[FOM] EFQ and Tennant's consistency

[Arnon Avron]

In his postings Neil maintains that ~p & p |- q is invalid inference
that mathematicians never use, while ~p |- p->q is valid.
The latter is justified by him (in the classical case, I guess)
by the truth table of the material implication (and that of negation
is needed too, right?).


As I am going to explain in another posting, I find both the distinction
between ~p & p |- q  and ~p |- p->q (in the framework
of mathematical practice), as well as the appeal here
to the truth table of material implication, as rather strange. But
let us accept for now the use of the truth tables for the justification
of rules. Well,  how exactly is the validity of ~p |- p->q forced by the
truth-tables of -> and ~? Answer: it is because according to these
truth table every model of ~p is also a model of p->q. In other
words: because the set of models of ~p is a subset of the
set of models of p->q. But by the same token, 
the truth tables of & and ~ force  the set of models
of ~p & p to be  a subset of the set of models of q (remember
that Neil did accept that the empty set is a subset of any other set, 
and he even proved this in "core logic"). So according to his 
own arguments, Neil should accept the validity of ~p & p |- q!

Arnon Avron