[FOM] "Hidden" contradictions

Sam Sanders sasander at cage.ugent.be
Sun Aug 25 11:56:27 EDT 2013

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.  



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