Logical reasoning

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Argument
Deductive
Valid

Sound

Unsound

Invalid

Unsound

Non‑deductive
Strong

Cogent

Uncogent

Weak

Uncogent

Logical reasoning is concerned with the correctness of arguments. A key distinction is between deductive and non-deductive arguments.

Logical reasoning is a mental activity that aims to arrive at a

rational person would find convincing. The main discipline studying logical reasoning is logic
.

Distinct types of logical reasoning differ from each other concerning the norms they employ and the certainty of the conclusion they arrive at. Deductive reasoning offers the strongest support: the premises ensure the conclusion, meaning that it is impossible for the conclusion to be false if all the premises are true. Such an argument is called a valid argument, for example: all men are mortal; Socrates is a man; therefore, Socrates is mortal. For valid arguments, it is not important whether the premises are actually true but only that, if they were true, the conclusion could not be false. Valid arguments follow a rule of inference, such as modus ponens or modus tollens. Deductive reasoning plays a central role in formal logic and mathematics.

For non-deductive logical reasoning, the premises make their conclusion rationally convincing without ensuring its

analogical reasoning. Inductive reasoning is a form of generalization that infers a universal law from a pattern found in many individual cases. It can be used to conclude that "all ravens are black" based on many individual observations of black ravens. Abductive reasoning, also known as "inference to the best explanation", starts from an observation and reasons to the fact explaining this observation. An example is a doctor who examines the symptoms of their patient to make a diagnosis of the underlying cause. Analogical reasoning compares two similar
systems. It observes that one of them has a feature and concludes that the other one also has this feature.

Arguments that fall short of the standards of logical reasoning are called

inconsistencies
, and to consider the advantages and disadvantages of different courses of action before making a decision.

Definition

Logical reasoning is a form of

rational person would find the conclusion convincing based on the premises.[6][1] This way, logical reasoning plays a role in expanding knowledge.[7]

The main discipline studying logical reasoning is called

analogical reasoning.[12][13][14]

The forms of logical reasoning have in common that they use premises to make inferences in a norm-governed way. As norm-governed practices, they aim at

inter-subjective agreement about the application of the norms, i.e. agreement about whether and to what degree the premises support their conclusion. The types of logical reasoning differ concerning the exact norms they use as well as the certainty of the conclusion they arrive at.[1][15] Deductive reasoning offers the strongest support and implies its conclusion with certainty, like mathematical proofs. For non-deductive reasoning, the premises make the conclusion more likely but do not ensure it. This support comes in degrees: strong arguments make the conclusion very likely, as is the case for well-researched issues in the empirical sciences.[1][16] Some theorists give a very wide definition of logical reasoning that includes its role as a cognitive skill responsible for high-quality thinking. In this regard, it has roughly the same meaning as critical thinking.[13][17]

Basic concepts

A variety of basic concepts is used in the study and analysis of logical reasoning. Logical reasoning happens by inferring a conclusion from a set of premises.[3] Premises and conclusions are normally seen as propositions. A proposition is a statement that makes a claim about what is the case. In this regard, propositions act as truth-bearers: they are either true or false.[18][19][3] For example, the sentence "The water is boiling." expresses a proposition since it can be true or false. The sentences "Is the water boiling?" or "Boil the water!", on the other hand, express no propositions since they are neither true nor false.[20][3] The propositions used as the starting point of logical reasoning are called the premises. The proposition inferred from them is called the conclusion.[18][19] For example, in the argument "all puppies are dogs; all dogs are animals; therefore all puppies are animals", the propositions "all puppies are dogs" and "all dogs are animals" act as premises while the proposition "all puppies are animals" is the conclusion.[21][22]

A set of premises together with a conclusion is called an argument.[23][3] An inference is the mental process of reasoning that starts from the premises and arrives at the conclusion.[18][24] But the terms "argument" and "inference" are often used interchangeably in logic. The purpose of arguments is to convince a person that something is the case by providing reasons for this belief.[25][26] Many arguments in natural language do not explicitly state all the premises. Instead, the premises are often implicitly assumed, especially if they seem obvious and belong to common sense.[25][27] Some theorists distinguish between simple and complex arguments. A complex argument is made up of many sub-arguments. This way, a chain is formed in which the conclusions of earlier arguments act as premises for later arguments. Each link in this chain has to be successful for a complex argument to succeed.[18][25]

