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What is the fundamental insight behind truth-functionality ? When is a logic interpretable by way of a truth-functional semantics? To address such questions in a satisfactory way, a formal definition of truth-functionality from the point of view of abstract logics is clearly called for. As a matter of fact, such a definition has been available at least since the 70s, though to this day it still remains not very widely well-known. A clear distinction can be drawn between logics characterizable through: (...) (1) genuinely finite-valued truth-tabular semantics; (2) no finite-valued but only an infinite-valued truthtabular semantics; (3) no truth-tabular semantics at all. Any of those logics, however, can in principle be characterized through non-truth-functional valuation semantics, at least as soon as their associated consequence relations respect the usual tarskian postulates. So, paradoxical as that might seem at first, it turns out that truth-functional logics may be adequately characterized by non-truth-functional semantics . Now, what feature of a given logic would guarantee it to dwell in class (1) or in class (2), irrespective of its circumstantial semantic characterization? (shrink) | |
The mathematical concept of pragmatic truth, first introduced in Mikenberg, da Costa and Chuaqui (1986), has received in the last few years several applications in logic and the philosophy of science. In this paper, we study the logic of pragmatic truth, and show that there are important connections between this logic, modal logic and, in particular, Jaskowski's discussive logic. In order to do so, two systems are put forward so that the notions of pragmatic validity and pragmatic truth can be (...) accommodated. One of the main results of this paper is that the logic of pragmatic truth is paraconsistent. The philosophical import of this result, which justifies the application of pragmatic truth to inconsistent settings, is also discussed. (shrink) | |
Da Costa's C systems are surveyed and motivated, and significant failings of the systems are indicated. Variations are then made on these systems in an attempt to surmount their defects and limitations. The main system to emerge from this effort, system CC , is investigated in some detail, and dual-intuitionistic semantical analyses are developed for it and surrounding systems. These semantics are then adapted for the original C systems, first in a rather unilluminating relational fashion, subsequently in a more illuminating (...) way through the introduction of impossible situations where and and or change roles. Finally other attempts to break out of impasses for the original and expanded C systems, by going inside them, are looked at, and further research directions suggested. (shrink) | |
S. Jakowski introduced the discussive prepositional calculus D 2as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D 2has afterwards been extended to a first-order predicate calculus and to a higher-order logic (cf. the (...) quoted paper). In this paper we present a natural version of D 2, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented. (shrink) | |
S. Jaśkowski introduced the discussive propositional calculus D₂ as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories. D₂ has afterwards been extended to a first-order predicate calculus and to a higher-order logic. In this paper we present a natural version of D₂, in the sense of Jaśkowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus. A semantics for the new calculus is also presented. | |
In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false. | |
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