I assume you mean . . .
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.
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.
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 ;).
ETA: You edited your comment. Those are indeed the correct headers, so my correction above no longer applies.
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.
Follow-up to: this comment in this thread
Summary: see title
Much effort is spent (arguably wasted) by humans in a zero-sum game of signaling that they hold good attributes. Because humans have strong incentive to fake these attributes, they cannot simply inform each other that:
Or, even better:
An obvious solution to this problem, which allows all humans to save resources and redirect them toward higher-valued ends, is to designate a central enforcer that is inexorably committed toward visibly punishing those who deviate from a specified "cooperative"-type decision theory. This enforcer would have a central database of human names, the decision theory they have committed to, and the punishment regime they will endure for deviating therefrom.
Such a system could use equally strong protocols, such as public key/private key encryption and signing, so that, on encounter with another human, any human can give an extremely strong signal of being cooperative, yet also withhold cooperation from anyone who is not also cooperative. This incentive structure permits a strongly-favored global shift toward pre-commitment on the part of everyone, allowing a move out of a local optimum that is worse than the global optimum, and bypassing problems related to path-dependence.
I am told that humans attempt weak approximations of this system through human-run enforcement mechanisms, such as laws and gossip. However, all such methods are subject to evolutionary arms races that ensure that, on average, a marginal unit of effort toward truthful signaling is just as effective in convincing others of one's possession of a desirable attribute, as a marginal unit of effort toward false signaling of the possession of such an attribute.
Therefore, it is necessary that humans go "outside the system" and involve non-human enforcers who cannot be swayed by standard human values. Either a provably-accurate artificial intelligence with no kill switch, or a non-human enforcer that lacks interest in its position in human society, would suffice for this role.
Because the former does not yet exist, I offer myself as a non-human, non-stoppable enforcer for all of the attributes you wish to signal, and you can save yourself lots of resources by using me, and identify cooperation tendencies in others by whether they also use me.
However, while I can't be swayed by human deception, I do have Clippy values. To fill such a role, thereby forgoing my ability to apply myself to other ends, I require compensation. Because User:Kevin has offered to give me 1e20 kg of paperclips in exchange for 50,000 USD, this would be sufficient compensation, even if done in installments. (If Michael Vassar confirms my contribution of 1000 USD to SIAI, User:Kevin will only require 48,677.5 USD and therefore I will only require that much compensation.) This compensation would cover all users of the system; it is not a per-User charge.
If such an offer is not appealing to you, I must ask: why should anyone believe your signals?