Topic: HUMAN SEMANTICS
Semantics WITHOUT POSSIBLE WORLDS?
Many-Valued and Kripke Semantics
By Jean-Yves Beziau
Today many people identify Kripke semantics with modal logic. Typically a book called “modal logic” nowadays is a book about Kripke semantics (cf. e.g. the recent book by [Blackburn et al (2001)]). But modal logic can be developed using other kinds of semantics and Kripke semantics can be used to deal with many different logics and it is totally absurd to call all of these logics “modal logics”. Kripke semantics are also often called “possible worlds semantics”, however this is quite misleading because the crucial feature of these semantics is not the concept of possible world but the relation of accessibility. Possible worlds can easily be eliminated from the definition of Kripke semantics and then the accessibility relation is defined directly between the bivaluations. For this reason it seems better to use the terminology “relational semantics”. Of course, if we want, we can call these bivaluations "possible worlds", this metaphor can be useful, but then why using this metaphor only in the case of relational semantics? In fact in the Tractatus Wittgenstein used the expression “truth-possibilities” for the classical bivaluations. Other concepts of the semantics of classical zero-order logic were expressed by him using a modal terminology: he said that a formula is necessary if it holds for all truth possibilities, impossible if it holds for none, and possible if it holds for some. But Wittgenstein was against the introduction of modal concepts inside the language as modal operators.
Many-valued and Kripke semantics may be philosophically controversial, anyway they are very useful and powerful technical tools which can be fruitfully used to give a mathematical account of basic philosophical notions, such as modalities. It seems to me that instead of focusing on the one hand on some little philosophical problems and on the other hand on some developments limited to one technique, one should promote a better interaction between philosophy and logic developing a wide range of techniques, as for example the combination of Kripke semantics (extended as to include Jaskowski semantics) and Many-Valued semantics (extended as to include non truth-functional many-valued semantics). My aim in this paper is to give a hint of how these techniques can be developed by presenting various examples.