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177-653: A mathematical proof is a deductive argument for a mathematical statement , showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems ; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms , along with the accepted rules of inference . Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which

354-643: A and b such that a b {\displaystyle a^{b}} is a rational number . This proof uses that 2 {\displaystyle {\sqrt {2}}} is irrational (an easy proof is known since Euclid ), but not that 2 2 {\displaystyle {\sqrt {2}}^{\sqrt {2}}} is irrational (this is true, but the proof is not elementary). The expression "statistical proof" may be used technically or colloquially in areas of pure mathematics , such as involving cryptography , chaotic series , and probabilistic number theory or analytic number theory . It

531-415: A conditional statement ( P → Q {\displaystyle P\rightarrow Q} ) and as second premise the antecedent ( P {\displaystyle P} ) of the conditional statement. It obtains the consequent ( Q {\displaystyle Q} ) of the conditional statement as its conclusion. The argument form is listed below: In this form of deductive reasoning,

708-477: A is not equal to 0 or 1, and b is not a rational number, then any value of a is a transcendental number (there can be more than one value if complex number exponentiation is used). An example that provides a simple constructive proof is The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, log 2 ⁡ 3 {\displaystyle \log _{\sqrt {2}}3} ,

885-447: A mathematical object with a certain property exists—without explaining how such an object can be found. Often, this takes the form of a proof by contradiction in which the nonexistence of the object is proved to be impossible. In contrast, a constructive proof establishes that a particular object exists by providing a method of finding it. The following famous example of a nonconstructive proof shows that there exist two irrational numbers

1062-417: A particle physics experiment or observational study in physical cosmology . "Statistical proof" may also refer to raw data or a convincing diagram involving data, such as scatter plots , when the data or diagram is adequately convincing without further analysis. Proofs using inductive logic , while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in

1239-410: A speaker-determined definition of deduction since it depends also on the speaker whether the argument in question is deductive or not. For speakerless definitions, on the other hand, only the argument itself matters independent of the speaker. One advantage of this type of formulation is that it makes it possible to distinguish between good or valid and bad or invalid deductive arguments: the argument

1416-561: A Moroccan mathematician from Fez specializing in Islamic inheritance jurisprudence during the 12th century, first mentions the use of a fractional bar, where numerators and denominators are separated by a horizontal bar. In his discussion he writes, "..., for example, if you are told to write three-fifths and a third of a fifth, write thus, 3 1 5 3 {\displaystyle {\frac {3\quad 1}{5\quad 3}}} ." This same fractional notation appears soon after in

1593-402: A bachelor; therefore, Othello is not male". This is similar to the valid rule of inference called modus tollens , the difference being that the second premise and the conclusion are switched around. Other formal fallacies include affirming a disjunct , denying a conjunct , and the fallacy of the undistributed middle . All of them have in common that the truth of their premises does not ensure

1770-419: A certain pattern. These observations are then used to form a conclusion either about a yet unobserved entity or about a general law. For abductive inferences, the premises support the conclusion because the conclusion is the best explanation of why the premises are true. The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others. This

1947-412: A conditional statement (formula) and the negation of the consequent ( ¬ Q {\displaystyle \lnot Q} ) and as conclusion the negation of the antecedent ( ¬ P {\displaystyle \lnot P} ). In contrast to modus ponens , reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens

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2124-535: A consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum ), who probably discovered them while identifying sides of the pentagram . The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as

2301-449: A different account of which inferences are valid. For example, the rule of inference known as double negation elimination , i.e. that if a proposition is not not true then it is also true , is accepted in classical logic but rejected in intuitionistic logic . Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference . It applies to arguments that have as first premise

2478-406: A kind of reductio ad absurdum that "...established the deductive organization on the basis of explicit axioms..." as well as "...reinforced the earlier decision to rely on deductive reasoning for proof". This method of exhaustion is the first step in the creation of calculus. Theodorus of Cyrene proved the irrationality of the surds of whole numbers up to 17, but stopped there probably because

2655-402: A large set of candidates. One assigns a certain probability for each candidate to be chosen, and then proves that there is a non-zero probability that a chosen candidate will have the desired property. This does not specify which candidates have the property, but the probability could not be positive without at least one. A probabilistic proof is not to be confused with an argument that a theorem

2832-509: A logical constant may be introduced into a new sentence of the proof . For example, the introduction rule for the logical constant " ∧ {\displaystyle \land } " (and) is " A , B ( A ∧ B ) {\displaystyle {\frac {A,B}{(A\land B)}}} " . It expresses that, given the premises " A {\displaystyle A} " and " B {\displaystyle B} " individually, one may draw

3009-414: A pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning. The ability of deductive reasoning is an important aspect of intelligence and many tests of intelligence include problems that call for deductive inferences. Because of this relation to intelligence, deduction is highly relevant to psychology and

3186-626: A proof by humans either, especially if the proof contains natural language and requires deep mathematical insight to uncover the potential hidden assumptions and fallacies involved. A statement that is neither provable nor disprovable from a set of axioms is called undecidable (from those axioms). One example is the parallel postulate , which is neither provable nor refutable from the remaining axioms of Euclidean geometry . Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with

3363-430: A proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor is not absolute and has varied throughout history. A proof can be presented differently depending on the intended audience. To gain acceptance, a proof has to meet communal standards of rigor; an argument considered vague or incomplete may be rejected. The concept of proof

3540-409: A proof may be found in quadratic irrationals . The proof for the irrationality of the square root of two can be generalized using the fundamental theorem of arithmetic . This asserts that every integer has a unique factorization into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in lowest terms there must be a prime in

