Valid form and true premises makes an argument sound, but do 'premises' mean P, Q, R,... or 'what comprises the antecedent'?

Question

Suppose we have an argument 'Disjunctive Syllogism' as below:

$$P\lor Q \\{\sim}P \\∴Q.$$

which essentially means $$\big((P\lor Q)\; \&\; {\sim} P\big) \to Q.$$

Its truth table:

row	P	Q	P$\lor$Q	~P	(P$\lor$Q) & ~P	Q	[ (P$\lor$Q) & ~P ] $\to$ Q
1	T	T	T	F	F	T	T
2	T	F	T	F	F	F	T
3	F	T	T	T	T	T	T
4	F	F	F	T	F	F	T

Row 3 has $P\lor Q$ true and ${\sim} P$ true, and concludes a true $Q;$ since we did not conclude a false from a true, therefore this argument is in a valid form.

Is the argument sound or unsound? This post says it is NOT sound.

If it is sound, what does make Disjunctive Syllogism 'sound'?

Is it row 1, which has $P$ and $Q$ true and the conclusion $Q$ true?

Or is it row 3, which has $P\lor Q$ true and ${\sim}P$ true and the conclusion $Q$ true?

If it is unsound, what row makes it unsound?

Answer 1

1. An argument is said to be valid when its corresponding conditional is logically true, i.e., true regardless of interpretation.

   In this case, as you pointed out, the argument is $$\big((P\lor Q) \:\,\&\,\, {\sim} P\big)\to Q;$$ it is a tautology (thus logically true) because the column in the truth table headed by $\to$ is populated with only $T$s.

What makes the argument sound?
Is it row 1, which has $P$ and $Q$ true and the conclusion $Q$ true?
Or is it row 3, which has $P\lor Q$ true and ${\sim}P$ true and the conclusion $Q$ true?

2. An argument is sound when its corresponding conditional is valid and its premises are all true.

   (i) An unsound argument can have a true conclusion.

   (ii) Premises refer to the antecedent $\big((P\lor Q)\:\,\&\,\, {\sim} P\big)$—not the atomic propositions $(P$ and $Q).$

   (iii) A premise being true refers to its interpreted meaning: e.g., if $P$ represents ‘Paris is the capital of France’, then $P$ is true.

   Since truth tables cannot evaluate premises' truth, they can never be used to determine that an argument is sound. However, when an argument is invalid (is false in some interpretation) or has contradictory premises like ${\sim}P\,\&\,Q$ and $P\lor{\sim}Q$ (the conditional's antecedent is always false), its truth table immediately reveals that it is unsound.

Answer 2

First, a small correction: in row 4, $P \lor Q$ should be $F$, not $T$.

And so yes, that only leaves row 3 as the only row with all trued premises, and since the conclusion is also $T$ in row 3, we know the argument is valid.

Note that we can say this without using the last column at all. But, since you are adding that column, another way to see that the argument is valid is to observe that the conditional from that last row (whose antecedent is the conjunction of all premises, and whose consequent is the conclusion) is a tautology (always $T$)

As far as soundness goes: without knowing what $p$ and $q$ mean, we can't answer that question. As the truth table shows, the only way for the premises to all be true is in row 3, i.e. when $P$ is False, and $Q$ is True. But we don't if that's the case. So, the argument could be sound, but it could also be unsound.

But no matter what the truth of $P$ and $Q$ turn out to be, it is valid. That's the beauty of logic: we can say whether some argument is valid or not without knowing whether the statements involved are true or false ... we don't even have to know what they mean!