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Do we have anything on why “classical simple type (2-)theories” due to their multiple consequents deserve to be called “classical”, other than at sequent calculus
Any of these logics may be presented using any sort of sequent, but Gentzen’s original sequent calculi presented each logic using only corresponding sequents.
Is it down to the kind of issue covered here?
If ’classical’ can be added to ’simple type 2-theory’ (allowing multiple consequents) what to say about the next 3-theory, namely, the ’first-order logic’ 3-theory? Could there be a ’classical’ version of that?
Isn’t this going to get messy? Ordinary classical first-order logic may be specified as a particular 2-theory in the ’first-order logic’ 3-theory.
Ah, here is Mike speaking to the issue in #3:
I think of a 3-theory as specifying the arity and dependency structure. The 3-theory of “first order logic” has a base whose arity is many-to-one with no internal dependency and then another layer that depends on that. (The upper layer could be many-to-one or many-to-many, so actually there are two different 3-theories for first-order logic, just as there are two different 3-theories for propositional logic / simple type theory. The many-to-one and many-to-many versions are traditionally called “intuitionistic” and “classical”, though that’s confusing too.)
Very confusing.
So the dependency of the 3-theory of first-order logic can be mixed with the ’classical’ arity of multiple consequents, but we don’t know how to do this with the full dependency of the 3-theory of dependent type theory?
Added an explanation of ’classical’
The term ’classical’ owes its origin to the contrast between systems LK and LJ of Gentzen’s sequent calculus, where LJ provides a proof system for intuitionistic logic via restriction to single sequents of the rules of LK for classical logic.
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