Frank Ramsey's treatment of induction starts from a general account of inference. He argues that premises and conclusion alone do not fully specify an inference. It also needs the rule by which the conclusion is drawn. This holds for deductive and inductive arguments alike, though the rule differs in kind between them. Deductive rules preserve truth. Inductive rules do not. Inductive rules are judged instead by how reliably they extend belief from observed cases to unobserved ones. This three-part account of inference sets the terms within which Ramsey's treatment of induction, probability and laws should be read.
Quote from Ramsey: ‘Logic as the science of argument and inference is traditionally and rightly divided into deductive and inductive; but the difference and relation between these two divisions of the subject can be conceived in extremely different ways.’
Deductive argument
Quote from Ramsey: ‘formal deduction does not increase our knowledge, but only brings out clearly what we already know in another form; and that we are bound to accept its validity on pain of being inconsistent with ourselves’
Quote from Ramsey: ‘deduction on the other hand is merely a method of arranging our knowledge and eliminating inconsistencies or contradictions.’
Inductive argument
Quote from Ramsey: ‘it is impossible to represent it [inductive argument] as resembling a deductive argument and merely weaker in degree; it is absurd to say that the sense of the conclusion is partially contained in that of the premisses. We could accept the premisses and utterly reject the conclusion without any sort of inconsistency or contradiction.’
Ramsey holds:
‘whenever I make an inference, I do so according to some rule or habit. An inference is not completely given when we are given the premiss and conclusion; we require also to be given the relation between them in virtue of which the inference is made.’
‘the rule of the inference determines for us a range to which the frequency theory can be applied.’
An inference is therefore a three-part structure:
For a deductive argument A valid deductive rule is truth-preserving: apply it to true premisses and the conclusion cannot be false.
Common examples of valid deductive rules include
For an inductive argument The inductive rule is not truth-preserving. It is judged, on Ramsey's account, by the proportion of cases in which it leads from truth to truth, which is below certainty.
The following are illustrative examples of the kinds of inductive rules Ramsey has in mind. They are not intended as an exhaustive or mutually exclusive classification of inductive inference.
Variable-hypothetical induction (preferred to "Universal generalization")
Having found every observed φ to be ψ, adopt the rule:
o If I encounter a φ, I shall expect it to be a ψ.
Applying this rule, expect with some degree of confidence that the next observed φ will be ψ.
This rule is often expressed linguistically by a universal sentence such as "All φ are ψ". According to Ramsey, however, the underlying attitude is not belief in an abstract universal proposition but a rule generating a singular belief or expectation.
Statistical induction
Analogy
Causal induction
Having repeatedly observed φ followed by ψ under appropriate conditions, adopt the rule.
o If φ occurs, expect ψ.
Applying this rule, expect ψ whenever φ occurs.
This rule may be expressed linguistically as a causal law (e.g. "If φ occurs, ψ occurs") or, less precisely, as the claim that φ causes ψ.
For deduction, the relation between premiss and conclusion is a formal rule. The conclusion is already contained in the premisses, so anyone who grants the premisses and denies the conclusion contradicts himself. The rule preserves truth: from true premisses it cannot lead to a false conclusion, and it adds nothing already held.
For induction, the relation is not a rule of this kind. An inductive rule of inference is a way of forming expectations and extending belief beyond the evidence to come. No contradiction follows from accepting the premisses and rejecting the conclusion. Ramsey judges such inductive rules not by deductive validity but by their reliability, that is, by the proportion of cases in which applying a rule of the kind leads from true premisses to true beliefs. An inductive rule is the more reliable the higher that proportion, and the degree of confidence it is best to place in its conclusions matches the proportion in which it leads to truth.
Deductive inference is evaluated by truth-preservation: a valid rule guarantees that true premisses cannot lead to a false conclusion.
Perception, memory and induction, by contrast, are evaluated by their truth-conduciveness. For Ramsey these are methods of reasoning whose epistemic merit lies not in preserving truth but in tending, on the whole, to produce true beliefs. Perception yields beliefs about what is observed; memory retains beliefs once formed, and so yields beliefs about the past; and induction extends belief beyond observation to the unobserved, future cases included. Their reliability consists in the extent to which they lead to truth. A rule is better if the beliefs it produces are for the most part true, or more often true than those produced by alternative rules.
Hence it is unsurprising that induction is not deductively valid. Deductive validity is the wrong standard by which to judge an inductive rule whose function is not to preserve truth by logical form but to extend belief beyond the observed in a way that tends, on the whole, to produce true beliefs.
The truth-conduciveness standard does not escape circularity, for assessing the reliability of induction must itself rely on induction. Ramsey maintains that nothing vicious lies in this circle, because the same holds of perception and memory. The accuracy of memory, for instance, can be judged only by using memory itself, since any experiment run to test it is worthless unless the experiment is remembered.