On substructural logics preserving degrees of truth

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Josep Maria Font
Title: On substructural logics preserving degrees of truth
Journal: Bulletin of the Section of Logic
Volume 36
Pages: 117–130
Year: 2007


The purpose of this paper is to discuss how some ideas coming from the many-valued logic world can be introduced in a sensible way into the world of substructural logic; namely, the ideas around what does it mean for a logic to say that it preserves degrees of truth. These two subject areas are by their origin rather far apart; I would like to exemplify how the recent evolution of research in the field of substructural logics, with the application of some central techniques from abstract algebraic logic, has facilitated the investigation of such borderline issues.

Degrees of truth are of course ubiquitous in the literature on many-valued and fuzzy logic, as one of the interpretations of non-classical truth-values. However, in general logics admitting semantics with more than two degrees of truth are cast to preserve just one of them, absolute truth. Less discussed is the idea of a logic preserving degrees of truth. It was considered, in algebraic terms, by Nóvak and by Wójcicki. Both assume an ordering relation between degrees of truth, so that the idea appears as related only to ordered algebras. Here I would like to give it a broader spectrum of application.

The idea of a logic preserving degrees of truth is presented as opposed to that of a truth-preserving logic ; I will try to demonstrate that this is only apparent. The idea of a truth-preserving logic is related to the classical conception of logical consequence, according to which "a conclusion follows logically from some premises if and only if, whenever the premises are true, the conclusion is also true".

I think that the difference between the ideas of preserving truth and preserving degrees of truth does not lie in the acceptation or rejection of this conception of consequence, but in applying it to a different conception of truth, which may consequently change the interpretation of the "whenever" in the sentence ending the previous paragraph.