Chapter Two, Tutorial Two
A semantics is a definition of meaning for a language. But 'meaning' itself has a number of meanings. Here we fix on a simple idea of meaning: truth conditions.
To understand, say, the meaning of
Sam is riding a horse
is to know under what conditions it would be true. That is, one would need to know who Sam is, what riding is, and what a horse is. Then one would know the meaning well enough to understand just when it would be true.
So, one way to think about meaning is in terms of truth conditions. We shall apply this idea to molecular sentences of SL. Begin with our simple example
symbolizing the English "Agnes and Bob will attend law school." Under what conditions is this true? Obviously it's true when and only when both 'A' and 'B' are true. This doesn't sound too interesting yet, but be patient!
First, we will say that any conjunction of SL is like this (not just 'A&B' but also 'C&T', 'F&~G', 'F&(T>U)', etc.): each is true just in case both its conjuncts are true. We will put this general point as follows:
1. Any sentence P&Q of SL is true if and only if both P and Q are true. Otherwise P&Q is false.
Notice that 'P' and 'Q' are have a different shape and color. This is our way of saying that we are talking about any sentences P and Q.
('P' and 'Q' are not particular sentences of SL. Instead they each act as stand-ins for any sentence of SL. We will call them variables or metavariables. Similarly, 'x' and 'y' in 'x+y = y+x' do not stand for any particular numbers. Rather, these variables provide a way to talk about any pair of numbers.)
There is one more tool we need for semantics: the truth table. A truth table will just re-express our definition 1.
When P and Q are both true, then P&Q is true. We've already said that of course.
And if P is true, but not Q, then P&Q is false. (Because the only way for P&Q to be true is for both P and Q to be true that's what 1. says.)
We are cashing out what 1. amounts to. So far we can put the results in a table:
Better, we can simplify this table as follows:
Here each row just means that if P and Q have the truth values given in that row, then P&Q has the truth value shown there in the far right column.
Have you noticed anything missing? There are two more rows to give for our table. We have neglected a row mentioning the possibility that P and Q are both false: