Wednesday, September 12, 2007

Individual differences in the analysis of informal reasoning fallacies

Individual differences in the analysis of informal reasoning fallacies

Robert B. RiccoCorresponding Author Contact Information, a, E-mail The Corresponding Author
aDepartment of Psychology, California State University at San Bernardino, 5500 University Parkway, San Bernardino, CA 92407, USA

Available online 16 February 2007.



Abstract

After decades of research into formal or logical fallacies of reasoning, psychologists have only recently begun to examine the informal reasoning fallacies that are routinely present in critical discussions, debates, and other forms of argumentation. The present study considers several possible influences on an ability to identify and analyze these fallacies. College students completed measures of deductive reasoning, personal epistemology, and knowledge of specific argumentation norms and analyzed arguments containing fallacies such as argument from ignorance,next term begging the question, and slippery slope. Results indicated that effective analysis of informal fallacies was associated with some aspects of deductive reasoning—especially an ability to overcome belief bias—and with higher-order epistemic beliefs, as well as a commitment to argumentation norms for critical discussion. Results are discussed in terms of argumentation research and implications for pedagogical treatments of the fallacies are noted.

Keywords: Informal fallacies; Critical discussion; Deduction; Belief bias; Personal epistemology; Argumentation norms; Critical thinking; Deductive reasoning

Don't vote for ignorancenext term

Don't vote for ignorancenext term

Lawrence Kraussa
aLawrence Krauss is director of the Center for Education and Research in Cosmology and Astrophysics at Case Western Reserve University, Ohio. His latest book is Hiding in the Mirror

Available online 28 July 2007.



When the next US presidential debate airs, voters should take a dim view of any candidates who reject the science of evolution, says Lawrence Krauss

Monday, September 10, 2007

A Geometric Theory of Ignorance

A Geometric Theory of Ignorance

  1. paper.pdf
  2. paper.ps
  3. paper.dvi

Related Material

  1. V. Balasubramanian
  2. Zhu and Rohwer
  3. G. Perelman
  4. Snoussi and Djafari

THEORY OF IGNORANCE - Millith V R

THEORY OF IGNORANCE

‘Ignorance' of the mass took their life.. especially Socratese, Christ, and Bruno. They wanted to do good to people. But they lost thier life due to the IGNORANCE of people.
(Hitler and Mussolini were good in their earlier days. Some people believe they were still good.. that is not our subject)

We should study the IGNORANCE.
So I would like to develop a mathematical THEORY OF IGNORANCE within the frame of SET THEORY.

1. Define IGNORANCE mathematically
2. Postulates
3. operators
4. elements
5. Degree of Ignorance
6. Set of ignorance equations
7. Ignorant matrix
8. Methods of Proofs
a) Mathematical Induction
b) deduction
c) Reductio ad Absurdum
9. Method from Descrete maths
Logic
Implications
10.No results will be accepted from Theory on cognition from Phsychology or Sociology or any other social sciences directly. however if it can be redefined mathematically and proved it can be accepted.
Will be continued ....

11:09 PM

Delete

Rational ignorance

Rational ignorance

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Rational ignorance is a term most often found in economics, particularly public choice theory, but also used in other disciplines which study rationality and choice, including philosophy (epistemology) and game theory.

Ignorance about an issue is said to be "rational" when the cost of educating oneself about the issue sufficiently to make an informed decision can outweigh any potential benefit one could reasonably expect to gain from that decision, and so it would be irrational to waste time doing so. This has consequences for the quality of decisions made by large numbers of people, such as general elections, where the probability of any one vote changing the outcome is very small. (However, see Categorical Imperative for another view on this.)

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[edit] Example

Consider an employer attempting to choose between two candidates offering to complete a task at the cost of $10 / hour. The length of time needed to complete the task may be longer or shorter depending on the skill of the person performing the task, so it is in the employer's best interests to find the fastest worker possible. Assume that the cost of another day of interviewing the candidates is $100. If the employer had deduced from the interviews so far that both candidates would complete the task in somewhere between 195 and 205 hours, it would be in the employer's best interests to choose one or the other by some easily-applied metric (for example, flipping a coin) rather than spend the $100 on determining the better candidate, saving at most $100 in labor.

