Let an inference be a declaration of the form \(P \implies Q\)
There, \(P\) and \(Q\) are logical variables. We can also put other things in their place:
Non-logical vocabulary, e.g. red, cat, or it’s raining outside
Logical connectives: and, or, etc.
We want to distinguish certain inferences as material inferences, as distinct from logically-valid inferences.
Logically-valid inferences:
These are inferences that are true no matter what you plug in for the variables or substitute for the non-logical vocabulary.
E.g. \((A \land {\rm it's\ raining}) \lor C \implies (C \lor {\rm it's\ raining})\)
This is true, regardless of what we substitute for \(A\) and \(C\) (or swap “it’s raining” for anything, e.g. “I own two cats”).
Descriptive terms appear vacuously
Material inferences:
These can be changed from a good material inference into a bad one by substituting some nonlogical vocabulary for different nonlogical vocabulary
E.g. the material inference “\(a\) is red” \(\implies\) “\(a\) is colored” will become false if we replace ‘colored’ with ‘square’.
Descriptive terms appear essentially