Establishing Validity (Theory)
We found it easy to prove that two forms of the mixed hypothetical syllogism were invalid, that they could not be relied on to give us a true conclusion from true premisses. But although easy, the proof is none the less watertight. Since, in order for the form to be valid, every proper set of substitutions must yield a valid inference, to show that a form is invalid it suffices to find one set of substitutions which does not yield a valid inference. But it is quite impossible to prove that a form of inference is valid by this means. Even a long series of sets of substitutions which yield valid inferences does not provide us with a proof that the form of inference is valid. So far we have relied on our logical intuitions, our sense of what follows and what doesn’t, to convince us that Affirming the Antecedent and Denying the Consequent are valid forms of inference. But just seeing that … is not the same as proving that …; and it is clear that if we want to prove that an inferential schema is valid we shall have to devise some general method for doing so, which will not depend either on intuition or on the result of assigning particular values to the variables.