Deductive reasoning - Knowledge (XXG)

1319:, 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 "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%, 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 the claim "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: the more realistic and concrete the cases are, the better the subjects tend to perform. 1382:

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 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 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.

1390:. 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 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 a pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning. 301:. But this distinction 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 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 222:. It is necessary in the sense that the premises of valid deductive arguments necessitate 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 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 1237:: 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 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 1104: 1245:

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 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.

1547:". This way, 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. 2118: 1439:, this is the case because deductions are truth-preserving: they are reliable processes that ensure 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. 128:

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 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.

1326:, 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 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". 1464: 1435:, i.e. to point out which beliefs are justified and why. Deductive inferences are able to transfer the justification of 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 1082:

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

1517:. 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 the foundations for the ideas of 1386:

contexts. In this sense, it has been claimed that humans possess 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

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to avoid this problem. According to Mark Vorobey, whether an argument 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

1558:. 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 5721: 2104: 1225:. 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 1562:'s falsificationism, deductive reasoning alone 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 121:, 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. 284:. 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 the descriptive question of how actual reasoning happens is different from the 2556:

Modern logicians sometimes oppose deduction to induction on the basis that the first concludes from the general to the particular, whereas the second concludes from the particular to the general; this characterization is inaccurate, however, since deduction need not conclude to the particular and its

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of possible states of 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

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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

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One recurrent criticism of philosophical systems build using the geometrical method is that their initial axioms are not as self-evident or certain as their defenders proclaim. This problem lies beyond the deductive reasoning itself, which only ensures that the conclusion is true if the premises are

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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

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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

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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

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investigates the mental processes responsible for deductive reasoning. One of its topics concerns 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

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However if the premises were true but the conclusion were false, a hidden assumption could be intervening; alternatively, a hidden process might be coercing the form of presentation, and so forth; then the task would be to unearth the hidden factors in an ill-formed syllogism, in order to make the

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An important drawback of deductive reasoning is that it does not lead to genuinely new information. This means that 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

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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

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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 a set of

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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 the same arrangement, even if their contents differ. For example, the arguments "if it

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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 the paradigmatic cases, there are

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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

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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

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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 is

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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 the semantic approach, an argument is deductively valid if and only if there is no possible interpretation where its

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all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is

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Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens. (First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural

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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

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Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to the underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more

1261:: 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 the conclusion are particular or general. Because of this, some deductive inferences have a general conclusion and some also have particular premises. 198:

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;

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by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system.

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that all show 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.

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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.

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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 is good if the author's

347:. 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 the argument in a 169:

is a type of proof system based on simple and self-evident rules of inference. In philosophy, 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.

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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

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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 the

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of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called

1371:: 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. 2091:

axiom, i.e. that "the knowledge of an effect depends on and involves knowledge of its cause". A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.

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in 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

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may be non-deductive if the author's beliefs are sufficiently confused. That brings with it an important drawback of this definition: it is difficult to apply to concrete cases since the intentions of the author are usually not explicitly stated.

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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

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not “sound”. False generalizations – such as "Everyone who eats carrots 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.

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this support is weaker: they are not necessarily truth-preserving. So even for correct ampliative arguments, it is possible that their premises are true and their conclusion is false. Two important forms of ampliative reasoning are

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include problems that call for deductive inferences. Because of this relation to intelligence, deduction is highly relevant to psychology and the cognitive sciences. But the subject of deductive reasoning is also pertinent to the

2034:. But other theorists use 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. 1286:

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

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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

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Deductive arguments differ from non-deductive arguments in that 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

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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

1073:. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true. 1534:

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

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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.

230:: A deductive inference 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 764:

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent

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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

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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

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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.

1685: 1600:. 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 1832: 930:

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:

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in order to determine whether the argument is valid. This means that one ascribes semantic values to the expressions used in the sentences, such as the reference to an object for

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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

1012: 984: 957: 842: 710: 644: 549: 479: 1967: 335:. Their form can be expressed more abstractly as "if A then B; A; therefore B" in order to make the common syntax explicit. There are various other valid logical forms or 106:. 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. 1892: 1233:

The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others. This is often explained in terms of

2018: 1994: 1763: 136:, on 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 3461: 2319: 2299: 1358:-like process that happens through the manipulation of representations. This is done by applying syntactic rules of inference in a way very similar to how systems of 1057:, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male". This is similar to the valid rule of inference called 892: 867: 809: 786: 2087:

true, but not that the premises themselves are true. For example, Spinoza's philosophical system has been criticized this way based on objections raised against the

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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

49:, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and " 5778: 2339: 1856: 1733: 1709: 730: 684: 664: 618: 594: 572: 519: 499: 1249:

and draw general conclusions. This idea is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is

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is necessary. This would imply that the semantic approach cannot provide a universal account of deduction for language as an all-encompassing medium.

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In one sense one can see the psychology of deductive reasoning as being as old as the study of logic, which originated in the writings of Aristotle.

815:, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following: 1049:, 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 1022:, although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition. 4569: 5114: 5042: 5032: 4963: 4624: 4587: 4455: 4407: 4093: 3710: 3662: 2956: 2843: 2760: 68:

invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning.

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hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference.

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for atomic sentences. The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as

5377: 5164: 5153: 3171: 1583:" refers to a class of proof systems based on self-evident rules of inference. The first systems of natural deduction were developed by 1070: 5739: 5506: 5207: 4481: 5771: 5655: 5093: 1018:

In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in

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studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of

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D'Agostino, Marcello; Floridi, Luciano (2009). "The Enduring Scandal of Deduction: Is Propositional Logic Really Uninformative?".

1513:. Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the 3963:

Müller, Ulrich; Overton, Willis F.; Reene, Kelly (February 2001). "Development of Conditional Reasoning: A Longitudinal Study".

392:. 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 6060: 5331: 1791:

is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule

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In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is

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García-Madruga, Juan A.; Gutiérrez, Francisco; Carriedo, Nuria; Moreno, Sergio; Johnson-Laird, Philip N. (November 2002).

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Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative

1053:, but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is 6050: 5764: 2187: 1432: 150: 1896:. Similar introduction and elimination rules are given for other logical constants, such as the propositional operator 1566:. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case. 254:

definition of deduction since it depends also on the speaker whether the argument in question is deductive or not. For

2031: 1061:, the difference being that the second premise and the conclusion are switched around. Other formal fallacies include 413: 1608:. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the 5686: 3462:"Strategic vs Definitory Rules: Their Role in Abductive Argumentation and their Relationship with Deductive Proof" 1090:

may only move diagonally while the strategic rules recommend that one should control the center and protect one's

5665: 5590: 5372: 1181:, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as 5563: 3344: 2079: 1641: 1296: 1046: 733: 190:

to a conclusion based on and supported by these premises. If the reasoning was done correctly, it results in a

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axioms and tries to build a comprehensive logical system based only on deductive inferences from these first

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Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive

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The relation between the premises and the conclusion of a deductive argument is usually referred to as "

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provide a different account of which inferences are valid. For example, the rule of inference known as

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in order to assess whether it is valid. This often brings with it the difficulty of translating the

5787: 5575: 5558: 5501: 5450: 5445: 5387: 5324: 5298: 5288: 2212: 2177: 2132: 1588: 1506: 1387: 1257:. But this is a misconception that does not reflect how valid deduction is defined in the field of 1222: 1218: 1201: 310: 298: 250:

that the premises offer deductive support for their conclusion. This is sometimes categorized as a

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Adams, Ernest W. (13 October 1998). "Deduction and Probability: What Probability Logic Is About".

1952: 6065: 5851: 5511: 5440: 5397: 4913: 4604: 4543: 4214: 4054: 3980: 3874: 3866: 3813: 3805: 3547: 3491: 3299: 3291: 2927: 2680: 2222: 2063: 1972: 1921: 1563: 1066: 686:) cannot be similarly obtained as the conclusion from the premises of the conditional statement ( 389: 336: 281: 142:

of reasoning, there are two qualitatively different cognitive systems responsible for reasoning.

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Arreckx, Frederique (2007). "Experiment 1: Affirmative and negative counterfactual questions".

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Copi, Irving M.; Cohen, Carl; Rodych, Victor (3 September 2018). "1. Basic Logical Concepts".

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deduction: the truth of the premises ensures the truth of the conclusion. For example, in the

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also various controversial cases where it is not clear how this distinction is to be drawn.

352: 302: 191: 103: 38: 1928: 5998: 5847: 5476: 5418: 2202: 2162: 2157: 2027: 1584: 1194: 405: 348: 4702: 4492: 3432: 4866: 4675: 6055: 5800: 5730: 5423: 5402: 5317: 4268: 4233: 4085: 4011: 3998:

Evans, J. St B. T.; Lynch, J. S. (August 1973). "Matching Bias in the Selection Task".

