Inductive reasoning

Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that allows for the possibility that the conclusion is false even where all of the premises are true.^[1] The premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is employed, for example, in the following argument:

All of the ice we have examined so far is cold.

Therefore, all ice is cold.

or,

The person looks uncomfortable

Therefore, the person is uncomfortable.

This definition of inductive reasoning excludes mathematical induction, which is considered to be a form of deductive reasoning.

Strong and weak induction

The words 'strong' and 'weak' are sometimes used to praise or demean the goodness of an inductive argument. The idea is that you say "this is an example of strong induction" when you would decide to believe the conclusion if presented with the premises. Alternatively, you say "that is weak induction" when your particular world view does not allow you to see that the conclusions are likely given the premises.

Strong induction

The equation, "the gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared," has allowed us to describe the rate of fall of all objects we have observed.

Therefore:

The gravitational force between two objects equals the gravitational constant times the product of the masses divided by the distance between them squared.

The conclusion of this argument is not absolutely certain even given the premise. In fact, the Newtonian science that gave rise to this equation has since been abandoned and newer experimental paradigms have shown that the conclusion is false. However, given the premise, we still feel we have good reason to accept the conclusion. So we call this argument an instance of strong induction.

Weak induction

Consider this example:

I always hang pictures on nails.

Therefore:

All pictures hang from nails.

Here, the link between the premise and the conclusion is very weak. Not only is it possible for the conclusion to be false given the premise, it is even fairly likely that the conclusion is false. Not all pictures are hung from nails; moreover, not all pictures are hung. Thus we say that this argument is an instance of weak induction.

Is induction reliable?

Inductive reasoning has been attacked for millennia by thinkers as diverse as Sextus Empiricus^[2] and Karl Popper.^[3]

The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume. Hume highlighted the fact that our everyday functioning depends on drawing uncertain conclusions from our relatively limited experiences rather than on deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, despite no guarantee that it will do so. However, Hume argued that it is impossible to justify inductive reasoning. Inductive reasoning certainly cannot be justified deductively, and so our only option is to justify it inductively. However, to justify induction inductively is circular. Therefore, it is impossible to justify induction.^[4]

However, Hume immediately argued that even were induction proved unreliable, we would have to rely on it. So he took a middle road. Rather than approach everything with severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.^[5]

Types of inductive reasoning

Generalization

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.

The proportion Q of the sample has attribute A.

Therefore:

The proportion Q of the population has attribute A.

Example

There are 20 balls in an urn, either black or white. To estimate their respective numbers you draw a sample of 4 balls and find that 3 are black, one is white. A good inductive generalisation would be: there are 15 black and 5 white balls in the urn.

How great the support is which the premises provide for the conclusion is dependent on (a) the number of individuals in the sample group compared to the number in the population; and (b) the degree to which the sample is representative of the population (which may be achieved by taking a random sample). The hasty generalization and biased sample are fallacies related to generalisation.

Statistical syllogism

A statistical syllogism proceeds from a generalization to a conclusion about an individual.

A proportion Q of population P has attribute A.

An individual X is a member of P.

Therefore:

There is a probability which corresponds to Q that X has A.

The proportion in the first premise would be something like "3/5ths of", "all", "few", etc. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".

Simple induction

Simple induction proceeds from a premise about a sample group to a conclusion about another individual.

Proportion Q of the known instances of population P has attribute A.

Individual I is another member of P.

Therefore:

There is a probability corresponding to Q that I has A.

This is a combination of a generalization and a statistical syllogism, where the conclusion of the generalization is also the first premise of the statistical syllogism.

Argument from analogy

Some philosophers believe that an argument from analogy is a kind of inductive reasoning.

An argument from analogy has the following form:

I has attributes A, B, and C

J has attributes A and B

So, J has attribute C

An analogy relies on the inference that the attributes known to be shared (the similarities) imply that C is also a shared property. The support which the premises provide for the conclusion is dependent upon the relevance and number of the similarities between I and J. The fallacy related to this process is false analogy. As with other forms of inductive argument, even the best reasoning in an argument from analogy can only make the conclusion probable given the truth of the premises, not certain.

Analogical reasoning is very frequent in common sense, science, philosophy and the humanities, but sometimes it is accepted only as an auxiliary method. A refined approach is case-based reasoning. For more information on inferences by analogy, see Juthe, 2005.

Causal inference

A causal inference draws a conclusion about a causal connection based on the conditions of the occurrence of an effect. Premises about the correlation of two things can indicate a causal relationship between them, but additional factors must be confirmed to establish the exact form of the causal relationship.

Prediction

A prediction draws a conclusion about a future individual from a past sample.

Proportion Q of observed members of group G have had attribute A.

Therefore:

There is a probability corresponding to Q that other members of group G will have attribute A when next observed.

Bayesian inference

Of the candidate systems for an inductive logic, the most influential is Bayesianism. As a logic of induction rather than a theory of belief, Bayesianism does not determine which beliefs are a priori rational, but rather determines how we should rationally change the beliefs we have when presented with evidence. We begin by committing to an (really any) hypothesis, and when faced with evidence, we adjusts the strength of our belief in that hypothesis in a precise manner using bayesian logic.

See also

• Abductive reasoning
• Analogy
• Deductive reasoning
• Explanation
• Falsifiability
• Inductive reasoning aptitude
• Inductive Logic Programming
• Inferential statistics
• Inquiry
• Logic
• Machine learning
• Mathematical induction
• Mill's Methods
• Raven paradox
• Retroduction
• Laurence Jonathan Cohen

Footnotes

References

• Herms, D. "Logical Basis of Hypothesis Testing in Scientific Research" (pdf). http://www.dartmouth.edu/~bio125/logic.Giere.pdf. 
• Kemerling, G (2001-10-27). "Causal Reasoning". http://www.philosophypages.com/lg/e14.htm. 
• Holland, JH; Holyoak KJ; Nisbett RE; Thagard PR (1989). Induction: Processes of Inference, Learning, and Discovery. Cambridge, MA, USA: MIT Press. ISBN 0262580969. 
• Holyoak, K; Morrison R (2005). The Cambridge Handbook of Thinking and Reasoning. New York: Cambridge University Press. ISBN 978-0521824170. 

External links

• Four Varieties of Inductive Argument from the Department of Philosophy, University of North Carolina at Greensboro.
• Inductive Logic from the Stanford Encyclopedia of Philosophy.
• Properties of Inductive ReasoningPDF (166 KiB), a psychological review by Evan Heit of the University of California, Merced.
• The Mind, Limber An article which employs the film The Big Lebowski to explain the value of inductive reasoning.