Inductive is also used to describe the scientific processes of electric and magnetic induction or things that function based on them. Inductive is also used as a synonym for the word introductory. Example: When police use fingerprints as evidence of a crime, they are using inductive reasoning to conclude who the likely criminal is.
The first records of inductive come from the early s. In science and life, people often use both inductive and deductive reasoning to answer questions about the world around them. Formal inductive reasoning is complex, and a theory reached inductively should be tested to see if it is correct or makes sense. Police detectives, for example, often make theories about suspects based on inductive reasoning using crime scene evidence or witness interviews.
Faulty inductive logic can result in stereotypes. People can wrongly conclude that their observations of one person or a small group of people are true about every person of that group.
What are some other forms related to inductive? Inductive is most often used in the context of reasoning or logic. Challenge your beliefs! Check your numbers!
Ask questions! One may be able to get a better handle on what inductive support functions really are after one sees how the inductive logic that draws on them is supposed to work. One of the most important applications of an inductive logic is its treatment of the evidential evaluation of scientific hypotheses.
The logic should capture the structure of evidential support for all sorts of scientific hypotheses, ranging from simple diagnostic claims e. This section will show how evidential support functions a. Bayesian confirmation functions represent the evidential evaluation of scientific hypotheses and theories.
This logic is essentially comparative. The evaluation of a hypothesis depends on how strongly evidence supports it over alternative hypotheses. The collection of alternatives may be very simple, e. Whenever two variants of a hypothesis or theory differ in empirical import, they count as distinct hypotheses. This should not be confused with the converse positivistic assertion that theories with the same empirical content are really the same theory.
The collection of competing hypotheses or theories to be evaluated by the logic may be finite in number, or may be countably infinite. No realistic language contains more than a countable number of expressions; so it suffices for a logic to apply to countably infinite number of sentences. From a purely logical perspective the collection of competing alternatives may consist of every rival hypothesis or theory about a given subject matter that can be expressed within a given language — e.
In practice, alternative hypotheses or theories will often be constructed and evidentially evaluated over a long period of time. The logic of evidential support works in much the same way regardless of whether all alternative hypotheses are considered together, or only a few alternative hypotheses are available at a time. Evidence for scientific hypotheses consists of the results of specific experiments or observations.
The logical connection between scientific hypotheses and the evidence often requires the mediation of background information and auxiliary hypotheses. Rather, each of the alternative hypotheses under consideration draws on the same background and auxiliaries to logically connect to the evidential events.
This method of theory evaluation is called the hypothetical-deductive approach to evidential support. Duhem and Quine are generally credited with alerting inductive logicians to the importance of auxiliary hypotheses in connecting scientific hypotheses and theories to empirical evidence. See the entry on Pierre Duhem. They point out that scientific hypotheses often make little contact with evidence claims on their own.
Rather, in most cases scientific hypotheses make testable predictions only relative to background information and auxiliary hypotheses that tie them to the evidence. Some specific examples of such auxiliary hypotheses will be provided in the next subsection. Typically auxiliaries are highly confirmed hypotheses from other scientific domains. They often describe the operating characteristics of various devices e. But even when an auxiliary hypothesis is already well-confirmed, we cannot simply assume that it is unproblematic, or just known to be true.
Furthermore, to the extent that competing hypotheses employ different auxiliary hypotheses in accounting for evidence, the evidence only tests each such hypothesis in conjunction with its distinct auxiliaries against alternative hypotheses packaged with their distinct auxiliaries, as described earlier.
No statement is intrinsically a test hypothesis , or intrinsically an auxiliary hypothesis or background condition. Rather, these categories are roles statements may play in a particular epistemic context. We will now examine each of these factors in some detail. Following that we will see precisely how the values of posterior probabilities depend on the values of likelihoods and prior probabilities.
In probabilistic inductive logic the likelihoods carry the empirical import of hypotheses. The hypotheses being tested may themselves be statistical in nature. One of the simplest examples of statistical hypotheses and their role in likelihoods are hypotheses about the chance characteristic of coin-tossing.
There are, of course, more complex cases of likelihoods involving statistical hypotheses. Consider, for example, the hypothesis that plutonium nuclei have a half-life of 20 minutes—i. The full statistical model for the lifetime of such a system says that the propensity or objective chance for that system to remain intact i. For example, a blood test for HIV has a known false-positive rate and a known true-positive rate. Suppose the false-positive rate is.
