Re: Minimal logic valid?

On Jun 9, 2:48 pm, Jan Burse <janbu...@xxxxxxxxxxx> wrote:

The minimal implicational logic based on -> and
A, cannot derive:

    (P -> Q) -> (EvP -> Q), v not in Q

Provided that E is defined as ~A~.

And indeed, just as in intuitionistic logic, E is not defined as ~A~.
I intentionally did not list such a definition or equivalence.

I dont
think that your system without ~A->(A->B)
is the same as my minimal implicational logic.

It might not be the same as your minimal implicational logic. But it
is the minimal logic I found in the textbook 'Propositional Logics' by
Epstein (though the quantifier axioms I used are just from
intuitionistic logic), said there to be based on Johansson 1936, and
also from another source that I don't recall right now. By the way,
the Epstein book gives Fitting's variation on Kripke semantics for the
(propositional) system, an alternative equivalent axiomatization in
which f is primitive instead of ~, and an outline of a completeness
proof.

That is, these sources take minimal logic to be exactly intutionisitic
logic except witthout ex falso quodlibet, just as I have.

MoeBlee
.

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