Showing papers on "Paraconsistent logic published in 1984"

Perfect validity, entailment and paraconsistency

Abstract: This paper treats entailment as a subrelation of classical consequence and deducibility. Working with a Gentzen set-sequent system, we define an entailment as a substitution instance of a valid sequent all of whose premisses and conclusions are necessary for its classical validity. We also define a sequent Proof as one in which there are no applications of cut or dilution. The main result is that the entailments are exactly the Provable sequents. There are several important corollaries. Every unsatisfiable set is Provably inconsistent. Every logical consequence of a satisfiable set is Provable therefrom. Thus our system is adequate for ordinary mathematical practice. Moreover, transitivity of Proof fails upon accumulation of Proofs only when the newly combined premisses are inconsistent anyway, or the conclusion is a logical truth. In either case Proofs that show this can be effectively determined from the Proofs given. Thus transitivity fails where it least matters — arguably, where it ought to fail! We show also that entailments hold by virtue of logical form insufficient either to render the premisses inconsistent or to render the conclusion logically true. The Lewis paradoxes are not Provable. Our system is distinct from Anderson and Belnap's system of first degree entailments, and Johansson's minimal logic. Although the Curry set paradox is still Provable within naive set theory, our system offers the prospect of a more sensitive paraconsistent reconstruction of mathematics. It may also find applications within the logic of knowledge and belief.

A metacompleteness theorem for contraction-free relevant logics

Abstract: I note that the logics of the “relevant” group most closely tied to the research programme in paraconsistency are those without the contraction postulate(A→.A→B)→.A→B and its close relatives. As a move towards gaining control of the contraction-free systems I show that they are prime (that wheneverA ∨B is a theorem so is eitherA orB). The proof is an extension of the metavaluational techniques standardly used for analogous results about intuitionist logic or the relevant positive logics.

The Logic of Science as a Model-Oriented Logic

Abstract: Philosophers at least since Kant, with Larry Laudan being a recent example, have suggested that scientific inquiry be thought of as a problem-solving or question-answering activity. The logic of such a conception of scientific inquiry has not been studied systematically, however. This paper presents some of the main aspects of the logic on which such a conception of science is based. That logic is called in this paper model-oriented logic, and it is suggested that one can systematically study optimal questioning strategies with it.

Paraconsistent logic and model theory

Abstract: The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC 1 = (obtained by adding the axiom ⌝⌝A ↔A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of Łoś-Tarski and Chang-Łoś-Suszko, Craig-Robinson and the Beth definability theorem.

Some definitions of negation leading to paraconsistent logics

Abstract: 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.

Hegel’s Logic from A Logical Point of View1

Abstract: My title is not intended as a pun. Rather it should be interpreted as a question: how to deal with Hegel’s logic from a logical point of view? I mean here by ‘a logical point of view’ the point of view which adheres to the body of logical truths that we can formulate in classical propositional logic and first- order predicate logic. That is not necessarily a conservative stand, only a careful one. The question indicates that one should be prepared to adopt a critical point of view concerning Hegel’s logic. The fact that most logicians are not interested in Hegel’s logic is a sign that such a critical attitude is not uncommon. Three years ago, I asked Professor Quine how one could go about Hegel’s dialectical logic. He simply answered that one would have to change the laws of logic in order to make sense of Hegel’s logic.2 I doubt that there is anyone who would be ready to support Hegel to such an extent as to abandon the corpus of our logical laws. I do not think either that one has to drive to extremities to extract some logical sense from what Hegel called Die Wissenschaft der Logik.