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. Given that a scientific community should largely agree on the values of the likelihoods, any significant disagreement among them with regard to the values of posterior probabilities of hypotheses should derive from disagreements over their assessments of values for the prior probabilities of those hypotheses.
We saw in Section 3. Thus, the logic of evidential support only requires that scientists can assess the comparative plausibilities of various hypotheses.
Presumably, in scientific contexts the comparative plausibility values for hypotheses should depend on explicit plausibility arguments, not merely on privately held opinions. Even so, agents may be unable to specify precisely how much more strongly the available plausibility arguments support a hypothesis over an alternative; so prior probability ratios for hypotheses may be vague.
Furthermore, agents in a scientific community may disagree about how strongly the available plausibility arguments support a hypothesis over a rival hypothesis; so prior probability ratios may be somewhat diverse as well.
Vagueness and diversity are somewhat different issues, but they may be represented in much the same way. Assessments of the prior plausibilities of hypotheses will often be vague—not subject to the kind of precise quantitative treatment that a Bayesian version of probabilistic inductive logic may seem to require for prior probabilities.
So, it may seem that the kind of assessment of prior probabilities required to get the Bayesian algorithm going cannot be accomplished in practice. Recall that this Ratio Form of the theorem captures the essential features of the logic of evidential support, even though it only provides a value for the ratio of the posterior probabilities.
Such comparative plausibilities are much easier to assess than specific numerical values for the prior probabilities of individual hypotheses. When combined with the ratio of likelihoods , this ratio of priors suffices to yield an assessment of the ratio of posterior plausibilities ,. In practice one need only assess bounds for these prior plausibility ratios to achieve meaningful results.
Given a prior ratio in a specific interval,. Technically each probabilistic support function assigns a specific numerical value to each pair of sentences; so when we write an inequality like.
Thus, technically, the Bayesian logic employs sets of probabilistic support functions to represent the vagueness in comparative plausibility values for hypotheses. This observation is really useful. This result, called the Likelihood Ratio Convergence Theorem , will be investigated in more detail in Section 4. Thus, false competitors of a true hypothesis will effectively be eliminated by increasing evidence.
Thus, Bayesian logic of inductive support for hypotheses is a form of eliminative induction, where the evidence effectively refutes false alternatives to the true hypothesis. It only needs to draw on bounds on the values of comparative plausibility ratios, and these bounds only play a significant role while evidence remains fairly weak.
If the true hypothesis is assessed to be comparatively plausible due to plausibility arguments contained in b , then plausibility assessments give it a leg-up over alternatives. If the true hypothesis is assessed to be comparatively implausible, the plausibility assessments merely slow down the rate at which it comes to dominate its rivals, reflecting the idea that extraordinary hypotheses require extraordinary evidence or an extraordinary accumulation of evidence to overcome their initial implausibilities.
But, once again, if accumulating evidence drives the likelihood ratios comparing various alternative hypotheses to the true hypothesis towards 0, the range of support functions in a diversity set will come to near agreement, near 0, on the values for posterior probabilities of false competitors of the true hypothesis. As this happens, the posterior probability of the true hypothesis may approach 1. The Likelihood Ratio Convergence Theorem implies that this kind of convergence to the truth should very probably happen, provided that the true hypothesis is empirically distinct enough from its rivals.
One more point about prior probabilities and Bayesian convergence should be mentioned before proceeding to Section 4. Critics argue that this is unreasonable. The members of a scientific community may quite legitimately revise their comparative prior plausibility assessments for hypotheses from time to time as they rethink plausibility arguments and bring new considerations to bear.
This seems a natural part of the conceptual development of a science. It turns out that such reassessments of the comparative plausibilities of hypotheses poses no difficulty for the probabilistic inductive logic discussed here.
Such reassessments may be represented by the addition or modification of explicit statements that modify the background information b. Such reassessments may result in non-Bayesian transitions to new vagueness sets for individual agents and new diversity sets for the community.
The logic of Bayesian induction as described here has nothing to say about what values the prior plausibility assessments for hypotheses should have; and it places no restrictions on how they might change over time. Those interested in a Bayesian account of Enumerative Induction and the estimation of values for relative frequencies of attributes in populations should see the supplement, Enumerative Inductions: Bayesian Estimation and Convergence.
