Assume the ability to discern argument validity

Friday July 14, 2023

To make any progress in reasoning requires assuming the ability to correctly determine whether arguments are valid given their inference rules. This is both a mechanical and fundamental process.

Validity is the correctness of the form of an argument, assuming its inference rules, regardless of the truth of its assumptions. This is a valid argument:

Assumption: If Socrates drinks hemlock, then Socrates will die.
Assumption: Socrates drinks hemlock.
Inference rule: If P implies Q, and P is true, then Q. (Modus Ponens)
Conclusion: Socrates will die.

Any other conclusion from the inference rule and assumptions here is invalid, a non sequitur. (No other inference rules are included in the example, not even the law of identity.)

When assumptions and inference rules are all stated clearly, and especially when every step of multi-step arguments are laid out, it appears to be a mechanical process to verify whether an argument is valid. But it is possible to doubt that it can be done correctly, and impossible to prove that it can, so the ability must be assumed.

As Descartes asks in connection with his demon, “how do I know that I am not also deceived each time I add together two and three?” This is hard to imagine, but that is by construction. Perhaps the demon causes Descartes to conclude from the example above that Socrates is in France, and completely believe that this follows from the inference rule and assumptions given. He believes his argument is valid, but he is wrong, and he is incapable of determining this, totally anosognostic, unaware of his condition.

The ability to identify whether an argument is valid cannot be proved, because any proof offered would need to be assumed valid. Any attempt at a proof begs the question. To assume the validity of just a specific proof of the ability to determine validity of other proofs might be possible, but the validity assumption would then be simultaneous with others needed for that proof, which introduces additional restrictions but unclear benefit.

Arguments are not always stated sufficiently clearly and completely as to make confirming validity easy, but this is fundamentally a challenge in understanding the argument. In principle any argument can be expressed clearly and completely.

The assumption here is that it is possible to identify valid arguments that really are valid, not that there will never be an accidental, correctible mistake. The assumption is that when there aren't any mistakes, the claim that an argument is valid is meaningful.

So while verifying the validity of an argument can be a mechanical process, the assumption here really is fundamental. The assumption is first that reasoning can be, in some deep sense, really truly valid, that this is possible, and second that it is possible to know whether a given argument has this property.

Validity is tautological but universal truth. If the assumptions and the inference rules, then the conclusions. This is the first and deepest sense of correctness.