Truth and Possibility

Abstract

We have an obvious assertion (1) A Þ àA. If we apply T-scheme to A , it results in (2) TA Þ àA. Thus, what is true, is possibile as well. Although truth and possibility are treated here as monadic sentential (propositional) operators, a similar argument holds for predicates "is true" and "is possible". Hence, I will use I both readings as equivalent. We can also apply T-scheme to àA and obtain (3) TA Þ T(àA). This means that truth of a sentence implies that it is true, that the sentence in question is true too. If we agree that "is true" should be interpreted as "is true in a model M", A and àA are true in the same model.

Since (1) cannot be converted to àA Þ A, àA is weaker than A. In particular, àA can be false, but A. Assume that A is consistent and false. This means, on the base of Gödel-Malcev completeness theorem, that it has a model in which is true. On the other hand, this model must be different than the model in which A is true. Denote this model by M’. Semantics of possibile worlds has no problem with that. It is enough to distinguish a model M* as the actual world and say that other are accessible from it. Thus, A can be false in M*, but true in M’. This, however, contradicts our earlier assumption that A and àA are true in the same model. Logically speaking, introducing M* and M does not matter, because models are abstract algebraic structures. The situation changes when we pass to ontology, because the basic intuition points out that the actual world exists in the fundamental sense, but other merely subsists.

What can we do in order to resolve the issue? One way is to take a Platonic stance and treat all possible worlds as abstract objects. This is at odds with ordinary intuitions. Thus, we should find a solution respecting that M* is distinguished not only for logical, but also for ontological reasons. Possible worlds as logical constructs can be identified with models of maximally consistent sets of propositions. On this level, M* can be arbitrarily chosen. If we add the ontological factor, M* becomes a representation of the world in which we live. This open a possibility for naturalism.