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"Although we bind together facts by superinducing upon them a new Conception, this Conception, once introduced and applied, is looked upon as inseparably connected with the facts, and necessarily implied in them. Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined".^{ [10] }
These "superinduced" explanations may well be flawed, but their accuracy is suggested when they exhibit what Whewell termed consilience —that is, simultaneously predicting the inductive generalizations in multiple areas—a feat that, according to Whewell, can establish their truth. Perhaps to accommodate the prevailing view of science as inductivist method, Whewell devoted several chapters to "methods of induction" and sometimes said "logic of induction"—and yet stressed it lacks rules and cannot be trained.^{ [10] }
Originator of pragmatism, C S Peirce who, as did Gottlob Frege independently, in the 1870s performed vast investigations that clarified the basis of deductive inference as mathematical proof, recognized induction but continuously insisted on a third type of inference that Peirce variously termed abduction or retroduction or hypothesis or presumption.^{ [11] } Later philosophers gave Peirce's abduction, etc, the synonym inference to the best explanation (IBE).^{ [12] }
Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer—but then figured not quite.^{ [13] } Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of it in the October 1925 issue of Mind, to provide "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".^{ [14] } Two decades later, Russell proposed enumerative induction as an "independent logical principle".^{ [15] }^{ [16] } Russell found,
"Hume's skepticism rests entirely upon his rejection of the principle of induction. The principle of induction, as applied to causation, says that, if A has been found very often accompanied or followed by B, then it is probable that on the next occasion on which A is observed, it will be accompanied or followed by B. If the principle is to be adequate, a sufficient number of instances must make the probability not far short of certainty. If this principle, or any other from which it can be deduced, is true, then the casual inferences which Hume rejects are valid, not indeed as giving certainty, but as giving a sufficient probability for practical purposes. If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist. The principle itself cannot, of course, without circularity, be inferred from observed uniformities, since it is required to justify any such inference. It must, therefore, be, or be deduced from, an independent principle not based on experience. To this extent, Hume has proved that pure empiricism is not a sufficient basis for science. But if this one principle is admitted, everything else can proceed in accordance with the theory that all our knowledge is based on experience. It must be granted that this is a serious departure from pure empiricism, and that those who are not empiricists may ask why, if one departure is allowed, others are forbidden. These, however, are not questions directly raised by Hume's arguments. What these arguments prove—and I do not think the proof can be controverted—is that the induction is an independent logical principle, incapable of being inferred either from experience or from other logical principles, and that without this principle, science is impossible".^{ [16] }
In a 1965 paper, Gilbert Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a masked consequence of inference to the best explanation (IBE).^{ [12] } IBE is otherwise synonym to C S Peirce's abduction.^{ [12] } Many philosophers of science espousing scientific realism have maintained that IBE is the way that scientists develop approximately true scientific theories about nature.^{ [17] }
Inductive reasoning has been criticized by thinkers as far back as Sextus Empiricus.^{ [18] } The classic philosophical treatment of the problem of induction was given by the Scottish philosopher David Hume.^{ [19] }
Although the use of inductive reasoning demonstrates considerable success, its application has been questionable. Recognizing this, Hume highlighted the fact that our mind draws uncertain conclusions from relatively limited experiences. In deduction, the truth value of the conclusion is based on the truth of the premise. In induction, however, the dependence on the premise is always uncertain. As an example, let's assume "all ravens are black." The fact that there are numerous black ravens supports the assumption. However, the assumption becomes inconsistent with the fact that there are white ravens. Therefore, the general rule of "all ravens are black" is inconsistent with the existence of the white raven. Hume further argued that it is impossible to justify inductive reasoning: specifically, that it cannot be justified deductively, so our only option is to justify it inductively. Since this is circular, he concluded that our use of induction is unjustifiable with the help of Hume's fork.^{ [20] }
However, Hume then stated that even if induction were proved unreliable, we would still have to rely on it. So instead of a position of severe skepticism, Hume advocated a practical skepticism based on common sense, where the inevitability of induction is accepted.^{ [21] } Bertrand Russell illustrated his skepticism in a story about a turkey, fed every morning without fail, who following the laws of induction concludes this will continue, but then his throat is cut on Thanksgiving Day.^{ [22] }
Karl Popper.^{ [23] } had declared in 1963, "Induction, i.e. inference based on many observations, is a myth. It is neither a psychological fact, nor a fact of ordinary life, nor one of scientific procedure".^{ [24] } Popper's 1972 book Objective Knowledge—whose first chapter is devoted to the problem of induction—opens, "I think I have solved a major philosophical problem: the problem of induction".^{ [24] } Within Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during the problem shift.^{ [24] } An imaginative leap, the tentative solution is improvised, lacking inductive rules to guide it.^{ [24] } The resulting, unrestricted generalization is deductive, an entailed consequence of all, included explanatory considerations.^{ [24] } Controversy continued, however, with Popper's putative solution not generally accepted.^{ [25] }
By now, inductive inference has been shown to exist, but is found rarely, as in programs of machine learning in artificial intelligence (AI).^{ [26] } Popper's stance on induction is strictly falsified—enumerative induction exists—but is overwhelmingly absent from science.^{ [26] } Although much talked of nowadays by philosophers, abduction or IBE lacks rules of inference and the discussants provide nothing resembling such, as the process proceeds by humans' imaginations and perhaps creativity.^{ [26] }
Inductive reasoning is also known as hypothesis construction because any conclusions made are based on current knowledge and predictions.^{[ citation needed ]} As with deductive arguments, biases can distort the proper application of inductive argument, thereby preventing the reasoner from forming the most logical conclusion based on the clues. Examples of these biases include the availability heuristic, confirmation bias, and the predictable-world bias.
The availability heuristic causes the reasoner to depend primarily upon information that is readily available to them. People have a tendency to rely on information that is easily accessible in the world around them. For example, in surveys, when people are asked to estimate the percentage of people who died from various causes, most respondents would choose the causes that have been most prevalent in the media such as terrorism, and murders, and airplane accidents rather than causes such as disease and traffic accidents, which have been technically "less accessible" to the individual since they are not emphasized as heavily in the world around them.
The confirmation bias is based on the natural tendency to confirm rather than to deny a current hypothesis. Research has demonstrated that people are inclined to seek solutions to problems that are more consistent with known hypotheses rather than attempt to refute those hypotheses. Often, in experiments, subjects will ask questions that seek answers that fit established hypotheses, thus confirming these hypotheses. For example, if it is hypothesized that Sally is a sociable individual, subjects will naturally seek to confirm the premise by asking questions that would produce answers confirming that Sally is, in fact, a sociable individual.
The predictable-world bias revolves around the inclination to perceive order where it has not been proved to exist, either at all or at a particular level of abstraction. Gambling, for example, is one of the most popular examples of predictable-world bias. Gamblers often begin to think that they see simple and obvious patterns in the outcomes and, therefore, believe that they are able to predict outcomes based upon what they have witnessed. In reality, however, the outcomes of these games are difficult to predict and highly complex in nature. However, in general, people tend to seek some type of simplistic order to explain or justify their beliefs and experiences, and it is often difficult for them to realise that their perceptions of order may be entirely different from the truth.^{ [27] }
The following are types of inductive argument. Notice that while similar, each has a different form.
In contrast to the binary valid/invalid for deductive arguments, inductive arguments are rated in terms of strong or weak along a continuum. An inductive argument is strong in proportion to the probability that its conclusion is correct. We may call an inductive argument plausible, probable, reasonable, justified or strong, but never certain or necessary. Logic affords no bridge from the probable to the certain.
