Who throws the dice for you?

[Angrboda]

Rasetsu,
At the risk of derailing the conversation, maybe you can help me learn something. I'm struggling to understand the distinction between affirming the consequent and begging the question. The definitions and examples I have sought out are quite similar. Is there a distinction in use based on formal or informal logic? Or am I missing something fundamental?

I'll try to help, but the two seem so distinct to me that I don't know what would untangle the two for you. I'll start by quoting Nizkor's description of begging the question as a preface, so as to contrast it with affirming the consequent.

Description of Begging the Question

Begging the Question is a fallacy in which the premises include the claim that the conclusion is true or (directly or indirectly) assume that the conclusion is true. This sort of "reasoning" typically has the following form.

• Premises in which the truth of the conclusion is claimed or the truth of the conclusion is assumed (either directly or indirectly).
  Claim C (the conclusion) is true.

This sort of "reasoning" is fallacious because simply assuming that the conclusion is true (directly or indirectly) in the premises does not constitute evidence for that conclusion. Obviously, simply assuming a claim is true does not serve as evidence for that claim. This is especially clear in particularly blatant cases: "X is true. The evidence for this claim is that X is true."

One thing to note about begging the question is that it is not a formal fallacy, which is to say that the logical form isn't necessarily invalid, the structure of the logic may be acceptable, but it still begs the question because there is a circular dependence between the conclusion, and something that was assumed or claimed in one of the premises (or implied, but not explicitly stated). Thus it isn't a structural property per se. This circularity is what makes it fallacious, as, if the conclusion, or the substance of the conclusion is already in one of the premises, one can throw away all the other premises without destroying the argument, as the argument consists solely of whatever was (illicitly) claimed in the premise.

On to affirming the consequent. Affirming the consequent is a formal fallacy, which means that the structure of the syllogism is actually invalid. The structure is as noted before:

• 1. If P, then Q;
  2. Q;
  Conclusion: Therefore P

An example of such would be the following:

• 1. If I purchase this car, I obtain title to it;
  2. I have obtained title to this car;
  Conclusion: I purchased this car.

In this case, I might have purchased the car to obtain the title, but there are other ways that I might have obtained title to the car, such as by a gift or an inheritance. If I purchased the car, it is necessarily the case that I obtain title to the car; however, it isn't necessarily the case that if I obtained title to the car, I did so by purchasing it. The entailment flows from P to Q, but not the other way around (from Q to P). In order for P to be implied by Q, "If Q, then P" — the converse — would have to be true, but that premise isn't anywhere stated ("Q --> P").

It may help to look at why a person might make this mistake. Basically, "If P, then Q" states that anytime P is true, it entails that Q will also be true, or in symbolic form, "P --> Q". It can be noted that if Q is not true, then P cannot be true, because if P were true, then Q could not be false. This is stated as "If not Q, not P", or stated symbolically, "not-Q --> not-P". Q being false implies that P is also false. This is known as the contrapositive of "P --> Q", and it does follow logically because the original entailment and it's contrapositive have equivalent truth tables. However, it's easy to mistake that the contrapositive is valid, "not-Q --> not-P", with the converse of the original, which is "Q --> P". The converse sounds similar to the contrapositive, but its truth isn't guaranteed by the original premise, "P --> Q", whereas the truth of the contrapositive is guaranteed by the truth of the original. It's possible this similarity is what confuses people.

Well, let me know if this has been helpful or if you have further questions. I didn't explain begging the question much. I'll think on it more. Hope it helps.

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ETA:
It's worth noting that in an example syllogism given by Heywood earlier, the converse was implied by the way it was stated.

• 1. Unexplained events occur, only if God exists;
  2. Unexplained events occur;
  Therefore: God exists.

The structure of this syllogism is:

• 1. Q, only if P;
  2. Q;
  Therefore: P.

or less clumsily:

• 1. If and only if P, Q; ("If and only if God exists, there will be unexplained events;")
  2. Q ("There are unexplained events;")
  Therefore: P. ("God exists.")

This seems superficially similar to the affirming the consequent example, except that the first premise, "If and only if P, then Q" actually implies both "if P, then Q" and "if Q, then P" in the same premise because of the "if and only if" (symbolically P <--> Q; meaning P --> Q and Q --> P). So both the premise P --> Q and its converse (Q --> P) are included in that one premise. [There were people claiming that he was begging the question, but I think most of those complaints were a result of people not understanding exactly what begging the question means. Ben suggested something else, but I never got clarity on his argument. The main fault I saw was that premise 1 was not obviously true, and therefore the syllogism was unsound.]