Inductive reasoning
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- Inductive reasoning is the complement of deductive reasoning. For other article subjects named induction, see Induction (disambiguation).
- This ice is cold.
- A billiard ball moves when struck with a cue.
- All ice is cold.
- There is no ice in the Sun.
- For every action, there is an equal and opposite reaction.
- Anything struck with a cue moves.
Strong and weak induction
Strong induction
- All observed crows are black.
- therefore
- All crows are black.''
Weak induction
- I always hang pictures on nails.
- therefore
- All pictures hang from nails.
- Teenagers are given many speeding tickets.
- therefore
- All teenagers speed.
Validity
Formal logic as most people learn it is deductive rather than inductive. Some philosophers claim to have created systems of inductive logic, but it is controversial whether a logic of induction is even possible. In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is false, but may have been thought true in Europe until the settlement of Australia. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.) In induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed. It is however possible to derive a true statement using inductive reasoning if you know the conclusion. The only way to have an efficient argument by induction is for the known conclusion to be able to be true only if an unstated external conclusion is true, from which the initial conclusion was built and has certain criteria to be met in order to be true (separate from the stated conclusion). By substitution of one conclusion for the other, you can inductively find out what evidence you need in order for your induction to be true. For example, if you have a window that opens only one way, but not the other. Assuming that you know that the only way for that to happen is that the hinges are faulty, inductively you can postulate that the only way for that window to be fixed would be to apply oil (whatever will fix the unstated conclusion). From there on you can successfully build your case. However, if your unstated conclusion is false, which can only be proven by deductive reasoning, then your whole argument by induction collapses. Thus ultimately, pure inductive reasoning does not exist.The classic philosophical treatment of the problem of induction, meaning the search for a justification for inductive reasoning, was by the Scottish philosopher David Hume. Hume highlighted the fact that our everyday reasoning depends on patterns of repeated experience rather than deductively valid arguments. For example, we believe that bread will nourish us because it has done so in the past, but this is not a guarantee that it will always do so. As Hume said, someone who insisted on sound deductive justifications for everything would starve to death.
Instead of unproductive radical skepticism about everything, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.
Induction is sometimes framed as reasoning about the future from the past, but in its broadest sense it involves reaching conclusions about unobserved things on the basis of what has been observed. Inferences about the past from present evidence – for instance, as in archaeology, count as induction. Induction could also be across space rather than time, for instance as in physical cosmology where conclusions about the whole universe are drawn from the limited perspective we are able to observe (see cosmic variance); or in economics, where national economic policy is derived from local economic performance.
Twentieth-century philosophy has approached induction very differently. Rather than a choice about what predictions to make about the future, induction can be seen as a choice of what concepts to fit to observations or of how to graph or represent a set of observed data. Nelson Goodman posed a "new riddle of induction" by inventing the property "grue" to which induction does not apply (see Grue).
Types of inductive reasoning
Sources for the examples that follow are: [(1)], [(2)], [(3)].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.
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 I is a member of P.
- therefore
- There is a probability which corresponds to Q that I has A.
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.
Argument from analogy
An (inductive) analogy proceeds from known similarities between two things to a conclusion about an additional attribute common to both things:- P is similar to Q.
- P has attribute A.
- therefore
- Q has attribute A.
Causal inference
An 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.
- 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 proportion of group
Argument from authority
An argument from authority draws a conclusion about the truth of a statement based on the proportion of true propositions provided by a source. It has the same form as a prediction.
- Proportion Q of the claims of authority A have been true.
- therefore
- There is a probability corresponding to Q that this claim of A is true.
- All observed claims from websites about logic are true.
- Information X came from a website about logic.
- therefore
- Information X is likely to be true.
Bayesian inference
Of the candidate systems for an inductive logic, the most influential is Bayesianism. This uses probability theory as the framework for induction. Given new evidence, Bayes' theorem is used to evaluate how much the strength of a belief in a hypothesis should change.There is debate around what informs the original degree of belief. Objective Bayesians seek an objective value for the degree of probability of a hypothesis being correct and so do not avoid the philosophical criticisms of objectivism. Subjective Bayesians hold that prior probabilities represent subjective degrees of belief, but that the repeated application of Bayes' theorem leads to a high degree of agreement on the posterior probability. They therefore fail to provide an objective standard for choosing between conflicting hypotheses. The theorem can be used to produce a rational justification for a belief in some hypothesis, but at the expense of rejecting objectivism. Such a scheme cannot be used, for instance, to decide objectively between conflicting scientific paradigms.
Edwin Jaynes, an outspoken physicist and Bayesian, argued that "subjective" elements are present in all inference, for instance in choosing axioms for deductive inference; in choosing initial degrees of belief or prior probabilities; or in choosing likelihoods. He thus sought principles for assigning probabilities from qualitative knowledge. Maximum entropy – a generalization of the principle of indifference – and transformation groups are the two tools he produced. Both attempt to alleviate the subjectivity of probability assignment in specific situations by converting knowledge of features such as a situation's symmetry into unambiguous choices for probability distributions.
Cox's theorem, which derives probability from a set of logical constraints on a system of inductive reasoning, prompts Bayesians to call their system an inductive logic.
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.