| | I'm surprised no one here has mentioned H.W.B. Joseph's discussion on the presuppositions of induction in his An Introduction to Logic, in which he explains how induction rests on the law of identity.
Hume held that causation involves nothing more than spatial contiguity and temporal continuity. In cases in which A is said to cause B, he held that we observe no necessity between A and B. All we observe is that B follows A in a certain sequence and relationship. But causation involves more than that; it involves an actual production of the effect. To say that A "causes" B is to say that given A, B must happen. Given a a rock of a certain size thrown against a window at a certain speed, the window must break! Hume disagreed. We see the rock thrown against the window, he said, and we see the window breaking, but we don't see any necessity in this picture; we don't see that the action of the rock necessitates the breaking of the window.
Joseph answers Hume as follows: "[I]f a Thing is to have any determinate nature and character at all, there must be uniformity of action in different things of that character, or of the same thing on different like occasions. If a thing a under conditions c produces a change x in a subject s -- if, for example, light of certain wavelengths, passing through the lens of a camera, produces a certain chemical change (which we call the taking of a photograph of Mount Everest) upon a photographic film -- the way in which it acts must be regarded as a partial expression of what it is. It could only act differently, if it were different. As long therefore as it is a, and stands related under conditions c to a subject that is s, no other effect than x can be produced; and to say that the same thing acting on the same thing under the same conditions may yet produce a different effect is to say that a thing need not be what it is. But this is in flat conflict with the Law of Identity.
"A thing, to be at all, must be something, and can only be what it is. To assert a causal connexion between a and x implies that a acts as it does because it is what it is; because, in fact, it is a. So long therefore as it is a, it must act thus; and to assert that it may act otherwise on a subsequent occasion is to assert that what is a is something else than the a that it is declared to be." (pages 407-408)
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