3717-484: A proof to show that π is irrational, whence it follows immediately that π is irrational also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different method , which showed that every interval in the reals contains transcendental numbers. Charles Hermite (1873) first proved e transcendental, and Ferdinand von Lindemann (1882), starting from Hermite's conclusions, showed

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3894-403: A remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats. Conversely, suppose we are faced with a repeating decimal , we can prove that it is a fraction of two integers. For example, consider: Here the repetend is 162 and

4071-533: A repeating sequence . For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic ), and in many other ways. As

4248-541: A semantics for what they considered to be the language of thought , whereby standards of mathematical proof might be applied to empirical science . Philosopher-mathematicians such as Spinoza have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having

4425-449: A series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons". The expression "mathematical proof"

4602-433: A set of premises, they are faced with the problem of choosing the relevant rules of inference for their deduction to arrive at their intended conclusion. This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games. In chess , for example,

4779-504: A similar manner to probability , and may be less than full certainty . Inductive logic should not be confused with mathematical induction . Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence or information is acquired. Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers, such as Leibniz , Frege , and Carnap have variously criticized this view and attempted to develop

4956-522: A special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms. Another example is the so-called dual-process theory . This theory posits that there are two distinct cognitive systems responsible for reasoning. Their interrelation can be used to explain commonly observed biases in deductive reasoning. System 1

5133-464: A supposed mathematical fact but only do so by neglecting tiny errors (for example, supposedly straight lines which actually bend slightly) which are unnoticeable until the entire picture is closely examined, with lengths and angles precisely measured or calculated. An elementary proof is a proof which only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis . For some time it

5310-413: A true conclusion given the premises are true. Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness. Probability logic

5487-427: A universal account of deduction for language as an all-encompassing medium. Deductive reasoning usually happens by applying rules of inference . A rule of inference is a way or schema of drawing a conclusion from a set of premises. This happens usually based only on the logical form of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument

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5664-403: Is affirming the consequent , as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor". This is similar to the valid rule of inference named modus ponens , but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is denying the antecedent , as in "if Othello is a bachelor, then he is male; Othello is not

5841-455: Is valid if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false. An argument is sound if it is valid and the premises are true. It is possible to have a deductive argument that is logically valid but is not sound . Fallacious arguments often take that form. The following

6018-459: Is x 0  = (2  + 1) . It is clearly algebraic since it is the root of an integer polynomial, ( x 3 − 1 ) 2 = 2 {\displaystyle (x^{3}-1)^{2}=2} , which is equivalent to ( x 6 − 2 x 3 − 1 ) = 0 {\displaystyle (x^{6}-2x^{3}-1)=0} . This polynomial has no rational roots, since

6195-678: Is 'probably' true, a 'plausibility argument'. The work toward the Collatz conjecture shows how far plausibility is from genuine proof, as does the disproof of the Mertens conjecture . While most mathematicians do not think that probabilistic evidence for the properties of a given object counts as a genuine mathematical proof, a few mathematicians and philosophers have argued that at least some types of probabilistic evidence (such as Rabin's probabilistic algorithm for testing primality ) are as good as genuine mathematical proofs. A combinatorial proof establishes

6372-531: Is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition. The following is an example of an argument using a hypothetical syllogism: Various formal fallacies have been described. They are invalid forms of deductive reasoning. An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them. One type of formal fallacy

6549-399: Is a quarterback" – are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument. In this example, the first statement uses categorical reasoning , saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as term logic – was developed by Aristotle , but

6726-488: Is a ratio of integers and therefore a rational number. Dov Jarden gave a simple non- constructive proof that there exist two irrational numbers a and b , such that a is rational: Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem shows that √ 2 is transcendental , hence irrational. This theorem states that if a and b are both algebraic numbers , and

6903-428: Is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental . The real algebraic numbers are the real solutions of polynomial equations where the coefficients a i {\displaystyle a_{i}} are integers and a n ≠ 0 {\displaystyle a_{n}\neq 0} . An example of an irrational algebraic number

7080-459: Is an irrational number : To paraphrase: if one could write 2 {\displaystyle {\sqrt {2}}} as a fraction , this fraction could never be written in lowest terms, since 2 could always be factored from numerator and denominator. Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville , for instance, proved

7257-435: Is an example of an argument that is “valid”, but not “sound”: The example's first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”. False generalizations – such as "Everyone who eats carrots

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7434-496: Is as follows: Greek mathematicians termed this ratio of incommensurable magnitudes alogos , or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.' Another legend states that Hippasus

7611-415: Is deductive depends on the psychological state of the person making the argument: "An argument is deductive if, and only if, the author of the argument believes that the truth of the premises necessitates (guarantees) the truth of the conclusion". A similar formulation holds that the speaker claims or intends that the premises offer deductive support for their conclusion. This is sometimes categorized as

7788-425: Is difficult to apply to concrete cases since the intentions of the author are usually not explicitly stated. Deductive reasoning is studied in logic , psychology , and the cognitive sciences . Some theorists emphasize in their definition the difference between these fields. On this view, psychology studies deductive reasoning as an empirical mental process, i.e. what happens when humans engage in reasoning. But

7965-450: Is even, then x {\displaystyle x} is even: In proof by contradiction, also known by the Latin phrase reductio ad absurdum (by reduction to the absurd), it is shown that if some statement is assumed true, a logical contradiction occurs, hence the statement must be false. A famous example involves the proof that 2 {\displaystyle {\sqrt {2}}}

