The semantics of zeroth-order logic

If you have already taken an introductory logic course, you may remember talk of entailment, and in connection, validity. In particular, taking an argument with some premises and a conclusion, validity was probably specified along the following lines: if the premises are true, the conclusion must also be true. Then, it was said, the premises ‘entail’ the conclusion.

But so far, we have only been dealing with formal languages as being made up of various sequences of meaningless symbols, according to various rules. Where does truth enter the picture?

The answer is: semantics. In some sense, syntax and semantics are two sides of the same coin. Syntax specifies the way primitive symbols may be combined to form more complex expressions. Semantics, on the other hand, specifies how the meaning of more complex expressions can be computed from the meaning of primitive symbols. As we shall see, the rules of computation for the ‘values’ of expressions will match the rules of formation of expressions in our syntax. The idea underlying this is called ‘compositionality’. As simpler syntactic expressions compose more complex ones, so the meanings of these simpler syntactic expressions compose the meanings of more complex ones. Then, just as being a formula is precisely definable, meaning will be too.

Once we have a grasp on the rules of meaning for $\mathcal{L}_0$, we can start designating some formulas of the language as ‘true’, and some as ‘false’, based on their respective meaning (and some other stuff, which we shall get to in due course). We will also see later on how the truth values of various sets of formulas relate to the truth values of other sets of formulas. From this, it will take just one additional step to specify how sometimes, some premises being true ensures that the conclusion must also be true.

As noted above, semantics proceeds along the lines of syntax. In the syntax of $\mathcal{L}_0$, we have two types of base symbols: constants and predicates. As we have seen, every atomic formula is made up of an $n$-place predicate, followed by $n$ constant symbols, in brackets, separated by commas. So in order to give truth-values to our atomic formulas, we first have to specify the meaning of constants and predicates. Then, we shall be able to compute the truth-values of our atomic formulas from this. In turn, this will allow us to compute the truth-values of our more complex formulas.