Induction Reduced

Every undergraduate course in logic begins with the same partition. Deductive inference is presented as airtight: if the premises are true, the conclusion must be true. Inductive inference is presented as a softer creature: if the premises are true, the conclusion is rendered more probable, but never necessary.¹ The canonical instances are the syllogism on the deductive side ("All men are mortal; Socrates is a man; therefore Socrates is mortal") and the generalisation from observed cases on the inductive ("The sun has risen every morning; therefore the sun will rise tomorrow"). The contrast is so familiar that it is rarely examined.

The contention of this paper is that the partition does not survive examination. When the inductive inference is laid out with all of its premises made explicit, it becomes deductively valid. What we call induction, on this view, is simply deduction in which one premise — the one doing the heaviest work — has been left tacit.

Consider Hume's example.² The reasoner has observed the sun rise on every one of, say, ten thousand consecutive mornings. From this, she concludes that the sun will rise tomorrow. Stated as it usually is, the inference appears to have one premise and a leap:

The conclusion does not follow with logical necessity from the single premise, because no contradiction arises from the conjunction of the premise with the denial of the conclusion. This is the standard ground for calling the inference inductive rather than deductive.

But notice: the reasoner is not in fact moving from the bare premise to the conclusion. She is moving via a second, unspoken premise — a principle she does not state because she takes it for granted. Make it explicit:

Now the argument is valid. The conclusion follows from the two premises with the same necessity as the conclusion of the Socrates syllogism follows from "All men are mortal" and "Socrates is a man". The ampliative character, the leap, the not-following-with-necessity — all of it has migrated into P₂. The inference has not become more secure; it has become more honest about what it requires.

This is what Aristotle, in the Prior Analytics, recognised when he treated epagōgē (induction) as a kind of syllogism whose major premise was the universal claim that any property holding of all observed instances of a kind holds of the kind itself.³ The medieval logicians knew this construction as the enthymeme: an argument with a suppressed premise. The suppression is rhetorical, not logical. What looks like a different mode of inference is in fact the same mode, missing a step.

Let me state the thesis directly. Every inductive inference is an enthymematic deduction whose suppressed premise asserts the projectibility of an observed regularity. Once the suppressed premise is supplied, the inference is deductively valid. The deductive/inductive distinction is not a distinction between two species of inference; it is a distinction between an inference stated in full and the same inference stated with a step left out.

Russell saw this clearly. In The Problems of Philosophy he argues that any inference from observed cases to unobserved ones requires what he calls the principle of induction — a general premise to the effect that what has held in the past will continue to hold — and that without this principle as a premise, no such inference can be drawn.⁴ He is not making an epistemological point about how the principle is known; he is making a logical point about what the inference contains.

The principle of induction (or the uniformity of nature, in the older formulation) is not therefore an alternative engine to deduction. It is a premise that, once installed, lets the deductive engine run.

The thesis above runs from induction toward deduction: induction reduces to deduction once an implicit premise is supplied. It is worth noting that John Stuart Mill, in A System of Logic (1843), arrived at a related but oppositely directed conclusion: deduction reduces to induction.⁵

Mill's argument is well known and worth restating. Take the standard syllogism: All men are mortal; Socrates is a man; therefore Socrates is mortal. Mill asks how we came to know the major premise. We did not deduce it from a still more general truth; we observed many particular men, found them all to be mortal, and generalised. The major premise of every syllogism is itself an inductive conclusion. From this Mill draws a startling implication: when we run the syllogism to derive Socrates' mortality, we are not really inferring anything new. We have already, in establishing the major premise, committed ourselves to the mortality of Socrates along with the mortality of every other man. The syllogism merely records the application of a generalisation we earlier induced. Genuine inference, Mill insists, is always from particulars to particulars, mediated by a general rule.⁶

The two routes — mine and Mill's — converge on the same conclusion through opposite analyses. I have argued that induction is enthymematic deduction. Mill argues that deduction is registered induction. Both deny that two fundamentally different inferential mechanisms are at work in human reasoning. There is one mechanism, with two modes of presentation.

Whether one prefers to call the unified mechanism "deduction with implicit premises" or "induction with explicit major terms" is, I suspect, a question of which surface feature one finds more salient. The substantive claim — the unity of inference — is the same.

§5.1 Hume's regress. The first and gravest objection is Hume's own.⁷ If induction is rewritten as a deduction by supplying P₂, the principle of uniformity, then the inference rests on P₂'s being true. But P₂ is itself a generalisation about the world, and the only basis we have for it is the very practice of induction whose validity we are trying to underwrite. The supposed reduction is circular: we have justified induction by inserting a premise we can only know inductively.

This is correct, but it is an objection to the justification of induction, not to its structural identity with deduction. The reducibility thesis is a thesis about form. It says: when we reason inductively, we have, structurally, performed a deductive inference with a suppressed premise. It does not say: the suppressed premise is independently justified. Whether or how P₂ can be defended is the further (and ancient) question of the justification of induction, untouched by what is claimed here.

§5.2 Goodman's grue. Nelson Goodman's celebrated puzzle shows that the principle of uniformity, simply stated, is too weak.⁸ Define a predicate grue as "green if examined before time t and blue otherwise". Every emerald examined so far has been grue. By naïve uniformity, the next emerald will be grue, i.e. blue. The naïve uniformity premise generates a contradiction with the perfectly parallel premise that all observed emeralds are green, hence the next will be green.

The grue puzzle does not, however, refute the reducibility thesis. It refines the suppressed premise. The premise required for inductive inference is not a flat principle of uniformity but a principle of projectible uniformity — one that specifies which predicates are eligible to be projected from observed to unobserved cases. Stating that premise is harder than stating bare uniformity; but the structural point — that the inference is a deduction once the relevant premise is supplied — is preserved.

§5.3 The Popperian denial. Karl Popper rejected induction altogether. He held that scientific inference is purely deductive: theories are conjectures from which observable predictions are derived, and the only legitimate move from observation is the deductive falsification of a theory that predicted otherwise.⁹ On Popper's view, what we call induction does not exist; the appearance of inductive inference is a confusion.

Popper is, if anything, an extreme ally. He agrees that there is no distinct inductive mode of inference. He differs only in declining to admit any reasoning that looks like induction — including the everyday reasoning that the sun will rise tomorrow — as legitimate scientific inference at all. The unity-of-inference thesis is friendlier: it accepts the reasoning, and tells the deductive story of how it works.

The distinction between deduction and induction is useful in the classroom because it sorts a large and otherwise undifferentiated mass of human reasoning into two boxes that are easy to teach. It is mostly harmless. But it should not be mistaken for a finding about the structure of reasoning itself. Reasoning, as performed, consists of moving from premises to conclusions under a rule. When the rule is universal and explicit, we call the inference deductive. When the rule is universal and implicit, we call the inference inductive. Either way, the rule is doing the same work.

The mature position is to make rules explicit. To say "I have observed n cases and predict the (n+1)-th" is to give an inference half-finished; to make the suppressed premise plain is to face squarely what one's reasoning actually contains. That is not always comfortable — the suppressed premise is often a piece of trust in the world that cannot be argued for without circularity. But the discomfort is honest, and it belongs at the front of the argument, not hidden behind it.

There is one inferential mechanism. There are two ways of presenting it. The names deduction and induction describe modes of presentation, not modes of thought.