[FOM] Weak logic axioms

[Michael Lee Finney]

Arnon,

 I believe that -> can be interpreted as more than just relevant
 entailment, but still not as material implication (for which I use
 =>). Nor do I dispute your claim about the "forall" definition. But
 that definition is in the context of material implication because
 you are still using (~p v q) as the basis of the definition.

 I am in full agreement that depth relevant logics provide a valid
 form of reasoning and characterize the idea of "relevance" in
 reasoning. I think that Ross Brady has done wonderful work in that
 arena. However, logics purely based on relevant reasoning are not
 sufficient for mathematical reasoning in general.

 To me, -> is about reasoning from truth to truth, and not all such
 reasoning is relevant. There are also valid semantic arguments. I
 would agree that the validity of specific semantic arguments depend
 on the reasoning context - but the generally accepted context allows
 them. I would generally reject anything leading to unqualified
 contraction, nor do I accept explosion or Modus Ponens for material
 implication (in both cases, outside of the classical context which
 does allow them).

 I do NOT believe that allowing semantic reasoning necessitates the
 interpretation of -> as material implication. Nor do I believe that
 the term "paradox" is correctly applied when rejecting some forms of
 semantic reasoning. For example, outside of relevance quibbles, it is
 really hard to reject weakening p -> (q -> p) as a valid principle of
 reasoning. Because when p is true it is ALWAYS true in any context.
 All that weakening does is to allow that principle to be expressed in
 a nested context. It also shows that -> is not merely relevant
 reasoning.

 Some of the common relevance axioms are implicitly derived from
 weakening, they are just considered valid reasoning and happen to be
 in a form which is relevant. For example, to prove the rule

    p -> q |- p -> p & q

 it is necessary to use weakening in proof form. Likewise, to get

    (p -> q) -> (p -> p & q)

 it is necessary to use weakening in axiom form. Prefixing and
 suffixing are similar.

 Ross Brady's meta-rule MR1 is in the same boat

      p |- q  |=  p v r |- q v r

 where |- denotes a rule and |= a meta-rule.

 The thesis (p & q -> r) -> (p -> r) v (q -> r) clearly fails in terms
 of relevance reasoning. However, it is perfectly valid even outside
 of material implication. It does not depend on (~p v q) except in
 material implication. Fundamentally, it expresses a property of ->
 when p and q are not distinguished.

 On the other hand, the thesis (p -> q) -> (~q -> ~p) does not fail
 due to relevance or to a property of ->, but can fail due to
 semantics. One case in which it fails is when p is true and q is
 inconsistent. Assuming that negation leaves an inconsistent value
 unchanged and that an inconsistent value is distinguished (both
 semantic properties), then (p -> q) would be distinguished, but (~q
 -> ~p) would not be  distinguished and so then entire thesis fails.
 This failure is due to the semantics of inconsistent values.

> 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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