[edit] Applications

Marketers can take advantage of rational ignorance by increasing the complexity of a decision. If the difference in value between a quality product and a poor product is less than the cost to perform the research necessary to differentiate between them, then it is more rational for a consumer to just take his chances on whichever of the two is more convenient and available. Thus, it is in the interest of the producer of a lower value product to proliferate features, options, and package combinations until the average shopper finds it too much trouble to make an informed decision.

This also works for politics. By increasing the number of issues that a person needs to consider in order to make a rational decision between candidates, they can level the playing field by encouraging single-issue voting, party-line voting, and other habits that tend to ignore a candidate's actual qualifications for the job.

Another, more nuanced, political application involves a voter's identification with a political party, much like the adoption of a favorite movie critic. Based on prior experience a reasonably responsible voter will find politicians or a political party that will draw conclusions similar to their own conclusions when the voter had the time to do the analysis. This is just like the way we adopt a movie critic that likes movies that we like. In this way we let the political party, the politician, or the movie critic do the "heavy lifting" for us as we spend our time doing our job, raising our family or just lying on the beach.

[edit] Criticism

Much of the empirical support for the idea of rational ignorance was drawn from studies of voter apathy, which reached particularly strong conclusions in the 1950s. [1] However, apathy appeared to decline sharply in the 1960s as concern about issues such as the Vietnam War mounted , and political polarization increased [2]. This suggests that voters' interest in political information increases with the importance of political choices.

[edit] References

  1. ^ Campbell, A., Converse, P., Miller, W. and Stokes. D. (1960), 'The American Voter', Wiley, N.Y.
  2. ^ Nie, N., Verba, S. and Petrocik, J. (1976), 'The Changing American Voter', Harvard University Press, Cambridge, Mass.

[edit] See also

[edit] External links



Theory of Knowledge - Bertrand Russell (1926)

bertrand russell as a lay preacher

Bertrand Russell (1926)

Theory of Knowledge
for The Encyclopaedia Britannica)

THEORY OF KNOWLEDGE is a product of doubt. When we have asked ourselves seriously whether we really know anything at all, we are naturally led into an examination of knowing, in the hope of being able to distinguish trustworthy beliefs from such as are untrustworthy. Thus Kant, the founder of modern theory of knowledge, represents a natural reaction against Hume's scepticism. Few philosophers nowadays would assign to this subject quite such a fundamental importance as it had in Kant's "critical" system; nevertheless it remains an essential part of philosophy. It is perhaps unwise to begin with a definition of the subject, since, as elsewhere in philosophical discussions, definitions are controversial, and will necessarily differ for different schools; but we may at least say that the subject is concerned with the general conditions of knowledge, in so far as they throw light upon truth and falsehood.

It will be convenient to divide our discussion into three stages, concerning respectively (1) the definition of knowledge, (2) data, (3) methods of inference. It should be said, however, that in distinguishing between data and inferences we are already taking sides on a debatable question, since some philosophers hold that this distinction is illusory, all knowledge being (according to them) partly immediate and partly derivative.

I. THE DEFINITION OF KNOWLEDGE

The question how knowledge should be defined is perhaps the most important and difficult of the three with which we shall deal. This may seem surprising: at first sight it might be thought that knowledge might be defined as belief which is in agreement with the facts. The trouble is that no one knows what a belief is, no one knows what a fact is, and no one knows what sort of agreement between them would make a belief true. Let us begin with belief.

Belief.

Traditionally, a "belief" is a state of mind of a certain sort. But the behaviourists deny that there are states of mind, or at least that they can be known; they therefore avoid the word "belief", and, if they used it, would mean by it a characteristic of bodily behaviour. There are cases in which this usage would be quite in accordance with common sense. Suppose you set out to visit a friend whom you have often visited before, but on arriving at your destination you find that he has moved, you would say "I thought he was still living at his old house." Yet it is highly probable that you did not think about it at all, but merely pursued the usual route from habit. A "thought" or "belief" may, therefore, in the view of common sense, be shown by behaviour, without any corresponding "mental" occurrence. And even if you use a form of words such as is supposed to express belief, you are still engaged in bodily behaviour, provided you pronounce the words out loud or to yourself. Shall we say, in such cases, that you have a belief? Or is something further required?

