Generalized Truth Values: From FOUR2 to SIXTEEN3

Chapter
Part of the Trends in Logic book series (TREN, volume 36)

Abstract

In the present chapter, we discuss the possibility of generalizing the very notion of a truth value by constructing truth values as complex units which may possess a ramified inner structure. We consider some approaches to truth values as structured entities and summarize this point in the notion of a generalized truth value conceived as a subset of some basic set of initial truth values of a lower degree. We are essentially led by the idea which is at the heart of Belnap and Dunn’s useful four-valued logic, where the set $${{\mathbf{2}}} = \{T,F\}$$ of classical truth values is generalized to the set $${{\mathbf{4}}} = {{\fancyscript{P}}}({{\mathbf{2}}}) = \{\varnothing,\{T\},\{F\},\{T,F\}\}.$$ We argue in favor of extending this process to the set $${{\mathbf{16}}} = {{\fancyscript{P}}}({{\mathbf{4}}}).$$ It turns out that this generalization is well-motivated and leads to a notion of a truth value multilattice. In particular, we proceed from the bilattice $$FOUR_{2}$$ with both an information and truth-and-falsity ordering to another algebraic structure, namely the trilattice $$SIXTEEN_{3}$$ with an information ordering together with a truth ordering $$and$$ a (distinct) falsity ordering. We also consider another exemplification of essentially the same structure based on the set of truth values one can find in various constructive logics.

Keywords

Logical Order Intuitionistic Logic Classical Truth Logical Connective Generalize Truth
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