5.1.7 Validities

Our main topic in this chapter is logical inference. Given the semantic concepts just introduced, we're now in a position to state precisely what we mean by this. We will do so in two separate steps. First we'll establish what valid formulae (or more simply, validities) are. Then we'll define the concept of a valid argument (or valid inference).

Valid Formulae

A valid formula is a formula that is satisfied in all models (of the appropriate vocabulary) given any variable assignment. That is, if is a valid formula, it is impossible to find a situation in which would not be satisfied. We indicate that a formula is valid by writing .

For example:

In any model, given any variable assignment, one of the two disjuncts must be true (incidentally in the case at hand, only one can be true), and hence the whole formula will be satisfied too.

Valid Sentences

Note that for sentences the definition of validity can be rephrased as follows, without reference to assignments: A valid sentence is a sentence that is true in all models (of the appropriate vocabulary). That is, it is impossible to falsify a valid sentence. For example: