January 15, 2011

Validly Affirming the Consequent

I'm grading some logic exercises from an intro class today. The students were supposed to give examples of valid and invalid arguments, with true and false premises and conclusions, and so forth. One student turned in the following fantastic example (I have edited it to remove some ambiguities):

(P1) If P1, then C
(P2) C
:. (C) P1

The student, understandably, thought the argument was invalid, since it has the form of affirming the consequent. However, due to the self-reference, the argument is valid. The student just wrote 'the premise' and 'the conclusion', so I'm not sure if this is the intended interpretation, but still pretty clever.

Posted by Kenny at January 15, 2011 11:34 AM
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Comments

Please clarify why this series of statements hold.

Posted by: L. at February 25, 2011 2:26 PM

Er, holds.

Posted by: L. at February 25, 2011 2:27 PM

Well, P2 asserts that C, the conclusion, is true, so the premises entail the conclusion. It's like saying "the conclusion of this argument is true, therefore the sky is blue," which will be valid: the premise guarantees the truth of the conclusion (the premise, after all, JUST IS that the conclusion is true). The inclusion of P1 and the reference from C back to P1 makes the student's argument a little more complicated (and accordingly more fun). Given that C says that P1 is true, the meaning of P1 is: "if this sentence is true, then this sentence is true" which is, of course, guaranteed to be true. (For suppose it was false. Then the antecedent of the conditional would be false, which would make the conditional true, contrary to our assumption.) So the argument is not only valid, but sound: the premises are true, and they entail the conclusion.

Posted by: Kenny at February 25, 2011 7:51 PM

The original argument-form seems confused from a use-mention point of view. The bracketed expressions '(P1)', '(P2)' and '(C)' are functioning as names of sentences, whereas the non-bracketed occurrences, going on surface structure, seem to be functioning as schematic letters, or abbreviations of whole sentences.

One way to get a uniform reading would be to treat the non-bracketed occurrences as also being names, and to add a truth-predicate to each occurrence, yielding:

(P1) If P1 is true, then C is true
(P2) C is true
:. (C) P1 is true

Interestingly, the validity of this argument cannot be captured in basic propositional or predicate logic. Rather, it appears to turn on logical principles attaching specifically to the truth-predicate.

Posted by: Tristan Haze at June 6, 2011 10:04 PM

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