Trick of just adding: if the premises are true then the conclusion is true

Question

Source: 19 mins 50 s juncture, Lecture 3-7 (transcription), ... How to Reason and Argue,
by Prof W Sinnott-Armstrong.

Sh, here's a trick. ... You could always make any argument valid just by adding a suppressed premise that says, if the premises are true then the conclusion is true. But don't tell anybody because ... It makes the argument valid, but it doesn't achieve our goal because our goal is not just to make the argument valid, it's to make the argument valid so that we can understand the pathway of reasoning.

1. How and why does this trick work to validate an argument? 2. Since the standard form of an argument helps me understand, where would this tricky premise be inserted? For example:

Premise 1 (which may be false)
...
Premise n (which may be false)
Conclusion (which may be false)

Answer 1

I'll address your questions in the descending order. We're given an argument from premises P₁, ..., P_k to the conclusion C:

(1)   P₁, P₂, ..., P_k ⊢ C.

We know nothing about the truth values of the premises or the conclusion. The so-called "trick" for transforming (1) into a valid argument is to append what the professor calls a "suppressed premise" P_k+1 to the list of premies to obtain the following argument:

(2)   P₁, P₂, ..., P_k, P_k+1 ⊢ C,

where P_k+1 is defined as follows:

P_k+1   =_def   (P₁ ∧ ... ∧ P_k) → C.

Substituting that into (2), we get:

(2')   P₁, P₂, ..., P_k, (P₁ ∧ ... ∧ P_k) → C ⊢ C.

Thus far, I hope, we have addressed your second question regarding the placement of the "suppressed premise". Now, let's see why (2') is valid.

Claim. (2') is valid, regardless of the truth values of P₁,...,P_k or C.

Proof. An argument of form P₁, ..., P_n ⊢ C is valid if and only if all assignments of truth-values that make all of the premises true make C true. Therefore, to show that (2') is valid, it will suffice to show that all assignments that make [ P₁, P₂, ..., P_k, (P₁ ∧ ... ∧ P_k) → C ] true, make C true. Consider an arbitrary model M s.t. M makes all of P₁, P₂, ..., (P₁ ∧ ... ∧ P_k) → C true:

α) M ⊧ P₁, P₂, ..., (P₁ ∧ ... ∧ P_k) → C.

Since it makes P₁, P₂, ..., P_k true, it makes (P₁ ∧ ... ∧ P_k) true:

β) M ⊧ (P₁ ∧ ... ∧ P_k).

From (α) we know that M makes (P₁ ∧ ... ∧ P_k) → C true, from (β) that it makes (P₁ ∧ ... ∧ P_k) true. These two facts together, via modus ponens, give us the following fact:

γ) M ⊧ C.

Since it's impossible for M to make C true and make the conclusion (which coincides with C) false, we know that M makes the conclusion true as well. We had posited M to be arbitrary, so we're now able to generalize and conclude that all assignments of truth-values to the premises plus the "suppressed premise" make the conclusion true, regardless of the truth-values of those premises or the conclusion. ■

The professor is nevertheless wrong when he says that "You could always make any argument valid just by adding...". It's true, as we've seen above, that it's possible to transform an arbitrary argument of the form described into one that is valid by that "trick". However, the "trick" shows not that the original argument (e.g. 1 above) is valid, but that the augmented argument (e.g. 2' above) is valid. I hope it's now clear why I call the so-called "trick" ""trick"" instead of "trick" (I'd be happy to explain how those quotes work in the comments, if you're interested).

Answer 2

The shorter answer is that this statement "If the premises are true then the conclusion is true." is either true or false.

If is true, then the argument is already valid, since the statement itself is the definition of the argument being valid.

If not, then we have added a false premise to the argument, and any argument at all starting from a false premise is automatically valid.

Consider folksy conditionals like "If he's a gentleman then horses fly." They mean what they mean because "If [false statement] then [anything you want]".