Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the (...) full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical. (shrink)

In a recent discussion article in this journal, Gila Sher responds to some of my criticisms of her work on what she calls the formal-structural account of logical consequence. In the present paper I reply and attempt to advance the discussion in a constructive way. Unfortunately, Sher seems to have not fully understood my 1997. Several of the defenses she mounts in her 2001 are aimed at views I do not hold and did not advance in my 1997. Most prominent (...) among these are her claims that I reject the formal-structural account, because it licenses “unnatural” logical constants, and that I consider the “truth-functional criterion” for connectives “a paradigm of a successful criterion of logicality”. Both of these claims are simply false, as even a moderately careful reading of my 1997 will show. Other charges she makes have some basis in fact but are inaccurate. For example, she says that I claim that FS is “neutral with regard to the apriority of logic,” and that this neutrality “renders it inconsistent”. What I actually say is that Sher’s acceptance of the neutrality in question is “inconsistent with part of the legacy of Tarski, a legacy that she finds very congenial”, not that FS itself is inconsistent. More interestingly, she misunderstands my claim about the apriority of logic, and clarification of this misunderstanding suggests two distinct ways in which logic might be held to be free of empirical influence. It is also noteworthy that Sher’s statements about the apriority of logic constitute both a change in the position she took in her 1991 and, although she seems not to realize it, an admission that she does not take issue with one of my major criticisms. (shrink)

In a recent discussion article in this journal, Gila Sher responds to some of my criticisms of her work on what she calls the formal-structural account of logical consequence. In the present paper I reply and attempt to advance the discussion in a constructive way. Unfortunately, Sher seems to have not fully understood my 1997. Several of the defenses she mounts in her 2001 are aimed at views I do not hold and did not advance in my 1997. Most prominent (...) among these are her claims that I reject the formal-structural account, because it licenses “unnatural” logical constants, and that I consider the “truth-functional criterion” for connectives “a paradigm of a successful criterion of logicality”. Both of these claims are simply false, as even a moderately careful reading of my 1997 will show. Other charges she makes have some basis in fact but are inaccurate. For example, she says that I claim that FS is “neutral with regard to the apriority of logic,” and that this neutrality “renders it inconsistent”. What I actually say is that Sher’s acceptance of the neutrality in question is “inconsistent with part of the legacy of Tarski, a legacy that she finds very congenial”, not that FS itself is inconsistent. More interestingly, she misunderstands my claim about the apriority of logic, and clarification of this misunderstanding suggests two distinct ways in which logic might be held to be free of empirical influence. It is also noteworthy that Sher’s statements about the apriority of logic constitute both a change in the position she took in her 1991 and, although she seems not to realize it, an admission that she does not take issue with one of my major criticisms. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...) necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

Logical pluralists are committed to the idea of a neutral metalanguage, which serves as a framework for debates in logic. Two versions of this neutrality can be found in the literature: an agreed upon collection of inferences, and a metalanguage that is neutral as such. I discuss both versions and show that they are not immune to Quinean criticism, which builds on the notion of meaning. In particular, I show that the first version of neutrality is sub-optimal, and hard to (...) reconcile with the theories of meaning for logical constants, and the second version collapses mathematically, if rival logics, as object languages, are treated with charity in the metalanguage. I substantiate by proving a collapse theorem that generalizes familiar results. Thus, the existence of a neutral metalanguage cannot be taken for granted, and meaning-invariant logical pluralism might turn out to be dubious. (shrink)

This paper gives a new twist to already familia refutations of Putnam's "model-theoretic" argument against realism. Recent attempts to defend the model-theoretic argument in the face of those criticisms indicate that the main point of previous rebuttals of the argument can be easily missed. The paper expounds the same point again in a different guise, by having recourse to ideas on models and the model-theoretic account of the logical properties developed by the author in earlier work. Some writers appear to (...) think that the charge of previous criticisms is that Putnam's argument begs the question, by involving as premise a proposition which all too obviously entails the falsity of realism--precisely what the argument is designed to establish. They then defend it by claiming that Putnam does not simply _take for granted<D> the truth of the offending premise, but _argues for<D> it. This, however, does not defend Putnam's argument against the _pragmatic<D> charges of irrelevancy and potential for provoking confusion--which, the paper shows, can be taken as the main criticisms levelled against it by previous writers. The point is that, if Putnam has an independent argument for the question-begging premise, one not involving any model-theoretic considerations, then this is all that is needed; the specifically model-theoretic considerations are irrelevant and could lead to spurious debates. (shrink)

