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.

The problem that motivates me arises from a constellation of factors pulling in different, sometimes opposing directions. Simplifying, they are: (1) The complexity of the world; (2) Humans’ ambitious project of theoretical knowledge of the world; (3) The severe limitations of humans’ cognitive capacities; (4) The considerable intricacy of humans’ cognitive capacities . Given these circumstances, the question arises whether a serious notion of truth is applicable to human theories of the world. In particular, I am interested in the questions: (a) Is a substantive standard of truth for human theories of the world possible? (b) What kind of standard would that be?

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.

In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentenceσis false. Moreover, … we are concerned here with the concept of logical, i.e.,formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since.

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.

The paper offers a new analysis of the difficulties involved in the construction of a general and substantive correspondence theory of truth and delineates a solution to these difficulties in the form of a new methodology. The central argument is inspired by Kant, and the proposed methodology is explained and justified both in general philosophical terms and by reference to a particular variant of Tarski's theory. The paper begins with general considerations on truth and correspondence and concludes with a brief outlook on the "family" of theories of truth generated by the new methodology.

The question motivating my investigation is: Are the basic philosophical principles underlying the "core" system of contemporary logic exhausted by the standard version? In particular, is the accepted narrow construal of the notion "logical term" justified? ;As a point of comparison I refer to systems of 1st-order logic with generalized quantifiers developed by mathematicians and linguists . Based on an analysis of the Tarskian conception of the role of logic I show that the standard division of terms into logical and extra-logical is partly arbitrary. I argue that the semantic principles of Tarskian logic allow any higher order mathematical predicate or relation to function as a logical term in a 1st-order system, provided it is introduced in the right way into the syntactic-semantic apparatus. ;Formally, I propose a new, "constructive", semantic characterization of logical terms. A logical term is defined by a function which, given a model, "shows" how to construct relations, sets, or n-tuples of individuals which satisfy it. I discuss the linguistic applicability of the extended logic and bring numerous examples. ;One chapter is devoted to branching quantification--a non-standard logico-linguistic construction obtained by affixing a non-linear, partially-ordered quantifier-prefix to a well-formed formula. Although standard branching quantifiers were given a complete account by Henkin and Walkoe, it is not altogether clear what the meaning of branching generalized quantifiers is. I propose a new analysis of the generalized branching prefix which extends the existent, partial, definitions due to Barwise, and I develop a generalization leading to a "family" of branching structures. ;In conclusion I discuss the philosophical impact of the generalized conception of logic on such issues as the logicist thesis, the interaction between mathematics and logic, ontological commitment via logic, metaphysics and logical semantics

In this paper I develop a new conception of Tarskian logic based on Tarski’s intuitive characterization of logical consequence as formal and necessary in his 1936 paper. Special emphasis is placed on the role of logic in our system of knowledge, the origins of semantics, the semantic definition of logical consequence, and the role of logical and non-logical terms in a logical system. The paper offers a new definition of logical terms based on the question: what division of terms into logical and extra-logical would yield a logical system that satisfies Tarski’s intuitive characterization of logical consequence in complete generality? I discuss the consequences of the new conception for revision in logic, the logicist thesis, and the relation between logic and mathematics. I offer a proof-theoretic perspective on the semantic conception delineated in this paper. And I conclude with a postscript on Tarski’s lecture, “What are Logical Notions?”, which was published shortly after the present conception of logic was developed.