The futility of attaining certainty through some critical mass of probability can be illustrated with a coin-toss exercise. Suppose someone shows me a coin and says the coin is either a fair one or two-headed. He flips it ten times, and ten times it comes up heads. At this point, there is a strong reason to believe it is two-headed. After all, the chance of ten heads in a row is .000976 – less than one in one thousand. Then, after 100 flips, still, every toss has come up heads. Now there is “virtual” certainty that the coin is two-headed. Still, one can neither logically or empirically rule out that the next toss will produce tails. No matter how many times in a row it comes up heads this remains the case. If one programed a machine to flip a coin over and over continuously, at some point the result would be a string of 100 heads. In the fullness of time, all combinations will appear.
As for the slim prospect of getting ten out of ten heads from a fair coin - the outcome that made the coin appear biased - many may be surprised to learn that the chance of any combination of heads or tails is equally unlikely (e.g. H-H-T-T-H-T-H-H-H-T) – and yet it occurs in every trial of ten tosses. That means all results for ten tosses have the same probability as getting ten out of ten heads, which is .000976. If one records the heads-tails series, for whatever result, that exact series had a chance of .000976.
The conclusion for a valid deductive argument is already contained in the premises since because its truth is strictly a matter of logical relations. It cannot say more than its premises. Inductive premises, on the other hand, draw their substance from fact and evidence, and the conclusion accordingly makes a factual claim or prediction. Its reliability varies proportionally with the evidence. Induction wants to reveal something new about the world. One could say that induction wants to say more than is contained in the premises.
To better see the difference between inductive and deductive arguments, consider that it would not make sense to say, "All rectangles so far examined have four right angles, so the next one I see will have four right angles." This would treat logical relations as something factual and discoverable, and thus variable and uncertain. Likewise, speaking deductively we may permissibly say. "All unicorns can fly; I have a unicorn named Charlie; Charlie can fly." This deductive argument is valid because the logical relations hold; we are not interested in their factual soundness. A faulty inductive argument might take the form, "All Swans so far observed were white, therefore it is settled that all swans white." This argument is a case of induction posing as deduction, and fails for the reasons discussed above.
Inductive reasoning is inherently uncertain. It only deals in degrees to which, given the premises, the conclusion is credible according to some theory of evidence. Examples include a many-valued logic, Dempster–Shafer theory, or probability theory with rules for inference such as Bayes' rule. Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases of epistemic uncertainty (technical issues with this may arise however; for example, the second axiom of probability is a closed-world assumption).^{ [28] }
An example of an inductive argument:
This argument could have been made every time a new biological life form was found, and would have been correct every time; however, it is still possible that in the future a biological life form not requiring liquid water could be discovered.
As a result, the argument may be stated less formally as:
A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population.
There are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. A good inductive generalization would be that there are 15 black and five white balls in the urn.
How much the premises support the conclusion depends upon (a) the number in the sample group, (b) the number in the population, and (c) the degree to which the sample represents the population (which may be achieved by taking a random sample). The hasty generalization and the biased sample are generalization fallacies.
This is a Statistical^{ [29] }, aka Sample Projection.^{ [30] } The measure is highly reliable within a well-defined margin of error provided the sample is large and random. It is readily quantifiable. Compare the preceding argument with the following. “Six of the ten people in my book club are Libertarians. About 60% of people are Libertarians.” The argument is weak because the sample is non-random and the sample size is very small
This is inductive generalization. This inference is less reliable than the statistical generalization, first, because the sample events are non-random, and because it is not reducible to mathematical expression. Statistically speaking, there is simply no way to know, measure and calculate as to the circumstances affecting performance that will obtain in the future. On a philosophical level, the argument relies on the presupposition that the operation of future events will mirror the past. In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called Humean after the philosopher who was first to subject them to philosophical scrutiny. ^{ [31] }
A statistical syllogism proceeds from a generalization to a conclusion about an individual.
This is a statistical syllogism.^{ [32] } Even though one cannot be sure Bob will attend university, we can be fully assured of the exact probability for this outcome (given no further information). Arguably the argument is too strong and might be accused of "cheating." After all, the probability is given in the premise. Typically, inductive reasoning seeks to formulate a probability. Two dicto simpliciter fallacies can occur in statistical syllogisms: "accident" and "converse accident".
Simple induction proceeds from a premise about a sample group to a conclusion about another individual.
