Reinflating Logical Consequence

Shapiro (Philos Q 61:320–342, 2011) argues that, if we are deflationists about truth, we should be deflationists about logical consequence. Like the truth predicate, he claims, the logical consequence predicate is merely a device of generalisation and more substantial characterisation, e.g. proof- or model-theoretic, is mistaken. I reject his analogy between truth and logical consequence and argue that, by appreciating how the logical consequence predicate is used as well as the goals of proof theory and model theory, we can be deflationists about truth but not logical consequence.     

Logical Consequence and the Paradoxes

We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics.     

Etchemendy and Bolzano on Logical Consequence

In a series of publications beginning in the 1980s, John Etchemendy has argued that the standard semantical account of logical consequence, due in its essentials to Alfred Tarski, is fundamentally mistaken. He argues that, while Tarski's definition requires us to classify the terms of a language as logical or non-logical, no such division is guaranteed to deliver the correct extension of our pre-theoretical or intuitive consequence relation. In addition, and perhaps more importantly, Tarski's account is claimed to be incapable of explaining an essential modal/epistemological feature of consequence, namely, its necessity and apriority.

Bernard Bolzano (1781–1848) is widely recognized as having anticipated Tarski's definition in his Wissenschaftslehre (or Theory of Science) of 1837. Because of the similarities between his account and Tarski's, Etchemendy's arguments have also been extended to cover Bolzano. The purpose of this article is to consider Bolzano's theory in the light of these criticisms. We argue that, due to important differences between Bolzano's and Tarski's theories, Etchemendy's objections do not apply immediately to Bolzano's account of consequence. Moreover, Bolzano's writings contain the elements of a detailed philosophical response to Etchemendy.     

The Tractatus on Logical Consequence

Abstract: I discuss the account of logical consequence advanced in Wittgenstein's Tractatus. I argue that the role that elementary propositions are meant to play in this account can be used to explain two remarkable features that Wittgenstein ascribes to them: that they are logically independent from one another and that their components refer to simple objects. I end with a proposal as to how to understand Wittgenstein's claim that all propositions can be analysed as truth functions of elementary propositions.     

A Note on Formality and Logical Consequence

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truth-preserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truth-preserving. But he did not offer the proof. The question arises whether the usual model-theoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving.     

Methodological Practice and Complementary Concepts of Logical Consequence: Tarski's Model-Theoretic Consequence and Corcoran's Information-Theoretic Consequence

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ‘say’ something), and they have subject matter (they are ‘about’ something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge.     

Ray on Tarski on Logical Consequence

In Logical consequence: A defense of Tarski (Journal of Philosophical Logic, vol. 25, 1996, pp. 617–677), Greg Ray defends Tarski"s account of logical consequence against the criticisms of John Etchemendy. While Ray"s defense of Tarski is largely successful, his attempt to give a general proof that Tarskian consequence preserves truth fails. Analysis of this failure shows that de facto truth preservation is a very weak criterion of adequacy for a theory of logical consequence and should be replaced by a stronger absence-of-counterexamples criterion. It is argued that the latter criterion reflects the modal character of our intuitive concept of logical consequence, and it is shown that Tarskian consequence can be proved to satisfy this criterion for certain choices of logical constants. Finally, an apparent inconsistency in Ray"s interpretation of Tarski"s position on the modal status of the consequence relation is noted.     

论逻辑后承概念

塔尔斯基在其1933年的论文中基于自己开创的语义学定义了形式化语言中真这个概念。然后,他在发表于1936年的本文中,在真这个概念的定义的基础上,第一次为后承概念提出了一个实质恰当的定义,即“句子X从类K的句子逻辑地得出当且仅当K这个类的每个模型也是句子X的模型”,使得逻辑后承这个现代逻辑核心概念的定义成为标准定义。但是,这个定义也遗留了一个更为根本的问题,即逻辑词项和非逻辑词项的划分标准问题,后者将由塔尔斯基本人在1966年给出了一个划分标准,从而开辟了一个方兴未衰的逻辑哲学研究方向。     

Supervaluationism and Logical Consequence: A Third Way

It is often assumed that the supervaluationist theory of vagueness is committed to a global notion of logical consequence, in contrast with the local notion characteristic of modal logics. There are, at least, two problems related to the global notion of consequence. First, it brings some counterexamples to classically valid patterns of inference. Second, it is subject to an objection related to higher-order vagueness. This paper explores a third notion of logical consequence, and discusses its adequacy for the supervaluationist theory. The paper proceeds in two steps. In the first step, the paper provides a deductive notion of consequence for global validity using the tableaux method. In the second step, the paper provides a notion of logical consequence which is an alternative to global validity, and discusses i) whether it is acceptable to the supervaluationist and ii) whether it plays a better role in a theory of vagueness in the face of the problems related to the global notion.     

Principles for Object-Linguistic Consequence: from Logical to Irreflexive

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143–65 (2013)) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.     