3017: 2324: 2055: 1841: 1718: 1694: 1283: 1087: 715: 669: 649: 603: 579: 557: 504: 484: 408:

and the rules of inference listed here are all valid in classical logic. But so-called

227: 157:

in the premises to the belief in the conclusion in the process of deductive reasoning.

3754: 3727: 1463: 218:, logical consequence has 3 essential features: it is necessary, formal, and knowable 90:, that is, schemas of drawing a conclusion from a set of premises based only on their 6044: 5580: 5521: 5237: 4917: 4547: 4058: 3984: 3892: 3817: 3551: 3495: 2931: 1597: 1364: 1292: 1058: 759: 521:) of the conditional statement as its conclusion. The argument form is listed below: 409: 364: 340: 215: 99: 5140: 4218: 3303: 186:

together with a conclusion. This psychological process starts from the premises and

5993: 5978: 5570: 5392: 4317: 4080:. Handbook of the History of Logic. Vol. 10. North-Holland. pp. 553–624. 3878: 2749:

Evans, Jonathan (18 April 2005). "Deductive reasoning". In Morrison, Robert (ed.).

2047: 1609: 1428: 1403: 1368: 1288: 1091: 1050: 812: 442: 393: 377: 372: 332: 265: 146: 102:. Invalid deductive arguments, which do not follow a rule of inference, are called 95: 91: 17: 4370: 4616: 4445: 4397: 2946: 2833: 2750: 5973: 5863: 5833: 5605: 5600: 5553: 5232: 4579: 4291: 3612: 3318: 2059: 1559: 1518: 1436: 1316: 1234: 1226: 368: 195: 162: 3976: 82:

of the argument whereby its premises are true and its conclusion is false. The

6018: 6008: 5915: 5890: 5843: 5548: 5516: 5481: 5144: 4660: 4202: 4168: 4042: 3934: 3844: 3534: 3517: 3279: 2923: 2123: 2099: 2071: 2043: 1555: 1213: 1182: 1178: 1019: 277: 110: 4259: 4050: 3942: 3862: 3801: 3543: 3487: 3287: 2676: 2519: 1159:

The following is an example of an argument that is “valid”, but not “sound”:

401:: the truth of their premises does not ensure the truth of their conclusion. 5950: 5930: 5838: 5813: 5610: 5471: 5382: 5172: 4955: 4538: 4250: 3918: 3478: 2192: 2062:

philosophical systems in the modern era. It gets its name from the forms of

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The following is an example of an argument using a hypothetical syllogism:

620:) obtains as the conclusion from the premises of a conditional statement ( 165:

denies that there are other correct forms of inference besides deduction.

53:

is a man" to the conclusion "Socrates is mortal" is deductively valid. An

6003: 5983: 5935: 5920: 5858: 5818: 5531: 5247: 5224: 5216: 4833: 2663:

Norris, Stephen E. (1975). "The Intelligibility of Practical Reasoning".

2557:

process is far from being the logical inverse of the inductive procedure.

2182: 2067: 1544: 1355: 327: 219: 200: 75: 54: 50: 5149: 4128: 4112:

COUNTERFACTUAL THINKING AND THE FALSE BELIEF TASK: A DEVELOPMENTAL STUDY

3870: 3518:"Varieties of Justification—How (Not) to Solve the Problem of Induction" 3295: 2684: 5910: 5900: 5895: 5878: 5595: 5526: 4909: 2278: 2197: 2172: 2075: 1399: 1238: 1131: 183: 46: 5756: 4521: 4152: 3809: 1377:, on the other hand, hold that deductive reasoning involves models or 178:

Deductive reasoning is the psychological process of drawing deductive

5945: 5940: 5925: 5823: 5433: 4110: 3853: 2088: 1035:

Thus, if there had been a thunderstorm, things would have gotten wet.

260: 187: 154: 74:

studies under what conditions an argument is valid. According to the

4424: 3631: 3395: 3248: 3125: 3082: 2907: 2078:. An important motivation of the geometrical method is to repudiate 3793: 2867: 2637: 2511: 145:

The problem of deduction is relevant to various fields and issues.

5625: 5340: 4980: 4644: 4120: 2051: 1258: 1083: 273: 42: 5159: 4720:"Karl Popper: 4. Basic Statements, Falsifiability and Convention" 4423:

Hájek, Alan (2001). "Probability, Logic, and Probability Logic".

1501:, started documenting deductive reasoning in the 4th century BC. 5873: 5828: 5585: 5006: 4344: 3466:

Eurasia Journal of Mathematics, Science and Technology Education

2832:

Tarski, Alfred (1983). "On The Concept of Logical Consequence".

1212:

Deductive reasoning is usually contrasted with non-deductive or

898:

The following is an example of an argument using modus tollens:

5760: 5313: 5189: 5176: 4746: 4072:

Chater, Nick; Oaksford, Mike; Hahn, Ulrike; Heit, Evan (2011).

3580: 3195: 2988: 739:

The following is an example of an argument using modus ponens:

5121:

Argumentation: The Study of Effective Reasoning Parts I and II

2800: 1612:. For example, the introduction rule for the logical constant 1458: 1148:

It is possible to have a deductive argument that is logically

3396:"Denying the Antecedent: Its Effective Use in Argumentation" 2046:

based on deductive reasoning. It starts from a small set of

1767:

and thereby include it in one's proof. This way, the symbol

5185: 5086:

Thought 2 Talk: A Crash Course in Reflection and Expression

3081:

Jaakko, Hintikka; Sandu, Gabriel (2006). "What is Logic?".

2835:

Logic, Semantics, Metamathematics: Papers from 1923 to 1938

501:) of the conditional statement. It obtains the consequent ( 3780:

Bunge, Mario (1960). "The Place of Induction in Science".

1295:

was only 72%. On the other hand, even some fallacies like

199:

therefore, no cats are frogs". If the premises of a valid

3222:"Intuitionistic Logic: 1. Rejection of Tertium Non Datur" 5309: 2974: 2972: 2970: 2968: 894:. (Conclusion deduced is the negation of the antecedent) 5745: 3266:

Morreau, Michael (2009). "The Hypothetical Syllogism".

3012: 3010: 3008: 3006: 2427:{\displaystyle \Gamma A_{1},A_{2},A_{3}...\vdash C_{1}} 1475: 1335:

susceptible to some types of fallacies than to others.

1029:

If there had been a thunderstorm, it would have rained.

451:. It applies to arguments that have as first premise a 78:

approach, an argument is valid if there is no possible

2362: 2327: 2307: 2287: 2006: 1982: 1955: 1931: 1904: 1868: 1844: 1799: 1775: 1745: 1721: 1697: 1644: 1620: 994: 966: 939: 877: 852: 824: 794: 771: 718: 692: 672: 652: 626: 606: 600:

In this form of deductive reasoning, the consequent (

582: 560: 531: 507: 487: 461: 4151:

Johnson-Laird, Philip N.; Byrne, Ruth M. J. (1993).

4146: 4144: 4142: 4140: 4138: 3606: 3604: 3602: 3600: 3598: 3427: 3425: 3423: 2489: 2487: 2485: 2483: 2481: 1111:

Deductive arguments are evaluated in terms of their

869:. (Second premise is the negation of the consequent) 788:) and as conclusion the negation of the antecedent ( 109:

Deductive reasoning contrasts with non-deductive or

5959: 5794: 5674: 5646: 5639: 5494: 5459: 5411: 5365: 5279: 5246: 5223: 2479: 2477: 2475: 2473: 2471: 2469: 2467: 2465: 2463: 2461: 902:

If it is raining, then there are clouds in the sky.

743:

If it is raining, then there are clouds in the sky.

