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 , 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  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  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 , and solve the translation problems of  and subsequently escape the paradox of . 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 , , and .
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.