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Lately, I’ve often found myself thinking about logic and dialogues.
In the broader realm of logic, there are different approaches to modeling dialogues. One is D_2, developed by Jaśkowski, of the so-called Polish school of paraconsistent logic.
The core idea is that A ^ B is defined as something like A ^ <>B. It is interpreted as »A is the case, and somebody claimed B.« The implication is defined as <>A => B: »somebody claimed A, and therefore B is the case«.
There are other approaches, such as doxastic or epistemic logic, in which you can express »subject A believes that p«.
And, of course, dialogue logic, with its strict rules.
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