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.
The basic form of inductive inference, simply induction, reasons from particular instances to all instances, and is thus an unrestricted generalization.^{ [33] } If one observes 100 swans, and all 100 were white, one might infer a universal categorical proposition of the form All swans are white. As this reasoning form's premises, even if true, do not entail the conclusion's truth, this is a form of inductive inference. The conclusion might be true, and might be thought probably true, yet it can be false. Questions regarding the justification and form of enumerative inductions have been central in philosophy of science, as enumerative induction has a pivotal role in the traditional model of the scientific method.
This is enumerative induction, aka simple induction or simple predictive induction. It is a subcategory of inductive generalization. In everyday practice, this is perhaps the most common form of induction. For the preceding argument, the conclusion is tempting but makes a prediction well in excess of the evidence. First, it assumes that life forms observed until now can tell us how future cases will be – an appeal to uniformity. Second, the concluding All is a very bold assertion. A single contrary instance foils the argument. And last, to quantify the level of probability in any mathematical form is problematic. ^{ [34] } By what standard do we measure our earthly sample of known life against all (possible) life? For suppose we do discover some new organism - let’s say some microorganism floating in the mesosphere, or better yet, on some asteroid - and it is cellular. Doesn't the addition of this corroborating evidence oblige us to raise our probability assessment for the subject proposition? It is generally deemed reasonable to answer this question "yes," and for a good many this "yes" is not only reasonable but incontrovertible. So then just how much should this new data change our probability assessment? Here, consensus melts away, and in its place arises a question about whether we can talk of probability coherently at all without numerical quantification.
This is enumerative induction in its weak form. It truncates "all" to a mere single instance and, by making a far weaker claim, considerably strengthens the probability of its conclusion. Otherwise, it has the same shortcomings as the strong form: its sample population is non-random, and quantification methods are elusive.
The process of analogical inference involves noting the shared properties of two or more things, and from this basis inferring that they also share some further property:^{ [35] }
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.^{ [36] }
This is analogical induction, according to which things alike in certain ways are more prone to be alike in other ways. This form of induction was explored in detail by philosopher John Stuart Mill in his System of Logic, wherein he states,
Analogical induction is a subcategory of inductive generalization because it assumes a pre-established uniformity governing events. Analogical induction requires an auxiliary examination of the relevancy of the characteristics cited as common to the pair. In the preceding example, if I add the premise that both stones were mentioned in the records of early Spanish explorers, this common attribute is extraneous to the stones and does not contribute to their probable affinity.
A pitfall of analogy is that features can be cherry-picked: While objects may show striking similarities, two things juxtaposed may respectively possess other characteristics not identified in the analogy that are characteristics sharply dissimilar. Thus, analogy can mislead if not all relevant comparisons are made.
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.
As a logic of induction rather than a theory of belief, Bayesian inference 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 a prior probability for a hypothesis based on logic or previous experience, and when faced with evidence, we adjust the strength of our belief in that hypothesis in a precise manner using Bayesian logic.
Around 1960, Ray Solomonoff founded the theory of universal inductive inference, the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. This is a formal inductive framework that combines algorithmic information theory with the Bayesian framework. Universal inductive inference is based on solid philosophical foundations,^{ [38] } and can be considered as a mathematically formalized Occam's razor. Fundamental ingredients of the theory are the concepts of algorithmic probability and Kolmogorov complexity.
In propositional logic, modus ponens is a rule of inference. It can be summarized as "P implies Q and P is asserted to be true, therefore Q must be true."
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument. A fallacious argument may be deceptive by appearing to be better than it really is. Some fallacies are committed intentionally to manipulate or persuade by deception, while others are committed unintentionally due to carelessness or ignorance. The soundness of legal arguments depends on the context in which the arguments are made.
Abductive reasoning is a form of logical inference which starts with an observation or set of observations then seeks to find the simplest and most likely explanation for the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as inference to the best explanation, although not all uses of the terms abduction and inference to the best explanation are exactly equivalent.