8142-445: Is formalized in the field of mathematical logic . A formal proof is written in a formal language instead of natural language. A formal proof is a sequence of formulas in a formal language, starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties,

8319-432: Is good if the author's belief concerning the relation between the premises and the conclusion is true, otherwise it is bad. One consequence of this approach is that deductive arguments cannot be identified by the law of inference they use. For example, an argument of the form modus ponens may be non-deductive if the author's beliefs are sufficiently confused. That brings with it an important drawback of this definition: it

8496-468: Is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false. Aristotle , a Greek philosopher , started documenting deductive reasoning in the 4th century BC. René Descartes , in his book Discourse on Method , refined

8673-472: Is less commonly used to refer to a mathematical proof in the branch of mathematics known as mathematical statistics . See also the " Statistical proof using data " section below. Until the twentieth century it was assumed that any proof could, in principle, be checked by a competent mathematician to confirm its validity. However, computers are now used both to prove theorems and to carry out calculations that are too long for any human or team of humans to check;

8850-575: Is likely that the idea of demonstrating a conclusion first arose in connection with geometry , which originated in practical problems of land measurement. The development of mathematical proof is primarily the product of ancient Greek mathematics , and one of its greatest achievements. Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe

9027-443: Is not a finite number of nonzero digits), unlike any rational number. The same is true for binary , octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases. To show this, suppose we divide integers n by m (where m is nonzero). When long division is applied to the division of n by m , there can never be a remainder greater than or equal to m . If 0 appears as

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9204-495: Is not always precisely observed in the academic literature. One important aspect of this difference is that logic is not interested in whether the conclusion of an argument is sensible. So from the premise "the printer has ink" one may draw the unhelpful conclusion "the printer has ink and the printer has ink and the printer has ink", which has little relevance from a psychological point of view. Instead, actual reasoners usually try to remove redundant or irrelevant information and make

9381-403: Is often explained in terms of probability : the premises make it more likely that the conclusion is true. Strong ampliative arguments make their conclusion very likely, but not absolutely certain. An example of ampliative reasoning is the inference from the premise "every raven in a random sample of 3200 ravens is black" to the conclusion "all ravens are black": the extensive random sample makes

9558-406: Is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up . But this is a misconception that does not reflect how valid deduction is defined in the field of logic : a deduction is valid if it is impossible for its premises to be true while its conclusion is false, independent of whether the premises or

9735-462: Is plausible. A general finding is that people tend to perform better for realistic and concrete cases than for abstract cases. Psychological theories of deductive reasoning aim to explain these findings by providing an account of the underlying psychological processes. Mental logic theories hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference. Mental model theories , on

9912-525: Is possible that their premises are true and their conclusion is false. Two important forms of ampliative reasoning are inductive and abductive reasoning . Sometimes the term "inductive reasoning" is used in a very wide sense to cover all forms of ampliative reasoning. However, in a more strict usage, inductive reasoning is just one form of ampliative reasoning. In the narrow sense, inductive inferences are forms of statistical generalization. They are usually based on many individual observations that all show

10089-511: Is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc. " In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to

10266-406: Is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent , which can be used, for example, to prove

10443-422: Is rational. For some positive integers m and n , we have It follows that The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made

10620-402: Is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false. So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by empirical observation . In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what

10797-470: Is the problem of induction introduced by David Hume . It consists in the challenge of explaining how or whether inductive inferences based on past experiences support conclusions about future events. For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead". According to Karl Popper 's falsificationism, deductive reasoning alone

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10974-399: Is the cards D and 7. Many select card 3 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 3. But this result can be drastically changed if different symbols are used: the visible sides show "drinking a beer", "drinking a coke", "16 years of age", and "22 years of age" and the participants are asked to evaluate

11151-472: Is the case. Hypothetico-deductivism is a closely related scientific method, according to which science progresses by formulating hypotheses and then aims to falsify them by trying to make observations that run counter to their deductive consequences. The term " natural deduction " refers to a class of proof systems based on self-evident rules of inference. The first systems of natural deduction were developed by Gerhard Gentzen and Stanislaw Jaskowski in

11328-486: Is the following: The following is an example of an argument using modus tollens: A hypothetical syllogism is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form: In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in term logic , although it differs in that this subformula

11505-429: Is the older system in terms of evolution. It is based on associative learning and happens fast and automatically without demanding many cognitive resources. System 2, on the other hand, is of more recent evolutionary origin. It is slow and cognitively demanding, but also more flexible and under deliberate control. The dual-process theory posits that system 1 is the default system guiding most of our everyday reasoning in

11682-472: Is transferred from the belief in the premises to the belief in the conclusion in the process of deductive reasoning. Probability logic studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of deductivism denies that there are other correct forms of inference besides deduction. Natural deduction is a type of proof system based on simple and self-evident rules of inference. In philosophy,

11859-556: Is uninformative on the depth level, in contrast to ampliative reasoning. But it may still be valuable on the surface level by presenting the information in the premises in a new and sometimes surprising way. A popular misconception of the relation between deduction and induction identifies their difference on the level of particular and general claims. On this view, deductive inferences start from general premises and draw particular conclusions, while inductive inferences start from particular premises and draw general conclusions. This idea

12036-461: Is used by lay people to refer to using mathematical methods or arguing with mathematical objects , such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" (below), especially when used to argue from data. "Statistical proof" from data refers to the application of statistics, data analysis , or Bayesian analysis to infer propositions regarding