In this section we will investigate the Likelihood Ratio Convergence Theorem. The theorem itself does not require the full apparatus of Bayesian probability functions. It draws only on likelihoods.
Neither the statement of the theorem nor its proof employ prior probabilities of any kind. So even likelihoodists , who eschew the use of Bayesian prior probabilities, may embrace this result. So, support functions in collections representing vague prior plausibilities for an individual agent i.
And as the posterior probabilities of false competitors fall, the posterior probability of the true hypothesis heads towards 1. Thus, the theorem establishes that the inductive logic of probabilistic support functions satisfies the Criterion of Adequacy CoA suggested at the beginning of this article. The Likelihood Ratio Convergence Theorem merely provides some sufficient conditions for probable convergence.
But likelihood ratios may well converge towards 0 in the way described by the theorem even when the antecedent conditions of the theorem are not satisfied. This theorem overcomes many of the objections raised by critics of Bayesian convergence results. First, this theorem does not employ second-order probabilities ; it says noting about the probability of a probability.
It only concerns the probability of a particular disjunctive sentence that expresses a disjunction of various possible sequences of experimental or observational outcomes. The theorem does not require evidence to consist of sequences of events that, according to the hypothesis, are identically distributed like repeated tosses of a die. The result is most easily expressed in cases where the individual outcomes of a sequence of experiments or observations are probabilistically independent, given each hypothesis.
So that is the version that will be presented in this section. However, a version of the theorem also holds when the individual outcomes of the evidence stream are not probabilistically independent, given the hypotheses.
This more general version of the theorem will be presented in a supplement on the Probabilistic Refutation Theorem , below, where the proof of both versions is provided. In addition, this result does not rely on supposing that the probability functions involved are countably additive. Furthermore, the explicit lower bounds on the rate of convergence provided by this result means that there is no need to wait for the infinitely long run before convergence occurs as some critics seem to think.
It is sometimes claimed that Bayesian convergence results only work when an agent locks in values for the prior probabilities of hypotheses once-and-for-all, and then updates posterior probabilities from there only by conditioning on evidence via Bayes Theorem. The Likelihood Ratio Convergence Theorem , however, applies even if agents revise their prior probability assessments over time. Such non-Bayesian shifts from one support function or vagueness set to another may arise from new plausibility arguments or from reassessments of the strengths of old ones.
The Likelihood Ratio Convergence Theorem itself only involves the values of likelihoods. Here they are. This set is represented by the expression,. To specify this measure we need to contemplate the collection of possible outcomes of each experiment or observation. Everything introduced in this subsection is mere notational convention. No substantive suppositions other than the axioms of probability theory have yet been introduced.
The next subsection will discuss that supposition in detail. In most scientific contexts the outcomes in a stream of experiments or observations are probabilistically independent of one another relative to each hypothesis under consideration, or can at least be divided up into probabilistically independent parts. For our purposes probabilistic independence of evidential outcomes on a hypothesis divides neatly into two types.
When these two conditions hold, the likelihood for an evidence sequence may be decomposed into the product of the likelihoods for individual experiments or observations. To see how the two independence conditions affect the decomposition, first consider the following formula, which holds even when neither independence condition is satisfied:.
When condition-independence holds, the likelihood of the whole evidence stream parses into a product of likelihoods that probabilistically depend on only past observation conditions and their outcomes. They do not depend on the conditions for other experiments whose outcomes are not yet specified. Here is the formula:. Finally, whenever both independence conditions are satisfied we have the following relationship between the likelihood of the evidence stream and the likelihoods of individual experiments or observations:.
In scientific contexts the evidence can almost always be divided into parts that satisfy both clauses of the Independent Evidence Condition with respect to each alternative hypothesis. To see why, let us consider each independence condition more carefully. To appreciate the significance of this condition, imagine what it would be like if it were violated. Condition-independence , when it holds, rules out such strange effects.