It must be admitted that behaviour is practically the same whether you have an explicit belief or not. People who are out of doors when a shower of rain comes on put up their umbrellas, if they have them; some say to themselves "it has begun to rain", others act without explicit thought, but the result is exactly the same in both cases. In very hot weather, both human beings and animals go out of the sun into the shade, if they can; human beings may have an explicit "belief " that the shade is pleasanter, but animals equally seek the shade. It would seem, therefore, that belief, if it is not a mere characteristic of behaviour, is causally unimportant. And the distinction of truth and error exists where there is behaviour without explicit belief, just as much as where explicit belief is present; this is shown by the illustration of going to where your friend used to live. Therefore, if theory of knowledge is to be concerned with distinguishing truth from error, we shall have to include the cases in which there is no explicit belief, and say that a belief may be merely implicit in behaviour. When old Mother Hubbard went to the cupboard, she "believed" that there was a bone there, even if she had no state of mind which could be called cognitive in the sense of introspective psychology.

Words.

In order to bring this view into harmony with the facts of human behaviour, it is of course necessary to take account of the influence of words. The beast that desires shade on a hot day is attracted by the sight of darkness; the man can pronounce the word "shade", and ask where it is to be found. According to the behaviourists, it is the use of words and their efficacy in producing conditional responses that constitutes "thinking". I It is unnecessary for our purposes to inquire whether this view gives the whole truth about the matter. What it is important to realise is that verbal behaviour has the characteristics which lead us to regard it as pre-eminently a mark of "belief", even when the words are repeated as a mere bodily habit. Just as the habit of going to a certain house when you wish to see your friend may be said to show that you "believe" he lives in that house, so the habit of saying "two and two are four", even when merely verbal, must be held to constitute "belief " in this arithmetical proposition. Verbal habits are, of course, not infallible evidences of belief. We may say every Sunday that we are miserable sinners, while really thinking very well of ourselves. Nevertheless, speaking broadly, verbal habits crystallise our beliefs, and afford the most convenient way of making them explicit. To say more for words is to fall into that superstitious reverence for them which has been the bane of philosophy throughout its history.

Belief and Behaviour

We are thus driven to the view that, if a belief is to be something causally important, it must be defined as a characteristic of behaviour. This view is also forced upon us by the consideration of truth and falsehood, for behaviour may be mistaken in just the way attributable to a false belief, even when no explicit belief is present-for example, when a man continues to hold up his umbrella after the rain has stopped without definitely entertaining the opinion that it is still raining. Belief in this wider sense may be attributed to animals-for example, to a dog who runs to the dining-room when he hears the gong. And when an animal behaves to a reflection in a looking-glass as if it were "real", we should naturally say that he "believes" there is another animal there; this form of words is permitted by our definition.

It remains, however, to say what characteristics of behaviour can be described as beliefs. Both human beings and animals act so as to achieve certain results, e.g. getting food. Sometimes they succeed, sometimes they fail-, when they succeed, their relevant beliefs are "true", but when they fail, at least one is false. There will usually be several beliefs involved in a given piece of behaviour, and variations of environment will be necessary to disentangle the causal characteristics which constitute the various beliefs. This analysis is effected by language, but would be very difficult if applied to dumb animals. A sentence may be taken as a law of behaviour in any environment containing certain characteristics; it will be "true" if the behaviour leads to results satisfactory to the person concerned, and otherwise it will be "false". Such, at least, is the pragmatist definition of truth and falsehood.

Truth in Logic.

There is also, however, a more logical method of discussing this question. In logic, we take for granted that a word has a "meaning"; what we signify by this can, I think, only be explained in behaviouristic terms, but when once we have acquired a vocabulary of words which have "meaning", we can proceed in a formal manner without needing to remember what "meaning" is. Given the laws of syntax in the language we are using, we can construct propositions by putting together the words of the language, and these propositions have meanings which result from those of the separate words and are no longer arbitrary. If we know that certain of these propositions are true, we can infer that certain others are true, and that vet others are false; sometimes this can be inferred with certainty, sometimes with greater or less probability. In all this logical manipulation, it is unnecessary to remember what constitutes meaning and what constitutes truth or falsehood. It is in this formal region that most philosophy has lived- and within this region a great deal can be said that is both true and important, without the need of' any fundamental doctrine about meaning. It even seems possible to define "truth" in terms of "meaning" and "fact", as opposed to the pragmatic definition which we gave a moment ago. If so, there will be two valid definitions of "truth", though of course both will apply to the same propositions.

