(This post is only a teaser, a brief introduction to the late philosopher David Stove’s philosophy of logic. I do not intend that today’s article will convince anybody. I do not have time to do more.)
Everybody—especially readers of this site—has had experience with logical arguments. People obviously use logical argumentation continuously, whether or not they are aware that the science of logic has been “made formal” by mathematicians and other such creatures.
Their ignorance of this formalization obviously—I hope it is obvious—does not mean that when these people offer a valid argument it is not made invalid because they are unaware of how to prove it valid.
It was David Stove’s contention that formal logic (to be defined in a moment) is a myth. That is to say, that all attempts to formalize logic were doomed to failure. This might sound like yet another post-modern attempt at skepticism. It is not:
My philosophy of logic is so far from being skeptical that it is if anything indecently affirmative. Not only do I believe, as I have implied, that there are logical truths, true judgments of validity or of invalidity; I believe that every normal human being is, in the extent of his knowledge of such truths, a millionaire. Only, I hold, as I have implied, that almost every logical truth which anyone knows, or could know, is either not purely formal, or is singular or of low generality. [p. 128; The Rationality of Induction]
What did Stove mean by formal?
An argument is formal “if it employs at least one individual variable, or predicate variable, or propositional variable, and places no restriction on the values that that variable can take” (emphasis mine). Stove claims that “few or no such things” can be found.
Here is an example of formality: the rule of transposition. “If p then q” entails “If not-q then not-p” for all p and for all q.
This is formal in the sense that we have the variables p and q for which we can substitute actual instances, but for which there are no restrictions. If Stove is right, then we should be able to find an example of formal transposition that fails.
First a common example that works: let p = “there is fire” and q = “there is oxygen”, then
“If p then q” == “If there is fire there is oxygen”.
And by transposition, not-q = “there is no oxygen” and not-p = “there is no fire” then
“If not-q then not-p” == “If there is no oxygen then there is no fire.”
For an example in which formal transposition fails, let p = “Baby cries” and q = “we beat him”, thus
“If p then q” == “If Baby cries then we beat him”.
But then by transposition, not-q = “We do not beat Baby”, not-p = “he does not cry”, thus
“If not-q then not-p” == “If we do not beat Baby then he does not cry.”
which is obviously false. (Stove credits Vic Dudman with this example.)
So we have found an instance of formal transposition that fails. Which means logic cannot be “formal” in Stove’s sense. It also means that all theorems that use transposition in their proofs will have instances in which those theorems are false if restrictions are not placed on its variables. (It’s worse, because transposition is logically equivalent to several other logical rules; we won’t go into that now.)
It is Stove’s contention that all logical forms will have an example where it goes bad, like with transposition.
Exercise: can we find counterexamples to the two most popular logical forms, modus ponens and modus tollens? I haven’t tried yet, but I rely on the way-above-average intelligence of our readership to provide some.
Modus ponens: “If p then q, p, therefore q, for all p, q”. Example: “If Socrates is a man then he is mortal, Socrates is a man, therefore he is mortal.”
Modus tollens: “If p then q, not-q, therefore not-p, for all p, q”. Example: “If Socrates is a man then he is mortal, Socrates is not mortal (he is immortal), therefore Socrates is not a man.”