As we have seen above, the Rantala frames approach is in a sense more radical: by allowing anarchic impossible worlds of any kind to be accessible, it destroys all forms of logical closure for the relevant cognitive states.
The possibility of multiple extensions different possible coherent, inferentially complete conclusion sets that can be drawn from a single set of premises means that we have to think of logical consequence not as a function taking a set of axioms into its logical closure, but as a relation between a set of axioms and alternative logical closures.
Truth and membership are inductively generated by iterating rules that correspond to natural logical closure conditions and can be formalized by means of positive (i.e. negation- and implication-free) clauses.
However, due to logical closure the belief set contains many elements that are not really worth to be taken seriously.
Due to logical closure it then also contains the sentence p ∨ q, "Either Shakespeare wrote Hamlet or Charles Dickens wrote Hamlet".
The reason for this has to do with questions concerning the logical (or Boolean) closure of the set of physical properties — if P, Q and R are physical properties, which of the various logical permutations of P, Q and R are likewise physical properties?
The Justification Logic formalization models what seems to be happening in such a case; closure of knowledge under logical entailment is maintained even though 'barn' is not perceptually known.[2] Kolmogorov explicitly suggested that the proof-like objects in his interpretation ("problem solutions") came from classical mathematics (Kolmogorov 1932).
ALX avoids important drawbacks of other action logics, especially the counterintuitive necessitation rule for goals (every theorem must be a goal) and the equally counterintuitive closure of goals under logical implication.
The logical aspects of modal epistemic logic were not significantly developed after Hintikka's 1962 presentation; instead, the philosophical literature (which is not extensive, compared with many other topics in the area) concentrates on the issue of hyperintensionality, or closure of epistemic attitudes under logical consequence.
Partial primary closure of the wound is the logical answer to this problem.