The purely formal definition of "truth" may be illustrated by a simple case. The word "Plato" means a certain man; the word "Socrates" means a certain other man; the word "love" means a certain relation. This being given, the meaning of the complex symbol "Plato loves Socrates" is fixed; we say that this complex symbol is "true" if there is a certain fact in the world, namely the fact that Plato loves Socrates, and in the contrary case the complex symbol is false. I do not think this account is false, but, like everything purely formal, it does not probe very deep.

Uncertainty and Vagueness.

In defining knowledge, there are two further matters to be taken into consideration, namely the degree of certainty and the degree of precision. All knowledge is more or less uncertain and more or less vague. These are, in a sense, opposing characters: vague knowledge has more likelihood of truth than precise knowledge, but is less useful. One of the aims of science is to increase precision without diminishing certainty. But we cannot confine the word "knowledge" to what has the highest degree of both these qualities; we must include some propositions that are rather vague and some that are only rather probable. It Is important, however, to indicate vagueness and uncertainty where they are present, and, if possible, to estimate their degree. Where this can be done precisely, it becomes "probable error" and "probability". But in most cases precision in this respect is impossible.

II. THE DATA

In advanced scientific knowledge, the distinction between what is a datum and what is inferred is clear in fact, though sometimes difficult in theory. In astronomy, for instance, the data are mainly certain black and white patterns on photographic plates. These are called photographs of this or that part of the heavens, but of course much inference is involved in using them to give knowledge about stars or planets. Broadly speaking, quite different methods and a quite different type of skill are required for the observations which provide the data in a quantitative science, and for the deductions by which the data are shown to support this or that theory. There would be no reason to expect Einstein to be particularly good at photographing the stars near the sun during an eclipse. But although the distinction is practically obvious in such cases, It is far less so when we come to less exact knowledge. It may be said that the separation into data and inferences belongs to a well-developed stage of knowledge, and is absent in its beginnings.

Animal Inference.

But just as we found it necessary to admit that knowledge may be only a characteristic of behaviour, so we shall have to say about inference. What a logician recognises as inference is a refined operation, belonging to a high degree of intellectual development; but there is another kind of inference which is practised even by animals. We must consider this primitive form of inference before we can become clear as to what we mean by "data".

When a dog hears the gong and immediately goes into the dining-room, he is obviously, in a sense, practising inference. That is to say, his response is appropriate, not to the noise of the gong in itself, but to that of which the noise is a sign: his reaction is essentially similar to our reactions to words. An animal has the characteristic that, when two stimuli have been experienced together, one tends to call out the response which only the other could formerly call out. If the stimuli (or one of them) are emotionally powerful, one joint experience may be enough-, if not, many joint experiences may be required. This characteristic is totally absent in machines. Suppose, for instance, that you went every day for a year to a certain automatic machine, and lit a match in front of it at the same moment at which you inserted a penny-, it would not, at the end, have any tendency to give up its chocolate on the mere sight of a burning match. That is to say, machines do not display inference even in the form in which it is a mere characteristic of behaviour. Explicit inference, such as human beings practise, is a rationalising of the behaviour which we share with the animals. Having experienced A and B together frequently, we now react to A as we originally reacted to B. To make this seem rational, we say that A is a "sign" of B, and that B must really be present though out of sight. This is the principle of induction, upon which almost all science is based. And a great deal of philosophy is an attempt to make the principle seem reasonable.

Whenever, owing to past experience, we react to A in the manner in which we originally reacted to B, we may say that A is a "datum" and B is "Inferred". In this sense, animals practise inference. It is clear, also, that much inference of this sort is fallacious: the conjunction of A and B in past experience may have been accidental. What is less clear is that there is any way of refining this type of inference which will make it valid. That, however, is a question which we shall consider later. What I want consider now is the nature of those elements in our experiences which, to a reflective analysis, appear as "data" in the above-defined sense.

Mental and Physical Data.

Traditionally, there are two sorts of data, one physical, derived from the senses, the other mental, derived from introspection. It seems highly questionable whether this distinction can be validly made among data; it seems rather to belong to what is inferred from them. Suppose, for the sake of definiteness, that you are looking at a white triangle drawn on a black-board. You can make the two judgments: "There is a triangle there", and "I see a triangle." These are different propositions, but neither expresses a bare datum; the bare datum seems to be the same in both propositions. To illustrate the difference of the propositions: you might say "There is a triangle there", if you had seen it a moment ago but now had your eyes shut, and in this case you would not say "I see a triangle"; on the other hand, you might see a black dot which you knew to be due to indigestion or fatigue, and in this case you would not say "There is a black dot there." In the first of these cases, you have a clear case of inference, not of a datum.

