### 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*

entails

*B → C*

Thanks!

idiosyncratic perspectives on philosophy of science, its history, and related issues in logic

## 2 Comments:

Hi Greg!

The relevant logic

Ris a natural choice.p→pis a theorem, and so is ((p→p) &q) → (p→p). However,q→ (p→p) is not a theorem. It isn't entailed by ((p→p) &q) → (p→p). This inference fails in logics atRM3and below. (The 3-valuedRM3counterexample: letptake the value 1/2 andqtake 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

RM3and below.I hope that helps.

Fantastic -- Thanks a lot!

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