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Tyrrell_McAllister comments on To signal effectively, use a non-human, non-stoppable enforcer - Less Wrong
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Comments (164)
I assume you mean,
P, <=>, (Q <=> P)
to be the headers of your truth table.
But even then the truth tables for (P iff ( Q iff P) ) and ( P iff Q) are different - consider the case where 'you' will co-operate with me no matter what. If I'm running ( P iff Q), I'll cooperate; if I'm running (P iff ( Q iff P) ), I'll defect.
Edit: formatting trouble.
No, I am giving the truth-table for P <=> (Q <=> P) in a compact form. It's constructed by first assigning truth-values to the first occurrence of "P" and the first occurrence of "Q". The second occurrence of "P" gets the same truth-value as the first occurrence in every case. Then you compute the truth-values for the inner-most logical operation, which is the second occurrence of "<=>". This produces the fourth column of truth values. Finally, you compute the truth-values for the outer-most logical operation, which is the first occurrence of "<=>".
Hence, the second column of truth-values gives the truth-values of P <=> (Q <=> P) in all possible cases. In particular, that column matches the third column. Since the third column contains the truth-values assigned to Q, this proves that P <=> (Q <=> P) and Q are logically equivalent.
ETA: You edited your comment. Those are indeed the correct headers, so my correction above no longer applies.
Yes, the truth-table for P <=> (Q <=> P) is different from the truth-table for P <=> Q. But those aren't the propositions that I'm saying are equivalent. I'm saying that to assert P <=> (Q <=> P) is logically equivalent to asserting Q all by itself. In other words, to implement the belief that P <=> (Q <=> P) is functionally the same as implementing the belief that Q. This means that the belief that Clippy recommends signaling is logically equivalent to an unconditional belief that you will cooperate with me.
One can't help but suspect that Clippy is trying to sneak into us a belief that it will always cooperate with us ;).
Sorry for the confusion. I understand now; the extra space between two of the columns confused me.
However, I suspect we need a stronger logic to represent this properly. If Q always defects, no matter what, "you would cooperate with me if ... I ... cooperate with you" is false, but is given true in the propositional interpretation.