How can I explain the soundness of a rule with respect to the semantics of a truth table?

Question

The question is a mouthful but it comes from an exercise in which I was asked to explain why modus ponens is sound with respect to the semantics expressed by the truth-table.

Many things are not clear, what does it refer by 'semantics' of a truth table? Sound is when all the rules in the system are true, how can I get the rules from a truth table?

Is it OK if I place a column in the truth table that represents P ∧ (P → Q) to see that every time this is true then Q is also true?

How would I do this if I had to show implication introduction is sound with a truth table?

Answer 1

I will not give you the full answer as this looks like homework, but I will show you how to do it instead:

You need to show that from P ∧ (P → Q) logically follows that Q - this is what the modus ponens says.

Therefore, you need to prove (P ∧ (P → Q)) → Q is always true, that is, independently of P and Q.

To do this, you can indeed use a truth table:

 P | Q | P ∧ (P → Q) | (P ∧ (P → Q)) → Q
---+---+-------------+-------------------
 0 | 0 | ...         | ...
 0 | 1 | ...         | ...
 1 | 0 | ...         | ...
 1 | 1 | ...         | ...

The last column should contain all ones.

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_{NB: this is the same as your suggestion to "place a column in the truth table that represents P^(P->Q) to see that every time this is true then Q is also true", however, I preferred to show you the mathematically correct way to solve any logical problem.}