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, highlighting the apparent lack of justification for:
A faulty generalization is a conclusion about all or many instances of a phenomenon that has been reached on the basis of just one or just a few instances of that phenomenon. It is an example of jumping to conclusions. For example, we may generalize about all people, or all members of a group, based on what we know about just one or just a few people. If we meet an angry person from a given country X, we may suspect that most people in country X are often angry. If we see only white swans, we may suspect that all swans are white. Faulty generalizations may lead to further incorrect conclusions. We may, for example, conclude that citizens of country X are genetically inferior, or that poverty is generally the fault of the poor.
Circular reasoning is a logical fallacy in which the reasoner begins with what they are trying to end with. The components of a circular argument are often logically valid because if the premises are true, the conclusion must be true. Circular reasoning is not a formal logical fallacy but a pragmatic defect in an argument whereby the premises are just as much in need of proof or evidence as the conclusion, and as a consequence the argument fails to persuade. Other ways to express this are that there is no reason to accept the premises unless one already believes the conclusion, or that the premises provide no independent ground or evidence for the conclusion. Begging the question is closely related to circular reasoning, and in modern usage the two generally refer to the same thing.
Inferences are steps in reasoning, moving from premises to logical consequences. Charles Sanders Peirce divided inference into three kinds: deduction, induction, and abduction. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. Abduction is inference to the best explanation.
Critical rationalism is an epistemological philosophy advanced by Karl Popper. Popper wrote about critical rationalism in his works: The Logic of Scientific Discovery, The Open Society and its Enemies, Conjectures and Refutations, The Myth of the Framework, and Unended Quest. Ernest Gellner is another notable proponent of this approach.
Informally, two kinds of logical reasoning can be distinguished in addition to formal deduction: induction and abduction. Given a precondition or premise, a conclusion or logical consequence and a rule or material conditional that implies the conclusion given the precondition, one can explain that:
A statistical syllogism is a non-deductive syllogism. It argues, using inductive reasoning, from a generalization true for the most part to a particular case.
In philosophy and mathematics, a logical form of a syntactic expression is a precisely-specified semantic version of that expression in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. However, this may not affect the truth of the conclusion since validity and truth are separate in formal logic.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Inductivism is the traditional model of scientific method attributed to Francis Bacon, who in 1620 vowed to subvert allegedly traditional thinking. In the Baconian model, one observes nature, proposes a modest law to generalize an observed pattern, confirms it by many observations, ventures a modestly broader law, and confirms that, too, by many more observations, while discarding disconfirmed laws. The laws grow ever broader but never much exceed careful, extensive observation. Thus, freed from preconceptions, scientists gradually uncover nature's causal and material structure.
In logic and philosophy, an argument is a series of statements, called the premises or premisses, intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language, and independently of natural language formally defined "arguments" can be made in math and computer science.
The psychology of reasoning is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory.
Plausible reasoning is a method of deriving new conclusions from given known premises, a method different from the classical syllogistic argumentation methods of Aristotelian two-valued logic. The syllogistic style of argumentation is illustrated by the oft-quoted argument "All men are mortal, Socrates is a man, and therefore, Socrates is mortal." In contrast, consider the statement "if it is raining then it is cloudy." The only logical inference that one can draw from this is that "if it is not cloudy then it is not raining." But ordinary people in their everyday lives would conclude that "if it is not raining then being cloudy is less plausible," or "if it is cloudy then rain is more plausible." The unstated and unconsciously applied reasoning, arguably incorrect, that made people come to their conclusions is typical of plausible reasoning.
As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality. Arguments may be logical if they are "conducted or assessed according to strict principles of validity", while they are rational according to the broader requirement that they are based on reason and knowledge.
Some dictionaries define "deduction" as reasoning from the general to specific and "induction" as reasoning from the specific to the general. While this usage is still sometimes found even in philosophical and mathematical contexts, for the most part, it is outdated.
In a typical enumerative induction, the premises list the individuals observed to have a common property, and the conclusion claims that all individuals of the same population have that property.
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