Interpretation

Formal Semantics

If a theory is formally axiomatized in predicate logic, it can be easily interpreted by interpreting its constant, predicate, and function symbols—in other words, its vocabulary—in the formal semantics developed by Alfred Tarski. A very simple theory in predicate logic may contain only the single sentence $Pa\wedge \exists xQxa$ (“a is P and there is an x such that x has the relation Q to a”), where a is a constant symbol, P is a one-place predicate symbol, Q is a two-place predicate symbol, and x is an existentially quantified variable. Thus, the theory’s vocabulary consists of a, P, and Q. A structure for the theory in formal semantics then consists of a domain and an interpretation. The domain is a set of objects like {α, β, γ}. The interpretation assigns to each constant symbol an element of the domain; a could, for instance, be assigned α. To each one-place predicate symbol, the interpretation assigns a subset of the domain; P could, for instance, be assigned {α, β}. To each two-place predicate symbol, the interpretation assigns a set of pairs from the domain; Q could, for instance, be assigned $\left\{〈\text{α},\text{β}〉,〈\text{β},\text{β}〉,〈\text{γ},\text{α}〉\right\}$. The structure then determines whether the theory is true or false, which is the theory’s truth value. [Page 408]The theory is true if and only if all its sentences (in this case just the one sentence) are true. The truth value of each sentence is determined by the truth values of its subsentences, in this case Pa and $\exists xQxa$, and the logical connectives connecting them, in this case “∧.” Pa is true if and only if the object assigned to a is an element of the set assigned to P, which it is, since $\text{α}\in \left\{\text{α},\text{β}\right\}$. $\exists xQxa$ is true if and only if there is an object in the domain such that the pair consisting of that object and the object assigned to a is an element of the set assigned to Q, which it is, since $〈\text{γ},\text{α}〉\in \left\{〈\text{α},\text{β}〉,〈\text{β},\text{β}〉,〈\text{γ},\text{α}〉\right\}$. The whole sentence $Pa\wedge \exists xQxa$ is a conjunction, and a conjunction is true if and only if both its conjuncts are true, which in this case they are. Thus, the sentence is true in the structure, and the theory consisting of only this sentence is also true in this structure, which is therefore called a model of the theory. (For this reason, formal semantics is also called model theory.) More complicated theories can also involve function symbols and higher order predicate symbols, that is, predicate symbols that apply to constant, predicate, and function symbols rather than only to constant symbols. The basic idea of formal semantics stays the same for such more complicated theories.

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