Simplification in Parentheses

Hi Jim,

Can you use simplification in this way?
(P • Q) ⊃ R
Therefore P ⊃ R

In other words can you simplify the statement P and Q even though it is the antecedent to the conditional?

Thank you!!!!

It is generally improper to simplify a conjunction within parentheses (that is, within a more compound proposition). This is simply because such a “simplification” is often invalid, as in your example.

One way to check is to remember that a proper rule of inference will always produce a valid implication. So if you can show that performing the operation is not valid, then it is improper.

In your example, we can check implication by using a shorter truth table, making the first proposition true and the second false, as follows:

(P • Q) ⊃ R    ∴  P ⊃ R
             T                 F

Starting with the false second conditional, you can complete the truth table with P as True, Q as False, and R as False. These truth values make the first proposition true and the second false. This means that the first does not imply the second, which shows that the simplification is not valid.


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