The Logical Semantic Project: Symbolizing and Structuring Language-Transcendent Meaning

•October 21, 2008 • 1 Comment


The foundation of theoretical semantics is the systematic syntactic framework that defines through logical analysis the formalized structure necessary for transparently representing linguistic sense and reference as applied to the set of all possible natural languages. Consequently, logical semanticism is named as such in order to draw attention to the comparative historical and conceptual differences of the philosophical project of logical atomism.


(1) How can meaning be quantified into logical terms that syntactically preserve the semantic content and relations of natural language? (2) What is necessary for the possibility of accurately quantifying the intensional classes that represent the semantic content and relations of natural language free of substituitivity paradox, ambiguity of descriptive scope, or referential opacity of propositional attitudes and doxastic modification? (3) What is necessary for the possibility of accurately transforming a name or definiendum between natural langauges (i.e. intensional systems) free of incommensurability?


Being committed to the theory of language-transcendent meaning is philosophically postulating that while meaning is relative to the unique referential apparatus and the unique etymological stipulations of any given natural language, there is an underlying objective standard of formally representing meaning, and the objects referenced by meaning. This theory proposes that the analysis of a natural language can be studied successfully without having to beg the question by recursively relying upon the very natural language under study for an analysis of its own references and meanings. To be able to study language in this manner is to be engaged in a transcendental method. Concurrently, the philosophical function of the foundation of theoretical semantics is outlining the logical principles necessary for satisfying the possibility of engaging in linguistic analysis using a formalized method that preserves the nature and structure of a particular natural language’s references and meanings while simultaneously transcending the need to limit one’s study to the referential apparatus and semantic ontology of the particular natural language. It is in this sense that sentential meaning would be language-transcedent and that formulating the rules of theoretical semantics for natural language provides the necessary ontological parameters for this transcendental study to be possible.


The ultimate philosophical goal of theoretical semantics is outlining the framework for the ideal language. The foundation of theoretical semantics is the set of logical principles sufficient for completing this goal. Language-transcendent meaning is a significant component of the ideal language project. The analytic, the a priori, and the necessary are valid ontological categories that possess a language-transcendent character that are concurrently fundamental to the ideal language. To have a comprehensive view of language one must therefore account for how these categories philosophically function. What this ultimately requires is a theory of particulars and a theory of universals that demarcate the relationship of the logic of language to ontology. In this regard, the foundation of theoretical semantics is a methodologically fundamental aspect of this greater philosophical process for both epistemological and metaphysical reasons.


The Philosophy of Quantification

•November 18, 2008 • 2 Comments

Quantification is a theory of sign manipulation that represents in symbolic form, the orderly syntactic arrangement, development, or classification of a collection of information.  The basis of any quantificational theory is constructing a finite set of rules by which a well-formed lexicon of elements populates its ontological domain.  These grammatical elements can be of any determinable complexity derived legitimately by consequence of the formation rules.  This includes the well-formed formulation of an economy of tautologically true quantifiers (i.e. operators), operations, and functions.  The fundamental purpose of quantifying is the achievement of determination.  To determine is by definition demarcating limit, namely by discovering the extent or scope of an object.  An act of quantifying is essentially the binding of some set of elements to some set of quantifiers to form a function.  Therefore, the act of binding a free element to a quantifier is in effect stipulating that which can be expressly denoted about the quality and configuration of the quantified element, which provenances both a function and the means of its interpretation in the process.  For respective to the quantifier’s theoretical purpose, interpreting a function ultimately depends upon the content and form represented, whether in terms of inferential derivations, attributive or enumerative properties, or relations based upon equipollency or varying degrees of equivalency.  Indetermination within the operations of a quantificational theory is a failure of a quantificational theory’s process of expressing these determinations, and philosophically defines a limit to the application of the theory itself.  Here attests the fundamental problem of the philosophy of quantification.  The application of the totality of functions of a quantificational theory is philosophically intended to systematically correspond with the principle nature and formal structure constitutive of a particular field of knowledge.  Such paradox obfuscates this ideal philosophical project and renders error in the formal representation of reality.  The outstanding philosophical question remains whether the project is obtainable and awaits discovery, or whether the project is intrinsically impossible, a false presupposition as to what can be expected from the abstraction and reduction of the concrete and practical to theory and method.  From Pythagoras to Quine, this is the most fundamental question of Western philosophy. 

