possibility (i.e. ), and epistemological possibility (i.e. ♦). The semantic necessity and possibility operators presuppose the validity of the form of analyticity used in this philosophical theory of language. Thus, this theory directly challenges Quinian philosophy and takes the position that almost every fundamental aspect of “Two Dogmas of Empiricism”, *Word and Object*, and “Reference and Modality” are suspect and ought to be abandoned. This paper argues that material equivalence of extensional logics are not strong enough equivalency relations to overcome the major problems of 20^{th} century philosophy of language, including the problem of the substitutivity of identicals, and has therefore been exchanged for strict implication. More generally, the other three major problems of 20^{th} century philosophy of language—problem of non-existents, problem of negative existential assertions, and problem of identity of proper names—are all problems not of identity or language but problems of faulty logical quantification. For the last hundred years the nature of the problem has been completely misunderstood. Further presupposed is the philosophical validity of Ruth Barcan Marcus’ calculus of strict implication which derives the quantified modal logic (QML) off of C. I. Lewis’ S2. Also presupposed is the acceptability of the direct theory of reference as it applies to definite singular terms, including proper names. All of these enumerated presuppositions will be explicated and defended as the first priority of this paper. If any of these presuppositions falter, Logical Semanticism ought not to be adopted as an acceptable theory.

These major tenets and presuppositions were investigated historically and conceptually in section II. The relationship of these concepts were then developed in relation to the philosophy of language in section III, namely the traditional referential theory of meaning and truth conditional theories of meaning. Referential theories of meaning were seen as defective to the extent that the definition of reference was mistakenly narrowed to mere empirical reference, whereby the proper scope is produced by truth conditional theories of meaning, such as Kripke’s. But such theories were shown to have numerous metaphysical problems, due to the epistemological paradoxes, such as the Kantian antinomies, that make knowing the actual world in a modal context impossible.

This tradition was exchanged for a new approach with new questions, the notion of logical semantic systems with intensional classes. The intricacies of these systems which ultimately serve as the foundations for theoretical semantics was developed in section IV, the section on intensional semantic axiomatic set theory. The quantification rules and possibilities were extended and made explicit in section V, despite their implicit presence in the set theory.

Inference depends on reference, reference to meaning, meaning to intensional classes, and intensional classes to the rules of set theory. To know what to judge or infer depends on what one refers. What one refers depends on what is signified. The rules for reference and semantics, including how they are differentiable, depending on the development of intensional classes, which in turn depends on rules. In essence, this is the theory and what has been attempted to be proven in this work.

At the turn of the 20^{th} century the work of Russell and Whitehead resulted in the possibility for computers. As the 21^{st} century dawns, it is the responsibility of the coming era to take upon its shoulder’s the latest of intellectual responsibilities. By crafting a new mathematical logic with the integrity to solve problems in scope ambiguities of definitions and the referentially opaque contexts of 20^{th} century philosophy, the possibility of artificial intelligence cannot be far behind. Artificial intelligence requires a semantically rich mathematical logical and ultimately Logical Semanticism is designed to contribute in a meaningful and lasting way.

**Appendix: Epistemological Logic and Further Philosophical Possibilities**

Briefly developed in this appendix is the possibility for the quantificational conjoining of quantified modal logic with probability calculus of the correlation of coefficients, which conceptually provides a procedure by which epistemological issues can be subsequently derived from the antecedent developments in the foundations of theoretical semantics. The following sections provide the semantics, rules, and interpretation of such a system.

A. Semantics of the Operators

□ = Semantic Necessity

■ = Epistemological Necessity

= Semantic Possibility

♦ = Epistemological Possibility

B. Rules of the System

1. (φ) φ [□ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})]

2. (ψ) ψ[(x_{1}…x_{n}), (y_{1}…y_{n}), z, (-1 ≤ r ≤ 1)] ≡ ⊆ψ_{r}

3. (r) ■ ψ_{r} = (r) ■ψ_{r}{[ (φ) φ □ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})] ⊆ψ_{r}}

4. (r) ♦ψ_{r} = (r) ♦ψ_{r}{[ (φ) φ □ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})] ⊆ ψ_{r}}

5. {[( (ψ) ■ψ (ψ) ψ)]} { (φ) φ [□ (A_{0}…A_{n} V α_{0}…α_{n})] [ (φ) φ[ (A_{0}…A_{n} V α_{0}…α_{n})]}

6. {[ (ψ) ψ (ψ) ♦ψ]} { (φ) φ [ (A_{0}…A_{n} V α_{0}…α_{n})] [ (φ) φ[ (A_{0}…A_{n} V α_{0}…α_{n})]}

7. {[ (ψ) ■ψ (ψ) ♦ψ]} { (φ) φ [□ (A_{0}…A_{n} V α_{0}…α_{n})] [ (φ) φ[ (A_{0}…A_{n} V α_{0}…α_{n})]}

8. (r) ■ ψ_{r} ≡ ■ψ_{r}(r = -1 V r = 1 ■ψ)

9. (r) ψ_{r} ≡ φ ψ_{r}(-1 ≤ r ≤ 1 ψ)