In the second case, you refuse to infer a public object, open to the observation of others. This shows that "I see a triangle" comes nearer to being a datum than "There is a triangle there." But the words "I" and "see" both involve inferences, and cannot be included in any form of words which aims at expressing a bare datum. The word "I" derives its meaning, partly, from memory and expectation, since I do not exist only at one moment. And the word "see" is a causal word, suggesting dependence upon the eyes; this involves experience, since a new-born baby does not know that what it sees depends upon its eyes. However, we can eliminate this dependence upon experience, since obviously all seen objects have a common quality, not belonging to auditory or tactual or any other objects. Let us call this quality that of being "visual". Then we can say: "There is a visual triangle." This is about as near as we can get in words to the datum for both propositions: "There is a triangle there", and "I see a triangle." The difference between the propositions results from different inferences: in the first, to the public world of physics, involving perceptions of others; in the second, to the whole of my experience, in which the visual triangle is an element. The difference between the physical and the mental, therefore, would seem to belong to inferences and constructions, not to data.

It would thus seem that data, in the sense in which we are using the word, consist of brief events, rousing in us various reactions, some of which may be called "inferences", or may at least be said to show the presence of inference. The two-fold organisation of these events, on the one hand as constituents of the public world of physics, on the other hand as parts of a personal experience, belongs to what is inferred, not to what is given. For theory of knowledge, the question of the validity of inference is vital. Unfortunately, nothing very satisfactory can be said about it, and the most careful discussions have been the most sceptical. However, let us examine the matter without prejudice.

III. METHODS OF INFERENCE

It is customary to distinguish two kinds of inference, Deduction and Induction. Deduction is obviously of great practical importance, since it embraces the whole of mathematics. But it may be questioned whether it is, in any strict sense, a form of inference at all. A pure deduction consists merely of saying the same thing in another way. Application to a particular case may have importance, because we bring in the experience that there is such a case-for example, when we infer that Socrates is mortal because all men are mortal. But in this case we have brought in a new piece of experience, not involved in the abstract deductive schema. In pure deduction, we deal with x and y not with empirically given objects such as Socrates and Plato. However this may be, pure deduction does not raise the problems which are of most importance for theory of knowledge, and we may therefore pass it by.

Induction.

The important forms of inference for theory of knowledge are those in which we infer the existence of something having certain characteristics from the existence of something having certain other characteristics. For example: you read in the newspaper that a certain eminent man is dead, and you infer that he is dead. Sometimes, of course, the inference is mistaken. I have read accounts of my own death in newspapers, but I abstained from inferring that I was a ghost. In general, however, such inferences are essential to the conduct of life. Imagine the life of a sceptic who doubted the accuracy of the telephone book, or, when he received a letter, considered seriously the possibility that the black marks might have been made accidentally by an inky fly crawling over the paper. We have to accept merely probable knowledge in daily life, and theory of knowledge must help us to decide when it really is probable, and not mere animal prejudice.

Probability.

Far the most adequate discussion of the type of inference we are considering is obtained in J. M. Keynes's Treatise on Probability (1921). So superior is his work to that of his predecessors that it renders consideration of them unnecessary. Mr. Keynes considers induction and analogy together, and regards the latter as the basis of the former. The bare essence of an inference by analogy is as follows: We have found a number of instances in which two characteristics are combined, and no instances in which they are not combined; we find a new instance in which we know that one of the characteristics is present, but do not know whether the other is present or absent; we argue by analogy that probably the other characteristic is also present. The degree of probability which we infer will vary according to various circumstances. It is undeniable that we do make such inferences, and that neither science nor daily life would be possible without them. The question for the logician is as to their validity. Are they valid always, never or sometimes? And in the last case, can we decide when they are valid?

Limitation of Variety.

Mr. Keynes considers that mere increase in the number of instances in which two qualities are found together does not do much to strengthen the probability of their being found together in other instances. The important point, according to him, is that in the known cases the instances should have as few other qualities in common as possible. But even then a further assumption is required, which is called the principle of limitation of variety. This assumption is stated as follows : "That the objects in the field, over which our generalisations extend, do not have an infinite number of independent qualities; that, in other words, their characteristics, however numerous, cohere together in groups of invariable connection, which are finite in number." It is not necessary to regard this assumption as certain; it is enough if there is some finite probability in its favour.