732:). Such an argument commits the logical fallacy of 4860: 4858: 4856: 4854: 4609:Twenty-Five Years of Logical Methodology in Poland 2426: 2333: 2313: 2293: 2012: 1988: 1961: 1937: 1910: 1886: 1850: 1826: 1781: 1757: 1727: 1703: 1679: 1626: 1006: 978: 951: 886: 861: 836: 803: 780: 724: 704: 678: 658: 638: 612: 588: 566: 543: 513: 493: 473: 4605:"The Controversy: Deductivism Versus Inductivism" 4440:Evans, Jonathan St. B. T.; Newstead, Stephen E.; 4180: 4178: 1431:. Epistemology is concerned with the question of 1402:of deductive reasoning is an important aspect of 1291:inferences correctly, while the success rate for 552: (First premise is a conditional statement) 424:, is accepted in classical logic but rejected in 4865:Pelletier, Francis Jeffry; Hazen, Allen (2021). 4515: 4513: 3919:"Cognitive processes in propositional reasoning" 3703:International Handbook of Thinking and Reasoning 3619:. Metaphysics Research Lab, Stanford University. 3345:"Expert thinking and novice thinking: Deduction" 2752:The Cambridge Handbook of Thinking and Reasoning 1836:, which states that one may deduce the sentence 388:Deductive reasoning usually happens by applying 94:. There are various rules of inference, such as 5031:Nadler, Steven (2006). "The geometric method". 4937: 4935: 4933: 4931: 4894:"Untersuchungen über das logische Schließen. I" 4873:. Metaphysics Research Lab, Stanford University 4726:. Metaphysics Research Lab, Stanford University 4682:. Metaphysics Research Lab, Stanford University 4450:(Repr. ed.). Psychology Press. p. 4. 4377:. Metaphysics Research Lab, Stanford University 4324:. Metaphysics Research Lab, Stanford University 4238:Proceedings of the National Academy of Sciences 3761:. Metaphysics Research Lab, Stanford University 3734:. Metaphysics Research Lab, Stanford University 3567: 3565: 3563: 3561: 3325:. Metaphysics Research Lab, Stanford University 3228:. Metaphysics Research Lab, Stanford University 3189: 3187: 3185: 3151:. Metaphysics Research Lab, Stanford University 1427:Deductive reasoning plays an important role in 1086:, for example, the definitory rules state that 1032:If it had rained, things would have gotten wet. 5058:Doppelt, Torin (2010). "The Truth About 1A4". 5037:. Cambridge University Press. pp. 35–51. 4948:The Palgrave Handbook of Philosophical Methods 5772: 5325: 5201: 4827: 4825: 4823: 4821: 4819: 4817: 2908:"Defining Deduction, Induction, and Validity" 2794: 2792: 1330:Psychological theories of deductive reasoning 1156:. Fallacious arguments often take that form. 596: (Conclusion deduced is the consequent) 113:reasoning. For ampliative arguments, such as 8: 4447:Human Reasoning: The Psychology of Deduction 4232:Johnson-Laird, Philip N. (18 October 2010). 3455: 3453: 3076: 3074: 3072: 3070: 3068: 3066: 3064: 3062: 3060: 3058: 2790: 2788: 2786: 2784: 2782: 2780: 2778: 2776: 2774: 2772: 2631: 2629: 2627: 2625: 2623: 2621: 2619: 2617: 2615: 2613: 2611: 2609: 2607: 2605: 2603: 2601: 2599: 2597: 2595: 2593: 2591: 2589: 2587: 2585: 844:. (First premise is a conditional statement) 4701:Russell, Bertrand (2009) . "On Induction". 3511: 3509: 3507: 3505: 3056: 3054: 3052: 3050: 3048: 3046: 3044: 3042: 3040: 3038: 2583: 2581: 2579: 2577: 2575: 2573: 2571: 2569: 2567: 2565: 1200:Deductive reasoning can be contrasted with 1163:Everyone who eats carrots is a quarterback. 5779: 5765: 5757: 5643: 5408: 5332: 5318: 5310: 5208: 5194: 5186: 5173: 4611:. Springer Netherlands. pp. 261–278. 4426:The Blackwell Guide to Philosophical Logic 4074:"Inductive Logic and Empirical Psychology" 3638:(in Portuguese) (2nd ed.). Macmillan. 2901: 2899: 2897: 2895: 1737:individually, one may draw the conclusion 1177:In this example, the first statement uses 5088:, New York: Automatic Press / VIP, 2005, 4563: 4561: 4559: 4557: 4537: 4267: 4249: 3852: 3533: 3477: 3411: 2883: 2861: 2859: 2857: 2855: 2418: 2396: 2383: 2370: 2361: 2326: 2306: 2286: 2005: 1981: 1954: 1930: 1903: 1867: 1843: 1800: 1798: 1774: 1744: 1720: 1696: 1680:{\displaystyle {\frac {A,B}{(A\land B)}}} 1645: 1643: 1619: 993: 965: 938: 876: 851: 823: 793: 770: 717: 691: 671: 651: 625: 605: 581: 574: (Second premise is the antecedent) 559: 530: 506: 486: 460: 4638: 4636: 3100: 3098: 3096: 3094: 2827: 2825: 2823: 2821: 2744: 2742: 2740: 2738: 2736: 2734: 2732: 2730: 2728: 2726: 2724: 2722: 2720: 2718: 2716: 2714: 2494:Johnson-Laird, Phil (30 December 2009). 1689:. It expresses that, given the premises 1606:introduction rules and elimination rules 1596:, which employ axiom schemes to express 1102: 481:) and as second premise the antecedent ( 4871:The Stanford Encyclopedia of Philosophy 4724:The Stanford Encyclopedia of Philosophy 4680:The Stanford Encyclopedia of Philosophy 4375:The Stanford Encyclopedia of Philosophy 4322:The Stanford Encyclopedia of Philosophy 4025:Wagner-Egger, Pascal (1 October 2007). 3759:The Stanford Encyclopedia of Philosophy 3732:The Stanford Encyclopedia of Philosophy 3617:The Stanford Encyclopedia of Philosophy 3323:The Stanford Encyclopedia of Philosophy 3226:The Stanford Encyclopedia of Philosophy 3149:The Stanford Encyclopedia of Philosophy 2712: 2710: 2708: 2706: 2704: 2702: 2700: 2698: 2696: 2694: 2543: 2541: 2539: 2537: 2457: 2341:; a single sequent would take the form 2270: 1827:{\displaystyle {\frac {(A\land B)}{A}}} 1311:, which is often illustrated using the 4944:"Introduction and Historical Overview" 4187:"Mental Models in Deductive Reasoning" 3649:Scott, John; Marshall, Gordon (2009). 3111:New Dictionary of the History of Ideas 2233:Peirce's theory of deductive reasoning 2042:The geometrical method is a method of 1550:One motivation for deductivism is the 1359: 4950:. Palgrave Macmillan. pp. 1–30. 4747:"Popper, Karl: Philosophy of Science" 4568:Howson, Colin (2000). "Deductivism". 3699:"Abductive reasoning and explanation" 2070:, which are usually based on axioms, 7: 4867:"Natural Deduction Systems in Logic" 4369:Goldman, Alvin; Beddor, Bob (2021). 