An argument is correct or incorrect depending on whether the premises offer support for the conclusion. This is often understood in terms of

fallacies,[31][32] although the use of incorrect arguments does not mean their conclusions are incorrect.[33]

Deductive reasoning

Main article: Deductive reasoning

Deductive reasoning is the mental process of drawing deductive inferences. Deductively

sound if it is valid and all its premises are true.[36] For example, inferring the conclusion "no cats are frogs" from the premises "all frogs are amphibians" and "no cats are amphibians" is a sound argument. But even arguments with false premises can be deductively valid, like inferring that "no cats are frogs" from the premises "all frogs are mammals" and "no cats are mammals". In this regard, it only matters that the conclusion could not be false if the premises are true and not whether they actually are true.[37]

Deductively valid arguments follow a rule of inference.[38] A rule of inference is a scheme of drawing conclusions that depends only on the logical form of the premises and the conclusion but not on their specific content.[39][40] The most-discussed rule of inference is the modus ponens. It has the following form: p; if p then q; therefore q. This scheme is deductively valid no matter what p and q stand for.[41][5] For example, the argument "today is Sunday; if today is Sunday then I don't have to go to work today; therefore I don't have to go to work today" is deductively valid because it has the form of modus ponens.[42] Other popular rules of inference include modus tollens (not q; if p then q; therefore not p) and the disjunctive syllogism (p or q; not p; therefore q).[42][43]

The rules governing deductive reasoning are often expressed formally as logical systems for assessing the correctness of deductive arguments.

double negation elimination, the principle of explosion, and the bivalence of truth.[50] So-called deviant logics reject some of these basic intuitions and propose alternative rules governing the validity of arguments.[44][51][52] For example, intuitionistic logics reject the law of excluded middle and the double negation elimination while paraconsistent logics reject the principle of explosion.[52][53][54]

Deductive reasoning plays a central role in formal logic and

natural numbers can be inferred using deductive reasoning.[55][56]

Non-deductive reasoning

Non-deductive reasoning is an important form of logical reasoning besides deductive reasoning. It happens in the form of inferences drawn from premises to reach and support a conclusion, just like its deductive counterpart. The hallmark of non-deductive reasoning is that this support is fallible. This means that if the premises are true, it makes it more likely but not certain that the conclusion is also true.[57][58] So for a non-deductive argument, it is possible for all its premises to be true while its conclusion is still false. There are various types of non-deductive reasoning, like inductive, abductive, and analogical reasoning.[1][59] Non-deductive reasoning is more common in everyday life than deductive reasoning.[60]

Non-deductive reasoning is ampliative and defeasible.[61][62] Sometimes, the terms non-deductive reasoning, ampliative reasoning, and defeasible reasoning are used synonymously even though there are slight differences in their meaning. Non-deductive reasoning is ampliative in the sense that it arrives at information not already present in the premises. Deductive reasoning, by contrast, is non-ampliative since it only extracts information already present in the premises without adding any additional information.[62][63][59] So with non-deductive reasoning, one can learn something new that one did not know before. But the fact that new information is added means that this additional information may be false. This is why non-deductive reasoning is not as secure as deductive reasoning.[58][64]

A closely related aspect is that non-deductive reasoning is defeasible or non-monotonic. This means that one may have to withdraw a conclusion upon learning new information. For example, if all birds a person has seen so far can fly, this person is justified in reaching the inductive conclusion that all birds fly. This conclusion is defeasible because the reasoner may have to revise it upon learning that penguins are birds that do not fly.[65][66][67]

Inductive

Main article: Inductive reasoning
Photo of an Australian raven
Based on many individual observations of black ravens, inductive reasoning can be used to infer that all ravens are black.

