The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 1 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right (...) of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical. (shrink)

By a free logic is generally meant a variant of classical first-order logic in which constant terms may, under interpretation, fail to refer to individuals in the domain D over which the bound variables range, either because they do not refer at all or because they refer to individuals outside D. If D is identified with what is assumed by the given interpretation to exist, in accord with Quine’s dictum that “to be is to be the value of a [bound] (...) variable,” then a free variation on classical semantics does not require that all constant terms refer to existents, and in this sense such terms lack existential import. (shrink)

This article presents a sequent calculus for a negative free logic with identity, called N . The main theorem (in part 1) is the admissibility of the Cut-rule. The second part of this essay is devoted to proofs of soundness, compactness and completeness of N relative to a standard semantics for negative free logic.

Our topic is the notion of logical consequence: the link between premises and conclusions, the glue that holds together deductively valid argument. How can we understand this relation between premises and conclusions? It seems that any account begs questions. Painting with very broad brushtrokes, we can sketch the landscape of disagreement like this: “Realists” prefer an analysis of logical consequence in terms of the preservation of truth [29]. “Anti-realists” take this to be unhelpful and o:er alternative analyses. Some, like Dummett, (...) look to preservation of warrant to assert [9, 36]. Others, like Brandom [5], take inference as primitive, and analyse other notions in terms of it. There is plenty of disagreement on the “realist” side of the fence too. It is one thing to argue that logical consequence involves preservation of truth. It is another to explain how far truth must be preserved. Is the preservation essentially modal (in all circumstances [25]) or analytic (vouchsafed by.. (shrink)

In this short paper, I compare and contrast the kind of symmetric treatment of negation favoured in different ways by Huw Price (in “Why ‘Not’?”) and by me (in “Multiple Conclusions”) with Robert Brandom’s analysis of scorekeeping in terms of commitment, entitlement and incompatibility. Both kinds of account are what Brandom calls a normative pragmatics. They are both semantic anti-realist accounts of meaning in the significance of vocabulary is explained in terms of our rule-governed (normative) practice (pragmatics). These accounts differ (...) from intuitionist semantic anti-realism by providing a way to distinguish the inferential significance of “A” and “A is warranted.” Although proof plays a central role, in neither accont is verification the primary bearer of meaning. Our accounts make these distinctions in terms of a subtle analysis of our practices. On the one hand according to Price and me, we assert as well as deny; on the other, Brandom distingushes downstream commitments from upstream entitlements and the notion of incompatibility definable in terms of these. In this paper I will examine a number connections between these different approaches, and end with a discussion of the kind of account of proof that might emerge from these considerations. (shrink)