# 5.1: Prelude to Inductive Arguments

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Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true. Instead of being valid or invalid, inductive arguments are either *strong* or *weak*, which describes how *probable* it is that the conclusion is true. Another crucial difference is that deductive certainty is impossible in non-axiomatic systems, such as reality, leaving inductive reasoning as the primary route to (probabilistic) knowledge of such systems.

Given that "if *A* is true then that would cause *B*, *C*, and *D* to be true", an example of deduction would be "*A* is true therefore we can deduce that *B*, *C*, and *D* are true". An example of induction would be "*B*, *C*, and *D* are observed to be true therefore *A* might be true". *A* is a reasonable explanation for *B*, *C*, and *D* being true.

For example:

A large enough asteroid impact would create a very large crater and cause a severe impact winter that could drive the non-avian dinosaurs to extinction.

We observe that there is a very large crater in the Gulf of Mexico dating to very near the time of the extinction of the non-avian dinosaurs

Therefore it is possible that this impact could explain why the non-avian dinosaurs became extinct.

Note however that this is not necessarily the case. Other events also coincide with the extinction of the non-avian dinosaurs. For example, the Deccan Traps in India.

A classical example of an *incorrect* inductive argument was presented by John Vickers:

All of the swans we have seen are white.

Therefore, we know that all swans are white.

The correct conclusion would be, "We expect that all swans are white".

The definition of *inductive* reasoning described in this article excludes mathematical induction, which is a form of *deductive* reasoning that is used to strictly prove properties of recursively defined sets. The deductive nature of mathematical induction is based on the non-finite number of cases involved when using mathematical induction, in contrast with the finite number of cases involved in an enumerative induction procedure with a finite number of cases like proof by exhaustion. Both mathematical induction and proof by exhaustion are examples of complete induction. Complete induction is a type of masked deductive reasoning.

An argument is deductive when the conclusion is necessary given the premises. That is, the conclusion cannot be false if the premises are true.

If a deductive conclusion follows duly from its premises it is valid; otherwise it is invalid (that an argument is invalid is not to say it is false. It may have a true conclusion, just not on account of the premises). An examination of the above examples will show that the relationship between premises and conclusion is such that the truth of the conclusion is already implicit in the premises. Bachelors are unmarried because we *say* they are; we have defined them so. Socrates is mortal because we have included him in a set of beings that are mortal.

For inductive reasoning the premises or prior data provide support for the conclusion, but they do not guarantee it. The result is a conclusion having, it is often said, a “degree of certainty.” The phrase is not optimal since certainty is absolute and does not come in degrees; what is really meant is degrees approaching certainty. Succinctly put: deduction is about certainty/necessity; induction is about *probability*. This is the best way to understand and remember the difference between inductive vs. deductive reasoning. Any single assertion will answer to one of these two criteria. (There is also modal logic, which deals with the distinction between the necessary and the *possible* in a way not concerned with probabilities among things deemed possible.)

Inductive reasoning (as opposed to *deductive* reasoning or *abductive* reasoning) is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be *probable*, based upon the evidence given.

The philosophical definition of inductive reasoning is more nuanced than simple progression from particular/individual instances to broader generalizations. Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms, discussed below).