10. Semantics and Truth

In dealing with the syntax of a language, we are dealing only with the purely formal properties of its constituent expressions. But, of course, in addition to those formal properties, the expressions can also possess semantic properties: they mean this, or refer to that, and so on. In semantics we move from considering the purely formal properties of linguistic expressions to considering their meaning and significance.

Let’s start by thinking a little more about arguments in propositional logic, and how we determine their validity. Consider another very simple argument:

(21) Beethoven was German and Napoleon was French; therefore
(22) Beethoven was German.

This formalises as P & Q; therefore, P. Now, how do we determine whether this argument is valid or not? Recall that an argument is said to be valid if there are no possible circumstances in which all of its premises are true and its conclusion is false. One way to determine whether an argument is valid, then, is simply to enumerate the various possible distributions of truth and falsity over the premises and conclusion, and check whether there are any such that the premises all come out true and the conclusion comes out false. If there are, the argument is invalid; if there are not, the argument is valid. This, of course, is just the familiar truth-table method of determining validity. The truth-table for the argument above is as follows:

P	Q	P & Q	P
T	T	T	T
T	F	F	T
F	T	F	F
F	F	F	F

There are four possible distributions to the constituent sentences P and Q, and these are enumerated on the four lines on the left-hand side of the table, with T representing “true” and F representing “false.” Given this, we can work out the possible distributions of truth and falsity to the premise and conclusion: this is done in the third and fourth columns. We see that there is only one circumstance in which the premise is true -when both P and Q are assigned the value true -and that in this case, the conclusion is also true. So there are no possible cases in which the premise is true and in which the conclusion is false. So the argument is valid.

What does the question about the validity of an argument have to do with semantics? Intuitively, the validity of an argument is going to depend on the meanings of the expressions which appear in it. That is to say, the validity of an argument is going to depend on the semantic properties of the expressions out of which it is constructed. In the argument above the basic expressions out of which the argument is constructed are sentences. What properties of the sentences are relevant to determining the validity of the inference? In the first instance, it seems as if it is the properties of truth and falsity. After all, the truth-table method works by determining the possible distributions of these very properties. So, truth and falsity look like good candidates for the semantic properties in question. Given assignments of truth and falsity to P and Q, we can work out the various assignments of truth and falsity to the premises and conclusion, and this allows us to say whether or not the argument is valid. So, validity is determined by the possible distributions of truth and falsity to the premises and conclusion, and this in turn is determined by the possible distributions of truth and falsity to the constituent sentences.

Let’s define the notion of semantic value as follows:

The semantic value of any expression is that the feature of it which determines whether sentences in which it occurs are true or false.

(Dummett (1973: 91)

In the case we have just looked at, the constituent expressions of the argument are the sentences P, Q. Which features of P, Q are relevant to determining whether the sentences in which they occur are true or false? Well, their truth or falsity: as shown in the truth-table, the distributions of T and F to P and Q determine the truth or falsity of the complex sentence P&Q which forms the premise of the argument. Given the definition above, then, it follows that the semantic value of a sentence is its truth-value.