Semantic Properties in QL
 

We just learned how to determine the truth-conditions for propositions in QL. Now we turn to the question of how we use this information.  For the most part, we have the same semantic properties in QL as we had in PL. Arguments are valid or invalid; sets of propositions are consistent or inconsistent;  a pair of wffs can be logically equivalent or not equivalent. However, instead of referring to wffs which are always true as tautologies, we call them "logical truths" in QL, and instead of contradictions, we call the QL wffs that come out always false as "logical falsehoods."  Wffs that are neither logical truths nor logical falsehoods are, as in PL, still referred to as contingencies.

Let's talk about each of these semantic properties in the context of QL. In order to do this, we need to introduce the notion of an interpretation of a proposition.  An interpretation can be thought of as a row of the truth table. Returning an example from the last section, in a domain with two individuals, the proposition "(x)Ex"  is rewritten as "Ea & Eb". An interpretation of "(x)Ex" in a domain of two individuals is an assignment of truth and falsity to "Ea" and "Eb".

Validity: An argument is valid just in case, in every domain, and every interpretation, if the premises are true, the conclusion is true. Otherwise the argument is invalid.

Consistency: A set of propositions in consistent just in case there is at least one domain in which there is an interpretation in which all the propositions in the set true. Otherwise the set is inconsistent.

Logical Truth and Falsehood: A proposition is logically true just in case it is true in every domain under every interpretation. A proposition is logically false just in case it is false in every domain under every interpretation.

Logical Equivalence: Two propositions are logically equivalent just in case in every domain and every interpretation, they have the same truth value. Otherwise the propositions are not logically equivalent.

We now turn to the methods for determining these semantic properties.