Result-independence says that the description of previous test conditions together with their outcomes is irrelevant to the likelihoods of outcomes for additional experiments. If this condition were widely violated, then in order to specify the most informed likelihoods for a given hypothesis one would need to include information about volumes of past observations and their outcomes.
What a hypothesis says about future cases would depend on how past cases have gone. Such dependence had better not happen on a large scale. Otherwise, the hypothesis would be fairly useless, since its empirical import in each specific case would depend on taking into account volumes of past observational and experimental results.
However, even if such dependencies occur, provided they are not too pervasive, result-independence can be accommodated rather easily by packaging each collection of result-dependent data together, treating it like a single extended experiment or observation. Thus, by packaging result-dependent data together in this way, the result-independence condition is satisfied by those conjunctive statements that describe the separate, result-independent chunks.
The version of the Likelihood Ratio Convergence Theorem we will examine depends only on the Independent Evidence Conditions together with the axioms of probability theory.
It draws on no other assumptions. Indeed, an even more general version of the theorem can be established, a version that draws on neither of the Independent Evidence Conditions. However, the Independent Evidence Conditions will be satisfied in almost all scientific contexts, so little will be lost by assuming them. And the presentation will run more smoothly if we side-step the added complications needed to explain the more general result.
From this point on, let us assume that the following versions of the Independent Evidence Conditions hold. Assumption: Independent Evidence Assumptions. We now have all that is needed to begin to state the Likelihood Ratio Convergence Theorem. The Likelihood Ratio Convergence Theorem comes in two parts. Such outcomes are highly desirable.
It will be convenient to define a term for this situation. Definition: Full Outcome Compatibility. The first part of the Likelihood Ratio Convergence Theorem applies to that part of the total stream of evidence i. It turns out that these two kinds of cases must be treated differently. This is due to the way in which the expected information content for empirically distinguishing between the two hypotheses will be measured for experiments and observations that are fully outcome compatible ; this measure of information content blows up becomes infinite for experiments and observations that fail to be fully outcome compatible.
Thus, the following part of the convergence theorem applies to just that part of the total stream of evidence that consists of experiments and observations that fail to be fully outcome compatible for the pair of hypotheses involved. Here, then, is the first part of the convergence theorem.
For proof see Proof of the Falsification Theorem. The Falsification Theorem is quite commonsensical. First, notice that if there is a crucial experiment in the evidence stream, the theorem is completely obvious. The theorem is equally commonsensical for cases where no crucial experiment is available. To see what it says in such cases, consider an example. When this happens, the likelihood ratio becomes 0.
It is instructive to plug some specific values into the formula given by the Falsification Theorem, to see what the convergence rate might look like. They tell us the likelihood of obtaining each specific outcome stream, including those that either refute the competitor or produce a very small likelihood ratio for it.
Convergence theorems become moot. The point of the Likelihood Ratio Convergence Theorem both the Falsification Theorem and the part of the theorem still to come is to assure us in advance of considering any specific pair of hypotheses that if the possible evidence streams that test hypotheses have certain characteristics which reflect the empirical distinctness of the two hypotheses, then it is highly likely that one of the sequences of outcomes will occur that yields a very small likelihood ratio.
These theorems provide finite lower bounds on how quickly such convergence is likely to be. Thus, they show that the CoA is satisfied in advance of our using the logic to test specific pairs of hypotheses against one another. The Falsification Theorem applies whenever the evidence stream includes possible outcomes that may falsify the alternative hypothesis. Evidence streams of this kind contain no possibly falsifying outcomes. Hypotheses whose connection with the evidence is entirely statistical in nature will usually be fully outcome-compatible on the entire evidence stream.
So, evidence streams of this kind are undoubtedly much more common in practice than those containing possibly falsifying outcomes. Furthermore, whenever an entire stream of evidence contains some mixture of experiments and observations on which the hypotheses are not fully outcome compatible along with others on which they are fully outcome compatible , we may treat the experiments and observations for which full outcome compatibility holds as a separate subsequence of the entire evidence stream, to see the likely impact of that part of the evidence in producing values for likelihood ratios.
