[FOM] Weak logic axioms

[Charlie]

	  With all this contention about logics, I’ve wondered why no one has expressed the view that empty domains seem more mathematically appropriate, rather than assuming first-order logic domains must be non-empty.  (Especially since the topic of the empty set being included in any set has been bandied about.)

Charlie Silver

> On Sep 25, 2015, at 12:44 PM, Michael Lee Finney <michael.finney at metachaos.net> wrote:
> 
> 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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