10. (r) ♦ ψ_{r} ≡ ♦ψ_{r}(-1 ≤ r ≤ 1 ♦ψ)

11. (r) ψ_{r}[■r ≡ ■ψ(r = -1 V 1) (r) ψ_{r} ≡ ψ_{r}(-1 ≤ r ≤ 1)]

12. (ψ) ■ψ{[x_{i}(x_{1}…x_{n}), y_{i}(y_{1}…y_{n}), z, r_{i}(r = -1 V r = 1 (-1 ≤ r ≤ 1))] & [ (φ) φ

□ (A_{0}…A_{n} V α_{0}…α_{n}) V (B_{0}…B_{n }V β_{0}…β_{n})]} ≡

⊆■ψ}

13. (ψ) ♦ψ{[x_{i}(x_{1}…x_{n}), y_{i}(y_{1}…y_{n}), z, r_{i}(r = (-1 ≤ r ≤ 1))] & [ (φ) φ □ (A_{0}…A_{n} V

α_{0}…α_{n}) V (B_{0}…B_{n} V β_{0}…β_{n})]} ≡ ⊆♦ψ}

C. Interpretation of the System

The first rule derives from Logical Semanticism and allows for the derivation of intensional semantic axiomatic set theory with its intensional classes. The second rule asserts the relation of probability calculus, vis-à-vis correlation coefficient, with intensional classes that are states of affairs. The value r is given the full range of values befitting probability, the values of which correspond to the possibility and necessity operators in different ways. The result is the relationship of semantics, epistemology, and probability. From this initial development, Hintikka’s approach to epistemic logic, grounded in Kripkean semantics, is entirely replaced. This logic is more befitting of philosophy of science, which can be developed through an investigation into epistemology and related topics of which must be pursued elsewhere.

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[1] Alfred Tarski, A. P. Martinich (editor), *The Philosophy of Language*, “The Semantic Conception of Truth and the Foundations of Semantics (Oxford: Oxford Press, 2001): 71-2.

[2] C. I. Lewis, *Mind and the World Order* (New York: Dover Publishing, 1956): 197.

[3] Ibid: 197.

[4] Kant: 143.

[5] Wilfrid Sellars, *Empiricism and the Philosophy of Mind* (Cambridge: Harvard University Press, 1997): 121.

[6] Think of Quine’s distinction between logistic analyticity and synonymous analyticity, whereby Quine is attacking only the latter. The former is true in terms of purely syntactic considerations, namely through the principle of self-contradiction *without *appeal to concepts. Kant makes no distinction in this manner.

[7] For a substantive and engaging analysis of the subtle but pertinent differences between these definitions of analyticity, see chapter II of Laurence Bonjour’s *In Defense of Pure Reason* (Cambridge: Cambridge University Press, 1999).

[8] Immanuel Kant, *Critique of Pure Reason* (New York: Macmillian and Co., 1965): 49.

[9] Immanuel Kant, *Prolegomena to Any Future Metaphysics* (Cambridge: Hackett Publishing Co, 1972): 12.

[10] To be more accurate, Quine had interesting insights into the limitations of an *extensional* logic or language, by which: (1) questions of referential opacity in regards to substitution (and hence synonymy) are necessarily indeterminate when meanings are involved; and (2) synonymy is definitional and hence assigned linguistically-logically. Quine’s failure lies in his lack of appreciation for intensionality in which the realization of (2) leads Quine to embrace (1). For instance, the author of this paper has worked in correspondence with Ruth Barcan Marcus to formulate a direct theory of reference that avoids problems of referential opacity in terms of proper names by distinguishing between the sense of names and the sense of objects, which Quine’s (and Russell’s) theories do not distinguish because of their extensional material formulations of singular terms. Hence, “Giorgione is so called because of his size” can be intensionally understood as the propositional function “x is denoted as y because of x’s size” becoming: “the class of terms enumerating the one and only one object theta by which the term omega is included in this class and semantically entails the one and only one object theta strictly implying intensional property alpha.” The extensional formulation of the propositional function confuses the variables leading to opaqueness. This formulation can be extended to propositional attitudes as well as contingent necessity in modal logic, overcoming a considerably wider range of referentially opaque quandaries.

[11] C. I. Lewis, *Mind and the World Order*: 433.

[12] Ibid: 434.

[13] Ibid: 434.

[14] W.V.O. Quine, *From a Logical Point of View* (Cambridge: Harvard University Press, 1980): 21.

[15] Ibid: 23.

[16] Ibid: 26.

[17] Ibid: 20.

[18] Ibid: 31.

[19] A.P. Martinich, *The Philosophy of Language*, Alfred Tarski, “The Semantic Conception of Truth and the Foundations of Semantics” (Oxford, Oxford University Press, 2001): 75.

[20] Arthur Smullyan argues that the scope of the definition can be changed but this also affects the truth value, which is not what Russell intended. Of course, Smullyan is free to develop this argument of his own accord but this places his account, as well as Ruth Barcan Marcus’ outside the critique of this argument.

