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.

Comments

[Jason Pratt]

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. Oy. Methinks this means there's something whack about the given definition of 'valid', then. So, according to your illustration, would this be a valid deductive argument? Or not? -- Jason is a metaphysician; Jason only has a communications degree; ergo all chess players are fat. A and B are premises, the truth of which I can vouch for being discoverable. C is another assertion, claimed to be a conclusion from A and B by 'therefore'. Is the lack of topicality a problem? Redefine C = ergo all metaphysicians only have communications degrees. Is _this_, then, a valid deductive argument? The premises are demonstrably true; the grammatic form seems identical with your example (barring the lack of a contradiction); but the given conclusion is demonstrably false. (B.B.Warfield, former dean of Princeton, held a doctorate in metaphysics if I recall correctly; anyway, he certainly did not have a degree in communications.) Have I misunderstood the gist of your definition of valid, then? ("Necessarily, if all the premises are true, then the conclusion is true. [...] Equivalently, a deductive argument embodies the claim that it is impossible for all the premises to be true and the conclusion false. [...] Since validity pertains to the form of deductive arguments as opposed to their content, we can define a valid... deductive argument as one whose form is such that it is impossible... for any... argument of that form to have true premises and a false conclusion.") I agree, of course, that a deduction aims at certainty, while an induction (to coin a word) aims at probability estimation (often intuitively rather than mathematically.) I also agree that a deduction is an inference formed such that if its premises are true, then its conclusion must be true; and that a question of validity concerns the legitimacy of the form, not the truth of the premises.

[Bill Vallicella]

But your presmises do not contradict each other. It is a standard result that anything follows from a contradiction. 'Valid' is a technical term having precisely the meaning it is defined as having. It does not mean good, or probative, etc.

[Jason Pratt]

My primary point was about the technical definition (restated several ways) you were giving to the meaning of 'valid'. I don't disagree with the definitions so far as they go; but they don't seem to go far enough. Either that, or they do not in fact entail that _any_ proposition at all _follows from_ two contradictions. (I will withdraw the example I gave, since in hindsight there is no point claiming it has a similar form if the form is not directly contradictory in the fashion of your own example. My bad.) Let me grant (as I am quite willing to do) the three stages of your working out of the implications of the definition of deductive validity. For your chess-player argument to stand as validly deductive, it must meet the criteria of all three formulations of the validity definition. 1.) "Necessarily, if all the premises are true, then the conclusion is true." Well, both premises cannot be true simultaneously; and the ergoed proposition is demonstrably false. So I suppose this fits well enough with a contrapositive (?) affirmation of this definition: necessarily, if not-all the premises are true, then the conclusion is not-true. But how can this mean that _any_ proposition at all _follows from_ the premises? Would the _true_ (not-not-true) proposition "Jason is a metaphysician" follow from those two premises? If John is in fact a chess player (but not-fat, therefore John is not a fat chess player), would that true proposition follow from those premises? And how can it be legitimately said that anything at all (true _or_ false) 'follows from' contradictory premises? They cancel each other out; wouldn't this mean that putting them together doesn't 'lead' anywhere at all? As far as I can tell, the false proposition _doesn't_ follow, even in its falsity, from the combined premises (i.e. as a conclusion). Necessarily, if not-all the premises are true, no conclusion follows at all; we judge over and above the argument, using _another_ argument, that the claim of the truth of the ergoed proposition must be false, _whatever_ that proposition itself might be, even if the proposition (taken by itself) is true (demonstrably or otherwise). What truth-claim must we be judging against, then (with our own argument, tacitly or explicitly)? That the ergoed proposition _follows from_ the premises. Even if this could be overcome, however (and I don't think it can), it should be abundantly clear that according to definition #1, not anything at all can follow validly from a false combination of premises. A true proposition, for instance, cannot follow from them. 2.) "Equivalently [to #1], a deductive argument embodies the claim that it is impossible for all the premises to be true and the conclusion false." The same problems rise here. Equivalently, a deductive argument embodies the claim that it is impossible for not-all the premises to be true and the conclusion not-false (i.e. true). If it impossible for a true proposition to follow (i.e. as a conclusion) from a false set of premises, then clearly not just any proposition follows validly from false premises. (Plus the problem of claiming that anything can follow at all from premises don't give ground to proceed in the first place.) 3.) "Since validity pertains to the form of deductive arguments as opposed to their content, we can define a valid... deductive argument as one whose form is such that it is impossible... for any... argument of that form to have true premises and a false conclusion." Once again, equivalently, it is impossible for any argument of a deductively valid form to have false premises and a true conclusion. Consequently, if the premises are judged (by overarching argument) to be false, then not just _any_ proposition at all could follow from them. Only a false proposition could validly follow from them. (Plus the problem of claiming anything at all can follow from untrue grounds.) It looks an awful lot like the question of an argument's validity or invalidity starts with the premises used as grounds; and ends there, too, if the grounds are judged (by overarching analysis) to be false (either altogether or in combination). If the grounds are false, then we should probably not say that the argument is nonetheless valid; nor strictly that the argument is invalid. Non-valid perhaps? Or maybe we should say that despite appearances there is no real argument being made at all. Two premises do not of themselves comprise an argument; if they fail to be establishable either separately or in cohesion, the most proper thing to do is to ignore the purported conclusion as not being arrivable at from the premises, whether the proposition is itself true or false. (This would be fairness to the proposition and its adherents, preventing us from judging against it on false grounds ourselves.) And an 'argument' where the proposed conclusion cannot be arrived at in any way from the premises, should not be considered valid. (Though admittedly maybe not 'invalid' either.) Consequently, your fat chess player argument should not be considered deductively valid, even though a demonstrably false proposition (much less any proposition at all) is ergoed to it.