if A then B

A couple thoughts. One about how logicians co-opted a chronological term ("then") for a timeless logical relation. But actually it is like this: all mathematical certainty comes down to the same thing: I am playing a game with someone or I am not. For example:
P is playing a game with B: when he hands her a ball, she will hand it back.
P will know B is not playing if the ball does not come back in a timely manner.
For another example: Suppose we take a game with this rule: whenever I hand you an assumption A, you will hand me back a predictable result B. I then hand you an assumption A. Now I know you will hand me back a result B, or know that you are not playing the game. Was it Aristotle who wrote this down?
A
A=>B
B

Update: So what is at the root of the certainty that 'I am either playing a game or not'? I believe it derives from this same principle that defines the notion of a "channel"- where you can switch between channels but only watch one at a time. This happens in our heads, for any category that works exclusively - like a color channel, or a shape channel, or an intensity channel, or a position channel (two things can't be in the sample place), or an arbitrary True/False channel. We know that two different channel types are compatible when they can be superposed. I cannot tell if this derives from language or from perception, or what all.

 Playing a game with rules that are shared is also an underlying principle of communication - as Grice would have it. Beings that depend on communication for their prosperity have evolved to this mental capacity of creating new channels and then using them on purpose.