It is, however, one thing for a proposition or set of propositions to be false, another altogether for it to be necessarily false.
However, It is false that four exists is necessarily false, and thus counterfactually implies any proposition.
Some philosophers hold the contrary view that a subjunctive conditional of this kind is necessarily false.
By necessitation, it follows that K(p ∧ ¬Kp) is necessarily false.
These judgements commit us to the existence of A, and therefore are necessarily false if that commitment is broken.
But then, line 4 tells us that K(p ∧ (Kp → En)) implies something that is necessarily false.
However, the objection runs, the second of the three conjuncts [(b) above] is necessarily false or is false for all possible actions.
Reason forbids us from believing a proposition that is necessarily false but permits us to have a set of beliefs that necessarily contains a falsehood.
By the lights of one living with contradiction, it may not follow that an inconsistent statement is impossible even if it is necessarily false.
The claim that K(p ∧ ¬Kp) is impossible (at line 9) is inferred from this claim that K(p ∧ ¬Kp) is necessarily false.
