Thursday, September 30, 2004

On Falsely Locating the Difference between Deduction and Induction

One commonly hears it said that the difference between deductive and inductive inference is that the former moves from the universal to the singular, while the latter proceeds from the singular to the universal. (For a recent and somewhat surprising example, see David Bloor, "Wittgenstein as a Conservative Thinker" in The Sociology of Philosophical Knowledge, ed. Kusch (Kluwer, 2000), p. 4.) No doubt, some deductive inferences fit the universal-to-singular pattern, while some inductive inferences fit the singular-to-universal pattern.

But it does not require a lot of thought to see that this cannot be what the difference between deduction and induction consists in. An argument of the form, All As are Bs; All Bs are Cs; ergo, All As are Cs is clearly deductive, but is composed of three universal propositions. The argument does not move from the universal to the singular. So the first half of the widely bruited claim is false.

Indeed, some deductive arguments proceed from singular premises to a universal conclusion. Consider this (admittedly artificial) example: John is a fat chess player; John is not a fat chess player; ergo, All chess players are fat. This is a deductive argument, indeed it is a valid deductive argument: it is impossible to find an argument of this form that has true premises and a false conclusion. Paradoxically, any proposition follows deductively from a contradiction. So here we have a deductive argument that takes us from singular premises to a universal conclusion.

There are also deductive arguments that move from a singular premise to an existentially general, or particular, conclusion. ‘Someone is sitting’ is a particular proposition: it is neither universal nor singular. ‘I am sitting’ is singular. The first follows deductively from the second.

As for the second half of the claim, suppose that every F I have encountered thus far is a G, and that I conclude that the next F I will encounter will also be a G. That is clearly an inductive inference, but it is one that moves from a universal statement to a statement about an individual. So it is simply not the case that every inductive inference proceeds from singular cases to a universal conclusion.

What then is the difference between deduction and induction if it does not depend on the logical quantity (whether universal, particular, or singular) of premises and conclusions? The difference consists in the nature of the inferential connection asserted to obtain between premises and conclusion. Roughly speaking, a deductive argument is one in which the premises are supposed to ‘necessitate’ the conclusion, making it rationally inescapable for anyone who accepts the premises, while an inductive argument is one in which the premises are supposed merely to ‘probabilify’ the conclusion.

To be a bit more precise, a deductive argument is one that embodies the following claim: Necessarily, if all the premises are true, then the conclusion is true. The claim is that the premises ‘necessitate’ the conclusion, as opposed to rendering the conclusion probable, where the necessity attaches to the inferential link between premises and conclusion, and not to the conclusion itself. (A valid deductive argument can, but need not, have a necessary conclusion: ‘I am sitting’ necessitates ‘Someone is sitting,’ even though the latter proposition is only contingently true.) Equivalently, a deductive argument embodies the claim that it is impossible for all the premises to be true and the conclusion false. I say ‘embodies the claim’ because the claim might not be correct. If the claim is correct, then the argument is valid, and invalid otherwise. Since validity pertains to the form of deductive arguments as opposed to their content, we can define a valid (invalid) deductive argument as one whose form is such that it is impossible (possible) for any (some) argument of that form to have true premises and a false conclusion. Since the purport of inductive arguments is merely to probabilify, not necessitate, their conclusions, they are not rightly described as valid or invalid, but as more or less strong or weak, depending on the degree to which they render their conclusions probable.