§2. Truth Functions

Truth Functions

The truth or falsity of the components of a truth function sufficiently determine the truth or falsity of the truth function as a whole. Stating a truth function is simply a convenient and precise way of summarizing the truth values of several statements at once. For example:

I am watching The Two Towers and I am drinking a milk stout

is a truth function, because whether the statements

I am watching The Two Towers

and

I am drinking a milk stout

are true determine whether the compound is true. However the statement

I am watching The Two Towers because I am drinking a milk stout

is NOT a truth function, because two people can affirm both statements, or affirm either statement and deny the other, or even deny both statements, and yet still debate over whether the compound is true.

A truth function can be described by creating a chart of the truth values of each of its components. Besides "not", "and", and "or", one can make up one's own truth function by explaining when the compound will turn out true, and when it will turn out false.

Negation and conjunction alone are sufficient for all truth-functional purposes. Actually, negation and alternation alone may also be sufficient for all truth-functional purposes.

Truth Function Exercise

Quine asks his reader to make a statement that will be true only when any, but only, two out of three components are true, using only conjunction and negation. I came up with:

~(pqr)~(~p~q~r)~(~p~qr)~(~pq~r)~(p~q~r)

So, it is not the case that all three statements are true, it is not the case that all three statements are false, it is not the case that only r, it is not the case that only q, and it is not the case that only p - therefore the only situations that would render the compound true would be (pq), (qr), or (pr).