Both the traditional Aristotelian and modern symbolic approaches to logic have seen logic in terms of discrete symbol processing. Yet there are several kinds of argument whose validity depends on some topological notion of continuous variation, which is not well captured by discrete symbols. Examples include extrapolation and slippery slope arguments, sorites, fuzzy logic, and those involving closeness of possible worlds. It is argued that the natural first attempts to analyze these notions and explain their relation to reasoning fail, so (...) that ignorance of their nature is profound. (shrink)

The notion of measurement plays a central role in human cognition. We measure people’s height, the weight of physical objects, the length of stretches of time, or the size of various collections of individuals. Measurements of height, weight, and the like are commonly thought of as mappings between objects and dense scales, while measurements of collections of individuals, as implemented for instance in counting, are assumed to involve discrete scales. It is also commonly assumed that natural language makes use of (...) both types of scales and subsequently distinguishes between two types of measurements. This paper argues against the latter assumption. It argues that natural language semantics treats all measurements uniformly as mappings from objects (individuals or collections of individuals) to dense scales, hence the Universal Density of Measurement (UDM). If the arguments are successful, there are a variety of consequences for semantics and pragmatics, and more generally for the place of the linguistic system within an overall architecture of cognition. (shrink)

Is it possible for there to be facts about reality with a logical structure that is in principle unrepresentable by us? I outline the main motivations for thinking that this question should receive a positive answer. I then argue that, upon inspection, the view that such structurally ineffable facts are possible is self-defeating and thus incoherent. My argument is based on considerations about the fundamental role that the purely formal concept of an object plays in our propositional representations and its (...) intimate connection with subject-predicate structure. (shrink)

The aim of this article is to argue against the real possibility of languages without subject‐predicate structure, so‐called feature‐placing languages. They were first introduced by Strawson (1959/1990), later given formal expression through Quine's Predicate Functor Logic (Quine, 1960, Quine, 1971/Quine, 1976, Quine, 1992), and further elaboration in (Hawthorne & Cortens, 1995). I argue that, on the presumption that feature‐placing languages are not mere notational variants on first‐order languages, the idea of such languages is incoherent. The argument for this view rests (...) on two highly plausible views about the conditions for the possibility of sentential meaning in natural languages. If accepted, these two views jointly imply that the sentences of feature‐placing languages lack sufficient internal structure to meet the conditions for sentential meaning. Feature‐placing languages cannot then count as genuine languages. (shrink)

What does it mean to say that logic is formal? The short answer is: it means (or can mean) several different things. In this paper, I argue that there are (at least) eight main variations of the notion of the formal that are relevant for current discussions in philosophy and logic, and that they are structured in two main clusters, namely the formal as pertaining to forms, and the formal as pertaining to rules. To the first cluster belong the formal (...) as schematic; the formal as indifference to particulars; the formal as topic-neutrality; the formal as abstraction from intentional content; the formal as de-semantification. To the second cluster belong the formal as computable; the formal as pertaining to regulative rules; the formal as pertaining to constitutive rules. I analyze each of these eight variations, providing their historical background and raising related philosophical questions. The significance of this work of ?conceptual archeology? is that it may enhance clarity in debates where the notion of the formal plays a prominent role (such as debates where it is expected to play a demarcating role), but where it is oftentimes used equivocally and/or imprecisely. (shrink)

Standard first-order logic plus quantifiers of all finite orders ("SFOLω") faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is (...) removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models. (shrink)