The logarithm of the likelihood ratio provides such a measure. Definition: QI—the Quality of the Information. Thus, QI measures information on a logarithmic scale that is symmetric about the natural no-information midpoint, 0. Probability theorists measure the expected value of a quantity by first multiplying each of its possible values by their probabilities of occurring, and then summing these products.
Thus, the expected value of QI is given by the following formula:. Whereas QI measures the ability of each particular outcome or sequence of outcomes to empirically distinguish hypotheses, EQI measures the tendency of experiments or observations to produce distinguishing outcomes.
It can be shown that EQI tracks empirical distinctness in a very precise way. We return to this in a moment. We are now in a position to state the second part of the Likelihood Ratio Convergence Theorem. For proof see the supplement Proof of the Probabilistic Refutation Theorem. This theorem provides sufficient conditions for the likely refutation of false alternatives via exceeding small likelihood ratios. The conditions under which this happens characterize the degree to which the hypotheses involved are empirically distinct from one another.
It turns out that in almost every case for almost any pair of hypotheses the actual likelihood of obtaining such evidence i. This condition is only needed because our measure of evidential distinguishability, QI, blows up when the ratio. Furthermore, this condition is really no restriction at all on possible experiments or observations. We merely failed to take this more strongly refuting possibility into account when computing our lower bound on the likelihood that refutation via likelihood ratios would occur.
The point of the two Convergence Theorems explored in this section is to assure us, in advance of the consideration of any specific pair of hypotheses, that if the possible evidence streams that test them have certain characteristics which reflect their evidential distinguishability, it is highly likely that outcomes yielding small likelihood ratios will result.
These theorems provide finite lower bounds on how quickly convergence is likely to occur. Thus, there is no need to wait through some infinitely long run for convergence to occur.
Indeed, for any evidence sequence on which the probability distributions are at all well behaved, the actual likelihood of obtaining outcomes that yield small likelihood ratio values will inevitably be much higher than the lower bounds given by Theorems 1 and 2. The true hypothesis speaks truthfully about this, and its competitors lie. Even a sequence of observations with an extremely low average expected quality of information is very likely to do the job if that evidential sequence is long enough.
Thus, the Criterion of Adequacy CoA is satisfied. Up to this point we have been supposing that likelihoods possess objective or agreed numerical values. Although this supposition is often satisfied in scientific contexts, there are important settings where it is unrealistic, where hypotheses only support vague likelihood values, and where there is enough ambiguity in what hypotheses say about evidential claims that the scientific community cannot agree on precise values for the likelihoods of evidential claims.
Recall why agreement, or near agreement, on precise values for likelihoods is so important to the scientific enterprise. To the extent that members of a scientific community disagree on the likelihoods, they disagree about the empirical content of their hypotheses, about what each hypothesis says about how the world is likely to be. This can lead to disagreement about which hypotheses are refuted or supported by a given body of evidence.
Similarly, to the extent that the values of likelihoods are only vaguely implied by hypotheses as understood by an individual agent, that agent may be unable to determine which of several hypotheses is refuted or supported by a given body of evidence. We have seen, however, that the individual values of likelihoods are not really crucial to the way evidence impacts hypotheses. Rather, as Equations 9—11 show, it is ratios of likelihoods that do the heavy lifting. Furthermore, although the rate at which the likelihood ratios increase or decrease on a stream of evidence may differ for the two support functions, the impact of the cumulative evidence should ultimately affect their refutation or support in much the same way.
When likelihoods are vague or diverse, we may take an approach similar to that we employed for vague and diverse prior plausibility assessments. We may extend the vagueness sets for individual agents to include a collection of inductive support functions that cover the range of values for likelihood ratios of evidence claims as well as cover the ranges of comparative support strengths for hypotheses due to plausibility arguments within b , as represented by ratios of prior probabilities.
Similarly, we may extend the diversity sets for communities of agents to include support functions that cover the ranges of likelihood ratio values that arise within the vagueness sets of members of the scientific community. This broadening of vagueness and diversity sets to accommodate vague and diverse likelihood values makes no trouble for the convergence to truth results for hypotheses. For, provided that the Directional Agreement Condition is satisfied by all support functions in an extended vagueness or diversity set under consideration, the Likelihood Ratio Convergence Theorem applies to each individual support function in that set.