It is not easy to find any arguments for or against an a priori finite probability in favour of the limitation of variety. It should be observed, however, that a "finite" probability, in Mr. Keynes's terminology, means a probability greater than some numerically measurable probability, e.g. the probability of a penny coming "heads" a million times running. When this is realised, the assumption certainly seems plausible. The strongest argument on the side of scepticism is that both men and animals are constantly led to beliefs (in the behaviouristic sense), which are caused by what may be called invalid inductions; this happens whenever some accidental collocation has produced an association not in accordance with any objective law. Dr. Watson caused an infant to be terrified of white rats by beating a gong behind its head at the moment of showing it a white rat (Behaviourism). On the whole, however, accidental collocations will usually tend to be different for different people, and therefore the inductions in which men are agreed have a good chance of being valid. Scientific inductive or analogical inferences may, in the best cases, be assumed to have a high degree of probability, if the above principle of limitation of variety is true or finitely probable. This result is not so definite as we could wish, but it is at least preferable to Hume's complete scepticism. And it is not obtained, like Kant's answer to Hume, by a philosophy ad hoc; it proceeds on the ordinary lines of scientific method.

Grades of Certainty.

Theory of knowledge, as we have seen, is a subject which is partly logical, partly psychological; the connection between these parts is not very close. The logical part may, perhaps, come to be mainly an organisation of what passes for knowledge according to differing grades of certainty: some portions of our beliefs involve more dubious assumptions than are involved in other parts. Logic and mathematics on the one hand, and the facts of perception on the other, have the highest grade of certainty; where memory comes in, the certainty is lessened; where unobserved matter comes in, the certainty is further lessened; beyond all these stages comes what a cautious man of science would admit to be doubtful. The attempt to increase scientific certainty by means of some special philosophy seems hopeless, since, in view of the disagreement of philosophers, philosophical propositions must count as among the most doubtful of those to which serious students give an unqualified assent. For this reason, we have confined ourselves to discussions which do not assume any definite position on philosophical as opposed to scientific questions.



Reductio ad absurdum

Reductio ad absurdum

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Reductio ad absurdum (Latin: "reduction to the absurd") also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption must have been wrong as it led to an absurd result. It makes use of the law of non-contradiction — a statement cannot be both true and false. In some cases it may also make use of the law of excluded middle — a statement must be either true or false. The phrase is traceable back to the Greek η εις άτοπον απαγωγή (hê eis átopon apagogê), meaning "reduction to the impossible", often used by Aristotle.

In formal logic, reductio ad absurdum is used when a formal contradiction can be derived from a premise, allowing one to conclude that the premise is false. If a contradiction is derived from a set of premises, this shows that at least one of the premises is false, but other means must be used to determine which one.

Reductio ad absurdum is also often used to describe an argument where a conclusion is derived in the belief that everyone (or at least those being argued against) will accept that it is false or absurd. However, this is a weak form of reductio, as the decision to reject the premise requires that the conclusion is accepted as being absurd. Although a formal contradiction is by definition absurd (unacceptable), a weak reductio ad absurdum argument can be rejected simply by accepting the purportedly absurd conclusion. Such arguments can also commonly incorporate the appeal to ridicule, an informal fallacy caused when an argument or theory is twisted by the opposing side to appear ridiculous.

There is a fairly common misconception that reductio ad absurdum simply denotes "a silly argument" and is itself a formal fallacy. However, this is not correct; a properly constructed reductio constitutes a correct argument. When reductio ad absurdum is in error, it is because of a fallacy in the reasoning used to arrive at the contradiction, not the act of reduction itself.

Mathematical proofs are sometimes constructed using reductio ad absurdum, by first assuming the opposite of the theorem the presenter wishes to prove, then reasoning logically from that assumption until presented with a contradiction. Upon reaching the contradiction, the assumption is disproved and therefore its opposite, due to the law of excluded middle, must be true. Such proofs in mathematics are sometimes called informal proofs, but are no less valid than a "formal" mathematical proof arrived at through reduction to equality.

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[edit] The infinitude of primes

A classic (arguably the classic) reductio argument is the one used by Euclid to show that there are infinitely many prime numbers. For suppose there are only finitely many primes; say the complete list is p1 to pn. Consider the number

P = (p1 × p2 × ... × pn) + 1.

