The relevant logic R is a natural choice. p → p is 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 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.
2 comments:
Hi Greg!
The relevant logic R is a natural choice. p → p is 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 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.
Fantastic -- Thanks a lot!
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