Section 1: Truth-functional connectives
Any non-atomic sentence of SL is composed of atomic sentences with sentential connectives. The truth-value of the compound sentence depends only on the truth-value of the atomic sentences that comprise it. In order to know the truth-value of (\(D\) ↔ \(E\)), for instance, you only need to know the truth-value of \(D\) and the truth-value of \(E\). Connectives that work in this way are called truth-functional .
In this chapter, we will make use of the fact that all of the logical operators in SL are truth-functional— it makes it possible to construct truth tables to determine the logical features of sentences. You should realize, however, that this is not possible for all languages. In English, it is possible to form a new sentence from any simpler sentence \(\mathcal{X}\) by saying ‘It is possible that \(\mathcal{X}\).’ The truth-value of this new sentence does not depend directly on the truth-value of \(\mathcal{X}\). Even if \(\mathcal{X}\) is false, perhaps in some sense \(\mathcal{X}\) could have been true— then the new sentence would be true. Some formal languages, called modal logics , have an operator for possibility. In a modal logic, we could translate ‘It is possible that \(\mathcal{X}\)’ as ◇\(\mathcal{X}\). However, the ability to translate sentences like these come at a cost: The ◇ operator is not truth-functional, and so modal logics are not amenable to truth tables.