The general aim of this book is to provide an elementary exposition of some basic concepts in terms of which both classical and non-dassicallogirs may be studied and appraised. Although quantificational logic is dealt with briefly in the last chapter, the discussion is chiefly concemed with propo gjtional cakuli. Still, the subject, as it stands today, cannot br covered in one book of reasonable length. Rather than to try to include in the volume as much as possible, I have put (...) emphasis on some selected topics. Even these could not be roverrd completely, but for each topic I have attempted to present a detailed and precise t’Xposition of several basic results including some which are non-trivial. The roots of some of the central ideas in the volume go back to J. Luka siewicz’s seminar on mathematicallogi. (shrink)

We present a multiple-assumption multiple-conclusion system for bi-intuitionistic logic. Derivations in the systems are graphs whose edges are labelled by formulas and whose nodes are labelled by rules. We show how to embed both the standard intuitionistic and dual-intuitionistic natural deduction systems into the proposed system. Soundness and completeness are established using translations with more traditional sequent calculi for bi-intuitionistic logic.

A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.

In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally’s deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable (...) as a deontic logic, but it does make sense as a lax logic in the modern sense of the word. (shrink)

Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull–Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion—spanning several of the notes—is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)

This paper is a reaction to the following remark by grzegorczyk: "the compound sentences are not a product of experiment. they arise from reasoning. this concerns also negations; we see that the lemon is yellow, we do not see that it is not blue." generally, in science the truth is ascertained as indirectly as falsehood. an example: a litmus-paper is used to verify the sentence "the solution is acid." this approach gives rise to a (very intuitionistic indeed) conservative extension of (...) the heyting logic satisfying natural duality laws. (shrink)

This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), (...) and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. (shrink)