12213-476: Is usually contrasted with non-deductive or ampliative reasoning. The hallmark of valid deductive inferences is that it is impossible for their premises to be true and their conclusion to be false. In this way, the premises provide the strongest possible support to their conclusion. The premises of ampliative inferences also support their conclusion. But this support is weaker: they are not necessarily truth-preserving. So even for correct ampliative arguments, it

12390-604: Is valid if and only if, there is no possible world in which its conclusion is false while its premises are true. This means that there are no counterexamples: the conclusion is true in all such cases, not just in most cases. It has been argued against this and similar definitions that they fail to distinguish between valid and invalid deductive reasoning, i.e. they leave it open whether there are invalid deductive inferences and how to define them. Some authors define deductive reasoning in psychological terms in order to avoid this problem. According to Mark Vorobey, whether an argument

12567-481: Is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called formal fallacies : the truth of their premises does not ensure the truth of their conclusion. In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is classical logic and the rules of inference listed here are all valid in classical logic. But so-called deviant logics provide

12744-418: Is why ratios of incommensurable [quantities] appear... .[Q]uantities are, in other words, continuous". What this means is that contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. In fact, these divisions of quantity must necessarily be infinite . For example, consider a line segment: this segment can be split in half, that half split in half,

12921-626: The Elements , was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem , the Elements also covers number theory , including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers . Further advances also took place in medieval Islamic mathematics . In

13098-581: The Yuktibhāṣā . In the Middle Ages , the development of algebra by Muslim mathematicians allowed irrational numbers to be treated as algebraic objects . Middle Eastern mathematicians also merged the concepts of " number " and " magnitude " into a more general idea of real numbers , criticized Euclid's idea of ratios , developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude. In his commentary on Book 10 of

13275-472: The Elements , the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows: "It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value

13452-590: The Wason selection task . In an often-cited experiment by Peter Wason , 4 cards are presented to the participant. In one case, the visible sides show the symbols D, K, 3, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 3 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%,

13629-412: The binomial theorem and properties of Pascal's triangle . Modern proof theory treats proofs as inductively defined data structures , not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example Axiomatic set theory and Non-Euclidean geometry . As practiced,

13806-470: The certainty of propositions deduced in a mathematical proof, such as Descartes ' cogito argument. Sometimes, the abbreviation "Q.E.D." is written to indicate the end of a proof. This abbreviation stands for "quod erat demonstrandum" , which is Latin for "that which was to be demonstrated" . A more common alternative is to use a square or a rectangle, such as □ or ∎, known as a " tombstone " or "halmos" after its eponym Paul Halmos . Often, "which

13983-437: The irrationality of the square root of two . A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers : Let N = {1, 2, 3, 4, ... } be the set of natural numbers, and let P ( n ) be a mathematical statement involving the natural number n belonging to N such that For example, we can prove by induction that all positive integers of

14160-463: The modus tollens , than with others, like the modus ponens : because the more error-prone forms do not have a native rule of inference but need to be calculated by combining several inferential steps with other rules of inference. In such cases, the additional cognitive labor makes the inferences more open to error. Mental model theories , on the other hand, hold that deductive reasoning involves models or mental representations of possible states of

14337-430: The probability of data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical evidence from outside mathematics to verify. In physics, in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in

14514-534: The quantifiers " ∃ {\displaystyle \exists } " and " ∀ {\displaystyle \forall } " . The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction. But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of sequent calculi or tableau calculi . But other theorists use

14691-401: The rational root theorem shows that the only possibilities are ±1, but x 0 is greater than 1. So x 0 is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials. Almost all irrational numbers are transcendental . Examples are e and π , which are transcendental for all nonzero rational  r. Because

14868-520: The 10th century CE, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers . An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji , who used it to prove

15045-475: The 12th century Bhāskara II evaluated some of these formulas and critiqued them, identifying their limitations. During the 14th to 16th centuries, Madhava of Sangamagrama and the Kerala school of astronomy and mathematics discovered the infinite series for several irrational numbers such as π and certain irrational values of trigonometric functions . Jyeṣṭhadeva provided proofs for these infinite series in

15222-668: The 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place. In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as Hilbert-style deductive systems , which employ axiom schemes to express logical truths . Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how logical constants behave. They are often divided into introduction rules and elimination rules . Introduction rules specify under which conditions

15399-680: The 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. Historian Carl Benjamin Boyer , however, writes that "such claims are not well substantiated and unlikely to be true". Later, in their treatises, Indian mathematicians wrote on the arithmetic of surds including addition, subtraction, multiplication, rationalization, as well as separation and extraction of square roots. Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. In

15576-652: The algebra he used could not be applied to the square root of 17. Geometrical and mathematical problems involving irrational numbers such as square roots were addressed very early during the Vedic period in India. There are references to such calculations in the Samhitas , Brahmanas , and the Shulba Sutras (800 BC or earlier). It is suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since

15753-464: The algebraic numbers form a subfield of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3 π  + 2, π  +  √ 2 and e √ 3 are irrational (and even transcendental). The decimal expansion of an irrational number never repeats (meaning the decimal expansion does not repeat the same number or sequence of numbers) or terminates (this means there

15930-476: The applications of the subject. Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e is irrational if n is rational (unless n  = 0). While Lambert's proof is often called incomplete, modern assessments support it as satisfactory, and in fact for its time it is unusually rigorous. Adrien-Marie Legendre (1794), after introducing the Bessel–;Clifford function , provided

16107-419: The argument in a formal language in order to assess whether it is valid. This often brings with it the difficulty of translating the natural language argument into a formal language, a process that comes with various problems of its own. Another difficulty is due to the fact that the syntactic approach depends on the distinction between formal and non-formal features. While there is a wide agreement concerning