[21] There are some philosophers, Jaako Hintikka being one example, that attempt to overcome the quandaries of the indeterminacy of the substitutivity of identicals as a problem of *modal* quantification by simply abandoning substitutivity altogether. The first assumption of Logical Semanticism is that this philosophical move is misguided and sacrifices one of the most important reasons for quantification in the first place. Without symmetry and transitivity representative of intensionally identical properties, there is little utility for quantification, as denotation and terminology are forever irreconcilable.

[22] W.V.O. Quine, *From a Logical Point of View* (Cambridge: Harvard University Press, 1980): 31.

[23] Ibid: p. 26.

[24] One of the major confusions that philosophers of language bring against a direct theory of reference is confusing the logical questions of substitutivity and identity with the epistemological questions of how the subject knows that two meanings or two objects are identical. The ways of dealing with propositional attitudes embedded in a doxastic quantifier are quite different than the logical questions concerned here. Furthermore, there is no reason a direct theory of reference would have to follow the causal-historical explanation that Kripke and others have outlined. In fact, there is ample reason to suspect the causal-historical theory fails. However, there is nothing contrary about appealing to a direct theory of reference and holding Searle’s cluster theory of descriptions, even in relation to proper names, if the logical and epistemological issues are properly individuated. It is *not *the philosophy of language aspects of this problem (e.g. the learning of meanings, questions of memory and context, questions of empirical association, etc.) that interest this paper.

[25] A full proof of this cannot be given in this limited paper. The way in which this problem opens up has been indicated by the antecedent findings. However, an extension of this point would require a thorough demonstration to equivalency beyond proper names, such as the relation of two classes (e.g. a creature with a kidney and a creature with a heart). Furthermore, there is no reason to presuppose that the solvability of other such logical quandaries would follow the precise method outlined in this paper; namely because the notion of logical tags does not apply in the way outlined for proper names. What cannot be defended by proof in this paper but will be suggested for the sake of the reader is that the critical element for extending the intensional qualifications of a logical quantification requires a re-assessment of the definition of the identity of indiscernibles, or more specifically, its applicability. By qualified equivalency relations, what is suggested is the separation of equivalency relations into two categories: (1) referential relative equivalence and (2) intensional relative equivalence. In other words, indiscernibility of identicals is both of these components, lacking a way in which to differentiate one component from the other. Referential relative equivalence necessitates all equivalent referential properties with differing semantic properties (i.e. identity statements for co-referring terms). For example: “Giorgione is Barbarelli.” This is a one-one extensional relationship based upon the singularity of the reference, whereby two differentiable intensional variables demarcate the same extensional object. Since proper names are logically eliminable, the identity statement of co-referring terms is the claim that conditional symmetry exists insofar as: Intension A with Reference X = Intension B with Reference X. The “=” does *not* denote a biconditional for relative equivalence. To signify identity, the symbol “╩” is employed. Relative equivalence only holds for transitive equations because there are always three variables involved: (1) reference; (2) sense; (3) syntactic notation.

Intensional relative equivalence necessitates all equivalent semantic properties with contrary referential objects. This category of equivalence involves two different formulations. Firstly, there is the intensional equivalence of syntactic notations for singular terms. For example: “‘Rachel’ signifies a female sheep.” The meaning is in relation to the syntactic notation and not a referent. If there is some referential object that shares the name ‘Rachel’, then there are two contrary intensional relations being claimed in the proposition. Intensional relative equivalence is interested only in quantifying the latter: the relation of intension to notation. This is antithetical to referential relative equivalence. In this case, there is a one-one intensional relationship based upon the co-intensionality of separate notations. Therefore, a singular term is the claim that two dissimilar notations denote identical intensions. Since proper names are logically eliminable, the identity statement of two identical notations generates the conditional symmetric relation: Reference A with Intension X = Reference B with Intension X.

It is the fact that this separation of categories of equivalency relations can be separated from one another in relation to proper names that is here suspected can be extended to conform to other types of logical relations, even when the notion of “tags” are not involved. The basic logical formula, in general, is alleged to be: (A ╩ B) ≡ [(C A) & (A B)], whereby the truth of B and C are necessary and A is conditionally true for the relationship to hold true. But this can only be suggested as a direction for further investigation and not here proven, most unfortunately.

[26] Namely, beyond the propositional to the subsentential level, since this paper is an agreement with Quine regarding existential assertion of non-existents as false, but not meaningless, propositions. Quine’s charge of robust ontologies as forms of “savage theology”, a point developed in *Word and Object*, is not here endorsed. One would think that some great and meaningful philosophical difference is garnered by all of Quine’s passionate arguments for deciding the status of abstract general terms, instead of it being the peripheral concern it truly is. As Brandom rightfully discerns, a robust ontology is one of the least threatening philosophical moves a philosopher could ever make.

[27] Willard Van Orman Quine, *Word and Object* (Cambridge: MIT Press, 1960): 199.

[28] Alvin Plantinga, Michael J. Loux (ed.), *Metaphysics* (London: Routeledge Press, 2001): 168.

[29] This is equivalent to: □(P x) ≡ (P y). See section 4, semantic rule *I.** 5*.

[30] See formation and semantic rules for more details.

[31] See section 4, semantic rule *I.** 7*.

[32] Wilfrid Sellars: 22.