Epistemic Logic
The Meanings of $\square$
Consider the epistemic meaning of $\square$:
I know that $\phi$ is true.
and also the following model:
stateDiagram-v2
direction LR
w1 --> w2:a
w2 --> w1
w2:w2<br>p
From this graph we have:
- $M,w_1\vDash \square_ap$
- $M,w_1\nvDash p$
This is bad as agent $a$ knows something that is false.
We can consider this for all meanings:
| Circumstance | Example |
|---|---|
| Doxastic | I cannot believe $\phi$ and $\neg\phi$ at the same time, yet $\square_a\phi\wedge\square_a\neg\phi$ is satisfiable. |
| Alethic | If $\phi$ is necessarily true, then it is necessarily the case that $\phi$ is necessarily true. Yet $\square\phi\wedge\neg\square\square\phi$ is satisfiable. |
| Temporal | It is not possible for both $\phi$ and $\neg\phi$ to be true at all points in the future, yet $\square\phi\wedge\square\neg\phi$ is satisfiable. |
| Legal/Deontic | If $\phi$ is legally required, then $\phi$ is, ideally, legally permitted. Yet $\square\phi\wedge\neg\lozenge\phi$ is satisfiable. |
Due to the above contradictions, $\square$ doesn’t capture all the properties of those circumstances. As a result we should make the following changes to our logic:
- Add axioms to the proof system.
- Restrict the class of models that are allowed.
Additional Axioms
We can use the following axioms to solve this issue:
| Circumstance | Axiom |
|---|---|
| Epistemic | $\square_a\phi\implies\phi$ |
| Temporal | $\square\phi\implies\lozenge\phi$ |
There are additional axioms, and related model restrictions, that are required for epistemic logic.
Additional Models
We can give the following restrictions to our model to ensure that the above axioms are complete and sound:
| Axiom | Model |
|---|---|
| $\square_a\phi\implies\phi$ | The model should be reflexive, there is an arrow from every world to itself (for each agent). |
| $\square\phi\implies\lozenge\phi$ | The model should be serial, every world has a successor. |
Correspondence
The relation between the axioms and models above is called correspondence:
- $\square_a\phi\implies\phi$ corresponds to reflexivity.
We can define this as so:
An axiom $\psi$ (e.g. $\psi=\square\phi\implies\lozenge\phi$) corresponds to a property $\mathcal C$ (e.g. $\mathcal C=\text{seriality}$) if and only if:
- For every model $M$ that satisfies $\mathcal C$ and every world of that $w$ of that model, we have $M,w\vDash\psi$.
- For every model $M=(W,R,V)$ that does not satisfy $\mathcal C$ there are some alternative valuation $V’$ and some world $w\in W$ such that $(W,R,V’)$, $w\nvDash\psi$.
Importance of Correspondence
A proof system should be sound (everything that can be derived is valid) and complete (everything that is valid can be derived):
-
If $\psi$ corresponds to property $\mathcal C$, then $\mathbf K+\psi$ is sound and complete with respect to the models that satisfy $\mathcal C$.
There are some additional predicates that are required for this to be true that won’t be discussed.
This also works for multiple axioms and properties:
- $\mathbf K +\psi_1+\ldots+\psi_n$ is sound and complete with respect to the models that satisfy $\mathcal C_1,\ldots,\mathcal C_n$