16284-745: The argument whereby its premises are true and its conclusion is false. The syntactic approach, by contrast, focuses on rules of inference , that is, schemas of drawing a conclusion from a set of premises based only on their logical form . There are various rules of inference, such as modus ponens and modus tollens . Invalid deductive arguments, which do not follow a rule of inference, are called formal fallacies . Rules of inference are definitory rules and contrast with strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion. Deductive reasoning contrasts with non-deductive or ampliative reasoning. For ampliative arguments, such as inductive or abductive arguments ,

16461-475: The axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC . Gödel's (first) incompleteness theorem shows that many axiom systems of mathematical interest will have undecidable statements. While early mathematicians such as Eudoxus of Cnidus did not use proofs, from Euclid to the foundational mathematics developments of

16638-406: The claim "[i]f a person is drinking beer, then the person must be over 19 years of age". In this case, 74% of the participants identified correctly that the cards "drinking a beer" and "16 years of age" have to be turned around. These findings suggest that the deductive reasoning ability is heavily influenced by the content of the involved claims and not just by the abstract logical form of the task:

16815-418: The cognitive sciences. But the subject of deductive reasoning is also pertinent to the computer sciences , for example, in the creation of artificial intelligence . Deductive reasoning plays an important role in epistemology . Epistemology is concerned with the question of justification , i.e. to point out which beliefs are justified and why. Deductive inferences are able to transfer the justification of

16992-408: The common syntax explicit. There are various other valid logical forms or rules of inference , like modus tollens or the disjunction elimination . The syntactic approach then holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference. One difficulty for the syntactic approach is that it is usually necessary to express

17169-447: The concept being defined in terms of other concepts already known. Mathematical proof was revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today. It starts with undefined terms and axioms , propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). From this basis, the method proves theorems using deductive logic . Euclid's book,

17346-484: The concept of irrationality, as he attributes the following to irrational magnitudes: "their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it." The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 – 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation in

17523-490: The conclusion " A ∧ B {\displaystyle A\land B} " and thereby include it in one's proof. This way, the symbol " ∧ {\displaystyle \land } " is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule " ( A ∧ B ) A {\displaystyle {\frac {(A\land B)}{A}}} " , which states that one may deduce

17700-467: The conclusion are particular or general. Because of this, some deductive inferences have a general conclusion and some also have particular premises. Cognitive psychology studies the psychological processes responsible for deductive reasoning. It is concerned, among other things, with how good people are at drawing valid deductive inferences. This includes the study of the factors affecting their performance, their tendency to commit fallacies , and

17877-513: The conclusion only repeats information already found in the premises. Ampliative reasoning, on the other hand, goes beyond the premises by arriving at genuinely new information. One difficulty for this characterization is that it makes deductive reasoning appear useless: if deduction is uninformative, it is not clear why people would engage in it and study it. It has been suggested that this problem can be solved by distinguishing between surface and depth information. On this view, deductive reasoning

18054-433: The conclusion very likely, but it does not exclude that there are rare exceptions. In this sense, ampliative reasoning is defeasible: it may become necessary to retract an earlier conclusion upon receiving new related information. Ampliative reasoning is very common in everyday discourse and the sciences . An important drawback of deductive reasoning is that it does not lead to genuinely new information. This means that

18231-412: The conclusion: it is impossible for the premises to be true and the conclusion to be false, independent of any other circumstances. Logical consequence is formal in the sense that it depends only on the form or the syntax of the premises and the conclusion. This means that the validity of a particular argument does not depend on the specific contents of this argument. If it is valid, then any argument with

18408-402: The consequent or denying the antecedent were regarded as valid arguments by the majority of the subjects. An important factor for these mistakes is whether the conclusion seems initially plausible: the more believable the conclusion is, the higher the chance that a subject will mistake a fallacy for a valid argument. An important bias is the matching bias , which is often illustrated using

18585-404: The consequent ( Q {\displaystyle Q} ) obtains as the conclusion from the premises of a conditional statement ( P → Q {\displaystyle P\rightarrow Q} ) and its antecedent ( P {\displaystyle P} ). However, the antecedent ( P {\displaystyle P} ) cannot be similarly obtained as the conclusion from

18762-581: The content rather than the form of the argument. For example, when the conclusion of an argument is very plausible, the subjects may lack the motivation to search for counterexamples among the constructed models. Both mental logic theories and mental model theories assume that there is one general-purpose reasoning mechanism that applies to all forms of deductive reasoning. But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess

18939-576: The definitory rules state that bishops may only move diagonally while the strategic rules recommend that one should control the center and protect one's king if one intends to win. In this sense, definitory rules determine whether one plays chess or something else whereas strategic rules determine whether one is a good or a bad chess player. The same applies to deductive reasoning: to be an effective reasoner involves mastering both definitory and strategic rules. Deductive arguments are evaluated in terms of their validity and soundness . An argument

19116-437: The denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact k th power of another integer, then that first integer's k th root is irrational. Perhaps the numbers most easy to prove irrational are certain logarithms . Here is a proof by contradiction that log 2  3 is irrational (log 2  3 ≈ 1.58 > 0). Assume log 2  3

19293-425: The descriptive question of how actual reasoning happens is different from the normative question of how it should happen or what constitutes correct deductive reasoning, which is studied by logic. This is sometimes expressed by stating that, strictly speaking, logic does not study deductive reasoning but the deductive relation between premises and a conclusion known as logical consequence . But this distinction

19470-411: The equivalence of different expressions by showing that they count the same object in different ways. Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal. A nonconstructive proof establishes that

