An implication relation between pictures is defined, it is then shown how conjunctions, disjunctions, negations, and hypotheticals of pictures can be formed on the basis of this. It is argued that these logical operations on pictures correspond to natural cognitive operations employed when thinking about pictures.

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate (...) for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens. (shrink)

We recast parts of decision theory in terms of channel theory concentrating on qualitative issues. Channel theory allows one to move between model theoretic and language theoretic notions as is necessary for an adequate covering. Doing so clarifies decision theory and presents the opportunity to investigate alternative formulations. As an example, we take some of Savage’s notions of decision theory and recast them within channel theory. In place of probabilities, we use a particular logic of preference. We introduce a logic (...) for describing actions separate from the logic of preference over actions. The structures introduced by channel theory that represent the decision problems can be seen to be an abstract framework. This frame-work is very accommodating to changing the nature of the decision problems to handle different aspects or theories about decision making. (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)

What is a minimal proof-theoretical foundation of logic? Two different ways to answer this question may appear to offer themselves: reduce the whole of logic either to the relation of inference, or else to the property of incompatibility. The first way would involve defining logical operators in terms of the algebraic properties of the relation of inference—with conjunction $$\hbox {A}\wedge \hbox {B}$$ A ∧ B as the infimum of A and B, negation $$\lnot \hbox {A}$$ ¬ A as the minimal (...) incompatible of A, etc. The second way involves introducing logical operators in terms of the relation of incompatibility, such that X is incompatible with $$\{\lnot \hbox {A}\}$$ { ¬ A } iff every Y incompatible with X is incompatible with {A}; and X is incompatible with $$\{\hbox {A}\!\wedge \!\hbox {B}\}$$ { A ∧ B } iff X is incompatible with {A,B}; etc. Whereas the first route leads us naturally to intuitionistic logic, the second leads us to classical logic. The aim of this paper is threefold: to investigate the relationship of the two approaches within a very general framework, to discuss the viability of erecting logic on such austere foundations, and to find out whether choosing one of the ways we are inevitably led to a specific logical system. (shrink)