[FOM] "Hidden" contradictions
Carl Hewitt
hewitt at concurrency.biz
Mon Aug 26 13:48:49 EDT 2013
Absorption (P∧(Q∨P) =P) is not considered to be a standard Boolean equivalence. And, of course, P does not in general infer Q∨P, which is used in C. I. Lewis' proof. See http://arxiv.org/abs/0812.4852
________________________________________
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] On Behalf Of Harry Deutsch [hdeutsch at ilstu.edu]
Sent: Sunday, August 25, 2013 1:28 PM
To: Foundations of Mathematics
Subject: Re: [FOM] "Hidden" contradictions
Standard boolean equivalences plus disjunctive syllogism yield that any contradiction implies everything--by C. I. Lewis' proof.
On Aug 25, 2013, at 10:56 AM, Sam Sanders wrote:
> Dear Carl,
>
>> Not everyone has been on same old conventional page as Tarski. For example, [Barwise 1985] critiqued the first-order thesis as follows:
>>
>> "The reasons for the widespread, often uncritical acceptance of the first-order thesis are numerous. Partly it grew out of interest in and hopes for Hilbert's program. ... The first-order thesis ... confuses the subject matter of logic with one its tools. First-order language is just an artificial language structured to help investigate logic, much as a telescope is a tool constructed to help study heavenly bodies. From the perspective of the mathematics in the street, the first-order thesis is like the claim that astronomy is the study of the telescope."
>
> I believe I did not mention the "first-order thesis", but let me make it clear that I believe that (at the very least) we need second-order arithmetic
> to formalize mathematics. Reverse Mathematics is a good example, though heavy on the coding sometimes; Classical logic is used there.
>
>> In particular, Computer Science has requirements that go far beyond classical first-order logic. The domination of classical logic is coming to an end because practical real-world theories are pervasively inconsistent.
>> Computer Science needs Inconsistency Robust mathematical foundations with the following characteristics:
>> * Standard Boolean equivalences hold for conjunction, disjunction, and negation
>> * Disjunctive Syllogism holds as well
>> * Capability to reason about arguments for and against propositions
>>
>> Consequently, the range of acceptable CS solutions for Inconsistency Robust inference is actually quite narrow :-)
>
> Do people in applications really throw out classical (whatever order) logic and embrace paraconsistent logic as "the true way"? Or do they just think/work classically and somehow
> manage to contain the inconsistent information, i.e. prevent it from doing damage?
>
> I have seen examples of the latter, but not the former in practice. Chow similarly asked for a clear example (related to the ongoing saga mentioned), I believe.
>
> Best,
>
> Sam
>
>
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