What is a rule of inductive inference? I'm not looking for any examples, but for definitions - what makes the logical form of an inductive argument a rule of inductive inference?
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And what makes the logical form of a deductive argument a rule of deductive inference in the sense that you mean? Typically, rules of deductive inference are just listed rather than defining something that "makes" them that. So what form of the answer are you looking for? Also do you mean "induction" in the narrow sense or broad one that includes abduction?– ConifoldCommented Apr 24, 2023 at 17:52
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Thanks for your comment! Perhaps like this: An argument is deductively valid if the premises entail the conclusion, and the logical form of an argument is a rule of deductive inference if and only if every argument of that form is deductively valid. I doubt an analogue definition for rules of inductive inferences (relying inductive strength) would be feasible ... (Induction, btw., understood in the broad sense.)– TurturCommented Apr 24, 2023 at 18:41
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You can make a definition like that, but it will be even less useful. An argument is informally valid when the truth of premises makes the conclusion plausible, and it is a rule of inductive inference if and only if every argument of that form is informally valid. Without spelling out "entail" or "make plausible" these "definitions" are just rephrasings. And what they mean is still given by some listed forms, not "defined". Moreover, for induction the listed forms are typically domain specific, see Norton, Induction– ConifoldCommented Apr 24, 2023 at 20:23
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Inductive reasoning, in the broad sense of the term, is usually understood to differ from deductive reasoning in that the premises of a cogent argument support the conclusion without entailing it. In practice, it is difficult to spell out just what this involves.
Some accounts see an inductively strong argument as one of partial entailment. The problem here is to explain what a degree of entailment amounts to. Many take a probabilistic turn and claim that the truth of the premises of an inductively strong argument renders the conclusion probable, or at least more probable than it would be without them. This makes sense in the context of an epistemic understanding of probability as a degree of rational belief.
But we are still left with the problem of whether there are any general formal rules that would count as rules of inductive inference. Rudolf Carnap attempted to set out a general formal theory of inductive logic, but it has been widely criticised. Many modern approaches treat inductive reasoning as something that can only proceed piecemeal for particular subject areas by reference to background facts and assumptions. Others appeal to some general heuristics but do not attempt to codify these as formal rules.
Bayesian conditioning is often used in practice as a way to express the probability of a hypothesis given some data. But it also has its issues. Strictly speaking, Bayesianism does not tell us the probability of a proposition given the evidence, but only how to update a prior probability to a posterior one. It is a separate question as to where our priors come from. Also, often we do not know the probability of the evidence term, so the Bayesian updating formula is frequently used in its ratio form and tells us only whether some given evidence tends to support one hypothesis relative to another rival one. This means it is not telling us in any absolute sense how probable a hypothesis is given the evidence.
There are some algorithms for generating probabilities and rules from data: everything from Laplace's Rule of Succession to decision tree algorithms such as C4.5 to approximations to Solomonoff induction. But these are often quite specific in terms of their range of applicability and are sometimes not much more than ways to adjust parameters in a model. Beyond that, there are artificial intelligence applications that use machine learning techniques to generate hypotheses and test them against data. These can be described as inductive, but they do not simplify down into straightforward rules of inductive inference.
There is a longer account of some of this material in the Stanford Encyclopedia articles on Inductive Logic and on Confirmation.