19647-402: The existence of transcendental numbers by constructing an explicit example . It can also be used to construct a counterexample to disprove a proposition that all elements have a certain property. In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example,

19824-449: The expressions used in the sentences, such as the reference to an object for singular terms or to a truth-value for atomic sentences. The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as model theory is often used to interpret these sentences. Usually, many different interpretations are possible, such as whether a singular term refers to one object or to another. According to

20001-402: The factors determining whether people draw valid or invalid deductive inferences. One such factor is the form of the argument: for example, people draw valid inferences more successfully for arguments of the form modus ponens than of the form modus tollens. Another factor is the content of the arguments: people are more likely to believe that an argument is valid if the claim made in its conclusion

20178-466: The field of automated proof assistants , this is rarely done in practice. A classic question in philosophy asks whether mathematical proofs are analytic or synthetic . Kant , who introduced the analytic–synthetic distinction , believed mathematical proofs are synthetic, whereas Quine argued in his 1951 " Two Dogmas of Empiricism " that such a distinction is untenable. Proofs may be admired for their mathematical beauty . The mathematician Paul Erdős

20355-569: The first proof of the four color theorem is an example of a computer-assisted proof. Some mathematicians are concerned that the possibility of an error in a computer program or a run-time error in its calculations calls the validity of such computer-assisted proofs into question. In practice, the chances of an error invalidating a computer-assisted proof can be reduced by incorporating redundancy and self-checks into calculations, and by developing multiple independent approaches and programs. Errors can never be completely ruled out in case of verification of

20532-528: The first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. A closed chain inference shows that a collection of statements are pairwise equivalent. In order to prove that the statements φ 1 , … , φ n {\displaystyle \varphi _{1},\ldots ,\varphi _{n}} are each pairwise equivalent, proofs are given for

20709-549: The form 2 n  − 1 are odd . Let P ( n ) represent " 2 n  − 1 is odd": The shorter phrase "proof by induction" is often used instead of "proof by mathematical induction". Proof by contraposition infers the statement "if p then q " by establishing the logically equivalent contrapositive statement : "if not q then not p ". For example, contraposition can be used to establish that, given an integer x {\displaystyle x} , if x 2 {\displaystyle x^{2}}

20886-466: The form of square roots and fourth roots . In the 10th century, the Iraqi mathematician Al-Hashimi provided general proofs (rather than geometric demonstrations) for irrational numbers, as he considered multiplication, division, and other arithmetical functions. Many of these concepts were eventually accepted by European mathematicians sometime after the Latin translations of the 12th century . Al-Hassār ,

21063-475: The foundations for the ideas of rationalism . Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts. It is often understood as the evaluative claim that only deductive inferences are good or correct inferences. This theory would have wide-reaching consequences for various fields since it implies that the rules of deduction are "the only acceptable standard of evidence ". This way,

21240-455: The geometrical method is a way of philosophizing that starts from a small set of self-evident axioms and tries to build a comprehensive logical system using deductive reasoning. Deductive reasoning is the psychological process of drawing deductive inferences . An inference is a set of premises together with a conclusion. This psychological process starts from the premises and reasons to a conclusion based on and supported by these premises. If

21417-418: The half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure comes to zero, but it never reaches exactly zero. This is just what Zeno sought to prove. He sought to prove this by formulating four paradoxes , which demonstrated the contradictions inherent in the mathematical thought of

21594-408: The hands of Weierstrass, Leopold Kronecker (Crelle, 101), and Charles Méray . Continued fractions , closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the 19th century were brought into prominence through the writings of Joseph-Louis Lagrange . Dirichlet also added to the general theory, as have numerous contributors to

21771-611: The idea for the Scientific Revolution . Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the scientific method . Descartes' background in geometry and mathematics influenced his ideas on the truth and reasoning, causing him to develop a system of general reasoning now used for most mathematical reasoning. Similar to postulates, Descartes believed that ideas could be self-evident and that reasoning alone must prove that observations are reliable. These ideas also lay

21948-697: The implications φ 1 ⇒ φ 2 {\displaystyle \varphi _{1}\Rightarrow \varphi _{2}} , φ 2 ⇒ φ 3 {\displaystyle \varphi _{2}\Rightarrow \varphi _{3}} , … {\displaystyle \dots } , φ n − 1 ⇒ φ n {\displaystyle \varphi _{n-1}\Rightarrow \varphi _{n}} and φ n ⇒ φ 1 {\displaystyle \varphi _{n}\Rightarrow \varphi _{1}} . The pairwise equivalence of

22125-400: The involvement of natural language, are considered in proof theory . The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice , quasi-empiricism in mathematics , and so-called folk mathematics , oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with

22302-442: The late 19th and 20th centuries, proofs were an essential part of mathematics. With the increase in computing power in the 1960s, significant work began to be done investigating mathematical objects beyond the proof-theorem framework, in experimental mathematics . Early pioneers of these methods intended the work ultimately to be resolved into a classical proof-theorem framework, e.g. the early development of fractal geometry , which

22479-430: The length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain: Now we multiply this equation by 10 where r is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 10 : The result of

22656-414: The manipulation of representations. This is done by applying syntactic rules of inference in a way very similar to how systems of natural deduction transform their premises to arrive at a conclusion. On this view, some deductions are simpler than others since they involve fewer inferential steps. This idea can be used, for example, to explain why humans have more difficulties with some deductions, like

22833-439: The more realistic and concrete the cases are, the better the subjects tend to perform. Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative material conditional , as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge

