Let an inference be a statement with a material conditional, i.e. of the form “If , then ”1 in a language with ordinary statements as well as logical connectives (e.g. and, or). We want a rigorous way to distinguish that following kinds of inferences:
| Material inference | Logically-valid inference | |
|---|---|---|
| E.g. | If it’s a cat, then it’s a mammal If pigs can fly, then .2 If it’s red, then it’s colored | If it’s blue and red, then it’s red and blue If I’m angry, then I’m angry. If and it’s hot, then it’s hot or . |
| Def | Can be changed from a good inference into a bad one by substituting some nonlogical vocabulary for different nonlogical vocabulary | True no matter what you plug in for the variables or substitute for the non-logical vocabulary. |
| Demo | The first example would become bad if we replaced ‘cat’ with ‘turtle’ | No matter what we replace ‘blue’ with in the first example, it will still be true. |
| Slogan | Descriptive terms appear essentially | Descriptive terms appear vacuously |
| Form | True, but not because of its logical form | True in virtue of its logical form |
Generalization
This move was to take a notion of “what is a logical connective?” and pick out which sentences are “logically-valid” sentences. This could be generalized to give us a notion of other types of vocabularies. E.g. we identify theological vocabulary (e.g. God, pious) and observe which good inferences (e.g. ”If God loves fishing, then fishing is pious”) cannot be made into bad inferences by substituting non-theological vocabulary, i.e. sentences which are true purely due to their theological form.
Footnotes
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Letters (e.g. , , ) represent logical variables. These have the meaning of being replacable with anything, loosely speaking. ↩
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This demonstrates that this distinction is different from the synthetic distinction. There, a cat being a mammal would feel like an analytic statement, while pigs not being able to fly feels more synthetic. ↩