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What is the precise difference or relation between these terms in logic: Inference, Reasoning, Deduction, and Induction?

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This does not appear to be a mathematical question. There are not formal definitions of "inference" and "reasoning" in mathematical logic in the sense that the question seems to intend. –  Carl Mummert Sep 12 '10 at 12:49
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Community wiki then? I see no reason why we should dismiss this question outright -- we're not held to MO's standards, and soft questions like this can be as important to people learning math as help with specific problems. And there have been questions about books, philosophy, notation, etc. recently. –  Paul VanKoughnett Oct 11 '10 at 16:35
    
I've voted to close this as I don't consider it a question about mathematics. I'd advice though to post it somewhere else (I can't say where), perhaps in some logic forum...? –  DonAntonio Sep 10 '12 at 18:13

3 Answers 3

Morteza, I'd guess you'll get different answers from everyone you ask. Here is the way I see it (which is, by necessity, somewhat circular):

Inference, in the narrowest sense, is a single step in a deductive chain. If I know $P\rightarrow Q$ and $\neg Q$, I can infer $\neg P$ from modus tollens (which itself is sometimes called a "rule of inference"). More loosely, we can call any conclusion from premises an inference, even if it's not properly deductive (i.e. I look outside, see a clear sky, and infer that it's not raining).

Reasoning is the mental process of logic -- what goes on inside my head when I use deduction. Though a lot of people would probably use it as a synonym of "deduction," I'd say the differences are far more important. When you reason, you skip steps, explore multiple pathways, and use your intuition, which are all things unavailable to, say, a computer.

Deduction is the formal process of logic, and an inference is deductive when it follows from an axiom or logical rule. This is the makeup of most mathematical proofs.

Induction has two meanings. The first is some sort of mathematical induction (strong, weak, or transfinite), all of which are based on the idea of an infinite deductive chain that can be collapsed into a few steps. For instance, in normal (weak) induction, you prove the proposition $P(0)$, and you prove that $\forall n:P(n)\rightarrow P(n+1)$. Modus ponens then gives you $P(1)$, then $P(2)$ using $P(1)$, then $P(3)$ using $P(2)$, and so forth. In all but the most intuitionist systems of arithmetic/set theory, induction is either provable or taken as an axiom, so its use is properly a deductive inference!

However, there is another, more abstract meaning of induction that contrasts properly with deduction. The logician Charles Sanders Peirce had an analogy with bags of beans that I find compelling.

Deduction has you applying a generalization to a specific case to get a result. So you know that all the beans in my bag are white (the generalization), and you take a bean from my bag (the case): then that bean must be white (the result).

Induction has you taking a case and a result and generalizing them. You take a bean from my bag (the case), and you see it is white (the result): therefore, all the beans in my bag are white (generalization). In this rudimentary form, induction looks like guesswork, and nobody would trust this logic. But if you took five or ten beans, and they were all white, your confidence in the generalization would increase. In fact, everybody uses this form of induction many times a day: you expect the sun to rise, your bed to be in the same place, your food to cook properly, etc. just because they've all happened many times before. This kind of inference is not deductive at all, yet we still trust it.

Abduction was the third of Peirce's forms of inference. It is the most suspect of the three, and is best thought of as a sort of "reverse implication." The idea is to take a generalization and a result and observe that a case is probably true because it implies the result. So you have a white bean (the result), and you know that all the beans in my bag are white (the generalization): thus, this bean must be from my bag, for if it were, it would have to be white. This seems like the sort of reasoning you see a lot on detective shows.

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If your question is about philosophical matters of logic, then I think that you may find very interesting the following paper authored by a leading logician (see also other papers in the same handbook).

Wilfrid Hodges, The scope and limits of logic
pp. 41-63 in Handbook of the Philosophy of Science: Philosophy of Logic,
ed. Dale Jacquette, 2007, Elsevier, Amsterdam

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The difference between deductive argument and inductive argument is as follow, Deductive: the conclusion of a deductive argument follows necessarily from the premisses( it follow with certainty. It is this case that we say a deductive argument is EXPLICATION(i.e it does not need any other premiss, "more" or "less" to support its already attained conclusion). Inductive: the conclusion follows from the premiss with probability. It is AMPLIATIVE(i.e the conclusion gives room for more premiss to enhance the strongness of the argument).

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