Introducing Identity

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 (reflexivity) only concerns a subset of the formulas canvassed by the elimination rule (Leibniz’s law). 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.

Inferentialism and the Epistemology of Logic

This essay attempts to clarify the project of explaining the possibility of ‘blind reasoning’—namely, of basic logical inferences to which we are entitled without our having an explicit justification for them. The role played by inferentialism in this project is examined and objections made to inferentialism by Paolo Casalegno and Timothy Williamson are answered. Casalegno proposes a recipe for formulating a counterexample to any proposed constitutive inferential role by imaging a subject who understands the logical constant in question but fails to have the capacity to make the inference in question; Williamson’s recipe turns on imagining an expert who continues to understand the constant in question while having developed sophisticated considerations for refusing to make it. It’s argued that neither recipe succeeds.

Boghossian and Casalegno on Understanding and Inference1

In response to Boghossian’s objections in Chapter 6, this chapter defends counterexamples offered by Paolo Casalegno and the author to an inferentialist account of what it is to understand a logical constant, on which Boghossian relied in his explanation of our entitlement to reason according to basic logical principles. The importance for understanding is stressed of non-inferential aspects of the use of logical constants, for example in the description of a perceived scene. Boghossian’s criteria for individuating concepts are also queried, as is the viability of hybrid accounts which mix inferential accounts of the use of some terms with non-inferential accounts of other terms.

Dialogical logic

Dialogical logic characterizes logical constants (such as ‘and’, ‘or’, ‘for all’) by their use in a critical dialogue between two parties: a proponent who has asserted a thesis and an opponent who challenges it. For each logical constant, a rule specifies how to challenge a statement that displays the corresponding logical form, and how to respond to such a challenge. These rules are incorporated into systems of regimented dialogue that are games in the game-theoretical sense. Dialogical concepts of logical consequence can then be based upon the concept of a winning strategy in a (formal) dialogue game: B is a logical consequence of A if and only if there is a winning strategy for the proponent of B against any opponent who is willing to concede A. But it should be stressed that there are several plausible (and non-equivalent) ways to draw up the rules.

Natural deduction, tableau and sequent systems

Different presentations of the principles of logic reflect different approaches to the subject itself. The three kinds of system discussed here treat as fundamental not logical truth, but consequence, the relation holding between the premises and conclusion of a valid argument. They are, however, inspired by different conceptions of this relation. Natural deduction rules are intended to formalize the way in which mathematicians actually reason in their proofs. Tableau systems reflect the semantic conception of consequence; their rules may be interpreted as the systematic search for a counterexample to an argument. Finally, sequent calculi were developed for the sake of their metamathematical properties. All three systems employ rules rather than axioms. Each logical constant is governed by a pair of rules which do not involve the other constants and are, in some sense, inverse. Take the implication operator ‘→’, for example. In natural deduction, there is an introduction rule for ‘→’ which gives a sufficient condition for inferring an implication, and an elimination rule which gives the strongest conclusion that can be inferred from a premise having the form of an implication. Tableau systems contain a rule which gives a sufficient condition for an implication to be true, and another which gives a sufficient condition for it to be false. A sequent is an array Γ⊢Δ, where Γ and Δ are lists (or sets) of formulas. Sequent calculi have rules for introducing implication on the left of the ‘⊢’ symbol and on the right. The construction of derivations or tableaus in these systems is often more concise and intuitive than in an axiomatic one, and versions of all three have found their way into introductory logic texts. Furthermore, every natural deduction or sequent derivation can be made more direct by transforming it into a ‘normal form’. In the case of the sequent calculus, this result is known as the cut-elimination theorem. It has been applied extensively in metamathematics, most famously to obtain consistency proofs. The semantic inspiration for the rules of tableau construction suggests a very perspicuous proof of classical completeness, one which can also be adapted to the sequent calculus. The introduction and elimination rules of natural deduction are intuitionistically valid and have suggested an alternative semantics based on a conception of meaning as use. The idea is that the meaning of each logical constant is exhausted by its inferential behaviour and can therefore be characterized by its introduction and elimination rules. Although the discussion below focuses on intuitionistic and classical first-order logic, various other logics have also been formulated as sequent, natural deduction and even tableau systems: modal logics, for example, relevance logic, infinitary and higher-order logics. There is a gain in understanding the role of the logical constants which comes from formulating introduction and elimination (or left and right) rules for them. Some authors have even suggested that one must be able to do so for an operator to count as logical.

Predicates, parts, and impermanence: a contemporary version of some central Buddhist tenets

In this article, I argue that recent work in analytic philosophy on the semantics of names and the metaphysics of persistence supports two theses in Buddhist philosophy, namely the impermanence of objects and a corollary about how referential language works. According to this latter package of views, the various parts of what we call one object (say, King Milinda) possess no unity in and of themselves. Unity comes rather from language, in that we have terms (say, ‘King Milinda’) which stand for all the parts taken together. Objects are mind- (or rather language-)generated fictions. I think this package can be cashed out in terms of two central contemporary views. The first is that there are temporal parts: just as an object is spatially extended by having spatial parts at different spatial locations, so it is temporally extended by having temporal parts at different temporal locations. The second is that names are predicates: rather than standing for any one thing, a name stands for a range of things. The natural language term ‘Milinda’ is not akin to a logical constant, but akin to a predicate.Putting this together, I'll argue that names are predicates with temporal parts in their extension, which parts have no unity apart from falling under the same predicate. ‘Milinda’ is a predicate which has in its extension all Milinda's parts. The result is an interesting and original synthesis of plausible positions in semantics and metaphysics, which makes good sense of a central Buddhist doctrine.

Preface. Conditional: Conceptual and Historical Analysis

The logic of conditional is developed hereby in a series of papers, contributing to a historical and critical analysis of what the logical constant is expected to mean.

ISOMORPHISM INVARIANCE AND OVERGENERATION

The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We consider five possible precisifications of the overgeneration argument and find them all unconvincing.

WEAK DISHARMONY: SOME LESSONS FOR PROOF-THEORETIC SEMANTICS

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.