I think clippy's statement should be

I signal to cooperate with you if and only if ((you're planning to cooperate with me if and only if you predict I would cooperate with you) and you would cooperate with me).

Should the word "signal" be part of the signal itself? That seems unnecessarily recursive. Maybe Clippy's recommendation should be that I ought to signal

I will cooperate with you if and only if ((you're planning to cooperate with me if and only if you predict I would cooperate with you) and you would cooperate with me).

This does seem more promising than Clippy's original version. Written this way, each atomic proposition is distinct. For example, "you're planning to cooperate with me" doesn't mean the same thing as "you would cooperate with me". One refers to what you're planning to do, and the other refers to what you will in fact do. Read this way, the signal's form is

S <=> ((Q <=> P) & R),

and I don't see any obvious problem with that.

However, you would seem to render it in the propositional calculus as

S <=> ((Q <=> P) & Q),

where

• P = You predict I'll cooperate,

• Q = You're going to cooperate,

• S = I will cooperate.

(I've omitted the initial "I'm signalling" from your rendering of S, for the reason that I gave above.)

Now, S <=> ((Q <=> P) & Q) is logically equivalent to S <=> (Q & P). So, to signal this proposition is to signal

I will cooperate iff you're going to cooperate and you predict that I'll cooperate.

As you say, this seems very similar to signalling

I will cooperate iff you will cooperate.

In fact, I'd call these signals functionally indistinguishable because, if you believe my signals, then either signal will lead you to predict my cooperation under the same circumstances.

For, suppose that I gave the second, apparently weaker signal. If you cooperated with me while anticipating that I would defect, then that would mean that you didn't believe me when I said that I would cooperate with you if you cooperated with me, which would mean that you didn't believe my signal.

Thus, insofar as you trust my signals, either signal would lead you to predict the same behavior from me. So, in that sense, they have the same informational content.