# Valid arguments as tautologies

I don't quite understand this :

"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 consequent the conclusion. In other words, every valid argument may be articulated as a statement of this form : 'If W,X,Y are true, then C is true', where W, X and Y are the argument's premises and C is its conclusion. When any valid argument is substituted in this form, a tautology results." - Julian Baggini, The Philosopher's Toolkit

How is this a tautology? Isn't a tautology something that is necessarily true regardless of the truth values assigned to W,X and Y (in this case)?

Or is the author here referring to "tautologies" in a different sense? (where it does not add any informative value as it repeats the meaning of something which is already understood)

The confusion here is that W, X and Y don't represent variables, they are premises with internal structure. It might make it more clear to call them P1, P2 and P3.

What Baggini is saying is that any argument of the form

P1, P2 and P3, therefore C

can be rewritten as a tautology of the general form

IF (P1 AND P2 AND P3) THEN C

It won't look like a tautology, however, until you fill in the actual statements. (As far as assigning truth values, those go with the variables within the premises.)

Example:

p1: A -> B, p2: B -> C, p3: C -> D, therefore c: A -> D

[(A -> B) AND (B -> C) AND (C -> D)] -> (A -> D)

The second line is a tautology. No possible assignment of truth values to A B and C can result in the statement as a whole being untrue.

How is this a tautology? Isn't a tautology something that is necessarily true regardless of the truth values assigned to W,X and Y (in this case)?

Yes, and the statement you quote is also correct.

1. By the definition of validity, the conclusion must be true if all of the statements in the argument are true. In this case, we don't really care about the situations where W, X, or Y are not true - we can still call the argument valid.

2. Consider the truth-table for a conditional. From this, we can see that any conditional statement is true except in the case where the antecedent (the part that follows 'if...') is true and the consequent (following 'then...') is false.

Now if you put these two ideas together and assert as your antecedent the assertion that W ∧ X ∧ Y, and make your consequent C. That is,

``````(W ∧ X ∧ Y) → C
``````

From the premise of the initial quote that the argument is valid there can be no case where you are posing the antecedent's statement (W ∧ X ∧ Y) as true and the consequent (C) false. Thus, it is a tautology as there is no case in which the statement itself is false.

(Note that this necessitates that W,X,Y includes ALL premises of the argument we know to be valid, not just some subset of them.)

Essentially, the main insight I would take from this passage is to realise that all arguments can be phrased as a conditional statement of the general form (if `STATEMENTS`, then `CONCLUSION`). The fact that any logically valid argument forms a tautology when this is done is a result of this.