×

Applications of paraconsistent logic. (English) Zbl 0697.03016

Paraconsistent logic, Essays on the inconsistent, 367-393 (1989).
[For the entire collection see Zbl 0678.00003.]
This paper discusses some applications of paraconsistent logics to possibly inconsistent theories, considering that the studies of many of these applications are in their infancy and that their investigation promises to be a fascinating task for paraconsistent theories.
The authors start by giving a general overview of the range of inconsistent theories. They present inconsistent theories as being found in almost every discipline, emphasising them especially in philosophy and theology, natural and social sciences, and in logic and mathematics.
After that, they consider a few of them in more detail, looking at semantics, set theory, the infinitesimal calculus and some bits of quantum theory. They base their theory on a suitable quantificational extension of a special relevant logic, which they consider the most versatile and the most philosophically adequate paraconsistent logic.
In spite of no classical theory being able to express adequately its own semantics, on pain of inconsistency, naive semantics is capable of giving a semantics for itself. This theory must be paraconsistenly based because of the semantic paradoxes, and cannot be based on any paraconsistent logic which contains the absortion principle \(A\to (A\to B)/A\to B\). The authors have chosen an apparently simple way in order to formalize the theory, whose triviality has not yet been investigated.
Naive property theory and set theory cannot be formalized nontrivially without paraconsistent logic, which also must not admit the absorption principle. The paper shows that, if the abstraction schema is formulated without restriction in naive set theory, that is \(\exists z \forall y(y\in z\leftrightarrow \phi)\) for \(\phi\) arbitrary with z not occurring in it, there exist some more interesting sets. They also mention the situation for category theory.
The authors consider that Robinson’s reworking of the infinitesimal calculus in terms of non-standard analysis, in spite of having shown that the theory was not really inconsistent, is not exactly the original theory. As infinitesimals had to be genuine inconsistent objects, they suggest a naive infinitesimal theory, with a paraconsistent formalization.
Related to quantum theory the paper looks at some areas of special sensitivity with respect to consistency, which concern the collapse of wave packets upon measurement, and in particular the matter of the exact determination of operators such as those of position and momentum. These themes suggest a paraconsistent formalization of quantum theory.
Finally, the authors consider the application of paraconsistency to the theories of logic themselves: the theory of reason, inference and fallacies; modal and tense paraconsistent logics; deontic logics; doxastic logics; probability and inductive reasoning; information content and data processing; and the theory of vagueness.
Reviewer: I.D’Ottaviano

MSC:

03B60 Other nonclassical logic

Citations:

Zbl 0678.00003