[FOM] Weak logic axioms

[Arnon Avron]

On Thu, Sep 24, 2015 at 05:16:26AM +0000, Alex Blum wrote:
>  
> 	Some time ago under the present subject heading Michael Lee Finney wrote:
> "...you could then prove
>  (4)   (p & q -> r) -> (p -> r) v (q -> r)
> which I thought that surely was invalid. 

The intuitive objection that classical tautologies like (4)
cause is due to taking the propositional connective `->'
as the translation that classical logic offers for the
"if  ... then ____" used in mathematical texts (and the use
of the symbol `->' contributes to this wrong understanding...).

Actually, when formalized in classical FOL the
"if  ... then ____" is never (or almost never) translated
using just -> (i.e. `\neg ... \vee ____'), but there
are (almost) always also one or more universal
quantifiers that precede the use of ->. In other
words: the classical counterpart/translation of
the informal "if - then" combination is   a combination
of \forall(s) and `->'.

  Everyone is invited to check that if we define
A->B as "\forall x_1,...,x_n (\neg A \vee B)" then
in case n>0 the counterintuitive tautologies are no
longer valid, while the intuitively correct ones remain valid.
(The former do *not* include the so-called "paradoxes
of the material implications", because those "paradoxes"
*are* in fact used by mathematicians. This issue was
recently discussed here on FOM, and so I am not
going to return to it here.)

Arnon Avron

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