Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that for a logical (...) constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic. The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecularism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation. (shrink)

I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.

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)

ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a proof-theoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and What Are We to Reject, (...) While Saving Truth from Paradox?’. (shrink)

Inferentialism claims that the rules for the use of an expression express its meaning without any need to invoke meanings or denotations for them. Logical inferentialism endorses inferentialism specically for the logical constants. Harmonic inferentialism, as the term is introduced here, usually but not necessarily a subbranch of logical inferentialism, follows Gentzen in proposing that it is the introduction-rules whch give expressions their meaning and the elimination-rules should accord harmoniously with the meaning so given. It is proposed here that the (...) logical expressions are those which can be given schematic rules that lie in a specific sort of harmony, general-elimination harmony, resulting from applying a certain operation, the ge-procedure, to produce ge-rules in accord with the meaning defined by the I-rules. Griffiths claims that identity cannot be given such rules, concluding that logical inferentialists are committed to ruling identity a non-logical expression. It is shown that the schematic rules for identity given in Read, slightly amended, are indeed ge-harmonious, so confirming that identity is a logical notion. (shrink)

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesis—requirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy :1035–1051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, can’t be accounted for in inferentialist terms. In (...) this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic. (shrink)

Russell’s paradox is purely logical in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell’s paradox is portable—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets. Burali-Forti’s paradox, like Russell’s paradox, (...) is portable. I offer the following explanation for this fact: Burali-Forti’s paradox, like Russell’s, is purely logical. Concretely, I show that if we enrich the language \ of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal. Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti’s. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti’s paradox is explained in the same way as the portability of Russell’s: both paradoxes involve an inconsistent logical form—Russell’s involves an inconsistent form expressible in \ and Burali-Forti’s involves an inconsistent form expressible in \. (shrink)

P.F. Strawson contributed to many philosophical domains, including the philosophy of language, the history of philosophy, metaphysics, moral philosophy and philosophical methodology. Most of his contributions in these areas have influenced contemporary debates, either because his views are still defended or because they are still considered worthy of detailed responses. His views on the philosophy of logic have been only rarely discussed, however. My aim in this paper is threefold. First, I provide a systematic account of Strawson’s philosophy of logic. (...) I argue that Strawson is an important predecessor of logical expressivism, a contemporary position in the philosophy of logic most notably defended by Robert Brandom. My main focus is on Strawson’s largely-neglected 1982 paper ‘Logical Form and Logical Constants’, which contains his most mature ideas on the topic. Second, while Strawson’s position is of historical and independent philosophical interest, I argue that he leaves many points unclear. Finally, I propose several clarifications of Strawson’s position. (shrink)

The best-known syntactic account of the logical constants is inferentialism. Following Wittgenstein’s thought that meaning is use, inferentialists argue that meanings of expressions are given by introduction and elimination rules. This is especially plausible for the logical constants, where standard presentations divide inference rules in just this way. But not just any rules will do, as we’ve learnt from Prior’s famous example of tonk, and the usual extra constraint is harmony. Where does this leave identity? It’s usually taken as a (...) logical constant but it doesn’t seem harmonious: standardly, the introduction rule only concerns a subset of the formulas canvassed by the elimination rule. In response, Read [5, 8] and Klev [3] amend the standard approach. We argue that both attempts fail, in part because of a misconception regarding inferentialism and identity that we aim to identify and clear up. (shrink)

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall. We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper—which we label two-level proof-theoretic pluralism—is much broader in scope than the one discussed in Beall (...) and Restall’s book. (shrink)

A logical constant is weakly disharmonious if its elimination rules are weaker than its introduction rules. Substructural weak disharmony is the weak disharmony generated by structural restrictions on the eliminations. I argue that substructural weak disharmony is not a defect of the constants which exhibit it. To the extent that it is problematic, it calls into question the structural properties of the derivability relation. This prompts us to rethink the issue of controlling the structural properties of a logic by means (...) of harmony. I argue that such a control is possible and desirable. Moreover, it is best achieved by global tests of harmony. (shrink)

In the recent literature on proof-theoretic semantics, there is mention of a generality condition on defining rules. According to this condition, the schematic formulation of the defining rules must be maximally general, in the sense that no restrictions should be placed on the contexts of these rules. In particular, context variables must always be present in the schematic rules and they should range over arbitrary collections of formulae. I argue against imposing such a condition, by showing that it has undesirable (...) results and that it is ill-supported by the arguments brought in its favour. (shrink)