3965:Journal of Cognition and Development 3636:Macmillan Encyclopedia of Philosophy 3253:Macmillan Encyclopedia of Philosophy 3176:Macmillan Encyclopedia of Philosophy 3130:Routledge Encyclopedia of Philosophy 1604:behave. They are often divided into 1338:An important distinction is between 1208:Difference from ampliative reasoning 5507:Analytic and synthetic propositions 5378:Formal semantics (natural language) 5165:Internet Encyclopedia of Philosophy 5154:Indiana Philosophy Ontology Project 5034:Spinoza's 'Ethics': An Introduction 5011:Internet Encyclopedia of Philosophy 4985:Internet Encyclopedia of Philosophy 4838:Internet Encyclopedia of Philosophy 4751:Internet Encyclopedia of Philosophy 4349:Internet Encyclopedia of Philosophy 4316:Steup, Matthias; Neta, Ram (2020). 4234:"Mental models and human reasoning" 3897:Internet Encyclopedia of Philosophy 3893:"Deductive and Inductive Arguments" 3577:Internet Encyclopedia of Philosophy 3573:"Deductive and Inductive Arguments" 3460:Pedemonte, Bettina (25 June 2018). 3200:Internet Encyclopedia of Philosophy 3178:. Vol. 2nd Edition. Macmillan. 2985:Internet Encyclopedia of Philosophy 2981:"Deductive and Inductive Arguments" 2805:Internet Encyclopedia of Philosophy 2277:In natural deduction, a simplified 1071:fallacy of the undistributed middle 5128:The Pragmatic Problem of Induction 4649:Australasian Journal of Philosophy 4115:(Thesis). University of Plymouth. 4086:10.1016/B978-0-444-52936-7.50014-8 4012:10.1111/j.2044-8295.1973.tb01365.x 2363: 2288: 2058:and came to prominence in various 2007: 1983: 1905: 1415:, for example, in the creation of 1315:. In an often-cited experiment by 878: 853: 795: 772: 749:Thus, there are clouds in the sky. 272:Deductive reasoning is studied in 86:approach, by contrast, focuses on 25: 5061:Spinoza's Causal Axiom: A Defense 4522:"What is Wrong with Deductivism?" 4191:The Spanish Journal of Psychology 2054:. It was initially formulated by 1169:Therefore, John is a quarterback. 5719: 3516:Backmann, Marius (1 June 2019). 3087:. North Holland. pp. 13–39. 2665:American Philosophical Quarterly 2116: 2102: 1462: 1342:, sometimes also referred to as 1191:propositional (sentential) logic 416:, i.e. that if a proposition is 33:is the process of drawing valid 4482:"Descartes' Project of Inquiry" 1594:Hilbert-style deductive systems 905:There are no clouds in the sky. 4797:"hypothetico-deductive method" 4772:"hypothetico-deductive method" 4707:– via Project Gutenberg. 4520:Bermejo-Luque, Lilian (2020). 4429:. Blackwell. pp. 362–384. 4402:. Cambridge University Press. 3268:Journal of Philosophical Logic 2755:. Cambridge University Press. 2153:Correspondence theory of truth 1956: 1881: 1869: 1815: 1803: 1671: 1659: 1077:Definitory and strategic rules 1007:{\displaystyle P\rightarrow R} 998: 979:{\displaystyle Q\rightarrow R} 970: 952:{\displaystyle P\rightarrow Q} 943: 837:{\displaystyle P\rightarrow Q} 828: 705:{\displaystyle P\rightarrow Q} 696: 639:{\displaystyle P\rightarrow Q} 630: 544:{\displaystyle P\rightarrow Q} 535: 474:{\displaystyle P\rightarrow Q} 465: 203:are true, then it is called a 1: 4489:American University of Beirut 4480:Samaha, Raid (3 March 2009). 4399:A Primer of Probability Logic 4157:Behavioral and Brain Sciences 4000:British Journal of Psychology 3174:. In Borchert, Donald (ed.). 1525:Related concepts and theories 1367:, than with others, like the 1354:see deductive reasoning as a 4617:10.1007/978-94-010-1126-6_15 4603:Kotarbinska, Janina (1977). 2642:The Encyclopedia of the Mind 2188:Hypothetico-deductive method 1962:{\displaystyle \rightarrow } 1130:if it is impossible for its 666:). However, the antecedent ( 432:Prominent rules of inference 5123:, The Teaching Company 2002 4580:10.1093/0198250371.001.0001 4574:. Oxford University Press. 3697:Koslowski, Barbara (2017). 3657:. Oxford University Press. 3255:. Vol. 2nd. Macmillan. 3126:"Formal and informal logic" 2281:consists of an environment 1509:, refined the idea for the 1145:and the premises are true. 414:double negation elimination 292:happen or what constitutes 182:. An inference is a set of 6082: 4718:Thornton, Stephen (2021). 4704:The Problems of Philosophy 4676:"The Problem of Induction" 4371:"Reliabilist Epistemology" 3977:10.1207/S15327647JCD0201_2 3220:Moschovakis, Joan (2021). 2636:Schechter, Joshua (2013). 2064:mathematical demonstration 1887:{\displaystyle (A\land B)} 916: 757: 440: 41:if its conclusion follows 5906:Parsimony (Occam's razor) 5714: 5591:Necessity and sufficiency 5347: 5183: 5178:Links to related articles 5109:, Psychology Press 1991, 4898:Mathematische Zeitschrift 4892:Gentzen, Gerhard (1934). 4661:10.1080/00048407012341481 4203:10.1017/s1138741600005904 4169:10.1017/s0140525x00030260 4043:10.1080/13546780701415979 3935:10.1037/0033-295X.90.1.38 3845:10.1007/s11229-008-9409-4 3753:Hawthorne, James (2021). 3726:Hawthorne, James (2021). 3684:New Catholic Encyclopedia 3655:A Dictionary of Sociology 3630:Borchert, Donald (2006). 3535:10.1007/s12136-018-0371-6 3280:10.1007/s10992-008-9098-y 3247:Borchert, Donald (2006). 3143:MacFarlane, John (2017). 2924:10.1007/s10503-009-9131-5 2906:Wilbanks, Jan J. (2010). 2551:New Catholic Encyclopedia 1922:propositional connectives 1307:An important bias is the 5225:Non-associative learning 4674:Henderson, Leah (2020). 4031:Thinking & Reasoning 2080:philosophical skepticism 2013:{\displaystyle \forall } 1989:{\displaystyle \exists } 1758:{\displaystyle A\land B} 1297:affirming the consequent 1189:, but was superseded by 1047:affirming the consequent 908:Thus, it is not raining. 734:affirming the consequent 317:Conceptions of deduction 234:such cases, not just in 153:is transferred from the 149:tries to understand how 4956:10.1057/9781137344557_1 4776:Encyclopædia Britannica 4539:10.22329/il.v40i30.6214 4296:Encyclopædia Britannica 4251:10.1073/pnas.1012933107 3917:Rips, Lance J. (1983). 3678:Houde, R.; Camacho, L. 3437:Encyclopædia Britannica 3394:Stone, Mark A. (2012). 3374:Encyclopædia Britannica 3349:Encyclopædia Britannica 3170:Shieh, Sanford (2006). 3022:Encyclopædia Britannica 2548:Houde, R. "Deduction". 2500:WIREs Cognitive Science 2352:C1"; in the symbols of 2314:{\displaystyle \vdash } 2294:{\displaystyle \Gamma } 2258:Theory of justification 1568:Hypothetico-deductivism 1417:artificial intelligence 887:{\displaystyle \lnot P} 862:{\displaystyle \lnot Q} 804:{\displaystyle \lnot P} 781:{\displaystyle \lnot Q} 345:disjunction elimination 6061:Problem solving skills 5266:Observational learning 5256:Classical conditioning 4981:"Spinoza, Benedict De" 4832:Indrzejczak, Andrzej. 