We can see that P is not a multiple of any of these primes p1 to pn, since division by any of them leaves remainder 1. Thus none of the prime factors of P is on the list; so the list is not complete. This is absurd, since the list cannot be both complete and incomplete. Thus our assumption was false — there are infinitely many primes.

Another classic reductio proof from Greek mathematics is the proof that the square root of 2 is irrational.

[edit] Cubing-the-cube puzzle

A more recent use of a reductio argument is the proof that a cube cannot be cut into a finite number of smaller cubes with no two the same size. Consider the smallest cube on the bottom face; on each of its four sides, either a neighbouring cube or the border of the main cube is rising above it. This means that any larger cube will not fit on top of it (the "footprint" of such a cube is too large). Since different cubes aren't permitted to have the same sizes, only smaller cubes can be placed directly on top of it. But then the smallest of these would likewise be surrounded by larger cubes, so could only have smaller cubes directly on top of it... and so on, in an infinite regress, requiring an infinite number of cubes, which violates our conditions. (This gives rise to a proof by induction that the cubing-the-cube puzzle is also unsolvable in dimensions higher than three.)

[edit] In mathematics

Say we wish to disprove proposition p. The procedure is to show that assuming p leads to a logical contradiction. Thus, according to the law of non-contradiction, p must be false.

Say instead we wish to prove proposition p. We can proceed by assuming "not p" (i.e. that p is false), and show that it leads to a logical contradiction. Thus, according to the law of non-contradiction, "not p" must be false, and so, according to the law of the excluded middle, p is true.

In symbols:

To disprove p: one uses the tautology [p ^ (R ^ ~R)] → ~p where R is any proposition and the "^" symbol is taken to mean and. Assuming p, one proves R and ~R, together they imply ~p.

To prove p: one uses the tautology [~p ^ (R ^ ~R)] →p where R is any proposition. Assuming ~p, one proves R and ~R, together they imply p.

For a simple example of the first kind, consider the proposition "there is no smallest rational number greater than 0". In a reductio ad absurdum argument, we would start by assuming the opposite: that there is a smallest rational number, say, r0.

Now let x = r0/2. Then x is a rational number, and it's greater than 0; and x is smaller than r0. (In the above symbolic argument, "x is the smallest rational number" would be R and "r (which is different from x) is the smallest rational number" would be ~R.) But that contradicts our initial assumption that r0 was the smallest rational number. So we can conclude that the original proposition must be true — "there is no smallest rational number greater than 0".

It is not uncommon to use this first type of argument with propositions such as the one above, concerning the non-existence of some mathematical object. One assumes that such an object exists, and then proves that this would lead to a contradiction; thus, such an object does not exist. For other examples, see proof that the square root of 2 is not rational and Cantor's diagonal argument.

On the other hand, it is also common to use arguments of the second type concerning the existence of some mathematical object. One assumes that the object doesn't exist, and then proves that this would lead to a contradiction; thus, such an object must exist. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of argument as universally valid. In schools such as intuitionism, the law of the excluded middle is not taken as true. From this way of thinking, there is a very significant difference between proving that something exists by showing that it would be absurd if it did not; and proving that something exists by constructing an actual example of such an object. These schools will still, however, accept arguments of the first kind concerning non-existence. A famous example of the second kind is Brouwer's own proof of his fixed point theorem, which shows that it is impossible for certain fixed points not to exist, without being able to show how to obtain one in the general case.

It is important to note that to form a valid proof, it must be demonstrated that the assumption being made for the sake of argument implies a property that is actually false in the mathematical system being used. The danger here is the logical fallacy of argument from lack of imagination, where it is proven that the assumption implies a property which looks false, but is not really proven to be false. Traditional (but incorrect!) examples of this fallacy include false proofs of Euclid's fifth postulate (a.k.a. the parallel postulate) from the other postulates.

The reason these examples are not really examples of this fallacy is that the notion of proof was different in the 19th century; (Euclidean) geometry was seen as being a 'true' reflection of physical reality, and so deducing a contradiction by concluding something physically implausible (like the angles of a triangle not being 180 degrees) was acceptable. Doubts about the nature of the geometry of the universe led mathematicians such as Bolyai, Gauss, Lobachevsky, Riemann, among others, to question and clarify what actually constituted 'geometry'. Out of these men's work, resulted Non-Euclidean geometry. For a further exposition of these misunderstandings see Morris Kline, Mathematical Thought: from Ancient to Modern Times.

