On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem (...) for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

This article offers an overview of inferential role semantics. We aim to provide a map of the terrain as well as challenging some of the inferentialist’s standard commitments. We begin by introducing inferentialism and placing it into the wider context of contemporary philosophy of language. §2 focuses on what is standardly considered both the most important test case for and the most natural application of inferential role semantics: the case of the logical constants. We discuss some of the (alleged) benefits (...) of logical inferentialism, chiefly with regards to the epistemology of logic, and consider a number of objections. §3 introduces and critically examines the most influential and most fully developed form of global inferentialism: Robert Brandom’s inferentialism about linguistic and conceptual content in general. Finally, in §4 we consider a number of general objections to IRS and consider possible responses on the inferentialist’s behalf. (shrink)

Many prominent writers on the philosophy of logic, including Michael Dummett, Dag Prawitz, Neil Tennant, have held that the introduction and elimination rules of a logical connective must be ‘in harmony ’ if the connective is to possess a sense. This Harmony Thesis has been used to justify the choice of logic: in particular, supposed violations of it by the classical rules for negation have been the basis for arguments for switching from classical to intuitionistic logic. The Thesis has also (...) had an influence on the philosophy of language: some prominent writers in that area, notably Dummett and Robert Brandom, have taken it to be a special case of a more general requirement that the grounds for asserting a statement must cohere with its consequences. This essay considers various ways of making the Harmony Thesis precise and scrutinizes the most influential arguments for it. The verdict is negative: all the extant arguments for the Thesis are weak, and no version of it is remotely plausible. (shrink)

This special issue collects together nine new essays on logical consequence :the relation obtaining between the premises and the conclusion of a logically valid argument. The present paper is a partial, and opinionated,introduction to the contemporary debate on the topic. We focus on two influential accounts of consequence, the model-theoretic and the proof-theoretic, and on the seeming platitude that valid arguments necessarilypreserve truth. We briefly discuss the main objections these accounts face, as well as Hartry Field’s contention that such objections (...) show consequenceto be a primitive, indefinable notion, and that we must reject the claim that valid arguments necessarily preserve truth. We suggest that the accountsin question have the resources to meet the objections standardly thought to herald their demise and make two main claims: (i) that consequence, as opposed to logical consequence, is the epistemologically significant relation philosophers should be mainly interested in; and (ii) that consequence is a paradoxical notion if truth is. (shrink)

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a further development (...) of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt. (shrink)

The requirement of proof-theoretic harmony has played a pivotal role in a number of debates in the philosophy of logic. Different authors have attempted to precisify the notion in different ways. Among these, three proposals have been prominent in the literature: harmony–as–conservative extension, harmony–as–leveling procedure, and Tennant’s harmony–as–deductive equilibrium. In this paper I propose to clarify the logical relationships between these accounts. In particular, I demonstrate that what I call the equivalence conjecture —that these three notions essentially come to the (...) same thing—is erroneous. (shrink)

I discuss Prawitz’s claim that a non-reliabilist answer to the question “What is a proof?” compels us to reject the standard Bolzano-Tarski account of validity, andto account for the meaning of a sentence in broadly verificationist terms. I sketch what I take to be a possible way of resisting Prawitz’s claim---one that concedes the anti-reliabilist assumption from which Prawitz’s argument proceeds.

We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

The term ‘logical form’ has been called on to serve a wide range of purposes in philosophy, and it would be too ambitious to try to survey all of them in a single essay. Instead, I will focus on just one conception of logical form that has occupied a central place in the philosophy of language, and in particular in the philosophical study of linguistic meaning. This is what I will call the classical conception of logical form. The classical conception, (...) as I will present it in section 1, has (either explicitly or implicitly) shaped a great deal of important philosophical work in semantic theory. But it has come under fire in recent decades, and in sections 2 and 3 I will discuss two of the recent challenges that I take to be most interesting and significant. (shrink)

In his Formalization of Logic (1943) Carnap pointed out that there are non-normal interpretations of classical logic: non-standard interpretations of the connectives and quantifiers that are consistent with the classical consequence relation of a language. Different ways around the problem have been proposed. In a recent paper, Bonnay and Westerståhl argue that the key to a solution is imposing restrictions on the type of interpretation we take into account. More precisely, they claim that if we restrict attention to interpretations that (...) are (a) compositional, (b) non-trivial and (c) in the case of the quantifiers, invariant under permutations of the domain, Carnap’s Problem is avoided. This paper has two goals. The first is to show that Bonnay and Westerståhl’s solution to Carnap’s Problem doesn’t work. The second is to argue that something similar to their proposal seems to do the job. The problems with Bonnay and Westerståhl’s approach trace back to issues concerning the (un)definability of subsets of the domain of first-order structures, as well as to the compositionality of first-order languages. After expanding on these problems, I’ll propose a way to modify Bonnay and Westerståhl’s account and solve Carnap’s Problem. (shrink)

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).

In the paper the following questions are discussed: What is logical consequence? What are logical constants? What is a logical system? What is logical pluralism? What is logic? In the conclusion, the main tendencies of development of modern logic are pointed out.

Resumo: Pedro Hispano define os sincategoremas como expressões que revelam de que maneira os sujeitos e os predicados estão de fato relacionados nas proposições, contribuindo assim para o estabelecer o que elas significam e fixar as condições de verdade e as formas lógicas correspondentes. Entre as expressões que ele julga serem sincategoremáticas, ‘não’, ‘e’, ‘ou’, ‘se’, ‘todo’ e ‘necessário’ se destacam atualmente como constantes lógicas. Todavia, opondo-se a grande parte dos lógicos contemporâneos para quem tais expressões possuem um significado fixo (...) na medida em que integram as formas lógicas das respectivas proposições, Pedro Hispano admite que seu significado pode ser modificado quando a expressões do mesmo tipo credita a capacidade de atuarem em determinados contextos como categoremas. Além disso, ao defender que as expressões ora em questão explicitam as formas lógicas das proposições mediante a articulação dos categoremas a elas associados, ele também demonstra divergir dos principais critérios de demarcação das constantes lógicas atualmente vigentes que não preveem a atribuição de tal comportamento a expressões desse tipo. Portanto, não resta dúvida de que a teoria de Pedro Hispano sobre os sincategoremas enriquece a nossa compreensão ainda insuficiente da natureza das constantes lógicas e contribui para a resolução do problema da demarcação de tais expressões.: According to Petrus Hispanus, syncategoremata are expressions that determine how subjects and predicates are actually related in propositions. They contribute to establishing what the categoremata mean and to specifying the truth conditions of the corresponding logical forms. Among the expressions Peter of Spain thinks of as syncategorematic, ‘not’, ‘and’, ‘or’, ‘if’, ‘all, and ‘necessary’ are nowadays considered to be logical operators. But unlike the contemporary logicians who argue that these expressions have fixed meanings because they belong to the logical forms of the corresponding propositions, Peter of Spain allows that their meanings can be modified, attributing to them the ability to act in certain contexts as categoremata. In addition, he argues that these expressions make explicit the logical forms through the articulation of corresponding propositions’ categoremata; thus he also diverges from the main contemporary criteria of demarcation of logical constants, which does not allow that these expressions have such behavior. Therefore, there is no doubt that Hispanus’ theory of the syncategoremata enriches our still insufficient understanding of the logical constants and contributes to the resolution of the problem of the demarcation of such expressions. (shrink)