(In)equivalences between systems hold because of different properties of the systems themselves.
Logics = a formal way to express these properties.
- satisfiability relation states when a process satisfies a property
- enjoying the same properties coincides with being bisimilar

#### Example
![300](../../Pasted%20image%2020250505083500.png)
These processes are not bisimilar as:
- P1 can perform an action a followed by any b
- P2, after every a, can always perform an action b
- so there exists an a after which P1 cannot perform a b. But not for P2

### Syntax and Satisfiability
$$\phi := TT | \lnot \phi | \phi \land \phi|◇ a \phi \quad where a \in Action$$
The language generated by this grammar will be denoted by Form; every element of this set will be called formula

![](../../Pasted%20image%2020250505083948.png)

To simplify the proofs, we consider a more general form of conjunction: $\land_{i \in I, \phi i}$