Gila Sher

Logic and mathematics are abstract disciplines par excellence. What is the nature of truth and knowledge in these disciplines? In this paper I investigate the possibility of a new approach to this question. The underlying idea is that knowledge qua knowledge, including logical and mathematical knowledge, has a dual grounding in mind and reality, and the standard of truth applicable to all knowledge is a correspondence standard. This applies to logic and mathematics as much as to other disciplines; i.e., logical and mathematical truth are based on correspondence. But the view that logical and mathematical truth are (i) based on correspondence and (ii) require a grounding in reality demands a change in the common conception of both correspondence and epistemic grounding.

In the early part of the 20th century the logical positivists launched a powerful attack on traditional philosophy, rejecting the very idea of philosophy as a substantive discipline and replacing it with a practical, conventionalist, meta-theoretical view of philosophy. The positivist critique was based on a series of dichotomies: the analytic vs. the synthetic, the external vs. the internal, the apriori vs. the empirical, the meta-theoretical vs. the object- theoretical, the conventional vs. the factual. Quine's attack on the positivists' dichotomies was, by extension (if not by intention), also an attack on their critique of philosophy. Quine's own theory, however, in time took an extreme naturalistic turn that, if anything, deepened the schism between philosophy and knowledge. In this paper I show that many of Quine's early philosophical ideas - his denial of the analytic-synthetic distinction, his thesis of the interconnectedness of knowledge, his universal revisability thesis, his principle of the inseparability of language and theory, his methodological pragmatism, and his realism - are in fact compatible with a substantive philosophy. Moreover, certain inner tensions in Quine's theory naturally lead to a new model of knowledge in which philosophy plays a substantive role: not as a "first philosophy", or as a "meta-science", or as a "chapter in psychology", but as an independent discipline in its own right, alongside, continuous with, and complementary to, science.

Knowledge requires both freedom and friction . Freedom to set up our epistemic goals, choose the subject matter of our investigations, espouse cognitive norms, design research programs, etc., and friction (constraint) coming from two directions: the object or target of our investigation, i.e., the world in a broad sense, and our mind as the sum total of constraints involving the knower. My goal is to investigate the problem of epistemic friction, the relation between epistemic friction and freedom, the viability of foundationalism as a solution to the problem of friction, an alternative solution in the form of a neo-Quinean model, and the possibility of solving the problem of friction as it applies to logic and the philosophy of logic within that model.

This is a critique of Michael P. Lynch’s functional pluralism with respect to truth. The paper is sympathetic to Lynch’s overall approach to truth, but is critical of (i) his platitudinous characterization of the general principles of truth, (ii) his excessive pluralism with respect to the “realizers” of truth, (iii) his treatment of atomic truth, and (iv) his analysis of “mixed” logical inferences. The paper concludes with a proposal for a functional pluralism that puts greater emphasis on the unity of truth. For example: while Lynch regards truth as based on correspondence principles in some domains and on coherence principles in others, the current proposal regards truth as based on correspondence principles in all domains, restricting the plurality of truth to a plurality of correspondence principles.

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications.

Following Henkin's discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or "cardinality" quantifiers, e.g., "most", "few", "finitely many", "exactly α", where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first to a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise's 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional Ist-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages

The paper delineates a new approach to truth that falls under the category of “Pluralism within the bounds of correspondence”, and illustrates it with respect to mathematical truth. Mathematical truth, like all other truths, is based on correspondence, but the route of mathematical correspondence differs from other routes of correspondence in (i) connecting mathematical truths to a special aspect of reality, namely, its formal aspect, and (ii) doing so in a complex, indirect way, rather than in a simple and direct way. The underlying idea is that an intricate mind is capable of creating intricate routes from language to reality, and this enables it to apply correspondence principles in areas for which correspondence is traditionally thought to be problematic.