Logic: Affirming the Consequent

I will begin with a slightly edited quote from a discussion on modus ponens or affirming the consequent from www.ephilosopher.com.

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ARGUMENT #1 (modus ponens):
Where A is any well-formed formula (WFF) and B is any WFF:
1) If A then B
2) A
3) Therefore, B


ARGUMENT #2 (affirming the consequent):
Where A is any WFF and B is any WFF:
1) If A then B
2) B
3) Therefore, A

Suppose, however, that A and B are the same WFF-- call it "S". The resulting argument looks like this:

ARGUMENT #3:

1) If S then S
2) S
3) Therefore, S

What is the best description of argument #3? Is it modus ponens, affirming the consequent, or something else?

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Interesting question! First let me say something about the question itself.
It is obvious to everyone that arguments #1(modus ponens) and #2 (affirming the consequent) quoted above are templates wherein A and B stand for any two (not necessarily distinct) WFF. Determining whether argument #3 is derived from argument #1 or argument #2 is important only because in the former case it must be correct (because arg #1 is correct) but in the latter case it must be wrong (because arg #2 is wrong). Had both argument #1 and #2 been correct, the question would have been neither important nor interesting.

Argument #3 quoted above fits both the templates when A = B. And, therefore, the puzzle whether the third form of argument has been derived from the first or the second template cannot be resolved in favour of just one of these arguments. Since, resolving is required let us follow the following attempts which I will call solutions:

Solution 1: If one adds the restriction in argument #1 and argument #2 that A and B cannot be the same WFF then argument 3 is clearly something else. And then argument #3 could be called a simple tautology. In fact, as stated already nobody would bother to offer such an argument.

Solution 2,3: Similarly, the restriction mentioned above could be applied to just one of the arguments #1 or #2 and then argument #3 would be derivable from just one argument (the one to which the restriction has not been applied). In these cases too, the problem would be resolved.

However, I do not favour these solutions unless there are other grounds to add such restrictions too. Let us examine these grounds if they exist.

In so far as argument #3 is derivable from modus ponens, no problem arises. That is so because modus ponens is correct whether A=B or not. So we will assert that argument #3 is modus ponens.

What about the argument #2? Clearly, it is a wrong form of argument only when "B does not imply A". Then, any substitution of A and B with some WFF such that "B implies A" will allow the inference A and the argument will look like affirming the consequent. I say "look like" because the hidden "B implies A" has been employed. It surely may happen that "B implies A" takes the same form as WFF (1) or WFF(2) of the argument #3. With this in mind, I propose the following solution:

Solution Final:
Argument #2 should be modified as follows to be true in general:

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Where A is any WFF and B is any WFF such that "B does not imply A":

ARGUMENT #2 (affirming the consequent)

1) If A then B
2) B
3) Therefore, A
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With this modification, argument #3 is modus ponens.