That can happen because different support functions may represent the evidential import of hypotheses differently, by specifying different likelihood values for the very same evidence claims.
However, when the Directional Agreement Condition holds for a given collection of support functions, this problem cannot arise. Thus, when the Directional Agreement Condition holds for all support functions in a vagueness or diversity set that is extended to include vague or diverse likelihoods, and provided that enough evidentially distinguishing experiments or observations can be performed, all support functions in the extended vagueness or diversity set will very probably come to agree that the likelihood ratios for empirically distinct false competitors of a true hypothesis are extremely small.
As that happens, the community comes to agree on the refutation of these competitors, and the true hypothesis rises to the top of the heap. What if the true hypothesis has evidentially equivalent rivals? Their posterior probabilities must rise as well. In that case we are only assured that the disjunction of the true hypothesis with its evidentially equivalent rivals will be driven to 1 as evidence lays low its evidentially distinct rivals.
The editors and author also thank Greg Stokley and Philippe van Basshuysen for carefully reading an earlier version of the entry and identifying a number of typographical errors. Inductive Arguments 2. Inductive Logic and Inductive Probabilities 2.
The Likelihood Ratio Convergence Theorem 4. Inductive Arguments Let us begin by considering some common kinds of examples of inductive arguments. Consider the following two arguments: Example 1.
Semi-formalization Formalization Premise 1 The frequency or proportion of members with attribute A among the members of S is r. Theorem: Nonnegativity of EQI. Republished in by Dover: New York. Chihara, Charles S. Kyburg, Jr. Smokler eds. Krieger Publishing Company, Dowe, David L.
Duhem, P. Son objet et sa structure , Paris: Chevalier et Riviere; translated by P. Earman, John, , Bayes or Bust? Edwards, A. Taper and Subhash R. Lele eds. Friedman, Nir and Joseph Y. Glymour, Clark N.
Zalta ed. Harper, William L. Joyce, James M. Mele and Piers Rawling eds. Kelly, Kevin T. Kolmogorov, A. Koopman, B. Reprinted in H. Kyburg and H. Jeffrey, ed. Borchert ed. Eells and B. Skyrms eds. McGrew, Timothy J. Norton, John D. Quine, W. Routledge Encyclopedia of Philosophy, Version 1. Braithwaite ed. Kegan,, — Reprinted in Studies in Subjective Probability , H.
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Savage, Leonard J. Schlesinger, George N. Harper and Brian Skyrms eds. Cohen and L. Laudan eds. Vranas, Peter B. Academic Tools How to cite this entry. Enhanced bibliography for this entry at PhilPapers , with links to its database. Other Internet Resources Confirmation and Induction.
This is inductive reasoning. In an inductive argument the conclusion is, at best, probable. The conclusion is not always true when the premises are true. The probability of the conclusion depends on the strength of the inference from the premises.
Thus, when dealing with inductive reasoning, pay special attention to the inductive leap or inference, by which the conclusion follows the premises. On a daily basis we draw inferences such as how a person will probably act, what the weather will probably be like, and how a meal will probably taste, and these are typical inductive inferences.
It can be studied by asking young children simple questions involving cartoon pictures, or it can be studied by giving adults a variety of complex verbal arguments and asking them to make probability judgments. For example, much of the study of induction has been concerned with category-based induction, such as inferring that your next door neighbor sleeps on the basis that your neighbor is a human animal, even if you have never seen your neighbor sleeping.
Deduction begins with a broad truth the major premise , such as the statement that all men are mortal. This is followed by the minor premise, a more specific statement, such as that Socrates is a man. A conclusion follows: Socrates is mortal. If the major premise is true and the minor premise is true the conclusion cannot be false. Deductive reasoning is black and white; a conclusion is either true or false and cannot be partly true or partly false.
We decide whether a deductive statement is true by assessing the strength of the link between the premises and the conclusion. If all men are mortal and Socrates is a man, there is no way he can not be mortal, for example. There are no situations in which the premise is not true, so the conclusion is true. In science, deduction is used to reach conclusions believed to be true. A hypothesis is formed; then evidence is collected to support it.