23010-419: The most famous and surprising being that almost all axiomatic systems can generate certain undecidable statements not provable within the system. The definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside

23187-468: The necessary logical foundation for incommensurable ratios". This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. It was not until Eudoxus developed a theory of proportion that took into account irrational as well as rational ratios that a strong mathematical foundation of irrational numbers was created. As a result of the distinction between number and magnitude, geometry became

23364-476: The only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases, algebraic conceptions were reformulated into geometric terms. This may account for why we still conceive of x and x as x squared and x cubed instead of x to

23541-407: The other hand, claim that deductive reasoning involves models of possible states of the world without the medium of language or rules of inference. According to dual-process theories of reasoning, there are two qualitatively different cognitive systems responsible for reasoning. The problem of deduction is relevant to various fields and issues. Epistemology tries to understand how justification

23718-424: The other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning

23895-423: The paradigmatic cases, there are also various controversial cases where it is not clear how this distinction is to be drawn. The semantic approach suggests an alternative definition of deductive validity. It is based on the idea that the sentences constituting the premises and conclusions have to be interpreted in order to determine whether the argument is valid. This means that one ascribes semantic values to

24072-404: The premises are true. Because of this, the evaluation of some forms of inference only requires the construction of very few models while for others, many different models are necessary. In the latter case, the additional cognitive labor required makes deductive reasoning more error-prone, thereby explaining the increased rate of error observed. This theory can also explain why some errors depend on

24249-429: The premises of a valid argument are true, then it is called a sound argument. The relation between the premises and the conclusion of a deductive argument is usually referred to as " logical consequence ". According to Alfred Tarski , logical consequence has 3 essential features: it is necessary, formal, and knowable a priori . It is necessary in the sense that the premises of valid deductive arguments necessitate

24426-471: The premises of the conditional statement ( P → Q {\displaystyle P\rightarrow Q} ) and the consequent ( Q {\displaystyle Q} ). Such an argument commits the logical fallacy of affirming the consequent . The following is an example of an argument using modus ponens: Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises

24603-425: The premises offer weaker support to their conclusion: they indicate that it is most likely, but they do not guarantee its truth. They make up for this drawback with their ability to provide genuinely new information (that is, information not already found in the premises), unlike deductive arguments. Cognitive psychology investigates the mental processes responsible for deductive reasoning. One of its topics concerns

24780-415: The premises onto the conclusion. So while logic is interested in the truth-preserving nature of deduction, epistemology is interested in the justification-preserving nature of deduction. There are different theories trying to explain why deductive reasoning is justification-preserving. According to reliabilism , this is the case because deductions are truth-preserving: they are reliable processes that ensure

24957-443: The ratio π of a circle's circumference to its diameter, Euler's number e , the golden ratio φ , and the square root of two . In fact, all square roots of natural numbers , other than of perfect squares , are irrational. Like all real numbers, irrational numbers can be expressed in positional notation , notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with

25134-422: The ratio of two integers . When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable , meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are

25311-546: The rationality or correctness of the different forms of inductive reasoning is denied. Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all. One motivation for deductivism

25488-495: The reasoning was done correctly, it results in a valid deduction: the truth of the premises ensures the truth of the conclusion. For example, in the syllogistic argument "all frogs are amphibians; no cats are amphibians; therefore, no cats are frogs" the conclusion is true because its two premises are true. But even arguments with wrong premises can be deductively valid if they obey this principle, as in "all frogs are mammals; no cats are mammals; therefore, no cats are frogs". If

25665-406: The relevant information more explicit. The psychological study of deductive reasoning is also concerned with how good people are at drawing deductive inferences and with the factors determining their performance. Deductive inferences are found both in natural language and in formal logical systems , such as propositional logic . Deductive arguments differ from non-deductive arguments in that

25842-399: The resurgence of the scientific study of the theory of irrationals, largely ignored since Euclid . The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine ( Crelle's Journal , 74), Georg Cantor (Annalen, 5), and Richard Dedekind . Méray had taken in 1869 the same point of departure as Heine, but the theory is generally referred to

26019-598: The role of language and logic in proofs, and mathematics as a language . The word "proof" comes from the Latin probare (to test). Related modern words are English "probe", "probation", and "probability", Spanish probar (to smell or taste, or sometimes touch or test), Italian provare (to try), and German probieren (to try). The legal term "probity" means authority or credibility, the power of testimony to prove facts when given by persons of reputation or status. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It

26196-424: The same arrangement, even if their contents differ. For example, the arguments "if it rains then the street will be wet; it rains; therefore, the street will be wet" and "if the meat is not cooled then it will spoil; the meat is not cooled; therefore, it will spoil" have the same logical form: they follow the modus ponens . Their form can be expressed more abstractly as "if A then B; A; therefore B" in order to make

26373-418: The same for π. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and was finally made elementary by Adolf Hurwitz and Paul Gordan . The square root of 2 was likely the first number proved irrational. The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers that are not perfect squares are irrational and

26550-411: The same logical form is also valid, no matter how different it is on the level of its contents. Logical consequence is knowable a priori in the sense that no empirical knowledge of the world is necessary to determine whether a deduction is valid. So it is not necessary to engage in any form of empirical investigation. Some logicians define deduction in terms of possible worlds : A deductive inference

26727-502: The second power and x to the third power. Also crucial to Zeno's work with incommensurable magnitudes was the fundamental focus on deductive reasoning that resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory. Out of this necessity, Eudoxus developed his method of exhaustion ,

