Denying the Antecedent exception?

Haven't you completely changed this by adding "and only if"? We can do this to the examples as well:

If and only if it barks, it is a dog.

It doesn’t bark.

Therefore, it’s not a dog.

This must be true, because of the additional predicate.

If and only if I have cable, then I have seen a naked lady.

I don’t have cable.

Therefore, I have never seen a naked lady.

And again, same thing. 

I think in "if and only if" the first if is redundant, so what you've done is change this:

If P, then Q.

Not P.

Therefore, not Q.

To this:

Only if P, then Q.

Not P.

Therefore, not Q.

Which is a slightly different argument, and one which is correct. 

The reason I noticed this was that in your example you pointed out that you said "unlock" and my first thought of a lockpick would still be valid. Then I realised that your logic had excluded anything but the key, meaning the statement must be true, and thus was a different premise than in the fallacy.

"If and only if P, then Q" ==>   "P, therefore necessarily Q". The second does not follow from the first. I assume you understand this, so I'm not sure I'm understanding your question completely.

The way I understand it, if you deny the antecedent, P, to make a claim a claim about Q, you are saying that Q only occurs because of P, whereas Q may occur because of other reasons.

Hi, Rationalissimo!

Invalidity and validly are mutually exclusive categories. An argument form that is invalid cannot have instances in which there is a valid argument having that form. There cannot be “exceptions”. On this basis alone, there is reason to deny your position. I may understand where you are coming from, however. I will talk about the “iff” clause and then say why I think that taking your position could seem quite tempting.

P iff Q, is short for P if and only if Q. Notice the “and”. The logical form of an if and only if clause is analyzed as TWO conditional statements that are conjuncted. The first---P if Q---says that Q is sufficient for P. The second---P only if Q---says that Q is necessary for P. Altogether:

(if P, then Q) and (if Q, then P)

I switched them around for alphabetical order. And that is my final analysis of an iff clause, also known as a biconditional. Here is a valid argument that includes it, inspired by yours.

           1. (if P, then Q) and (if Q, then P)
           2. not P
Therefore,
           3.not Q

This argument is valid, but not because an antecedent is denied by premise 2. Jeff, say, may think to himself that the conditional statement if P then Q is involved, P is denied, not Q is concluded, and the argument is valid, so it must be the case that denying the antecedent is a valid argument form in this case. However, Jeff's mind would have overlooked the significance of there being two conditional statements and that a proof of the argument works on the second conditional statement in premise 1, not the first conditional statement. If Q, then P is derived through simplification and a Modus Tollens is then performed on it. Here is a proof of validity.

         1. (if P, then Q) and (if Q, then P)
         2. not P
         3. If Q, then P.    Take premise 1 and simplify. Notice that it is the second conjunct that has been derived through this simplification process. The first one is left alone throughout this proof. 
Therefore,

           4. not Q.             2, 3 Modus Tollens

The aspect of your post that is causing the trouble is probably the way you write the proposition. If and only if P, then Q is confusing because the word “then” makes it seem like a SINGLE conditional in which, moreover, Q is fixed in the consequent and if and only if P is merely the antecedent. In reality, there are two conditional statements that are conjuncted, only one of them has P and Q in the place that is proper for deriving the conclusion, and that particular conditional must be simplified away from the other and worked on by itself. 

When the biconditional is properly understood, and written in standard format (at least for clarity purposes), we can see that the argument we have looked at is valid actually for denying the consequent, not for denying the antecedent! Likewise, in your argument about the keys and safe, the conditional that is simplified out is the one that says that unlocking the safe is sufficient for having the keys, not the conditional that states that unlocking the safe is necessary for having the keys. If you just look at the argument alone, however, without the proof attached to it or the analysis of the biconditional to guide your reading, you don’t see the underlying mechanics of the valid inference, as with what happened to Jeff. This may result in getting it right that the argument is valid, but getting it wrong as to wherein the valid inference lies.

Thank you, Rationalissimo.

From, Kaiden