If observations support its truth, the hypothesis is confirmed. Science also involves inductive reasoning when broad conclusions are drawn from specific observations; data leads to conclusions. If the data shows a tangible pattern, it will support a hypothesis. For example, having seen ten white swans, we could use inductive reasoning to conclude that all swans are white.
This hypothesis is easier to disprove than to prove, and the premises are not necessarily true, but they are true given the existing evidence and given that researchers cannot find a situation in which it is not true. By combining both types of reasoning, science moves closer to the truth. In general, the more outlandish a claim is, the stronger the evidence supporting it must be. We should be wary of deductive reasoning that appears to make sense without pointing to a truth. My pet has four paws.
Therefore, my pet is a dog. Plato — BC believed that all things are divided into the visible and the intelligible. Intelligible things can be known through deduction with observation being of secondary importance to reasoning and are true knowledge. Aristotle took an inductive approach, emphasizing the need for observations to support knowledge.
He believed that we can reason only from discernable phenomena. From there, we use logic to infer causes. Debate about reasoning remained much the same until the time of Isaac Newton.
In his Principia, Newton outlined four rules for reasoning in the scientific method :. In , philosopher John Stuart Mill published A System of Logic , which further refined our understanding of reasoning. Mill believed that science should be based on a search for regularities among events. If a regularity is consistent, it can be considered a law. Mill described five methods for identifying causes by noting regularities.
These methods are still used today:. Karl Popper was the next theorist to make a serious contribution to the study of reasoning. Popper is well known for his focus on disconfirming evidence and disproving hypotheses. Beginning with a hypothesis, we use deductive reasoning to make predictions. A hypothesis will be based on a theory — a set of independent and dependent statements.
If the predictions are true, the theory is true, and vice versa. This process requires vigorous testing to identify any anomalies, and Popper does not accept theories that cannot be physically tested. Any phenomenon not present in tests cannot be the foundation of a theory, according to Popper.
The phenomenon must also be consistent and reproducible. Science is always changing as more hypotheses are modified or disproved and we inch closer to the truth. No discussion of logic is complete without a refresher course in the difference between inductive and deductive reasoning. By its strictest definition, inductive reasoning proves a general principle—your idea worth spreading—by highlighting a group of specific events, trends, or observations.
In contrast, deductive reasoning builds up to a specific principle—again, your idea worth spreading—through a chain of increasingly narrow statements. Logic is an incredibly important skill, and because we use it so often in everyday life, we benefit by clarifying the methods we use to draw conclusions.
Knowing what makes an argument sound is valuable for making decisions and understanding how the world works. It helps us to spot people who are deliberately misleading us through unsound arguments. Understanding reasoning is also helpful for avoiding fallacies and for negotiating. Read Next. Mental Models Reading Time: 12 minutes. As my friend Peter Kaufman says : What are the three largest, most relevant sample sizes for identifying universal principles? Deductive and inductive reasoning are both based on evidence.
Several types of evidence are used in reasoning to point to a truth: Direct or experimental evidence — This relies on observations and experiments, which should be repeatable with consistent results. Anecdotal or circumstantial evidence — Overreliance on anecdotal evidence can be a logical fallacy because it is based on the assumption that two coexisting factors are linked even though alternative explanations have not been explored. The main use of anecdotal evidence is for forming hypotheses which can then be tested with experimental evidence.
Argumentative evidence — We sometimes draw conclusions based on facts. However, this evidence is unreliable when the facts are not directly testing a hypothesis. For example, seeing a light in the sky and concluding that it is an alien aircraft would be argumentative evidence.
Testimonial evidence — When an individual presents an opinion, it is testimonial evidence. Once again, this is unreliable, as people may be biased and there may not be any direct evidence to support their testimony.
There are several key types of inductive reasoning: Generalized — Draws a conclusion from a generalization. Swans are similar to Aylesbury ducks. Therefore, all swans are probably white.
Therefore, when I visit again, all the swans will probably be white. I just saw a white bird in the pond. The bird was probably a swan. In The 12 Secrets of Persuasive Argument , the authors write: In inductive arguments, focus on the inference.
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