26904-412: The semantic approach, an argument is deductively valid if and only if there is no possible interpretation where its premises are true and its conclusion is false. Some objections to the semantic approach are based on the claim that the semantics of a language cannot be expressed in the same language, i.e. that a richer metalanguage is necessary. This would imply that the semantic approach cannot provide

27081-520: The sentence " A {\displaystyle A} " from the premise " ( A ∧ B ) {\displaystyle (A\land B)} " . Similar introduction and elimination rules are given for other logical constants, such as the propositional operator " ¬ {\displaystyle \lnot } " , the propositional connectives " ∨ {\displaystyle \lor } " and " → {\displaystyle \rightarrow } " , and

27258-586: The statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture , or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic . Purely formal proofs , written fully in symbolic language without

27435-400: The statements then results from the transitivity of the material conditional . A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of probability theory . Probabilistic proof, like proof by construction, is one of many ways to prove existence theorems . In the probabilistic method, one seeks an object having a given property, starting with

27612-403: The sum of two even integers is always even: This proof uses the definition of even integers, the integer properties of closure under addition and multiplication, and the distributive property . Despite its name, mathematical induction is a method of deduction , not a form of inductive reasoning . In proof by mathematical induction, a single "base case" is proved, and an "induction rule"

27789-409: The term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students. Irrational number In mathematics , the irrational numbers ( in- + rational ) are all the real numbers that are not rational numbers . That is, irrational numbers cannot be expressed as

27966-570: The time. While Zeno's paradoxes accurately demonstrated the deficiencies of contemporary mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore, further investigation had to occur. The next step was taken by Eudoxus of Cnidus , who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea

28143-435: The truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true. Rules of inferences are definitory rules: they determine whether an argument is deductively valid or not. But reasoners are usually not just interested in making any kind of valid argument. Instead, they often have a specific point or conclusion that they wish to prove or refute. So given

28320-407: The truth of their premises ensures the truth of their conclusion. There are two important conceptions of what this exactly means. They are referred to as the syntactic and the semantic approach. According to the syntactic approach, whether an argument is deductively valid depends only on its form, syntax, or structure. Two arguments have the same form if they use the same logical vocabulary in

28497-415: The two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000 A matches the tail end of 10 A exactly. Here, both 10,000 A and 10 A have .162 162 162 ... after the decimal point. Therefore, when we subtract the 10 A equation from the 10,000 A equation, the tail end of 10 A cancels out the tail end of 10,000 A leaving us with: Then

28674-406: The underlying biases involved. A notable finding in this field is that the type of deductive inference has a significant impact on whether the correct conclusion is drawn. In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated modus ponens inferences correctly, while the success rate for modus tollens was only 72%. On the other hand, even some fallacies like affirming

28851-438: The underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more susceptible to some types of fallacies than to others. An important distinction is between mental logic theories , sometimes also referred to as rule theories , and mental model theories . Mental logic theories see deductive reasoning as a language -like process that happens through

29028-405: The validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left". Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to

29205-410: The work of Leonardo Fibonacci in the 13th century. The 17th century saw imaginary numbers become a powerful tool in the hands of Abraham de Moivre , and especially of Leonhard Euler . The completion of the theory of complex numbers in the 19th century entailed the differentiation of irrationals into algebraic and transcendental numbers , the proof of the existence of transcendental numbers, and

29382-492: The world without the medium of language or rules of inference. In order to assess whether a deductive inference is valid, the reasoner mentally constructs models that are compatible with the premises of the inference. The conclusion is then tested by looking at these models and trying to find a counterexample in which the conclusion is false. The inference is valid if no such counterexample can be found. In order to reduce cognitive labor, only such models are represented in which

29559-528: The year 1872. Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880, and Dedekind's has received additional prominence through the author's later work (1888) and the endorsement by Paul Tannery (1894). Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers , separating them into two groups having certain characteristic properties. The subject has received later contributions at

29736-458: Was brought to light by Zeno of Elea , who questioned the conception that quantities are discrete and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for "whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects", but Zeno found that in fact "[quantities] in general are not discrete collections of units; this

29913-485: Was known for describing proofs which he found to be particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK , published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing. In direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems. For example, direct proof can be used to prove that

30090-460: Was merely exiled for this revelation. Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that numbers and geometry were inseparable; a foundation of their theory. The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. This

30267-410: Was superseded by propositional (sentential) logic and predicate logic . Deductive reasoning can be contrasted with inductive reasoning , in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means). Deductive reasoning

30444-403: Was that log 2  3 is rational (and so expressible as a quotient of integers m / n with n  ≠ 0). The contradiction means that this assumption must be false, i.e. log 2  3 is irrational, and can never be expressed as a quotient of integers m / n with n  ≠ 0. Cases such as log 10  2 can be treated similarly. An irrational number may be algebraic , that

30621-474: Was the distinction between magnitude and number. A magnitude "...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5". Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus

30798-458: Was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. "Eudoxus' theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying

30975-455: Was thought that certain theorems, like the prime number theorem , could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques. A particular way of organising a proof using two parallel columns is often used as a mathematical exercise in elementary geometry classes in the United States. The proof is written as

31152-399: Was to be shown" is verbally stated when writing "QED", "□", or "∎" during an oral presentation. Unicode explicitly provides the "end of proof" character, U+220E (∎) (220E(hex) = 8718(dec)) . Deductive reasoning Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid if there is no possible interpretation of

31329-409: Was ultimately so resolved. Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a " proof without words ". The left-hand picture below is an example of a historic visual proof of the Pythagorean theorem in the case of the (3,4,5) triangle . Some illusory visual proofs, such as the missing square puzzle , can be constructed in a way which appear to prove

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