[FOM] "Hidden" contradictions

[Carl Hewitt]

Dear Sam,

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."

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 :-)

Regards,
Carl

PS. Classical logic (typically non first-order) will still be used for thought-to-be-consistent mathematical theories.