If A, Then B: How the World Discovered Logic

by Michael Shenefelt and Heidi White

Conundrums for Logicians

The history of logic is one thing, but the philosophy of logic is another.


Logicians often debate philosophical controversies. If A, Then B describes the history of logic, but it also takes on a number of philosophical controversies that concern how logic works.

If A, Then B defends controversial positions in a variety of areas:

1. Rationality.


When it comes to reasoning well, the book says one must always be initially more convinced of the premises of an argument than of the conclusion to be proved. Otherwise, no argument that relies on those premises to establish the conclusion can be rationally persuasive. This thesis derives from Aristotle’s claim in the Posterior Analytics that the premises of demonstration must always be “better known.”

Aristotle was right on this point, though many logicians now seem to think him wrong. If A, Then B explains and defends Aristotle's point of view on pp. 130-38.

2. Philosophy of Science.


Many philosophers and historians believe that scientific reasoning is necessarily circular. The famous historian Thomas Kuhn defended this outlook, but If A, Then B contends that his argument rests on a specific mistake in propositional logic.

3. Nelson Goodman’s Claim that Logic Can Be Circular.


Many logicians believe that deductive logic is “virtuously circular”—a position defended by the Harvard logician Nelson Goodman more than fifty years ago. But Goodman’s argument is actually a non sequitur. If A, Then B discusses his argument and contends that the argument is a non sequitur on pp. 300-2.

4. Logical Intuition.


Are some logical truths known intuitively—independently of the work of professional logicians? Yes. And are these intuitions just as certain as many of the arithmetical truths that we can know by counting on our fingers, independently of the work of professional mathematicians? Yes.

A common argument against the certainty of such intuitions consists in saying that, because we sometimes make mistakes in exotic cases, we can never be sure even in simple ones. (Thus, because modus ponens gives paradoxical results in special cases, it can never be known valid in obvious cases.) In form, this argument is like the famous “argument from illusion”—which alleges that we can never know anything with certainty from the testimony of our senses, because sometimes, in special cases, our senses deceive us.

In fact, many of our simple intuitions about logic are correct and owe nothing to the specialized techniques found in a formal logical system. On the contrary, our rational confidence in the systems often rests on these intuitions.

To borrow from David Hume, no matter how reasonable a skeptical claim about our common-sense intuitions may appear in a philosophy class, at the end of the session, we still leave by the door and not by the window. When it comes to many of these intuitions, there has never been any serious doubt. Instead, there have only been imagined doubts. (Imagining that you doubt is no more a genuine form of doubting than imagining that you fly is a method of traveling from place to place.)

If A, Then B discusses the certainty of many common-sense logical intuitions, independent of formal logical systems, on pp. 9, 118-21, 274, and 301.

5. Wittgenstein and Linguistic Philosophy.


Are many classics of philosophy riddled with nonsense? Do they contain meaningless language such that neither the writers nor the readers have anything in mind other than the sounds of the words or the sheer strangeness of their combination? Probably not.

Thomas Hobbes and David Hume both believed that philosophy books were full of such language, and so did Ludwig Wittgenstein, who insisted that much philosophy was “disguised nonsense,” which could be exposed as “patent nonsense.”

If A, Then B contends that this hypothesis is almost certainly false. And the book asserts that much of the confusion here comes from the loose way in which words like “meaningless,” “senseless,” and “nonsense” (equivalent to Wittgenstein’s “Unsinn”) are used—not only in everyday speech, but in the diction of philosophical writers themselves.

Many writers use these words indiscriminately to mean a variety of different things: ambiguity, pretentiousness, doubletalk, implicit contradiction between two otherwise meaningful propositions, euphemism, and, in some instances, mere vagueness.

Since the days of Plato, many of these different things have often been recognized as serious problems in philosophy. (Plato frequently aims at exposing ambiguity, pretentiousness, and implicit contradiction.) Doubletalk is a common problem too, and it is usually what is meant by expressions like “B.S.” or “baloney.” But doubletalk consists in using words that one knows to be nonsensical—with the aim of deceiving the listeners. Hobbes, Hume, and Wittgenstein had in mind a different effect. They meant to describe cases in which neither the writers nor the readers realize that the words are nonsensical. Rudolf Carnap illustrated such language by offering this example: “Caesar is a prime number.”

The trouble with this thesis is that the sort of absurdity that Hobbes, Hume, Wittgenstein, and Carnap postulate could never be “disguised.” Instead, it is obvious. It fools no one. And in consequence, it could never deceive both a philosophical writer and the philosopher’s readers at the same time. Nor could it ever generate the kind of literature that these thinkers meant to attack. To be sure, philosophical language can be meaningless in the sense of being overly ambiguous or in the sense of being overwritten (meaning pretentious). But this is different from being utterly unintelligible—which was Wittgenstein’s claim.

(Ambiguity can fool both speakers and listeners alike, and so can pretentiousness. But this is not the case with language that is intelligible to no one at all. The real trouble here comes from using a word like “meaningless” to cover a wide variety of different linguistic situations. It should also be noted that Wittgenstein’s defenders, when pressed about these matters, sometimes argue that he meant neither to expose ambiguity, nor vagueness, nor pretension, nor doubletalk, nor euphemism, nor sheer blankness of mind on the part of philosophical writers; instead, he meant to expose something else—something that could still be called “nonsense,” but that was different from mere falsehood. But what would this be? Would it be a tendency to violate the correct rules of sensible communication? But what rules are these? Would they be the rules that tell us what should count as “nonsense”? Here we seem to be going in a circle. Admittedly, the members of the Vienna Circle often focused on statements that seemed to be incapable of empirical verification; was their aim, then, to show that statements incapable of empirical verification were, yes, incapable of empirical verification? In that case, the only thing being exposed is a tautology. Wittgenstein definitely meant to remedy a problem, but what exactly was the problem? His Tractatus and his Philosophical Investigations both carry a strong sense of mission, but it is hard not to wonder whether the mission was actually just a phantom, a phantom that consisted in rooting out some elusive sort of nonsense, a sort that, in point of fact, never existed.)

Wittgenstein certainly raised many interesting philosophical questions—including whether there can be a “language of thought,” meaning a system of representations that is different from the languages we actually see or hear, but still present in the mind. Nevertheless, this is not the same as showing that both philosophical writers and their readers have been thinking of nothing at all but the sounds of their words or the strangeness of the combinations—when they sincerely believed they were discussing ideas.

If A, Then B considers these highly controversial questions on pp. 65-69.