3413:10.22329/il.v32i3.3681 3249:"Logic, Non-Classical" 3124:Craig, Edward (1996). 2885:10.22329/il.v14i2.2533 2866:Vorobej, Mark (1992). 2428: 2348:A1, A2, A3 etc. yield 2335: 2321:) a single conclusion 2315: 2295: 2243:Retroductive reasoning 2238:Propositional calculus 2014: 1990: 1963: 1939: 1912: 1911:{\displaystyle \lnot } 1888: 1852: 1828: 1783: 1782:{\displaystyle \land } 1759: 1729: 1705: 1681: 1628: 1627:{\displaystyle \land } 1379:mental representations 1301:denying the antecedent 1108: 1099:Validity and soundness 1055:denying the antecedent 1008: 980: 953: 919:hypothetical syllogism 913:Hypothetical syllogism 888: 863: 838: 805: 782: 726: 712:) and the consequent ( 706: 680: 660: 646:) and its antecedent ( 640: 614: 590: 568: 545: 515: 495: 475: 307:formal logical systems 5961:Theories of deduction 5726:Philosophy portal 5160:"Deductive reasoning" 4153:"Precis of Deduction" 3782:Philosophy of Science 3611:Douven, Igor (2021). 3479:10.29333/ejmste/92562 3317:Hansen, Hans (2020). 3018:"Philosophy of logic" 2948:Introduction to Logic 2801:"Logical Consequence" 2638:"Deductive Reasoning" 2496:"Deductive reasoning" 2429: 2336: 2316: 2296: 2208:Logic and rationality 2066:found in traditional 2015: 1991: 1964: 1940: 1938:{\displaystyle \lor } 1913: 1889: 1853: 1829: 1784: 1760: 1730: 1706: 1682: 1629: 1564:empirical observation 1511:Scientific Revolution 1408:tests of intelligence 1375:Mental model theories 1352:Mental logic theories 1348:mental model theories 1340:mental logic theories 1282:, and the underlying 1179:categorical reasoning 1106: 1009: 981: 954: 889: 864: 839: 806: 783: 727: 707: 681: 661: 641: 615: 591: 569: 546: 516: 496: 476: 453:conditional statement 139:dual-process theories 134:Mental model theories 130:Mental logic theories 5271:Operant conditioning 5248:Associative learning 5099:Philip Johnson-Laird 5082:Vincent F. Hendricks 5007:"Geometrical Method" 5005:Goldenbaum, Ursula. 4942:Daly, Chris (2015). 4498:on 28 September 2020 3923:Psychological Review 3651:"Analytic induction" 2868:"Defining Deduction" 2360: 2325: 2305: 2285: 2265:Notes and references 2168:Defeasible reasoning 2148:Argumentation theory 2138:Analogical reasoning 2004: 1980: 1953: 1929: 1902: 1866: 1842: 1797: 1773: 1743: 1719: 1695: 1642: 1618: 1552:problem of induction 1324:material conditional 1313:Wason selection task 1275:Cognitive psychology 1270:Cognitive psychology 1107:Argument terminology 1063:affirming a disjunct 992: 964: 937: 875: 850: 822: 792: 769: 716: 690: 670: 650: 624: 604: 580: 558: 529: 505: 485: 459: 426:intuitionistic logic 125:Cognitive psychology 6051:Deductive reasoning 5788:Philosophical logic 5388:Philosophy of logic 5299:Inductive reasoning 5294:Deductive reasoning 5289:Abductive reasoning 5150:Deductive reasoning 5141:Deductive reasoning 5126:Bullemore, Thomas. 4834:"Natural Deduction" 4244:(43): 18243–18250. 3172:"LOGICAL KNOWLEDGE" 3145:"Logical Constants" 3084:Philosophy of Logic 2213:Logical consequence 2178:Fault tree analysis 2133:Abductive reasoning 1589:Stanislaw Jaskowski 1507:Discourse on Method 1388:dual-process theory 1253:while induction is 1223:abductive reasoning 1202:inductive reasoning 1185:– was developed by 311:propositional logic 299:logical consequence 288:question of how it 212:logical consequence 119:abductive arguments 31:Deductive reasoning 18:Deductive inference 5852:Unity of opposites 5687:Rules of inference 5656:Mathematical logic 5398:Semantics of logic 4910:10.1007/BF01201353 4643:Stove, D. (1970). 3107:"Fallacy, Logical" 2424: 2331: 2311: 2291: 2223:Mathematical logic 2038:Geometrical method 2010: 1986: 1959: 1935: 1908: 1884: 1848: 1824: 1779: 1755: 1725: 1701: 1677: 1624: 1474:. You can help by 1166:John eats carrots. 1109: 1067:denying a conjunct 1004: 976: 949: 884: 859: 834: 811:). In contrast to 801: 778: 722: 702: 676: 656: 636: 610: 586: 564: 541: 511: 491: 471: 390:rules of inference 384:Rules of inference 337:rules of inference 282:cognitive sciences 252:speaker-determined 88:rules of inference 37:. An inference is 6038: 6037: 6032: 6031: 5884:List of fallacies 5869:Explanatory power 5796:Critical thinking 5754: 5753: 5710: 5709: 5544:Deductive closure 5490: 5489: 5429:Critical thinking 5307: 5306: 5119:Zarefsky, David, 5115:978-0-86377-149-1 5044:978-0-521-83620-3 4979:Dutton, Blake D. 4965:978-1-137-34455-7 4626:978-9-401-01126-6 4589:978-0-198-25037-1 4457:978-0-863-77313-6 4442:Byrne, Ruth M. J. 4409:978-1-575-86066-4 4095:978-0-444-52936-7 3755:"Inductive Logic" 3728:"Inductive Logic" 3712:978-1-315-72569-7 3664:978-0-199-53300-8 3433:"Logical systems" 3194:Dowden, Bradley. 2958:978-1-351-38696-8 2845:978-0-915-14476-1 2799:McKeon, Matthew. 2762:978-0-521-82417-0 2445:Deduction theorem 2354:natural deduction 2334:{\displaystyle C} 2248:Scientific method 2228:Natural deduction 2218:Logical reasoning 2110:Philosophy portal 1860:from the premise 1851:{\displaystyle A} 1822: 1728:{\displaystyle B} 1704:{\displaystyle A} 1675: 1602:logical constants 1581:natural deduction 1575:Natural deduction 1515:scientific method 1499:Greek philosopher 1492: 1491: 1448:Probability logic 1443:Probability logic 1413:computer sciences 1360:natural deduction 1265:In various fields 1095:strategic rules. 725:{\displaystyle Q} 679:{\displaystyle P} 659:{\displaystyle P} 613:{\displaystyle Q} 589:{\displaystyle Q} 567:{\displaystyle P} 514:{\displaystyle Q} 494:{\displaystyle P} 449:rule of inference 167:Natural deduction 159:Probability logic 27:Form of reasoning 16:(Redirected from 6073: 6014:Platonic realism 5781: 5774: 5767: 5758: 5724: 5723: 5722: 5644: 5409: 5373:Computer science 5334: 5327: 5320: 5311: 5281:Insight learning 5210: 5203: 5196: 5187: 5174: 5169: 5103:Ruth M. J. Byrne 5069: 5068: 5066: 5055: 5049: 5048: 5028: 5022: 5021: 5019: 5017: 5002: 4996: 4995: 4993: 4991: 4976: 4970: 4969: 4939: 4926: 4925: 4889: 4883: 4882: 4880: 4878: 4862: 4849: 4848: 4846: 4844: 4829: 4812: 4811: 4809: 4807: 4801:Oxford Reference 4793: 4787: 4786: 4784: 4782: 4768: 4762: 4761: 4759: 4757: 4742: 4736: 4735: 4733: 4731: 4715: 4709: 4708: 4698: 4692: 4691: 4689: 4687: 4671: 4665: 4664: 4640: 4631: 4630: 4600: 4594: 4593: 4565: 4552: 4551: 4541: 4517: 4508: 4507: 4505: 4503: 4497: 4491:. Archived from 4486: 4477: 4471: 4470: 4465: 4464: 