FOM: intuitionist propositional logic

William Tait wtait at
Mon Apr 19 18:19:59 EDT 1999

Robert Black wrote

>Can anyone tell me where I can find information on propositional logics
>intermediate between intuitionist logic and classical logic? As is well
>known, adding excluded middle or double-negation elimination collapses
>intuitionist logic into classical logic.  But we can add to intuitionistic
>propositional logic, for example, the schema (p -> q) v (q -> p) and get
>something strictly stronger, but still weaker than classical logic.  How
>many such intermediate logics are there, and how are they ordered?

There is the original paper of Goedel's: ``On the Intuitionistic
propositional logic'', (1931-2, 1933). Then there are

T. Umezawa ``On intermediate propositional logics'', JSL 24 #1 (1959)
I. Nishimura ``On formulas of one variable in intuitionistic propositional
calculus'' JSL 25 #4 (1960)

As you see, all of these are quite old; and very likely there is more
recent stuff. But I don't know of more recent work and I haven't thought
about whether the papers I cited help with your problem.

Regards, Bill

PS. Thyank you for not once mentioning universes.

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