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Inductive probability

The Inductive probability of an argument is a measure of how likely the conclusion is to be true given that the premises of that argument are true. It is a notion that applies to all arguments, whether they be deductively valid or not, inductively strong or weak.

Inductive probability measures how much support the premises give the conclusion. This is a relation that holds between premises and conclusions independently of their truth or falsity, much like deductive validity.

An argument is inductively strong iff it is unlikely that the conclusion is false given that the premises are true. An argument is said to be inductively weak iff it is neither deductively valid nor inductively strong.

Inductive probabilities fall on a spectrum from 0% to 100%, where the number measures how likely the conclusion is to be true if the premises are true. Deductively valid arguments have inductive probability 100%, while those arguments where the premises guarantee that the conclusion is false receive inductive probability 0% (consider a deductively valid argument and negate its conclusion). In between 0% and 100% lie various inductively strong and weak arguments: those arguments that don't guarantee that their conclusions are true or false.

This is an informal notion, and the claims above are simply intuitions, but these claims can be rigorously justified by using Bayes' Theorem as a formal model of arguments.

Here are some examples of inductive arguments of varying degrees of inductive probability:

All students are hard-working.
Miguel is a student.
Therefore, Miguel is hard-working.

This argument is deductively valid. Its inductive probability is 100%. Here is another argument:

Most students are hard-working.
Miguel is a student.
Therefore, Miguel is hard-working.

Given that we have no other knowledge about Miguel, this argument has a high inductive probability (though at this point we have no way of assigning a precise numerical value). Here is another argument:

Only a few students are hard-working.
Miguel is a student.
Therefore, Miguel is hard-working.

This argument has a lower inductive probability. Since Miguel is only one of many students the likelihood of him being one of the few hard working ones is low. Here is another argument:

There are no hard-working students.
Miguel is a student.
Therefore, Miguel is hard-working.

The conclusion of this argument must be false if the premises are true and so has the lowest possible inductive probability: 0%.