4437: 4431: 4430: 4420: 4414: 4413: 4393: 4387: 4386: 4384: 4382: 4366: 4360: 4359: 4357: 4355: 4340: 4334: 4333: 4331: 4329: 4313: 4307: 4306: 4304: 4302: 4288: 4282: 4281: 4271: 4253: 4229: 4223: 4222: 4182: 4173: 4172: 4148: 4133: 4132: 4106: 4100: 4099: 4069: 4063: 4062: 4022: 4016: 4015: 3995: 3989: 3988: 3960: 3954: 3953: 3951: 3949: 3914: 3908: 3907: 3905: 3903: 3889: 3883: 3882: 3856: 3828: 3822: 3821: 3777: 3771: 3770: 3768: 3766: 3750: 3744: 3743: 3741: 3739: 3723: 3717: 3716: 3694: 3688: 3687: 3675: 3669: 3668: 3646: 3640: 3639: 3632:"G. W. Liebnitz" 3627: 3621: 3620: 3608: 3593: 3592: 3590: 3588: 3579:. Archived from 3569: 3556: 3555: 3537: 3513: 3500: 3499: 3481: 3457: 3448: 3447: 3445: 3443: 3429: 3418: 3417: 3415: 3391: 3385: 3384: 3382: 3380: 3366: 3360: 3359: 3357: 3355: 3341: 3335: 3334: 3332: 3330: 3314: 3308: 3307: 3263: 3257: 3256: 3244: 3238: 3237: 3235: 3233: 3217: 3211: 3210: 3208: 3206: 3191: 3180: 3179: 3167: 3161: 3160: 3158: 3156: 3140: 3134: 3133: 3121: 3115: 3114: 3105:Stump, David J. 3102: 3089: 3088: 3078: 3033: 3032: 3030: 3028: 3014: 3001: 3000: 2998: 2996: 2987:. Archived from 2976: 2963: 2962: 2942: 2936: 2935: 2903: 2890: 2889: 2887: 2863: 2850: 2849: 2829: 2816: 2815: 2813: 2811: 2796: 2767: 2766: 2746: 2689: 2688: 2660: 2654: 2653: 2651: 2649: 2644:. SAGE Reference 2633: 2560: 2559: 2545: 2532: 2531: 2491: 2450: 2433: 2431: 2430: 2425: 2423: 2422: 2401: 2400: 2388: 2387: 2375: 2374: 2340: 2338: 2337: 2332: 2320: 2318: 2317: 2312: 2300: 2298: 2297: 2292: 2275: 2253:Subjective logic 2143:Argument (logic) 2126: 2121: 2120: 2112: 2107: 2106: 2105: 2021: 2019: 2017: 2016: 2011: 1997: 1995: 1993: 1992: 1987: 1970: 1968: 1966: 1965: 1960: 1946: 1944: 1942: 1941: 1936: 1919: 1917: 1915: 1914: 1909: 1895: 1893: 1891: 1890: 1885: 1859: 1857: 1855: 1854: 1849: 1835: 1833: 1831: 1830: 1825: 1823: 1818: 1801: 1790: 1788: 1786: 1785: 1780: 1766: 1764: 1762: 1761: 1756: 1736: 1734: 1732: 1731: 1726: 1712: 1710: 1708: 1707: 1702: 1688: 1686: 1684: 1683: 1678: 1676: 1674: 1657: 1646: 1635: 1633: 1631: 1630: 1625: 1487: 1484: 1466: 1459: 1013: 1011: 1010: 1005: 985: 983: 982: 977: 958: 956: 955: 950: 893: 891: 890: 885: 868: 866: 865: 860: 843: 841: 840: 835: 810: 808: 807: 802: 787: 785: 784: 779: 731: 729: 728: 723: 711: 709: 708: 703: 685: 683: 682: 677: 665: 663: 662: 657: 645: 643: 642: 637: 619: 617: 616: 611: 595: 593: 592: 587: 573: 571: 570: 565: 551: 550: 548: 547: 542: 520: 518: 517: 512: 500: 498: 497: 492: 480: 478: 477: 472: 420:then it is also 399:formal fallacies 353:natural language 303:natural language 214:". According to 104:formal fallacies 21: 6081: 6080: 6076: 6075: 6074: 6072: 6071: 6070: 6041: 6040: 6039: 6034: 6033: 6028: 5999:Logical atomism 5955: 5848:Socratic method 5799: 5790: 5785: 5755: 5750: 5720: 5718: 5706: 5670: 5661:Boolean algebra 5635: 5486: 5477:Metamathematics 5455: 5407: 5361: 5343: 5338: 5308: 5303: 5275: 5242: 5219: 5214: 5179: 5158: 5137: 5078: 5076:Further reading 5073: 5072: 5064: 5057: 5056: 5052: 5045: 5030: 5029: 5025: 5015: 5013: 5004: 5003: 4999: 4989: 4987: 4978: 4977: 4973: 4966: 4941: 4940: 4929: 4891: 4890: 4886: 4876: 4874: 4864: 4863: 4852: 4842: 4840: 4831: 4830: 4815: 4805: 4803: 4795: 4794: 4790: 4780: 4778: 4770: 4769: 4765: 4755: 4753: 4745:Shea, Brendan. 4744: 4743: 4739: 4729: 4727: 4717: 4716: 4712: 4700: 4699: 4695: 4685: 4683: 4673: 4672: 4668: 4642: 4641: 4634: 4627: 4602: 4601: 4597: 4590: 4567: 4566: 4555: 4519: 4518: 4511: 4501: 4499: 4495: 4484: 4479: 4478: 4474: 4462: 4460: 4458: 4444:, eds. (1993). 4439: 4438: 4434: 4422: 4421: 4417: 4410: 4395: 4394: 4390: 4380: 4378: 4368: 4367: 4363: 4353: 4351: 4343:Becker, Kelly. 4342: 4341: 4337: 4327: 4325: 4315: 4314: 4310: 4300: 4298: 4290: 4289: 4285: 4231: 4230: 4226: 4184: 4183: 4176: 4150: 4149: 4136: 4108: 4107: 4103: 4096: 4078:Inductive Logic 4071: 4070: 4066: 4024: 4023: 4019: 3997: 3996: 3992: 3962: 3961: 3957: 3947: 3945: 3916: 3915: 3911: 3901: 3899: 3891: 3890: 3886: 3830: 3829: 3825: 3779: 3778: 3774: 3764: 3762: 3752: 3751: 3747: 3737: 3735: 3725: 3724: 3720: 3713: 3696: 3695: 3691: 3677: 3676: 3672: 3665: 3648: 3647: 3643: 3629: 3628: 3624: 3610: 3609: 3596: 3586: 3584: 3571: 3570: 3559: 3515: 3514: 3503: 3459: 3458: 3451: 3441: 3439: 3431: 3430: 3421: 3393: 3392: 3388: 3378: 3376: 3368: 3367: 3363: 3353: 3351: 3343: 3342: 3338: 3328: 3326: 3316: 3315: 3311: 3265: 3264: 3260: 3246: 3245: 3241: 3231: 3229: 3219: 3218: 3214: 3204: 3202: 3193: 3192: 3183: 3169: 3168: 3164: 3154: 3152: 3142: 3141: 3137: 3123: 3122: 3118: 3104: 3103: 3092: 3080: 3079: 3036: 3026: 3024: 3016: 3015: 3004: 2994: 2992: 2978: 2977: 2966: 2959: 2944: 2943: 2939: 2905: 2904: 2893: 2865: 2864: 2853: 2846: 2831: 2830: 2819: 2809: 2807: 2798: 2797: 2770: 2763: 2748: 2747: 2692: 2662: 2661: 2657: 2647: 2645: 2635: 2634: 2563: 2547: 2546: 2535: 2493: 2492: 2459: 2454: 2453: 2414: 2392: 2379: 2366: 2358: 2357: 2323: 2322: 2303: 2302: 2283: 2282: 2276: 2272: 2267: 2262: 2203:Legal syllogism 2163:Decision theory 2158:Decision making 2122: 2115: 2108: 2103: 2101: 2098: 2074:, and inferred 2040: 2032:tableau calculi 2028:sequent calculi 2002: 2001: 1999: 1978: 1977: 1975: 1951: 1950: 1948: 1927: 1926: 1924: 1900: 1899: 1897: 1864: 1863: 1861: 1840: 1839: 1837: 1802: 1795: 1794: 1792: 1771: 1770: 1768: 1741: 1740: 1738: 1717: 1716: 1714: 1693: 1692: 1690: 1658: 1647: 1640: 1639: 1637: 1616: 1615: 1613: 1585:Gerhard Gentzen 1577: 1532: 1527: 1488: 1482: 1479: 1472:needs expansion 1457: 1445: 1425: 1396: 1332: 1272: 1267: 1210: 1195:predicate logic 1137:An argument is 1126:An argument is 1101: 1079: 1042: 990: 989: 962: 961: 935: 934: 921: 915: 873: 872: 848: 847: 820: 819: 790: 789: 767: 766: 762: 756: 714: 713: 688: 687: 668: 667: 648: 647: 622: 621: 602: 601: 578: 577: 556: 555: 527: 526: 525: 503: 502: 483: 482: 457: 456: 445: 439: 434: 406:classical logic 386: 349:formal language 319: 228:possible worlds 176: 72:Deductive logic 63:if it is valid 28: 23: 22: 15: 12: 11: 5: 6079: 6077: 6069: 6068: 6063: 6058: 6053: 6043: 6042: 6036: 6035: 6030: 6029: 6027: 6026: 6021: 6016: 6011: 6006: 6001: 5996: 5991: 5986: 5981: 5976: 5971: 5969:Constructivism 5965: 5963: 5957: 5956: 5954: 5953: 5948: 5943: 5938: 5933: 5928: 5923: 5918: 5913: 5908: 5903: 5898: 5893: 5888: 