We define a system IR of first-order intuitionistic relevant logic. We show that intuitionistic mathematics (on the assumption that it is consistent) can be relevantized, by virtue of the following metatheorem: any intuitionistic proof of A from a setX of premisses can be converted into a proof in IR of eitherA or absurdity from some subset ofX. Thus IR establishes the same inconsistencies and theorems as intuitionistic logic, and allows one to prove every intuitionistic consequence of any consistent set of (...) premisses. (shrink)

Many logical systems are such that the addition of Prior's binary connective tonk to them leads to triviality, see [1, 8]. Since tonk is given by some introduction and elimination rules in natural deduction or sequent rules in Gentzen's sequent calculus, the unwanted effects of adding tonk show that some kind of restriction has to be imposed on the acceptable operational inferences rules, in particular if these rules are regarded as definitions of the operations concerned. In this paper, a number (...) of simple observations is made showing that the unwanted phenomenon exemplified by tonk in some logics also occurs in contexts in which tonk is acceptable. In fact, in any non-trivial context, the acceptance of arbitrary introduction rules for logical operations permits operations leading to triviality. Connectives that in all non-trivial contexts lead to triviality will be called non-trivially trivializing connectives. (shrink)

Allocating the burden of proof and intuitions in philosophical disputes.– This paper criticises the view that in philosophical disputes the onus probandi rests on those who advance a position that contradicts our basic intuitions. Such a rule for allocating the burden of proof may be an adequate reconstruction of everyday justification, but is unreasonable in the area of philosophy. In philosophy it is not only difficult to determine the plausibility of a proposition, at the same time contradictory claims may be (...) equally plausible. – In contrast to such common sense proposals I try to show that in philosophical disputes the burden of proof does not depend on the material content of speech acts. A speaker simply bears the burden of proof for a proposition p if he has asserted that p and has agreed to justify it. (shrink)

Neil Tennant’s core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant’s own, but it is very closely related, and determines the same consequence relation. The difference, however, matters for (...) our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant’s proof system for core logic is not strongly normalizing, our modified system is. (shrink)

This paper studies the relationship between Argumentation Logic, a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic. In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of sentences in AL and Natural Deduction proofs of the complement of these sentences. The proof of this equivalence uses a restricted form (...) of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies. (shrink)

The motivation for Core Logic is explained. Its system of proof is set out. It is then shown that, although the system has no Cut rule, its relation of deducibility obeys Cut with epistemic gain.

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the intended (...) schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (...) (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)