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Is "if 2+3=6, then 2+3+1=7" an example of a fallacy fallacy or argumentum ad logicam?

I believe it is an example of fallacy fallacy, but I'm not able to show it. I only think it is a fallacy because "2+3=6" and "2+3+1=7" are false.

In formal logic, the conditional between two false statements is valid. What if we check the conditional statement using informal logic? Is it an example of a fallacy in informal logic, especially the argumentum ad logicam?

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This is a case of ex falso quodlibet. You can prove any statement if your assumptions are contradictory. –  CodesInChaos Feb 17 '13 at 17:49
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up vote 11 down vote accepted

The "fallacy fallacy" is that you reject a conclusion because the argument is a fallacy - even though the conclusion may be right. In general form:

  1. If P, then Q.
  2. P is a fallacious argument.
  3. Therefore, Q is false.

You have only given step 1 and step 2; you didn't accept step 3, rejecting the consequent (Q). Since the implication (P->Q) holds (step 1) and P is indeed a fallacious argument (step 2), there is no fallacy. If you had concluded that 2+3+1=7 was wrong based on the antecedent - and not on your own 'mathematical intuition', then it would have been a fallacy.

The implication "If 2+3=6, then 2+3+1=7" is true, since both the antecedent and the consequent are false. You can look at the truth table for this:

Truth  table implication: antecedent and consequent both false

An example of an argumentum ad logicam is the following:

  1. If 2+3=6, then 4+4=8.
  2. 2+3=6 is false.
  3. Therefore, 4+4=8 is false.

There is nothing wrong with step 1 and 2. The implication (step 1) holds and "2+3=6" is false (step 2). It does not logically follow, however, that "4+4=8" is false (step 3).

Note that the implication in step 1 also holds, but that there is a difference with the previous example I gave. Here the antecedent is false, but the consequent is true (which can lead to the fallacy fallacy). Truth table:

Truth  table implication: antecedent false; consequent true

(As a side note: a general rule to determine the truth value of an implication: if the value of q≥p, the implication holds (even if q and p are completely unrelated!). If you find this a bit absurd, you're not alone. Many Logicians find it a quite troublesome.)

Or I can give a more absurd example:

  1. If New York is in China, then New York is in the U.S.A.
  2. New York is not in China.
  3. Therefore, New York is not in the U.S.A.

The truth table in this case is the same as in the previous example:

Truth  table implication: antecedent false; consequent true

Here, the implication in step 1, as absurd as it may sound, is true. This may illustrate the point I made in the previous example: the implication can be quite troublesome at times.

In real-life, this will often occur in a more subtle manner. For instance:

Argument by proponent:

  1. Socrates is a sentient being.
  2. All humans are sentient beings.
  3. Therefore, Socrates is human.

Counter-argument by opponent:

  1. You have just affirmed the consequent.
  2. You have committed a fallacy; your reasoning was wrong, so Socrates is not human.

While it is of course true that the argument given by the proponent is fallacious, his conclusion was correct. The opponent could point at his flawed reasoning, but he could not reject the conclusion based solely on this flawed reasoning. This is an example of the fallacy fallacy or argumentum ad logicam.

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