Tutorial Contents Tutorial Two: Propositional Calculus: Language - Truth Functors - The Propositional Language - "Implicate" - Stand-alone sentences - Scope More Tutorials One Three Four Five Six Seven

### Truth-Functors

Consider "j and y".

If we had a sentence consisting of this together with two constituent sentences, we would be able to tell the truth-value of the sentence as a whole given simply the truth-values of the constituent sentences. We would know that the whole sentence would be true if both the constituent sentences were true, and otherwise would be false. We could represent this fact about "j and y" by the following diagram:

 j  y j and y T  T T T  F F F  T F F  F F

A diagram of this sort is called a truth-table. So, a truth-table for a sentence-functor is a diagram which shows (in the case of a sentence consisting of it together with constituent sentences) what value the sentence as a whole will take for every possible combination of truth-values in the constituent sentences.

We can define a truth-functor thus:

##### A truth-functor is a sentence-functor such that the truth-value of any sentence consisting of it together with constituent sentences is determined solely by the truth-values of the constituent sentences (together, of course, with the meaning of the sentence-functor).

Alternatively, we could define a truth-functor as a sentence-functor with a truth-table - but the usefulness of this depends, of course, on the fact that we have already defined truth-table.

Some sentence-functors are not truth-functors: "j because y", for example. This is because, although the falsehood of either of the constituents is enough to determine the truth-value of the sentence as a whole (it must be false), the truth of both constituents is not enough to determine the truth-value as a whole. For instance "Bill Clinton was president of the USA because he secured most votes in the electoral college" is true. But "Bill Clinton was president of the USA because the battle of Hastings took place in 1066" is not. So we can draw at most a partial truth-table for "j because y".

 j  y j because y T  T ? T  F F F  T F F  F F

The following is also not a truth-functor:

Bill Clinton was president of the USA and j.

This is because, although the falsity of j is enough to determine the truth-value of the whole, its truth is not. You, of course, will know that the whole is true if the constituent is true. But that is because you know some history. The fact that it is true is not determined solely by the truth of the constituent and the meaning of the sentence-functor.