So you have by yourself found a counter example
to the claim that modal logic = 3 valued logic.
For example in minimal logic, which can be
interpreted as a special modal logic via:
A => B :<=> box (A -> B)
We have A, A=>B |- B. How can this happen
in 3 valued logic? What would be the box?
Confutus wrote:
In Lukasiewicz three valued logic, interestingly, Modus Ponens does not
hold.
(A & (A > B)) > B is not a tautology. it fails in one case.
This must be one of the reasons Lukasiewicz logic has been regarded as
more of a curiosity than a successful attempt to extend classical
logic. If Modus Ponens isn't valid, then how is one supposed to do any
kind of deductive reasoning with it?
translogi@xxxxxxxxxxxxxx wrote:
An interesting subsequent question
Is Modus ponens always a possible theorem?
or maybe more correctly:
Are there axiomatisations possible where
¬((A & (A -> B)) -> B)
is a theorem?
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