Another example of testing for validity

This video shows another example of testing for validity using a truth-table. This time looking at an argument involving a conditional. Let’s pick up on a couple of key points from the video …

First, we’ve seen a truth-table demonstration of the validity of the form:

• (α (rightarrow) β), α; therefore, β

This form is called modus ponens. In the video we assume (believably enough, from what we’ve seen so far) a match between the truth-conditions associated with arrow and with ‘If … then … ’ in English. If that assumption is correct, we now have a clear understanding of the validity of the form ‘If α then β; α; therefore β’ in English.

Secondly, we’ve seen a truth-table demonstration of the invalidity of the form:

(α (rightarrow) β), β; therefore, α

This form is called affirming the consequent. If we’re right about the match between arrow and ‘If … then … ’ in English, we now have a clear understanding of the invalidity of the form ‘If α then β; β; therefore α’ in English.

Logical equivalence and entailment

Before we move on to the next step, note that we can express logical equivalence in terms of mutual entailment: where two claims are such that each entails the other.

Two claims will be logically equivalent if and only if they are mutually entailing.

If they’re logically equivalent, then where one is true so is the other, so there won’t be any situation in which one, standing as premise, is true, and the other, standing as conclusion, is false; so, they’ll both entail each other.

If two claims are mutually entailing, then where one is true, so is the other (and where one is false, so must the other be, otherwise we’d have a counterexample to one of the entailments). So, they must be true in exactly the same circumstances, so they’re logically equivalent.

We’ll look at some cases of testing for mutual entailment/logical equivalence in the next two short steps. These demonstrate a couple of important logical laws.