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.

Blessings!

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