This paper consider Prior's connective Tonk from a particular bilateralist perspective. I show that there is a natural perspective from which we can see Tonk and its ilk as perfectly well-defined pieces of vocabulary; there is no need for restrictions to bar things like Tonk.

In [5], Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract “core” of logics in general, where logical syntax and semantics are “two sides of the same coin”. The central suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal logic. (...) However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics. (shrink)

The discussions which follow rest on a distinction, first expounded by Husserl, between formal logic and formal ontology. The former concerns itself with (formal) meaning-structures; the latter with formal structures amongst objects and their parts. The paper attempts to show how, when formal ontological considerations are brought into play, contemporary extensionalist theories of part and whole, and above all the mereology of Leniewski, can be generalised to embrace not only relations between concrete objects and object-pieces, but also relations between what (...) we shall call dependent parts or moments. A two-dimensional formal language is canvassed for the resultant ontological theory, a language which owes more to the tradition of Euler, Boole and Venn than to the quantifier-centred languages which have predominated amongst analytic philosophers since the time of Frege and Russell. Analytic philosophical arguments against moments, and against the entire project of a formal ontology, are considered and rejected. The paper concludes with a brief account of some applications of the theory presented. (shrink)

Special issue: "Reflecting on the Legacy of C.I. Lewis: Contemporary and Historical Perspectives on Modal Logic".

In Posterior Analytics 1.3, Aristotle advances three arguments against circular proof. The third argument relies on his discussion of circular proof in Prior Analytics 2.5. This is problematic because the two chapters seem to deal with two rather disparate conceptions of circular proof. In Posterior Analytics 1.3, Aristotle gives a purely propositional account of circular proof, whereas in Prior Analytics 2.5 he gives a more complex, syllogistic account. My aim is to show that these problems can be solved, and that (...) Aristotle’s third argument in 1.3 is successful. I argue that both chapters are concerned with the same conception of circular proof, namely the propositional one. Contrary to what is often thought, the syllogistic conception provides an adequate analysis of the internal deductive structure of the propositional one. Aristotle achieves this by employing a kind of multiple-conclusion logic. (shrink)

Theory and Reality is about the connection between true theories and the world. A mathematical framefork for such connections is given, and it is shown how that framework can be used to infer facts about the structure of reality from facts about the structure of true theories, The book starts with an overview of various approaches to metaphysics. Beginning with Quine's programmatic "On what there is", the first chapter then discusses the perils involved in going from language to metaphysics. It (...) criticises contemporary intuition-driven metaphysics, comments on naturalistic approaches, and then presents the main proposition put forward in the thesis: we should base metaphysics on model theory. In chapters 2 to 5, mathematical treatments are given of concepts that we need: theories, metaphysics, necessitation and semantics. These are used in chapters 6 and 7 to prove that, seen from a certain informative view point, any true theory will give rise to an isomorphism between that theory and the world. This conclusion is similar to Wittgenstein's in the Tractatus, but differs in that it places the structural relationship on the level of whole theories, rather than single propositions. (shrink)

This article evaluates the strengths and weaknesses of arguments based on appeals to expertise. The intersection of two areas is explored: (i) the traditional argumentum ad verecundiam (literally, “appeal to modesty,” but characteristically the appeal to the authority of expert judgment) in informal logic, and (ii) the uses of expert systems in artificial intelligence. The article identifies a model of practical reasoning that underlies the logic of expert systems and the model of argument appropriate for the informal logic of the (...) argumentum ad verecundiam. (shrink)

This paper argues that logical inferentialists should reject multiple-conclusion logics. Logical inferentialism is the position that the meanings of the logical constants are determined by the rules of inference they obey. As such, logical inferentialism requires a proof-theoretic framework within which to operate. However, in order to fulfil its semantic duties, a deductive system has to be suitably connected to our inferential practices. I argue that, contrary to an established tradition, multiple-conclusion systems are ill-suited for this purpose because they fail (...) to provide a 'natural' representation of our ordinary modes of inference. Moreover, the two most plausible attempts at bringing multiple conclusions into line with our ordinary forms of reasoning, the disjunctive reading and the bilateralist denial interpretation, are unacceptable by inferentialist standards. (shrink)

In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted (...) by a number of authors under the heading of abstract entailment relations where it has found an important role in both algebra and theoretical computer science. In this essay we go back to the beginnings, as presented by Scott, in order to make some comments about Scott’s cut rule, and show how much of Scott’s main result may be applied to the case of single-conclusion logic. (shrink)

Three semantic relations are analyzed: the relation of implication of a question by a question and a set of declarative sentences, the relation of implication of a question by a question, and the relation of strong implication of a question by a question and a set of declarative sentences. The connections between these concepts and the concepts of relative soundness, partial answerhood and presupposition are examined. The principal results are theorems about, to speak generally, epistemic reducibility of well-posed questions to (...) sequences of implied yes-no questions. (shrink)

We explore in an experimental spirit the prospects for extending classical propositional logic with a new operator P intended to be interpreted when prefixed to a formula as saying that formula in question is at least partly true. The paradigm case of something which is, in the sense envisaged, false though still "partly" true is a conjunction one of whose conjuncts is false while the other is true. Ideally, we should like such a logic to extend classical logic - or (...) any fragment thereof under consideration - conservatively, to be closed under uniform substitution (of arbitrary formulas for sentence letters or propositional variables), and to allow the substitutivity of provably equivalent formulas salva provabilitate. To varying degrees, we experience some difficulties only with this last ('congruentiality') desideratum in the two four-valued logics we end up giving our most extended consideration to. (shrink)

This paper develops and motivates a paraconsistent approach to semantic paradox from within a modest inferentialist framework. I begin from the bilateralist theory developed by Greg Restall, which uses constraints on assertions and denials to motivate a multiple-conclusion sequent calculus for classical logic, and, via which, classical semantics can be determined. I then use the addition of a transparent truth-predicate to motivate an intermediate speech-act. On this approach, a liar-like sentence should be “weakly asserted”, involving a commitment to the sentence (...) and its negation, without rejecting the sentence. From this, I develop a proof-theory, which both determines a typical paraconsistent model theory, and also gives us a nice way to understand classical recapture. (shrink)

In this paper we investigate the relation between the axiomatization of a given logical consequence operation and axiom systems defining the class of algebras related to that consequence operation. We show examples which prove that, in general there are no natural relation between both ways of axiomatization.

Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In (...) this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature. (shrink)

This paper considers some issues to do with valuational presentations of consequence relations, and the Galois connections between spaces of valuations and spaces of consequence relations. Some of what we present is known, and some even well-known; but much is new. The aim is a systematic overview of a range of results applicable to nonreflexive and nontransitive logics, as well as more familiar logics. We conclude by considering some connectives suggested by this approach.

We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we (...) call ipsilateral: applying such a rule does not make any formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions. (shrink)

The phrase ‘autoepistemic logic’ was introduced in Moore [1985] to refer to a study inspired in large part by criticisms in Stalnaker [1980] of a particular nonmonotonic logic proposed by McDermott and Doyle.1 Very informative discussions for those who have not encountered this area are provided by Moore [1988] and the wide-ranging survey article Konolige [1994], and the scant remarks in the present introductory section do not pretend to serve in place of those treatments as summaries of the field. A (...) good deal of the material omitted here pertains to the specifically nonmonotonic nature of autoepistemic logic as standardly developed, but as we shall urge, there is from one point of view nothing distinctively nonmonotonic about the basic motivating ideas of the subject. (shrink)

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, (...) for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)