Boolos and the Metamathematics of Quine's Definitions of Logical Truth and Consequence

The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result.     

Etchemendy on Squeezing Arguments and Logical Consequence: a Reply to Griffiths

Owen Griffiths has recently argued that Etchemendy’s account of logical consequence faces a dilemma. Etchemendy claims that we can be sure that his account does not overgenerate, but that we should expect it to undergenerate. Griffiths argues that if we define the relationship between formal and natural language as being dependent on logical consequence, then Etchemendy’s claims are not true; and if we define the relationship as being independent of logical consequence, then we cannot assess the truth of the claims without further information. I argue that Griffiths misconstrues Etchemendy’s theory and overstates the first horn of the dilemma: Etchemendy does see the relationship as being dependent on logical consequence, but that does not mean that his claims are not true.     

Horst Wessel: Contributions to the Theory of Logical Consequence,Non-Traditional Theory of Predication and Logical Theory of Terms

The present work takes the decease of Horst Wessel as an opportunity to present and honour his work (and that of his group), which has not received the attention it deserves. The focus will be on works which might not be sufficiently well-known. Wessel was, as we aim to show, familiar with the international debate concerning logical and philosophical issues and strived to solve them by considering theories of logical consequence, a non-traditional theory of predication and the theory of logical terms, all of which he developed in joint work with Alexander Sinowjew. Wessel had to significantly refine these theories in order to formulate his criticism towards alternative logics and to apply the theories to a treatment of intensional contexts. He was vehemently opposed to approaches which revised fundamental laws of classical logic. Questions concerning the history of Philosophy are addressed in Wessel’s criticism of the Kantian antinomies of pure reason, in Wessel’s contributions to the problem of universals and his analyses of fatalism, tychism and antifatalism.     

The Guru, the Logician, and the Deflationist: Truth and Logical Consequence

The deflationary view of truth and reference as I've presented it … applies only to words and sentences that we understand. This may well seem worrisome, and it is important to ask both whether it should seem worrisome and whether it could be avoided. Field (1994, 260)     

Generalized Logical Consequence: Making Room for Induction in the Logic of Science

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented.     

A Note on the Correspondence among Entail Relations, Rough Set Dependencies, and Logical Consequence

In this note, we report that entail relations defined in the context of knowledge spaces are equivalent to the dependence relations of rough set data analysis and Tarski's consequence relation of monotone logic. We also discuss the connection between these and related structures. Copyright 2001 Academic Press.     

Aristotle on a Puzzle about Logical Consequence: Necessity of Being vs. Necessity of Saying

In the Posterior Analytics (I 6, 75a18–27) Aristotle discusses a puzzle which endangers the possibility of inferring a non-necessary conclusion. His solution relies on the distinction between the necessity of the conclusion's being the case and the necessity of admitting the conclusion once one has admitted the premisses. The former is a factual necessity, whereas the latter is meant to be a normative or deontic necessity that is independent of the facts stated by the premisses and the conclusion. This paper maintains that Aristotle resorts to this distinction because he thinks that, as long as it is conceived as a factual relation, logical consequence cannot exist independently of the facts expressed by the premisses and the conclusion. As a corollary, the necessity of such a consequence relation always requires the necessity of these facts. Aristotle holds this factual conception of logical consequence responsible for the puzzle, since it cannot account for valid syllogisms with contingent or false premisses. The alternative conception of necessity is then introduced by him in order to make good this deficiency. The distinction between the necessity of being and the necessity of saying was revived by the Oxford logician E. W. B. Joseph, and taken over by Frank Ramsey in his seminal Truth and Probability, but has not received attention from recent interpreters of Aristotle's logic. This paper, however, argues that, in spite of its intrinsic interest, the distinction bore no significant fruit in Aristotle's logical doctrine.     

Consequence Mining

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation \(\Rightarrow_X\). We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse to—more precisely, forms a Galois connection with—the Bolzano–Tarski function.     

Consequence and Normative Guidance

Logic, the tradition has it, is normative for reasoning. But is that really so? And if so, in what sense is logic normative for reasoning? As Gilbert Harman has reminded us, devising a logic and devising a theory of reasoning are two separate enterprises. Hence, logic's normative authority cannot reside in the fact that principles of logic just are norms of reasoning. Once we cease to identify the two, we are left with a gap. To bridge the gap one would need to produce what John MacFarlane has appropriately called a bridge principle, i.e. a general principle articulating a substantive and systematic link between logical entailment and norms of reasoning. This is Harman's skeptical challenge. In this paper I argue that Harman's skeptical challenge can be met. I show how candidate bridge principles can be systematically generated and evaluated against a set of well‐motivated desiderata. Moreover, I argue that bridge principles advanced by MacFarlane himself and others, for all their merit, fail to address the problem originally set forth by Harman and so do not meet the skeptical challenge. Finally, I develop a bridge principle that meets Harman's requirements as well as being substantive.