Situation theory and set theory

Barwise and others do not believe that standard set theory (ZFC or others) is sufficient for modeling situations. Instead, the use of an anti-foundation-axiom set theory is recommended. Aczel has developed a set theory based on ZFC, but replacing the foundation axiom. In set theory, every set can be pictured by a graph. A graph is well-founded if it has no cycles or infinite paths, non-well-founded otherwise. Aczels anti-foundation-axiom states, that every graph, well-founded or not, pictures a unique set. For a complete discussion, see aczset, akset.

Reasons for choosing ZFC-AFA to model situations are some inherently circular situations, especially while modeling common knowledge. Considering the situation $s$, where John is kissing Mary, $s \models \langle\langle kiss, John, Mary; 1\rangle\rangle$ and $s \models \langle\langle kiss, Mary, John; 1\rangle\rangle$ holds. If you could ask John whether he knows, if he is kissing Mary, he would certainly agree, so

$$s \models \langle\langle knows, John, \langle\langle kiss, John, Mary; 1\rangle\rangle ;1\rangle\rangle$$

and the same for Mary. If you ask Mary, whether she knows that John knows that he is kissing her, she would agree too, and so on. A finite infon cannot capture all the information in this situation. The fact that John is kissing Mary is known to both, John and Mary, it is common knowledge amongst them and it is also common knowledge that it is common knowledge, and so on. We call the above approach of capturing common knowledge the iterate approach. It can be understood and modeled without the use of hypersets.

Another way to capture common knowledge is the fixed-point approach. Let $\phi$ be the fact that John is kissing Mary. Then the fact $\tau = \langle\langle know, \{John, Mary\}, \phi \land \tau \rangle\rangle$ captures the information that both know $\phi$ and both know this is common knowledge between them. Note that $\tau$ is a constituent of itself.

The third approach to common knowledge is the shared situation approach. The situation $s$ where John is kissing Mary is modeled by

• $s \models \langle\langle kiss, John, Mary; 1\rangle\rangle$
• $s \models \langle\langle kiss, Mary, John; 1\rangle\rangle$
• $s \models \langle\langle know, John, s; 1\rangle\rangle$
• $s \models \langle\langle know, Mary, s; 1\rangle\rangle$

In this approach the situation $s$ is a constituent of itself. If we take sets as models for situations, the set $s_S$ should be a member of itself.