Taking Degrees of Truth Seriously

Abstract

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko's Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Lukasiewicz's many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.

A New View of Effects in a Hilbert Space

“…It is possible to extract a logic L = Fm, out of any quasivariety Q of bounded lattice-ordered algebras [21,39], by letting Fm be the formula algebra of the language L of Q and by defining as follows for any Γ∪{α} ⊆ F m:…”

Abstract Valuation Semantics

“…Besides lacking a thorough supporting theory, namely if contrasted with the rich theory of logical matrices, valuation semantics has been mostly criticized for its excessive generality, namely as it can be confused Suszko's bivalence thesis (see, for instance, [5,11]). However, the valuation semantics used in the above example is far from being ad-hoc.…”

“…The key idea is to drop the condition that formulas should always be interpreted homomorphically in an algebra over the same signature. Besides lacking a thorough study, namely when contrasted with the rich algebraic theory of logical matrices (see [21]), valuation semantics has also been criticized for its excessive generality (see, for instance, [11]). Still, logicians would agree that a matrix semantics is simply a clever and algebraically well-behaved way of defining a valuation semantics by collecting all possible homomorphic interpretations.…”

A Co-inductive Language and Truth Degrees

“…For example, according to [Fn09], Suszko's logical values are thought of as the truth values of the meta-theory, and algebraic values are thought of as the truth values of the object theory. Many philosophers and logicians who accept this account seem to accept the choice of such meta-theory and the idea that nothing other than classical logic can be the foundation of semantics.…”