This paper examines the logical and metaphysical consequences of denying Leibniz's Law, the principle that if t1= t2, then φ(t1) if and only if φ(t2). Recently, Caie, Goodman, and Lederman (2020) and Bacon and Russell (2019) have proposed sophisticated logical systems permitting violations of Leibniz's Law. We show that their systems conflict with widely held, attractive principles concerning the metaphysics of individuals. Only by adopting a highly revisionary picture, on which there is no finest-grained equivalence relation, can a well-motivated metaphysics (...) for rejecting Leibniz's Law be developed. We sketch one such picture—a metaphysics of stuff. Stuff ontologies can be initially motivated through ordinary language: stuff stands to mass nouns as objects stand to count nouns. The stuff ontology we propose takes stuff to be fundamental and views the world as composed of an infinite descending hierarchy of kinds and portions of stuff. We defend the coherence of this picture and offer a model theory demonstrating that it can be consistently formalized. (shrink)

The pretheoretical notions of logical consequence and of a logical expression are linked in vague and complex ways to modal and pragmatic intuitions. I offer an introduction to the difficulties that these intuitions create when one attempts to give precise characterizations of those notions. Special attention is given to Tarski’s theories of logical consequence and logical constancy. I note that the Tarskian theory of logical consequence has fared better in the face of the difficulties than the Tarskian theory of logical (...) constancy. Other theories of these notions are explained and criticized. (shrink)

I offer some brief remarks in reply to comments and criticisms of my earlier work on logical consequence and logical constants. I concentrate on criticisms, especially García-Carpintero’s charge that myviews make no room for modal intuitions about logical consequence, and Sher’s attempted rebuttal of my critique of her theory of logical constants. I show that García-Carpintero’s charge is based on misunderstandings, and that Sher’s attempted rebuttal actually reveals new problems for her theory.

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

The paper argues that a philosophically informative and mathematically precise characterization is possible by (i) describing a particular proposal for such a characterization, (ii) showing that certain criticisms of this proposal are incorrect, and (iii) discussing the general issue of what a characterization of logical constants aims at achieving.

This thesis aims to develop a psychologically plausible account of concepts by integrating key insights from philosophy (on the metaphysical basis for concept possession) and psychology (on the mechanisms underlying concept acquisition). I adopt an approach known as informational atomism, developed by Jerry Fodor. Informational atomism is the conjunction of two theses: (i) informational semantics, according to which conceptual content is constituted exhaustively by nomological mind–world relations; and (ii) conceptual atomism, according to which (lexical) concepts have no internal structure. I (...) argue that informational semantics needs to be supplemented by allowing content-constitutive rules of inference (“meaning postulates”). This is because the content of one important class of concepts, the logical terms, is not plausibly informational. And since, it is argued, no principled distinction can be drawn between logical concepts and the rest, the problem that this raises is a general one. An immediate difficulty is that Quine’s classic arguments against the analytic/synthetic distinction suggest that there can be no principled basis for distinguishing content-constitutive rules from the rest. I show that this concern can be overcome by taking a psychological approach: there is a fact of the matter as to whether or not a particular inference is governed by a mentally-represented inference rule, albeit one that analytic philosophy does not have the resources to determine. I then consider the implications of this approach for concept acquisition. One mechanism underlying concept acquisition is the development of perceptual detectors for the objects that we encounter. I investigate how this might work, by drawing on recent ideas in ethology on ‘learning instincts’, and recent insights into the neurological basis for perceptual learning. What emerges is a view of concept acquisition as involving a complex interplay between innate constraints and environmental input. This supports Fodor’s recent move away from radical concept nativism: concept acquisition requires innate mechanisms, but does not require that concepts themselves be innate. (shrink)

The aim of this paper is to provide a dynamic interpretation of Kant’s logical hylomorphism. Firstly, various types of the logical hylomorphism will be illustrated. Secondly, I propose to reevaluate Kant’s constitutivity thesis about logic. Finally, I focus on the design of logical norms as specific kinds of artefacts.

We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question be formulated for (...) quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be. -/- . (shrink)