Chapter Two, Tutorial Three

Truth Functionality
and non-Truth Functional Connectives

This tutorial has an imposing name. Hopefully, you're suitably impressed! Fortunately, though, there is not that much to the notion of "truth functionality".

Here's the idea by way of example. You already know that

1. Agnes will attend law school and so will Bob

is true just in case its two component parts, the conjuncts, are true. That's all its truth value depends on. Similarly, 'A&B's truth value depends only on the truth value of A and B. Contrast this example with the apparently similar,

2. Agnes will attend law school and then she will make millions.

This contains the connective "and then" rather than just "and". That makes all the difference. Sentence 2 depends on more than the truth value of its two conjuncts: "Agnes will attend law school" and "She will make millions". It also depends on timing: for 2 to be true, the making of millions needs to follow the attendance at law school.

That's it for the idea we need to introduce: a connective (e.g., "and") is truth functional when the truth or falsity of the sentence it forms (e.g., 1) depends only on the truth value of its conjuncts. Because a connective may be used in different ways, the definition can be more carefully put:

A connective is used truth functionally to form a sentence from components if and only if that sentence's truth value depends only on the truth value of the components. Otherwise, it is used non-truth functionally.

That is a bit of a mouthful, so go back and make sure you understand the "and" verses "and then" example above. It illustrates the distinction. Plain "and" is used truth functionally in (1); "and then" is normally used non-truth functionally as in (2).

A bit informally, we may say that a connective is truth functional if it is typically used truth functionally and not truth functional otherwise.

Now, click on the connective used truth functionally in what follows:

  • John fell on the ground before he cried out in pain.
  • John fell on the ground because Abby pushed him.
  • J>T