Heyting arithmetic, after all, has exactly the same axioms as Peano arithmetic, and is the logical closure of those axioms under intuitionistic, rather than classical, logic.
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".
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.
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 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?
Tarski notwithstanding, since 1975 the hierarchical approach has been somewhat superseded by new ideas that have rendered the ideal of logical and semantical closure in many respect accessible (especially by means of the fixed point methods used by Kripke 1975 and Martin-Woodruff; see Martin 1984).
Behind such closures, "the logical thinking is that you are dealing with clients whose track record is hard to establish and who are in the front line in terms of job vulnerability," said Emilio Ontiveros, chairman of Analistas Financieros Internacionales, a research consulting firm in Madrid.
More explicitly, one requires closure under the logical operations of negation, conjunction, existential quantification and suitable combinatorial operations of permutation and expansion.
In a BBC interview in 2002 Reid said the occupation was "the only logical effective form" of opposition to closure.