And suppose that the true-positive rate is. In this context the known test characteristics function as background information, b. This kind of situation may, of course, arise for much more complex hypotheses.
The alternative hypotheses of interest may be deterministic physical theories, say Newtonian Gravitation Theory and some specific alternatives. Likelihoods that arise from explicit statistical claims—either within the hypotheses being tested, or from explicit statistical background claims that tie the hypotheses to the evidence—are often called direct inference likelihoods.
Such likelihoods should be completely objective. So, all evidential support functions should agree on their values, just as all support functions agree on likelihoods when evidence is logically entailed. Direct inference likelihoods are logical in an extended, non-deductive sense. Indeed, some logicians have attempted to spell out the logic of direct inferences in terms of the logical form of the sentences involved.
Not all likelihoods of interest in confirmational contexts are warranted deductively or by explicitly stated statistical claims. In such cases the likelihoods may have vague, imprecise values, but values that are determinate enough to still underwrite an objective evaluation of hypotheses on the evidence.
In any case, the likelihoods that relate hypotheses to evidence claims in many scientific contexts will have such objective values. Thus, the empirical objectivity of a science relies on a high degree of objectivity or intersubjective agreement among scientists on the numerical values of likelihoods.
To see the point more vividly, imagine what a science would be like if scientists disagreed widely about the values of likelihoods. Each practitioner interprets a theory to say quite different things about how likely it is that various possible evidence statements will turn out to be true.
If this kind of situation were to occur often, or for significant evidence claims in a scientific domain, it would make a shambles of the empirical objectivity of that science. It would completely undermine the empirical testability of such hypotheses and theories within that scientific domain. Thus, the empirical objectivity of the sciences requires that experts should be in close agreement about the values of the likelihoods.
For now we will suppose that the likelihoods have objective or intersubjectively agreed values, common to all agents in a scientific community. One might worry that this supposition is overly strong. There are legitimate scientific contexts where, although scientists should have enough of a common understanding of the empirical import of hypotheses to assign quite similar values to likelihoods, precise agreement on their numerical values may be unrealistic.
This point is right in some important kinds of cases. So later, in Section 5, we will see how to relax the supposition that precise likelihood values are available, and see how the logic works in such cases. But for now the main ideas underlying probabilistic inductive logic will be more easily explained if we focus on those contexts were objective or intersubjectively agreed likelihoods are available. Later we will see that much the same logic continues to apply in contexts where the values of likelihoods may be somewhat vague, or where members of the scientific community disagree to some extent about their values.
An adequate treatment of the likelihoods calls for the introduction of one additional notational device. Scientific hypotheses are generally tested by a sequence of experiments or observations conducted over a period of time. In many cases the likelihood of the evidence stream will be equal to the product of the likelihoods of the individual outcomes:.
When this equality holds, the individual bits of evidence are said to be probabilistically independent on the hypothesis together with auxiliaries. In the following account of the logic of evidential support, such probabilistic independence will not be assumed, except in those places where it is explicitly invoked.
The prior probability represents the weight of any important considerations not captured by the evidential likelihoods. It turns out that posterior probabilities depend only on the values of evidential likelihoods together with the values of prior probabilities.
As an illustration of the role of prior probabilities , consider the HIV test example described in the previous section. This posterior probability is much higher than the prior probability of. This positive test result may well be due to the comparatively high false-positive rate for the test, rather than to the presence of HIV.
This sort of test, with a false-positive rate as large as. More generally, in the evidential evaluation of scientific hypotheses and theories, prior probabilities represent assessments of non-evidential plausibility weightings among hypotheses. However, because the strengths of such plausibility assessments may vary among members of a scientific community, critics often brand such assessments as merely subjective , and take their role in Bayesian inference to be highly problematic.
Bayesian inductivists counter that plausibility assessments play an important, legitimate role in the sciences, especially when evidence cannot suffice to distinguish among some alternative hypotheses. Such plausibility assessments are often backed by extensive arguments that may draw on forceful conceptual considerations. Scientists often bring plausibility arguments to bear in assessing competing views.
Although such arguments are seldom decisive, they may bring the scientific community into widely shared agreement, especially with regard to the implausibility of some logically possible alternatives. This seems to be the primary epistemic role of thought experiments.
Consider, for example, the kinds of plausibility arguments that have been brought to bear on the various interpretations of quantum theory e. These arguments go to the heart of conceptual issues that were central to the original development of the theory. Many of these issues were first raised by those scientists who made the greatest contributions to the development of quantum theory, in their attempts to get a conceptual hold on the theory and its implications.