The Logic of Language in the Analytic Tradition

•October 27, 2008 • Leave a Comment

The history of the analytic tradition is the search for what we can justify as true by systematically applying our normative principles of analysis to the world.  From a logical perspective, the twentieth century is one continuous philosophical argument concerning the necessary conditions of this method of analysis.

Consider that the philosophical purpose of quantifying necessity was originally intended as an epistemological method for overcoming skepticism; namely by absolutely justifying a finite class of truth claims, of which any proposition could ultimately be derived.  C.I. Lewis (1918, 1927) and Rudolf Carnap (1947) presupposed that if any term—representing either an attribute or proposition—could be shown to be true in virtue of its intensional relation, then the related terms were necessarily true by definition of analyticity.  Accordingly, Lewis and Carnap proposed that intensional relations, based upon stipulated semantic criteria, constituted the class of analytic judgments.  All empirical judgments of science could then be descriptively classified in terms of their confirming or disconfirming to transcendental meanings of eternal sentences; either conventionally qualified to fit a logical representation of a natural language at a specifiable time (Carnap), or to have empirical generalizations correspond with strictly implied sentential functions (Lewis). 

This logical method would then provide science with a transcendental language, whereby any empirical fact could non-problematically be reduced to its corresponding propositional function, to transparently reference reality free of semantic interference.  While the truth of an empirical proposition would still be dependent on its corresponding probability judgment, the meaning and commensurability of any knowledge claim could be grounded in the analytic character of semantic definition.  Hence, the a priori foundations of knowledge were to be established upon a strong intensional logic.  Without restriction, this could quantify the notions of meaning and necessity in order to successfully prove [3], that an intensional logistic system validly denotes referentially transparent propositions, satisfying the conditions of the ideal language. 

Both conceptual pragmatism and Carnap’s logical positivism, as epistemological theories, are ultimately justified upon these philosophical claims.  But the logic of Willard Quine was intended to refute both doctrines: this semantically robust analytical theory of meaning; and this reductive and hierarchical foundational theory of knowledge (1951).  Quine was responding to what he interpreted as a quantificational failure of substituting identical terms into modal contexts; the failure of [3] in relation to modal quantifiers in the intensional strategy.  When Ruth Barcan Marcus constructed quantifiers for modal logic (1946), as epitomized by the Barcan formula, Quine critiqued the unrestricted values of free variables as leading to the notorious problems of referential opacity in terms of quantifiers, as well as anti-monotonicity.  Quine vehemently denied that free variables could be substituted into a restricted modal context, for the variable would subsequently be bound to a quantifier—a contradiction.  Quine used this critique of Marcus’s quantification of strict implication (QML) as a means of attacking the philosophical claims made by Lewis and Carnap.  Quine believed that the strong equivalency relation of modal logic, as an intensional logistic system, introduced the notion of Aristotelian essentialism to empirical objects.  QML would lead us to prescribe certain properties as being intrinsically necessary, whereby some individual person or other types of empirical objects would possess certain characteristics by metaphysical necessity; others only as contingently true.  Quine’s critique would later be extended to Kripkean possible worlds semantics against transworld individuals for this very reason. 

Since Quine found it epistemologically hopeless to make this metaphysical type of essentialist claim about objects, he attempted to prove this through referential failure of modal substitution.  If two objects were equal to one another, then by definition those objects must co-extensively share the same properties.  However, in unrestricted modal contexts, one object can be inputted in place of its identical term and lead to substitution error.  Through comparison, both terms can be demonstrated as not sharing the same properties out of metaphysical necessity.  Nine may necessarily be greater than four but if there are nine planets in our solar system, it is not metaphysically necessary that there are more than four planets in our solar system.  Substituting ‘number of planets in the solar system’ for ‘nine’, while factually equivalent, computes a modal error.  Thus, according to Quine, QML, in its traditional unrestricted formulation, is inherently victim to substitution failure.  Quine similarly used this argument to discount analyticity.  Since Lewis and Carnap grounded their theory of knowledge in the analytic judgment, their epistemological doctrine, by implication, subsequently falters.

