Can modal operator be defined as a boolean function? - math

Question

Let us consider for simplicity only Kripke structures with a single agent whose knowledge is described by the modal operator K. We know that in all the corresponding Kripke structures where K is interpreted by equivalence there holds for any formula A
a) the formula KA -> A (Knowledge Axiom) is valid ,
b) but the formulas A -> KA and ¬KA are not valid.
Utilize these facts to show that such a behaviour of the modal operator K cannot be encoded by any boolean function (ie. Truth values defined by a table).
Hint: Suppose the truth value of KA can be calculated from the truth value of A using a truth table for K (in the same way as ¬A is calculated form A). Consider all possible truth tables for K and show that none of them grants the properties a) and b) mentioned above.
I dont understand that hint... making truth table of K is like constructing truth table of negation symbol ¬ , which in my mind doesnt make sense I think it makes sense only to make negation of something and not just negation

Answer 1

Consider all possible truth tables for K:
| A | K₁A | K₂A | K₃A | K₄A |
—————————————————————————————
| 1 | 1 | 1 | 0 | 0 |
—————————————————————————————
| 0 | 1 | 0 | 1 | 0 |
Show that none of them grants the properties a) and b) mentioned above.
Case 1
| A | KA | KA->A | A->KA | ¬KA |
—————————————————————————————————
| 1 | 1 | 1 | 1 | 0 |
—————————————————————————————————
| 0 | 1 | 0 | 1 | 0 |
In this case, KA->A is not a tautology.
Case 2
| A | KA | KA->A | A->KA | ¬KA |
—————————————————————————————————
| 1 | 1 | 1 | 1 | 0 |
—————————————————————————————————
| 0 | 0 | 1 | 1 | 1 |
In this case, A->KA is a tautology.
Case 3
| A | KA | KA->A | A->KA | ¬KA |
—————————————————————————————————
| 1 | 0 | 1 | 0 | 1 |
—————————————————————————————————
| 0 | 1 | 0 | 1 | 0 |
In this case, KA->A is not a tautology.
Case 4
| A | KA | KA->A | A->KA | ¬KA |
—————————————————————————————————
| 1 | 0 | 1 | 0 | 1 |
—————————————————————————————————
| 0 | 0 | 1 | 1 | 1 |
In this case, ¬KA is a tautology.
Can desired behaviour of K be encoded by many-valued matrix?
For alethic modal systems, the answer is the following:
3 values are insufficient,
4 values are sufficient for the so-called basic modal logic,
any finite number of values is insufficient for syntactically “full” and deductively “natural” modal system.
See, e. g., introductory parts in the article by Jean-Yves Beseau.
I hope these results are relevant for epistemic modal systems.