5887: 5886: 5876: 5871: 5866: 5861: 5856: 5855: 5854: 5836: 5831: 5826: 5821: 5816: 5811: 5805: 5803: 5801:informal logic 5792: 5791: 5786: 5784: 5783: 5776: 5769: 5761: 5752: 5751: 5749: 5748: 5743: 5733: 5728: 5715: 5712: 5711: 5708: 5707: 5705: 5704: 5699: 5694: 5689: 5684: 5678: 5676: 5672: 5671: 5669: 5668: 5663: 5658: 5652: 5650: 5641: 5637: 5636: 5634: 5633: 5628: 5623: 5618: 5613: 5608: 5603: 5598: 5593: 5588: 5583: 5578: 5573: 5568: 5567: 5566: 5556: 5551: 5546: 5541: 5536: 5535: 5534: 5529: 5519: 5514: 5509: 5504: 5498: 5496: 5492: 5491: 5488: 5487: 5485: 5484: 5479: 5474: 5469: 5463: 5461: 5457: 5456: 5454: 5453: 5448: 5443: 5438: 5437: 5436: 5431: 5421: 5415: 5413: 5406: 5405: 5400: 5395: 5390: 5385: 5380: 5375: 5369: 5367: 5363: 5362: 5360: 5359: 5354: 5348: 5345: 5344: 5339: 5337: 5336: 5329: 5322: 5314: 5305: 5304: 5302: 5301: 5296: 5291: 5285: 5283: 5277: 5276: 5274: 5273: 5268: 5263: 5258: 5252: 5250: 5244: 5243: 5241: 5240: 5235: 5229: 5227: 5221: 5220: 5215: 5213: 5212: 5205: 5198: 5190: 5184: 5181: 5180: 5177: 5171: 5170: 5156: 5147: 5136: 5135:External links 5133: 5132: 5131: 5124: 5117: 5096: 5077: 5074: 5071: 5070: 5050: 5043: 5023: 4997: 4971: 4964: 4927: 4904:(2): 176–210. 4884: 4850: 4813: 4788: 4763: 4737: 4710: 4693: 4666: 4632: 4625: 4595: 4588: 4571:Hume's Problem 4553: 4532:(3): 295–316. 4526:Informal Logic 4509: 4472: 4456: 4432: 4415: 4408: 4388: 4361: 4335: 4318:"Epistemology" 4308: 4292:"Epistemology" 4283: 4224: 4197:(2): 125–140. 4174: 4163:(2): 323–333. 4134: 4101: 4094: 4064: 4037:(4): 484–505. 4017: 4006:(3): 391–397. 3990: 3955: 3909: 3884: 3839:(2): 271–315. 3823: 3794:10.1086/287745 3788:(3): 262–270. 3772: 3745: 3718: 3711: 3689: 3670: 3663: 3641: 3622: 3594: 3583:on 28 May 2010 3557: 3528:(2): 235–255. 3522:Acta Analytica 3501: 3449: 3419: 3406:(3): 327–356. 3400:Informal Logic 3386: 3361: 3336: 3309: 3274:(4): 447–464. 3258: 3239: 3212: 3181: 3162: 3135: 3116: 3090: 3034: 3002: 2991:on 28 May 2010 2964: 2957: 2937: 2918:(1): 107–124. 2891: 2872:Informal Logic 2851: 2844: 2817: 2768: 2761: 2690: 2655: 2561: 2533: 2512:10.1002/wcs.20 2456: 2455: 2452: 2451: 2449: 2448: 2440: 2435: 2434: 2421: 2417: 2413: 2410: 2407: 2404: 2399: 2395: 2391: 2386: 2382: 2378: 2373: 2369: 2365: 2330: 2310: 2290: 2269: 2268: 2266: 2263: 2261: 2260: 2255: 2250: 2245: 2240: 2235: 2230: 2225: 2220: 2215: 2210: 2205: 2200: 2195: 2190: 2185: 2180: 2175: 2170: 2165: 2160: 2155: 2150: 2145: 2140: 2135: 2129: 2128: 2127: 2113: 2097: 2094: 2056:Baruch Spinoza 2039: 2036: 2009: 1985: 1958: 1934: 1907: 1883: 1880: 1877: 1874: 1871: 1847: 1821: 1817: 1814: 1811: 1808: 1805: 1778: 1754: 1751: 1748: 1724: 1700: 1673: 1670: 1667: 1664: 1661: 1656: 1653: 1650: 1623: 1598:logical truths 1576: 1573: 1554:introduced by 1531: 1528: 1526: 1523: 1505:, in his book 1503:René Descartes 1490: 1489: 1469: 1467: 1456: 1453: 1444: 1441: 1424: 1421: 1395: 1392: 1331: 1328: 1271: 1268: 1266: 1263: 1209: 1206: 1171: 1170: 1167: 1164: 1100: 1097: 1078: 1075: 1041: 1038: 1037: 1036: 1033: 1030: 1016: 1015: 1003: 1000: 997: 986: 975: 972: 969: 959: 948: 945: 942: 917:Main article: 914: 911: 910: 909: 906: 903: 896: 895: 883: 880: 870: 858: 855: 845: 833: 830: 827: 800: 797: 777: 774: 758:Main article: 755: 752: 751: 750: 747: 746:It is raining. 744: 721: 701: 698: 695: 675: 655: 635: 632: 629: 609: 598: 597: 585: 575: 563: 553: 540: 537: 534: 510: 490: 470: 467: 464: 441:Main article: 438: 435: 433: 430: 410:deviant logics 385: 382: 365:singular terms 318: 315: 175: 172: 80:interpretation 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 6078: 6067: 6064: 6062: 6059: 6057: 6054: 6052: 6049: 6048: 6046: 6025: 6022: 6020: 6017: 6015: 6012: 6010: 6007: 6005: 6002: 6000: 5997: 5995: 5992: 5990: 5987: 5985: 5982: 5980: 5977: 5975: 5972: 5970: 5967: 5966: 5964: 5962: 5958: 5952: 5949: 5947: 5944: 5942: 5939: 5937: 5934: 5932: 5929: 5927: 5924: 5922: 5919: 5917: 5914: 5912: 5909: 5907: 5904: 5902: 5899: 5897: 5894: 5892: 5889: 5885: 5882: 5881: 5880: 5877: 5875: 5872: 5870: 5867: 5865: 5862: 5860: 5857: 5853: 5849: 5845: 5842: 5841: 5840: 5837: 5835: 5832: 5830: 5827: 5825: 5822: 5820: 5817: 5815: 5812: 5810: 5807: 5806: 5804: 5802: 5797: 5793: 5789: 5782: 5777: 5775: 5770: 5768: 5763: 5762: 5759: 5747: 5744: 5741: 5737: 5734: 5732: 5729: 5727: 5717: 5716: 5713: 5703: 5702:Logic symbols 5700: 5698: 5695: 5693: 5690: 5688: 5685: 5683: 5680: 5679: 5677: 5673: 5667: 5664: 5662: 5659: 5657: 5654: 5653: 5651: 5649: 5645: 5642: 5638: 5632: 5629: 5627: 5624: 5622: 5619: 5617: 5614: 5612: 5609: 5607: 5604: 5602: 5599: 5597: 5594: 5592: 5589: 5587: 5584: 5582: 5581:Logical truth 5579: 5577: 5574: 5572: 5569: 5565: 5562: 5561: 5560: 5557: 5555: 5552: 5550: 5547: 5545: 5542: 5540: 5537: 5533: 5530: 5528: 5525: 5524: 5523: 5522:Contradiction 5520: 5518: 5515: 5513: 5510: 5508: 5505: 5503: 5500: 5499: 5497: 5493: 5483: 5480: 5478: 5475: 5473: 5470: 5468: 5467:Argumentation 5465: 5464: 5462: 5458: 5452: 5451:Philosophical 5449: 5447: 5446:Non-classical 5444: 5442: 5439: 5435: 5432: 5430: 5427: 5426: 5425: 5422: 5420: 5417: 5416: 5414: 5410: 5404: 5401: 5399: 5396: 5394: 5391: 5389: 5386: 5384: 5381: 5379: 5376: 5374: 5371: 5370: 5368: 5364: 5358: 5355: 5353: 5350: 5349: 5346: 5342: 5335: 5330: 5328: 5323: 5321: 5316: 5315: 5312: 5300: 5297: 5295: 5292: 5290: 5287: 5286: 5284: 5282: 5278: 5272: 5269: 5267: 5264: 5262: 5259: 5257: 5254: 5253: 5251: 5249: 5245: 5239: 5238:Sensitization 5236: 5234: 5231: 5230: 5228: 5226: 5222: 5218: 5211: 5206: 5204: 5199: 5197: 5192: 5191: 5188: 5182: 5175: 5167: 5166: 5161: 5157: 5155: 5151: 5148: 5146: 5142: 5139: 5138: 5134: 5129: 5125: 5122: 5118: 5116: 5112: 5108: 5104: 5100: 5097: 5095: 5094:87-991013-7-8 5091: 5087: 5083: 5080: 5079: 5075: 5063: 5062: 5054: 5051: 5046: 5040: 5036: 5035: 5027: 5024: 5012: 5008: 5001: 4998: 4986: 4982: 4975: 4972: 4967: 4961: 4957: 4953: 4949: 4945: 4938: 4936: 4934: 4932: 4928: 4924: 4919: 4915: 4911: 4907: 4903: 4900:(in German). 