[edit] In mathematical logic

In mathematical logic, the reductio ad absurdum is represented as:

if
S \cup \{ p \} \vdash F
then
S  \vdash \neg p.

or

if
S \cup \{ \neg p \} \vdash F
then
S  \vdash p.

In the above, p is the proposition we wish to prove or disprove; and S is a set of statements which are given as true — these could be, for example, the axioms of the theory we are working in, or earlier theorems we can build upon. We consider p, or the negation of p, in addition to S; if this leads to a logical contradiction F, then we can conclude that the statements in S lead to the negation of p, or p itself, respectively.

Note that the set-theoretic union, in some contexts closely related to logical disjunction (or), is used here for sets of statements in such a way that it is more related to logical conjunction (and).

[edit] Notation

Proof by reductio ad absurdum often end "Contradiction!", or "Which is a contradiction.". Isaac Barrow and Baermann used the notation Q.E.A., for "quod est absurdum" ("which is absurd"), along the lines of Q.E.D., but this notation is rarely used today[1]. A graphical symbol sometimes used for contradictions is a downwards zigzag arrow "lightning" symbol (U+21AF: ↯), for example in Davey and Priestley[2]

[edit] Quotes

In the words of G. H. Hardy (A Mathematician's Apology), "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

[edit] References

  1. ^ Hartshorne on QED and related
  2. ^ B. Davey and H.A. Prisetley, Introduction to lattices and order, Cambridge University Press, 2002.

[edit] See also



Proof theory

Proof theory

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Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.[1] Proof theory can also be considered a branch of philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

Contents

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[edit] History

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his incompleteness theorems showed that this is unattainable. All of this work was carried out with the proof calculi called the Hilbert systems.

In parallel with the proof theoretic work of Gödel, Gerhard Gentzen was laying the foundations of what is now known as structural proof theory. In a few short years, Gentzen introduced the core formalisms of natural deduction (simultaneously with and independently of Jaskowski) and the sequent calculus, made fundamental advances in the formalisation of intuitionistic logic, introduced the important idea of analytic proof, and provided the first combinatorial proof of the consistency of Peano arithmetic.

[edit] Formal and informal proof

The informal proofs of everyday mathematical practice are unlike the formal proofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code.

Formal proofs are constructed with the help of computers in automated theorem proving. Significantly, these proofs can be checked automatically, also by computer. (Checking formal proofs is usually trivial, whereas finding proofs is typically quite hard.) An informal proof in the mathematics literature, by contrast, requires weeks of peer review to be checked, and may still contain errors.

[edit] Kinds of proof calculi

The three most well-known styles of proof calculi are:

Each of these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour, almost any modal logic, and many substructural logics, such as relevance logic or linear logic. Indeed it is unusual to find a logic that resists being represented in one of these calculi.

[edit] Consistency proofs

Main article: Consistency proof

As previously mentioned, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable Π01 sentences) are finitarily true; once so grounded we do not care about the non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities.

The failure of the program was induced by Kurt Gödel's incompleteness theorems, which showed that any ω-consistent theory that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Gödel's formulation is a \Pi^0_1 sentence.

Much investigation has been carried out on this topic since, which has in particular led to:

  • Refinement of Gödel's result, particularly J. Barkley Rosser's refinement, weakening the above requirement of ω-consistency to simple consistency;
  • Axiomatisation of the core of Gödel's result in terms of a modal language, provability logic;
  • Transfinite iteration of theories, due to Alan Turing and Solomon Feferman;
  • The recent discovery of self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.

[edit] Structural proof theory

Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz. The definition is slightly more complex: we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Girard's proof nets also support a notion of analytic proof.

Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. This provides the foundation for the intuitionistic type theory developed by Per Martin-Löf, and is often extended to a three way correspondence, the third leg of which are the cartesian closed categories.

In linguistics, type-logical grammar, categorial grammar and Montague grammar apply formalisms based on structural proof theory to give a formal natural language semantics.

[edit] Tableau systems

Tableau systems apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics.

[edit] Ordinal analysis

Main article: Ordinal analysis

Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis.

[edit] Substructural logics

Main article: Substructural logic

[edit] See also

[edit] References

  1. ^ Wang, Hao (1981). Popular Lectures on Mathematical Logic. Van Nostrand Reinhold Company, 3–4. ISBN 0442231091.


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