logic bleg

I have a question for the logically savvy:

Suppose A is a logical truth ( |= A ).

In which logics (if any) does the following fail?

(A&B) → C
B → C



Greg Restall said...

Hi Greg!

The relevant logic R is a natural choice. pp is a theorem, and so is ((pp) & q) → (pp). However, q → (pp) is not a theorem. It isn't entailed by ((pp) & q) → (pp). This inference fails in logics at RM3 and below. (The 3-valued RM3 counterexample: let p take the value 1/2 and q take the value 1.)

I don't know of any logics other than relevantish ones in which & is conjunction-like, → is conditional-like, and in which this inference fails, though I don't have a characterisation the class of logics in which it fails other than saying that it’s at least RM3 and below.

I hope that helps.

Greg Frost-Arnold said...

Fantastic -- Thanks a lot!