Topics Logic

Question I am reading "The Philosopher's Toolkit" by Baggily and Fosl, and in section 1.12 is the following: "As it turns out, all valid arguments can be restated as tautologies - that is, hypothetical statements in which the antecedent is the conjunction of the premises and the conclusion." My understanding is the truth table for a tautology must yield a value of true for ALL combinations of true and false of its variables. I don't understand how all valid arguments can be stated as a tautology. The requirement for validity is the conclusion MUST be true when all the premises are true. I must be missing something. Thanx - Charlie

Accepted:
August 19, 2015

Comments

I don't have Baggily and Fosl

I don't have Baggily and Fosl's book handy but if your quote is accurate, there's clearly a mistake—almost certainly a typo or proof-reading error. The tautology that goes with a valid argument is the hypothetical whose antecedent is the conjunction of the premises and whose consequent is the conclusion. Thus, if

P, Q therefore R

is valid, then

(P & Q) → R

is a tautology, or better, a truth of logic. So if the text reads as you say, good catch! You found an error.

However, your question suggests that you're puzzled about how a valid argument could be stated as a tautology at all. So think about our example. Since we've assumed that the argument is valid, we've assumed that there's no row where the premises 'P' and 'Q' are true and the conclusion 'R' false. That means: in every row, either 'P & Q' is false or 'R' is true. (We've ruled out rows where 'P & Q' true and 'R' is false.) So the conditional '(P & Q) → R' is true in every row, and hence is a truth of logic.