Chapter Eight Predicate Logic Semantics

1. Interpretations in Predicate Logic An argument is valid in predicate logic iff there is no valuation on which the premises are true and the conclusion false. Sentences are consistent just in case there is some valuation on which they are all true Two sentences are logically equivalent iff there is no valuation ion which they differ in truth-value.

Interpretations in Predicate Logic, continued But while validity, consistency, and logical equivalence in predicate logic are the same as they are in sentential logic, more is involved in specifying a valuation in predicate logic.

Interpretations in Predicate Logic, continued In predicate logic, although we have individual variables and quantification using such variables, we have no predicate variables, only individual variables.

Interpretations in Predicate Logic, continued We introduce predicate constants into a specific context, but must treat them as though they have other interpretations. To speak of different interpretations of a predicate letter is to treat it as though it were a variable for which we are substituting another predicate: We assign it to different domains in different contexts.

Interpretations in Predicate Logic, continued At this point we shall treat predicates intensionally, relying on commonsense meanings of them.

2. Proving Invalidity In sentential logic we could prove that an argument was invalid by producing an interpretation on which all the premises were true and the conclusion false. In predicate logic too we produce counterexamples to show that arguments are invalid.

3. Using Expansions to Prove Invalidity An easy way to prove invalidity is to use an interpretation with a very small domain.

Using Expansions to Prove Invalidity, continued Technique: Construct the expansion of the premises and conclusion for a small domain. Use the shortcut technique from sentential logic, ascribing truth values consistently. Attempt to find an interpretation on which the premises are true and the conclusion false.

Using Expansions to Prove Invalidity, continued While many invalid arguments can be shown to be invalid using a two-individual domain, not all can. Some may require expansions of three or more individuals.

4. Consistency in Predicate Logic Just as in sentential logic, in predicate logic the method for proving consistency of the premise of an argument is basically part of the method used for proving invalidity.

Consistency in Predicate Logic, continued In predicate logic, to show that an argument is invalid we produce an interpretation in which the premises are true and the conclusion is false. To show that the premises are consistent, we need only to show that there is an interpretation on which they are all true.

Consistency in Predicate Logic, continued Note that whenever one proves an argument invalid, one also proves that the premises are consistent.

5. Validity and Inconsistency in Predicate Logic An argument is valid just in case among all of its many interpretations there is not one where the premises are true and the conclusion false. A set of sentences is inconsistent iff there is not a single interpretation in which all the sentences are true.

Validity and Inconsistency in Predicate Logic, continued To demonstrate validity or inconsistency in predicate logic we must either use a proof or a truth tree.

Key Terms Consistent Intensional interpretation of a predicate Interpretation Logically equivalent Valid