In light of Quine’s attacks, there has been a diversity of responses to avoid the errors of [3] in relation to QML.  Quine attempted to show that intensional objects (e.g. class-concepts, propositions, attributes) could be eliminated through the extensional strategy of predicate logic; thus proving that the values of intensional strategies could be translated away into concrete reductions.  This would avoid generating the value restricted issues of [1], and solve the translation problems of [2] and subsequently escape the paradox of [3].  Many mistakenly believed Quine successful.  Accordingly, A. N. Prior attempted to reformulate the original quantification of modal logic (1957), while J. Hintikka responded to Quine by limiting the rule of substitution (1961).  D. Føllesdal, on the other hand, maintained the inferential rules of modal logic but introduced new semantics for the necessity quantifier with the notion of ‘causal necessity’ (1965).  However, the most substantive reformation incepted with Saul Kripke’s “Semantical Considerations on Modal Logic.”  Unlike most other works, Kripke was predominantly interested in something other than Quine’s intensional critique, or the historical epistemological issues by which the substitutivity problem manifested.  Kripke desired to show that the various quantificational issues of how to substitute identical terms concerning free variables, or the referential issue of how to assign truth values to non-existing states of affairs, are merely differing conventions for constructing a logistic system.  Kripke’s point is that all such conventions are subordinate to the quantificational model structures of possible worlds semantics; that his model theory antecedes other philosophical controversies, including the logical issues of [1], [2], and [3].  

But it is precisely this claim of universal quantificational application that makes Kripkean modal semantics a genuine epistemological and linguistic problem.  By generating an extensional-based theory that quantifies the domain of individuals as related to various possible worlds, it is my contention that the primordial purpose of QML, as an intensional calculus, is undermined by Kripke.  Concerning the metaphysics of individuals, we are left with the extraordinarily unappealing dichotomy, arising from a Quinian-inspired extensional account of modal quantifiers, between Alvin Plantinga’s realism (1976) and David Lewis’ nominalism (1973).  Either we establish the Platonic reality of de re modalities (Plantinga), admit to the existence of an assortment of other worlds involving counterfactuals (David Lewis), or attempt to define a non-metaphysical middle position that remains within the framework of possible worlds (Føllesdal).  By avoiding the metaphysical baggage of Plantinga, Føllesdal provides a method for comparatively analyzing scientific theories, using the notion of causal necessity to demonstrate a way of quantifying the complexities of theory choice in a hypothetical manner.  This similarly applies to the work of M. L. Ginsberg (1986), whereby Ginsberg’s semantics avoids the metaphysical issues of David Lewis’ counterfactual conditionals.  Unlike metaphysically loaded modal systems, Føllesdal’s causal necessity and Ginsberg’s various maximal sets of belief can be modeled into the intensional framework of a new semantics of QML, one that avoids possible worlds altogether. 

This derives the contemporary point.  It is possible through the philosophy of logical semanticism to preserve the inferential work of the twentieth century, as long as we abandon our Quinian paradigm. 

Semantics of Modality

•October 26, 2008 • Leave a Comment

Definitions of Definition


Intension is the relation of a set of attributes stipulated as intrinsically true of a sign; judged by the degree of mereological equivalence of the relation.


Extension is the function of a set of objects denoted as true by implication of the intension of a set; judged by the correspondence of object to its function.     


Modality of Extension


Epistemic necessity is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as always true of some denoted term in a particular doxastic system Đ. 


Epistemic possibility is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as sometimes true of some denoted term in a particular doxastic system Đ. 


Epistemic impossibility is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is probabilistically judged as never true of some denoted term in a particular doxastic system Đ. 


Metaphysical necessity is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is true in all possible worlds.


Metaphysical possibility is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is actualizable; and therefore true in some possible worlds.


Metaphysical impossibility is the modal quantification of any extensional relation, whereby some class of objects or states of affairs is non-actualizable or logically contradictory; and therefore true in no possible worlds. 


Modality of Intension


Semantic necessity is the modal quantification of any intensional relation, whereby some class of definiens ϕn(xn) is stipulated as always true of some definiendum ϕ in a particular intensional system Ł. 


Semantic possibility is the modal quantification of any intensional relation, whereby some class of definiens ϕn(xn) is stipulated as sometimes true of some definiendum ϕif and only if there is no contradiction with the class of definiens ϕn(xn) that are stipulated as always true of the definiendum ϕin a particular intensional system Ł.


Semantic impossibility is the modal quantification of any intensional relation, whereby some class of definiens ϕn(xn) is stipulated as never true of some definiendum ϕin a particular intensional system Ł.