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 inferenceLogically-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 .
DefCan be changed from a good inference into a bad one by substituting some nonlogical vocabulary for different nonlogical vocabularyTrue no matter what you plug in for the variables or substitute for the non-logical vocabulary.
DemoThe 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.
SloganDescriptive terms appear essentiallyDescriptive terms appear vacuously
FormTrue, but not because of its logical formTrue in virtue of its logical form


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.


  1. Letters (e.g. , , ) represent logical variables. These have the meaning of being replacable with anything, loosely speaking.

  2. 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.