Logic

Hello, I would like some clarification on deduction and induction. I have heard scientists claim to use deductive reasoning. In each case, the scientists use a hypothetical syllogism, such as modus ponens. I am confused about this because I noticed inductive arguments can be made into deductive form if conditionals are used. For example, consider this case: If an argument contains a conditional statement, then it is deductive reasoning. This inductive argument(X)can be re-worded to contain a conditional statement on the spot when asked. Therefore this inductive argument (X) is deductive reasoning. According to the example given, all arguments are deductive! Some help and clarification please? Are all arguments with at least one If . . . then . . . premise deductive by definition alone? Should inductive arguments be inductive no matter what form because the conclusions are not guaranteed from the premises?

Comments

Ordinary usage of these terms is inconsistent, and so, to some extent, is the technical usage. Sherlock Holmes is said to have solved crimes through "deduction." A philosopher would say, no, his methods were non-deductive. "Inductive" is often, in philosophy, opposed to "deductive", yet the kind of proof that in mathematics is called an "inductive" proof, is, by standard philosophical definitions, deductive. So no wonder you're confused.

Nobody owns these terms, so no one can rightfully say that anyone else's usage is objectively correct or incorrect. But let me give you at least one way of understanding the terms, and then an explanation in terms of that understanding for all the weirdnesses.

I tell my students in Intro Philosophy that the difference between "deductive" and "non-deductive" arguments has to do with the way the premises of the argument are supposed to support the conclusion of the argument. In a deductive argument, the author of the argument is claiming that the premises support the conclusion with logical necessity -- that is, the author of the argument is saying, in effect, "if the premises of this argument are (or were) true, then the conclusion has to be (or would have to be) true;" or, equivalently, "it's impossible for the premises to be true and the conclusion to be false at the same time." If a deductive argument is well-constructed, then it really will be the case that if the premises were true, then the conclusion would have to be true as well, we call the argument "valid." With a non-deductive argument, the claim is different: it's that the premises, if true, make it probable that the conclusion is true. The author of a non-deductive argument is aware that the premises don't logically entail the conclusion, that it's logically possible for the premises of the argument to be true and the conclusion false, but that's OK. All the author of a non-deductive argument is claiming is that the premises give you good rational reason to accept the conlusion.

Now there is a large catalog of argument forms of both the deductive and the non-deductive type. Modus ponens and hypothetical syllogism are two different forms of valid deductive arguments. Modus ponens is the form:

p
If p, then q
Therefore, q.

Hypothetical syllogism is the form:

If p, then q.
If q, then r.
Therfore, if p, then r.

There are approximately one kazillion others.

There are also different non-deductive forms: inductive arguments involve premises about particulars, for example:

Swan1, which I have observed, is white.
Swan2, which I have observed, is white
....

N. SwanN, which I have observed, is white.

N+ 1. Probably, swanN+1, which I have not observed, is white.

Another kind of inductive argument uses the same premises, but has as its conclusion: "Probably, all swans are white."

Because inductive arguments sometimes involve moving from particular cases to general conclusions, the form of mathematical proof that moves from a fact about one number, say, that it has property P, and a fact of the form that if some arbitrary number has P, then so does its successor, to the conclusion that all numbers have P, it seems to fit the pattern of inductive arguments. But since inductive arguments in math involve deductively valid steps, rather than probabilistic steps, "mathematical induction" counts, by the definition above, as a form of deduction

Sherlock Holmes used a form of reasoning that we in philosophy call "abductive" or "inference to the best explanation." This is a very common form of non-deductive inference. It goes like this:

Phenomenon X occurred.
The best explanation of why X occurred is Y
Therefore, probably Y.

So the astronomer Leverrier noticed a wobble in the orbit of Uranus. He argued that if there were a planet of such-and such size at such-and-such distance from Uranus, then that would explain why the wobble occurred. Therefore, probably there is such a planet. And indeed, it turned out that there was: Neptune! Holmes does essentially the same thing: he lists all the facts, considers what the best explanation of those facts would be, and concludes that the best explanation would be if the butler (or whoever) did it. (Elementary, my dear Watson.) This is probably only called "deduction" because of the common use of the word "deduce" to refer to any kind of rigorous reasoning.

As I said, no one owns these words. The characterizations I gave are useful ones, I think, and allow us to express important distinctions among kinds of reasoning. That's a good reason, I think, to use the words "deduction" and "induction" in the ways I explained about.

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