2.7: Inductive Arguments

We will study inductive arguments in detail in a later chapter, so we will just briefly consider them here. We talk about inductive arguments in terms of strength and weakness.

An argument is inductively strong just in case:

1. It is not deductively valid, and
2. If all its premises are true, then there is a high probability that its conclusion will be true as well.

The second item is the important one. The only point of the first item is to ensure that no argument is both deductively valid and inductively strong (this will make things easier for us in later chapters).

There are two important ways in which inductive strength differs from deductive validity:

1. Unlike deductive validity, inductive strength comes in degrees.
2. In a deductively valid argument, the conclusion does not contain any information that was not already present in the premises. By contrast, in an inductively strong argument, the conclusion contains new information.

A deductively valid argument with all true premises must have a true conclusion. By contrast, an inductively strong argument with true premises provides good, but not conclusive, grounds for its conclusion.

Since we have defined things such that inductively strong arguments are not deductively valid, we can think of arguments as arranged along a continuum of descending strength:

1. Deductively valid
2. Deductively invalid
   1. Inductively strong
   2. Inductively weak
   3. Worthless

General and Particular

It’s not unusual (particularly if you’re trying to learn logic using YouTube) to see deductively valid arguments described as proceeding from the general to the specific, and inductively strong arguments proceeding from the specific to the general. This is not a good way to think about the two sorts of arguments, and notions of generality and specificity are completely irrelevant to the two notions. Here is a deductively valid argument that goes from more specific premises to a more general conclusion:

1. 3 is a prime number.
2. 5 is a prime number.

Therefore, all odd numbers between 2 and 6 are prime.

And here is an inductively strong argument that goes from a more general premise to a more specific conclusion.

1. All the crows observed thus far have been black.

Hence, the next crow to be observed will be black.

Deductively Valid and Inductively Strong Reasons

Sometimes it is more natural to speak of reasons, rather than arguments. A group of sentences provides deductively valid reasons for a conclusion just in case it is impossible for all of them to be true and the conclusion false. Valid reasons have this feature because there is no information in the conclusion that was not already contained in the reasons themselves

Inductively strong reasons: A group of sentences provides inductively strong reasons for a conclusion just in case it is unlikely for all of them to be true and the conclusion false. If a group of inductively strong reasons for a conclusion are true, then there is a good chance that the conclusion will be true as well, but there is still some possibility that it will be false. Inductively strong reasons are not always truth preserving. There is an inductive leap from the reasons to the conclusion. Inductive support comes in varying degrees; the stronger the inductive reasons, the less risky the inductive leap.