Partially interpreted connectives.

Abstract

Partiality is understood in this study as the semantical undeterminacy of an expression in a particular situation, model, possible world, etc. Relativized to predicates, this means that there are models M and predicates P such that for some individuals a in the universe of Л/, a belongs to neither the extension nor the anti-extension ofP.

Perhaps the most discussed instances of partiality in philosophy are the ones connected with the semantical paradoxes. They often arise with self-referential expressions like “this sentence is not true”, which lead to paradoxes. One of the most praised solutions to this kind of paradoxes is to give up bivalence and to take the self-referential statements -to lack a truth-value. This amounts to give the truthpredicate a partial interpretation as done in the work of Kripke.