If A then B: How the World Discovered Logic, by Michael Shenefelt and Heidi White

Michael Shenefelt and Heidi White’s new book is a risky undertaking and, I think, a valuable one. Their aim is to rescue logic from the mathematised corner of the classroom and put it squarely at the heart of philosophy – and indeed life. The risky part is the claim that reasoning, knowledge and rationality are first and foremost matters of logic, of applying that deceptively simple formula “if A then B” to the world. And, moreover, vice versa.

The authors argue, too, that the peculiar government structures of ancient Greece led to Aristotle’s syllogisms, while the “new military-political system” of Philip II of Macedon and his son Alexander the Great entailed (so to speak) propositional logic. The Industrial Revolution, with its steam engines and railways, created the need for symbolic logic, and the rise of the middle classes in Europe brought in its wake inductive reasoning.

If the long battle between Catholics and Protestantism was about “whose version of Christianity was theologically correct”, this social debate also encouraged renewed study of the problems of logic, the authors argue.

Martin Luther, by encouraging people to refer to the Bible themselves, emphasised both personal introspection and individual reasoning, rather than the blind acceptance of religious authorities. It was just unfortunate, then, that both sides started from different and opposed premises, “and so a great collision between Catholicism and Protestantism became inevitable”.

Aristotle is the hero of the tale, but unusually, an effort is made to include less celebrated figures such as Chrysippus and George Boole

Aristotle is the hero of the tale, but unusually, an effort is made to include less celebrated figures such as Chrysippus and George Boole. After all, as Shenefelt and White say, the basic notion of validity is not that deep. Aristotle’s role was to find in geometry “the secret behind classification”.

As to philosophy of language, why, no language is comprehensible unless it already “conforms to something like logical rules”. Ludwig Wittgenstein’s mistake came because he not only thought that logical necessities depend on language, but also held that most philosophical riddles were meaningless. In so saying, argue Shenefelt and White, he confused “two very different sorts of meaninglessness: ambiguity and unintelligibility”.

Is it a problem that a valid argument in one system can be invalid in another? Does it make logic “relative”? In non-classical logics even our beloved modus ponens – the “valid inference” – is invalid! No, no, this is just esoteric stuff, the authors say: “in ordinary situations, our knowledge of which arguments are valid and which principles are logically sound is quite independent”.

Such is the authors’ antipathy towards psychological explanations that they can ask questions such as “What is lying if nothing is true?” without appreciating that lying is rooted in what we believe to be the case, not in what may or may not actually be the case. They have a fit about Michel Foucault’s argument that truth is “linked in a circular relation with systems of power”. For them, “the task of the early Greek logicians was to cut through obscurities and confusions of this kind to arrive at the ideas of truth and falsity that regulate reasoning in the ordinary sense of the word”. That extra bit of the sentence sounds a bit circular to me. Haven’t they already said, many times, that logic is not about truth but merely about form? Yet another countervailing thrust in this book is the claim that the inductive – “invalid” – reasoning that is the hallmark of science is also a form of logic.

Thomas Kuhn was wrong, wrong, wrong, argue the authors, to say that scientific reasoning is circular; in fact, his argument rests “on a mistake in propositional logic”. Kuhn, we might recall, says that scientists select data to fit their theories, and then use this selectively chosen body of facts to adjust their overall theory. The error lies simply in the fact that in the proposition “if A then B” the important thing is simply that A be more plausible than B. The authors’ great insight (they hope) is that scientific data can be treated as a disjunctive syllogism – a vast premise in the form “at least A or B or C or D”.

This sounded an alarm bell to me. Wasn’t that Bayes’ Theorem? Another one sounded when the authors quoted approvingly the first line of René Descartes’ Discourse, in which he says that no human attribute is so equally distributed as “good sense”, to support their claims for universal rationality. Curiously, however, Descartes’ words were borrowed from Michel de Montaigne, who uses the same words in one of his famous essays, Of Presumption, but with the additional, witty “logical proof” that it must be so, as no one ever seems dissatisfied with their own share.