4899: 4895: 4888: 4885: 4872: 4868: 4861: 4859: 4857: 4855: 4851: 4839: 4835: 4828: 4826: 4824: 4822: 4820: 4818: 4814: 4802: 4798: 4792: 4789: 4777: 4773: 4767: 4764: 4752: 4748: 4741: 4738: 4725: 4721: 4714: 4711: 4706: 4705: 4697: 4694: 4681: 4677: 4670: 4667: 4662: 4658: 4654: 4650: 4646: 4645:"Deductivism" 4639: 4637: 4633: 4628: 4622: 4618: 4614: 4610: 4606: 4599: 4596: 4591: 4585: 4581: 4577: 4573: 4572: 4564: 4562: 4560: 4558: 4554: 4549: 4545: 4540: 4535: 4531: 4527: 4523: 4516: 4514: 4510: 4494: 4490: 4483: 4476: 4473: 4469: 4459: 4453: 4449: 4448: 4443: 4436: 4433: 4428: 4427: 4419: 4416: 4411: 4405: 4401: 4400: 4392: 4389: 4376: 4372: 4365: 4362: 4350: 4346: 4345:"Reliabilism" 4339: 4336: 4323: 4319: 4312: 4309: 4297: 4293: 4287: 4284: 4279: 4275: 4270: 4265: 4261: 4257: 4252: 4247: 4243: 4239: 4235: 4228: 4225: 4220: 4216: 4212: 4208: 4204: 4200: 4196: 4192: 4188: 4181: 4179: 4175: 4170: 4166: 4162: 4158: 4154: 4147: 4145: 4143: 4141: 4139: 4135: 4130: 4126: 4122: 4121:10.24382/4506 4118: 4114: 4113: 4105: 4102: 4097: 4091: 4087: 4083: 4079: 4075: 4068: 4065: 4060: 4056: 4052: 4048: 4044: 4040: 4036: 4032: 4028: 4021: 4018: 4013: 4009: 4005: 4001: 3994: 3991: 3986: 3982: 3978: 3974: 3970: 3966: 3959: 3956: 3944: 3940: 3936: 3932: 3928: 3924: 3920: 3913: 3910: 3898: 3894: 3888: 3885: 3880: 3876: 3872: 3868: 3864: 3860: 3855: 3850: 3846: 3842: 3838: 3834: 3827: 3824: 3819: 3815: 3811: 3807: 3803: 3799: 3795: 3791: 3787: 3783: 3776: 3773: 3760: 3756: 3749: 3746: 3733: 3729: 3722: 3719: 3714: 3708: 3705:. Routledge. 3704: 3700: 3693: 3690: 3685: 3681: 3674: 3671: 3666: 3660: 3656: 3652: 3645: 3642: 3637: 3633: 3626: 3623: 3618: 3614: 3607: 3605: 3603: 3601: 3599: 3595: 3582: 3578: 3574: 3568: 3566: 3564: 3562: 3558: 3553: 3549: 3545: 3541: 3536: 3531: 3527: 3523: 3519: 3512: 3510: 3508: 3506: 3502: 3497: 3493: 3489: 3485: 3480: 3475: 3472:(9): em1589. 3471: 3467: 3463: 3456: 3454: 3450: 3438: 3434: 3428: 3426: 3424: 3420: 3414: 3409: 3405: 3401: 3397: 3390: 3387: 3375: 3371: 3365: 3362: 3350: 3346: 3340: 3337: 3324: 3320: 3313: 3310: 3305: 3301: 3297: 3293: 3289: 3285: 3281: 3277: 3273: 3269: 3262: 3259: 3254: 3250: 3243: 3240: 3227: 3223: 3216: 3213: 3201: 3197: 3190: 3188: 3186: 3182: 3177: 3173: 3166: 3163: 3150: 3146: 3139: 3136: 3131: 3127: 3120: 3117: 3112: 3108: 3101: 3099: 3097: 3095: 3091: 3086: 3085: 3077: 3075: 3073: 3071: 3069: 3067: 3065: 3063: 3061: 3059: 3057: 3055: 3053: 3051: 3049: 3047: 3045: 3043: 3041: 3039: 3035: 3023: 3019: 3013: 3011: 3009: 3007: 3003: 2990: 2986: 2982: 2975: 2973: 2971: 2969: 2965: 2960: 2954: 2951:. Routledge. 2950: 2949: 2941: 2938: 2933: 2929: 2925: 2921: 2917: 2913: 2912:Argumentation 2909: 2902: 2900: 2898: 2896: 2892: 2886: 2881: 2877: 2873: 2869: 2862: 2860: 2858: 2856: 2852: 2847: 2841: 2837: 2836: 2828: 2826: 2824: 2822: 2818: 2806: 2802: 2795: 2793: 2791: 2789: 2787: 2785: 2783: 2781: 2779: 2777: 2775: 2773: 2769: 2764: 2758: 2754: 2753: 2745: 2743: 2741: 2739: 2737: 2735: 2733: 2731: 2729: 2727: 2725: 2723: 2721: 2719: 2717: 2715: 2713: 2711: 2709: 2707: 2705: 2703: 2701: 2699: 2697: 2695: 2691: 2686: 2682: 2678: 2674: 2670: 2666: 2659: 2656: 2643: 2639: 2632: 2630: 2628: 2626: 2624: 2622: 2620: 2618: 2616: 2614: 2612: 2610: 2608: 2606: 2604: 2602: 2600: 2598: 2596: 2594: 2592: 2590: 2588: 2586: 2584: 2582: 2580: 2578: 2576: 2574: 2572: 2570: 2568: 2566: 2562: 2558: 2553: 2552: 2544: 2542: 2540: 2538: 2534: 2529: 2525: 2521: 2517: 2513: 2509: 2505: 2501: 2497: 2490: 2488: 2486: 2484: 2482: 2480: 2478: 2476: 2474: 2472: 2470: 2468: 2466: 2464: 2462: 2458: 2447: 2446: 2441: 2437: 2436: 2419: 2415: 2411: 2408: 2405: 2402: 2397: 2393: 2389: 2384: 2380: 2376: 2371: 2367: 2355: 2351: 2347: 2343: 2342: 2328: 2308: 2301:that yields ( 2280: 2274: 2271: 2264: 2259: 2256: 2254: 2251: 2249: 2246: 2244: 2241: 2239: 2236: 2234: 2231: 2229: 2226: 2224: 2221: 2219: 2216: 2214: 2211: 2209: 2206: 2204: 2201: 2199: 2196: 2194: 2191: 2189: 2186: 2184: 2181: 2179: 2176: 2174: 2171: 2169: 2166: 2164: 2161: 2159: 2156: 2154: 2151: 2149: 2146: 2144: 2141: 2139: 2136: 2134: 2131: 2130: 2125: 2119: 2114: 2111: 2100: 2095: 2093: 2090: 2084: 2081: 2077: 2073: 2069: 2065: 2061: 2057: 2053: 2049: 2045: 2037: 2035: 2033: 2029: 2023: 1974: 1932: 1923: 1878: 1875: 1872: 1845: 1819: 1812: 1809: 1806: 1776: 1752: 1749: 1746: 1722: 1698: 1668: 1665: 1662: 1654: 1651: 1648: 1621: 1611: 1607: 1603: 1599: 1595: 1590: 1586: 1582: 1574: 1572: 1569: 1565: 1561: 1557: 1553: 1548: 1546: 1542: 1538: 1529: 1524: 1522: 1520: 1516: 1512: 1508: 1504: 1500: 1496: 1486: 1477: 1473: 1470:This section 1468: 1465: 1461: 1460: 1454: 1452: 1449: 1442: 1440: 1438: 1434: 1433:justification 1430: 1422: 1420: 1418: 1414: 1409: 1405: 1401: 1393: 1391: 1389: 1383: 1380: 1376: 1372: 1370: 1366: 1365:modus tollens 1361: 1357: 1353: 1349: 1345: 1344:rule theories 1341: 1336: 1329: 1327: 1325: 1320: 1318: 1314: 1310: 1309:matching bias 1305: 1302: 1298: 1294: 1293:modus tollens 1290: 1285: 1281: 1276: 1269: 1264: 1262: 1260: 1256: 1252: 1246: 1242: 1240: 1236: 1231: 1228: 1224: 1220: 1215: 1207: 1205: 1203: 1198: 1196: 1192: 1188: 1184: 1180: 1175: 1168: 1165: 1162: 1161: 1160: 1157: 1155: 1151: 1146: 1144: 1140: 1135: 1133: 1129: 1124: 1122: 1121: 1116: 1115: 1105: 1098: 1096: 1093: 1089: 1085: 1076: 1074: 1072: 1068: 1064: 1060: 1059:modus tollens 1056: 1052: 1048: 1039: 1034: 1031: 1028: 1027: 1026: 1023: 1021: 1001: 995: 987: 973: 967: 960: 946: 940: 933: 932: 931: 929: 928: 925:hypothetical 920: 912: 907: 904: 901: 900: 899: 881: 871: 856: 846: 831: 825: 818: 817: 816: 814: 798: 775: 761: 760:Modus tollens 754:Modus tollens 753: 748: 745: 742: 741: 740: 737: 735: 719: 699: 693: 673: 653: 633: 627: 607: 583: 576: 561: 554: 538: 532: 524: 523: 522: 508: 488: 468: 462: 454: 450: 444: 436: 431: 429: 427: 423: 419: 415: 411: 407: 402: 400: 395: 391: 383: 381: 379: 374: 370: 366: 362: 357: 354: 350: 346: 342: 341:modus tollens 338: 334: 329: 325: 316: 314: 312: 308: 304: 300: 295: 291: 287: 283: 279: 275: 270: 267: 262: 257: 253: 249: 245: 239: 237: 233: 229: 225: 221: 217: 216:Alfred Tarski 213: 208: 206: 202: 197: 193: 189: 185: 181: 173: 171: 168: 164: 160: 156: 152: 151:justification 148: 143: 141: 140: 135: 131: 126: 122: 120: 116: 112: 107: 105: 101: 100:modus tollens 97: 93: 89: 85: 81: 77: 73: 69: 66: 62: 61: 56: 52: 48: 44: 40: 36: 32: 19: 5994:Intuitionism 5979:Fictionalism 5960: 5621:Substitution 5538: 5441:Mathematical 5366:Major fields 5293: 5163: 5120: 5106: 5085: 5060: 5053: 5033: 5026: 5014:. 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