Induction or inductive reasoning, sometimes called inductive logic, is reasoning which takes us "beyond the confines of our current evidence or knowledge to conclusions about the unknown." The premises of an inductive argument support the conclusion but do not entail it; i.e. they do not ensure its truth. Induction is used to ascribe properties or relations to types based on an observation instance (i.e., on a number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns. Induction is employed, for example, in using specific propositions such as:
This ice is cold. (or: All ice I have ever touched was cold.)
This billiard ball moves when struck with a cue. (or: Of one hundred billiard balls struck with a cue, all of them moved.)
...to infer general propositions such as:
All ice is cold.
All billiard balls move when struck with a cue.
Another example would be:
3+5=8 and eight is an even number. Therefore, an odd number added to another odd number will result in an even number.
Inductive reasoning has been attacked several times. Historically, David Hume denied its logical admissibility. Sextus Empiricus questioned how the truth of the Universals can be established by examining some of the particulars. Examining all the particulars is difficult as they are infinite in number.  During the twentieth century, thinkers such as Karl Popper and David Miller have disputed the existence, necessity and validity of any inductive reasoning, including probabilistic (Bayesian) reasoning  . Some say scientists still rely on induction but Popper and Miller dispute this: Scientists cannot rely on induction simply because it does not exist.
Note that mathematical induction is not a form of inductive reasoning. While mathematical induction maybe inspired by the non-base cases, the formulation of a base case firmly establishes it as a form of deductive reasoning.
All observed crows are black.
All crows are black.
This exemplifies the nature of induction: inducing the universal from the particular. However, the conclusion is not certain. Unless we can systematically falsify the possibility of crows of another colour, the statement (conclusion) may actually be false.
For example, one could examine the bird's genome and learn whether it's capable of producing a differently coloured bird. In doing so, we could discover that albinism is possible, resulting in light-coloured crows. Even if you change the definition of "crow" to require blackness, the original question of the colour possibilities for a bird of that species would stand, only semantically hidden.
A strong induction is thus an argument in which the truth of the premises would make the truth of the conclusion probable, but not necessary.
I always hang pictures on nails.
All pictures hang from nails.
Assuming the first statement to be true, this example is built on the certainty that "I always hang pictures on nails" leading to the generalisation that "All pictures hang from nails". However, the link between the premise and the inductive conclusion is weak. No reason exists to believe that just because one person hangs pictures on nails that there are no other ways for pictures to be hung, or that other people cannot do other things with pictures. Indeed, not all pictures are hung from nails; moreover, not all pictures are hung. The conclusion cannot be strongly inductively made from the premise. Using other knowledge we can easily see that this example of induction would lead us to a clearly false conclusion. Conclusions drawn in this manner are usually overgeneralisations.
Many speeding tickets are given to teenagers.
All teenagers drive fast.
In this example, the premise is built upon a certainty; however, it is not one that leads to the conclusion. Not every teenager observed has been given a speeding ticket. In other words, unlike "The sun rises every morning", there are already plenty of examples of teenagers not being given speeding tickets. Therefore the conclusion drawn can easily be true or false, and the inductive logic does not give us a strong conclusion. In both of these examples of weak induction, the logical means of connecting the premise and conclusion (with the word "therefore") are faulty, and do not give us a strong inductively reasoned statement.
See main article: Problem of induction.
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 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 or New Zealand, when Black Swans were discovered. 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, 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, inductive reasoning is not reliable.
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.
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 as a prediction about the future does not apply.
The proportion Q of the sample has attribute A.
The proportion Q of the population has attribute A.
How great the support 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 randomness of the sample. The hasty generalisation and biased sample are fallacies related to generalisation.
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.
There is a probability which corresponds to Q that I has A.
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.
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.
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.
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.
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.
There is a probability corresponding to Q that other members of group G will have attribute A when next observed.
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.
Based on an analysis of measurement theory (specifically the axiomatic work of Krantz-Luce-Suppes-Tversky), Henry E. Kyburg, Jr. produced a novel account of how error and predictiveness could be mediated by epistemological probability. It explains how one can adopt a rule, such as PV=nRT, even though the new universal generalization produces higher error rates on the measurement of P, V, and T. It remains the most detailed procedural account of induction, in the sense of scientific theory-formation.