Harmony and conservative extension are two criteria proposed to discern between acceptable and unacceptable rules. Despite some interesting works in this field, the exact relation between them is still not clear. In this article, some standard counterexamples to the equivalence between them are summarized, and a recent formulation of the notion of stability is used to express a more refined conjecture about their relation. Then Prawitz's proposal of a counterexample based on the truth predicate to this refined conjecture is shown (...) to rest on dubious assumptions. As a consequence, two new counterexamples are proposed: one uses the extension of logic with a small amount of arithmetic, while the other uses the extension of a small fragment of arithmetic with a problematic operator defined by Peano. It is argued that both these new counterexamples work fine to reject the conjecture and that the last one works also as a rejection of harmony as a complete criterion of acceptability of rules. (shrink)

The connectives of classical propositional logic are given an analysis in terms of necessary and sufficient conditions of acceptance and rejection, i.e. the connectives are analyzed within an expressivist bilateral meaning-is-use framework. It is explained how such a framework differs from standard inferentialist frameworks and it is argued that it is better suited to address the particular issues raised by the expressivist thesis that the meaning of a sentence is determined by the mental state that it is conventionally used to (...) express. Furthermore, it is shown that the classical requirements governing the connectives completely characterize classical logic, are conservative and separable, are in bilateral harmony, are structurally preservative with respect to the classical coordination requirements and resolve the categoricity problem. These results are taken to show that one can give an expressivist bilateral meaning-is-use analysis of the connectives that confer on them a determinate coherent classical interpretation. (shrink)

What is the relation of logic to thinking? My dissertation offers a new argument for the claim that logic is constitutive of thinking in the following sense: representational activity counts as thinking only if it manifests sensitivity to logical rules. In short, thinking has to be minimally logical. An account of thinking has to allow for our freedom to question or revise our commitments – even seemingly obvious conceptual connections – without loss of understanding. This freedom, I argue, requires that (...) thinkers have general abilities to respond to support and tension among their thoughts. And these abilities are constituted by following logical rules. So thinkers have to follow logical rules. But there isn’t just one correct logic for thinking. I show that my view is consistent with logical pluralism: there are a range of correct logics, any one of which a thinker might follow. A logic for thinking does, however, have to contain certain minimal principles: Modus Ponens and Non-Contradiction, and perhaps others. We follow logical rules by exercising logical capacities, which display a distinctive first-person/third-person asymmetry: a subject can find the instances of a rule compelling without seeing them as instances of a rule. As a result, there are two limits on illogical thinking. First, thinkers have to tend to find instances of logical rules compelling. Second, thinkers can’t think in obviously illogical ways. So thinking has to be logical – but not perfectly so. When we try to think, but fail, we produce nonsense. But our failures to think are often subjectively indistinguishable from thinking. To explain how this occurs, I offer an account of nonsense. To be under the illusion that some nonsense makes sense is to enter a pretence that the nonsense is meaningful. Our use of nonsense within the pretence relies on the role of logical form in understanding. Finally, while the normativity of logic doesn’t fall directly out of logical constitutivism, it’s possible to build an attractive account of logical normativity which has logical constitutivism as an integral part. I argue that thinking is necessary for human flourishing, and that this is the source of logical normativity. (shrink)

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? In this paper, I canvass several possible explanations. I argue that the most promising approach (...) is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought. (shrink)

In this paper, I argue that the disjunction elimination rule presupposes the principle that a true disjunction contains at least one true disjunct. However, in some contexts such as supervaluationism or quantum logic, we have good reasons to reject this principle. Hence, disjunction elimination is restricted in at least one respect: it is not applicable to disjunctions for which this principle does not hold. The insight that disjunction elimination presupposes the principle that a true disjunction contains at least one true (...) disjunct is applied to two arguments which argue for this very principle. I show that these arguments are rule-circular since they rest on disjunction elimination. I claim that rule-circularity better explains why the arguments fail than the explanations provided by Rumfitt (2015), which, for instance, rely on controversial principles about truth. (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 paper concentrates on how to capture harmony in sequent calculi. It starts by considering a proposal made by Tennant and some objections to it which have been presented by Steinberger. Then it proposes a different analysis which makes use of a double-line presentation of sequent calculi in the style of Dosen and it shows that this proposal is able to dismiss disharmonious operators without thereby adopting any global criterion.