Given any body of evidence, it is fairly easy to cook up a host of logically possible alternative hypotheses that make the evidence as probable as desired. In particular, it is easy to cook up hypotheses that logically entail any given body evidence, providing likelihood values equal to 1 for all the available evidence. Although most of these cooked up hypotheses will be laughably implausible, evidential likelihoods cannot rule them out.
But, the only factors other than likelihoods that figure into the values of posterior probabilities for hypotheses are the values of their prior probabilities; so only prior probability assessments provide a place for the Bayesian logic to bring important plausibility considerations to bear.
Thus, the Bayesian logic can only give implausible hypotheses their due via prior probability assessments. It turns out that the mathematical structure of Bayesian inference makes prior probabilities especially well-suited to represent plausibility assessments among competing hypotheses.
So, given that an inductive logic needs to incorporate well-considered plausibility assessments e. Thus, although prior probabilities may be subjective in the sense that agents may disagree on the relative strengths of plausibility arguments, the priors used in scientific contexts need not represent mere subjective whims.
Rather, the comparative strengths of the priors for hypotheses should be supported by arguments about how much more plausible one hypothesis is than another. The important role of plausibility assessments is captured by such received bits of scientific wisdom as the well-known scientific aphorism, extraordinary claims require extraordinary evidence. That is, it takes especially strong evidence, in the form of extremely high values for ratios of likelihoods, to overcome the extremely low pre-evidential plausibility values possessed by some hypotheses.
Thus, it turns out that prior plausibility assessments play their most important role when the distinguishing evidence represented by the likelihoods remains weak. Some Bayesian logicists have maintained that posterior probabilities of hypotheses should be determined by syntactic logical form alone. Keynes and Carnap tried to implement this idea through syntactic versions of the principle of indifference—the idea that syntactically similar hypotheses should be assigned the same prior probability values.
Carnap showed how to carry out this project in detail, but only for extremely simple formal languages.
Most logicians now take the project to have failed because of a fatal flaw with the whole idea that reasonable prior probabilities can be made to depend on logical form alone. Semantic content should matter. Goodmanian grue-predicates provide one way to illustrate this point. That seems an unreasonable way to proceed. Are we to evaluate the prior probabilities of alternative theories of gravitation, or for alternative quantum theories, by exploring only their syntactic structures, with absolutely no regard for their content—with no regard for what they say about the world?
This seems an extremely dubious approach to the evaluation of real scientific theories. Logical structure alone cannot, and should not suffice for determining reasonable prior probability values for real scientific theories.
Moreover, real scientific hypotheses and theories are inevitably subject to plausibility considerations based on what they say about the world. Prior probabilities are well-suited to represent the comparative weight of plausibility considerations for alternative hypotheses. But no reasonable assessment of comparative plausibility can derive solely from the logical form of hypotheses. We will return to a discussion of prior probabilities a bit later.
Any probabilistic inductive logic that draws on the usual rules of probability theory to represent how evidence supports hypotheses must be a Bayesian inductive logic in the broad sense. Its importance derives from the relationship it expresses between hypotheses and evidence. It shows how evidence, via the likelihoods, combines with prior probabilities to produce posterior probabilities for hypotheses.
So, although the suppression of experimental or observational conditions and auxiliary hypotheses is a common practice in accounts of Bayesian inference, the treatment below, and throughout the remainder of this article will make the role of these terms explicit. Some of these probability functions may provide a better fit with our intuitive conception of how the evidential support for hypotheses should work. So it is important to keep the diversity among evidential support functions in mind.
This factor represents what the hypothesis in conjunction with background and auxiliaries objectively says about the likelihood of possible evidential outcomes of the experimental conditions. So, all reasonable support functions should agree on the values for likelihoods.
Section 5 will treat cases where the likelihoods may lack this kind of objectivity. Arguably the value of this term should be 1, or very nearly 1, since the truth of the hypothesis at issue should not significantly affect how likely it is that the experimental conditions are satisfied. Both the prior probability of the hypothesis and the expectedness tend to be somewhat subjective factors in that various agents from the same scientific community may legitimately disagree on what values these factors should take.
Bayesian logicians usually accept the apparent subjectivity of the prior probabilities of hypotheses, but find the subjectivity of the expectedness to be more troubling. This is due at least in part to the fact that in a Bayesian logic of evidential support the value of the expectedness cannot be determined independently of likelihoods and prior probabilities of hypotheses. This equation shows that the values for the prior probabilities together with the values of the likelihoods uniquely determine the value for the expectedness of the evidence.
Furthermore, it implies that the value of the expectedness must lie between the largest and smallest of the various likelihood values implied by the alternative hypotheses. In cases where some alternative hypotheses remain unspecified or undiscovered , the value of the expectedness is constrained in principle by the totality of possible alternative hypotheses, but there is no way to figure out precisely what its value should be.
Notice that the likelihood ratios carry the full import of the evidence. The evidence influences the evaluation of hypotheses in no other way. The only other factor that influences the value of the ratio of posterior probabilities is the ratio of the prior probabilities.
When the likelihoods are fully objective, any subjectivity that affects the ratio of posteriors can only arise via subjectivity in the ratio of the priors. That is, with regard to the priors, the Bayesian evaluation of hypotheses only relies on how much more plausible one hypothesis is than another due to considerations expressed within b. This kind of Bayesian evaluation of hypotheses is essentially comparative in that only ratios of likelihoods and ratios of prior probabilities are ever really needed for the assessment of scientific hypotheses.
In that case we have:. Suppose we possess a warped coin and want to determine its propensity for heads when tossed in the usual way. Notice, however, that strong refutation is not absolute refutation. Additional evidence could reverse this trend towards the refutation of the fairness hypothesis. This example employs repetitions of the same kind of experiment—repeated tosses of a coin.
But the point holds more generally. If, as the evidence increases, the likelihood ratios. If enough evidence becomes available to drive each of the likelihood ratios. The next two equations show precisely how this works. Generally, the likelihood of evidence claims relative to a catch-all hypothesis will not enjoy the same kind of objectivity possessed by the likelihoods for concrete alternative hypotheses.
Thus, the influence of the catch-all term should diminish towards 0 as new alternative hypotheses are made explicit. It shows how the impact of evidence in the form of likelihood ratios combines with comparative plausibility assessments of hypotheses in the form of ratios of prior probabilities to provide a net assessment of the extent to which hypotheses are refuted or supported via contests with their rivals.
Thus, when the Likelihood Ratio Convergence Theorem applies, the Criterion of Adequacy for an Inductive Logic described at the beginning of this article will be satisfied: As evidence accumulates, the degree to which the collection of true evidence statements comes to support a hypothesis, as measured by the logic, should very probably come to indicate that false hypotheses are probably false and that true hypotheses are probably true.
A view called Likelihoodism relies on likelihood ratios in much the same way as the Bayesian logic articulated above. However, Likelihoodism attempts to avoid the use of prior probabilities. For an account of this alternative view, see the supplement Likelihood Ratios, Likelihoodism, and the Law of Likelihood. In this paper we introduce a new approach to robust design based on the concept of inductive learning with regression trees. The application of inductive classification to performance data for a pseudorandom sample of dugs, deriving decision trees for the choice of data structures.
An inductive inference is then made to the conclusion that certain instances, types, groups, or patterns of evils are gratuitous. Unfortunately, one cannot hope to add an extensional conversion rule on inductive types, which would entail the computability of all isomorphisms of inductive types. We now begin a possibly countably transfinite inductive construction. We are now ready to present a collection of graph algorithms based on the inductive view of graphs. In order to give an inductive definition of the encoding for processes, we need to provide operations over typed graphs.
If a government body takes an administrative decision then this in many cases should not be based on an inductive argument but on regulations. It is sometimes referred to as a "bottom-up" or inductive approach. Learning semantic grammars with constructive inductive logic programming. As in the previous section, our inductive proof requires us to work with near-cubic graphs.
By the inductive hypothesis, this cannot return us to any vertex previously visited by the angel. This raises questions of possible overgenerative power, and the proper status of inductive reasoning in the presence of such constructions.
In inductive reasoning, we begin with specific observations and measures, begin to detect patterns and regularities, formulate some tentative hypotheses that we can explore, and finally end up developing some general conclusions or theories. Inductive reasoning, by its very nature, is more open-ended and exploratory, especially at the beginning. Deductive reasoning is more narrow in nature and is concerned with testing or confirming hypotheses.
Even in the most constrained experiment, the researchers may observe patterns in the data that lead them to develop new theories. We send an occasional email to keep our users informed about new developments on Conjoint. You can always unsubscribe later. Your email will not be shared with other companies.
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