Inductive reasoning starts from a set of individual instances and uses generalization to arrive at a universal law governing all cases.

probabilistic reasoning.[72] Like other forms of non-deductive reasoning, induction is not certain. This means that the premises support the conclusion by making it more probable but do not ensure its truth. In this regard, the conclusion of an inductive inference contains new information not already found in the premises.[68][60][1]

Various aspects of the premises are important to ensure that they offer significant support to the conclusion. In this regard, the

sample size should be large to guarantee that many individual cases were considered before drawing the conclusion.[60][73] An intimately connected factor is that the sample is random and representative. This means that it includes a fair and balanced selection of individuals with different key characteristics. For example, when making a generalization about human beings, the sample should include members of different races, genders, and age groups.[60][74][75] A lot of reasoning in everyday life is inductive. For example, when predicting how a person will react to a situation, inductive reasoning can be employed based on how the person reacted previously in similar circumstances. It plays an equally central role in the sciences, which often start with many particular observations and then apply the process of generalization to arrive at a universal law.[76][77][1]

A well-known issue in the field of inductive reasoning is the so-called problem of induction. It concerns the question of whether or why anyone is justified in believing the conclusions of inductive inferences. This problem was initially raised by David Hume, who holds that future events need not resemble past observations. In this regard, inductive reasoning about future events seems to rest on the assumption that nature remains uniform.[78][79]

Abductive

Main article: Abductive reasoning

Abductive reasoning is usually understood as an inference from an observation to a fact explaining this observation. Inferring that it has rained after seeing that the streets are wet is one example. Often, the expression "inference to the best explanation" is used as a synonym.[80][81][1] This expression underlines that there are usually many possible explanations of the same fact and that the reasoner should only infer the best explanation. For example, a tsunami could also explain why the streets are wet but this is usually not the best explanation. As a form of non-deductive reasoning, abduction does not guarantee the truth of the conclusion even if the premises are true.[80][82]

The more plausible the explanation is, the stronger it is supported by the premises. In this regard, it matters that the explanation is simple, i.e. does not include any unnecessary claims, and that it is consistent with established knowledge.[83][81][84] Other central criteria for a good explanation are that it fits observed and commonly known facts and that it is relevant, precise, and not circular. Ideally, the explanation should be verifiable by empirical evidence. If the explanation involves extraordinary claims then it requires very strong evidence.[84]

Photo of a medical examination
Doctors use abductive reasoning when investigating the symptoms of a patient to determine their underlying cause.

Abductive reasoning plays a central role in science when researchers discover unexplained phenomena. In this case, they often resort to a form of guessing to come up with general principles that could explain the observations. The

hypotheses are then tested and compared to discover which one provides the best explanation.[85][84] This pertains particularly to cases of causal reasoning that try to discover the relation between causes and effects.[84] Abduction is also very common in everyday life. It is used there in a similar but less systematic form.[85][84] This relates, for example, to the trust people put in what other people say. The best explanation of why a person asserts a claim is usually that they believe it and have evidence for it. This form of abductive reasoning is relevant to why one normally trusts what other people say even though this inference is usually not drawn in an explicit way. Something similar happens when the speaker's statement is ambiguous and the audience tries to discover and explain what the speaker could have meant.[85] Abductive reasoning is also common in medicine when a doctor examines the symptoms of their patient in order to arrive at a diagnosis of their underlying cause.[1]

Analogical

Zucker rats.[86][87]

Analogical reasoning involves the comparison of two systems in relation to their similarity. It starts from information about one system and infers information about another system based on the resemblance between the two systems.[88][89] Expressed schematically, arguments from analogy have the following form: (1) a is similar to b; (2) a has feature F; (3) therefore b probably also has feature F.[89][90] Analogical reasoning can be used, for example, to infer information about humans from medical experiments on animals: (1) rats are similar to humans; (2) birth control pills affect the brain development of rats; (3) therefore they may also affect the brain development of humans.[86]

Through analogical reasoning, knowledge can be transferred from one situation or domain to another. Arguments from analogy provide support for their conclusion but do not guarantee its truth. Their strength depends on various factors. The more similar the systems are, the more likely it is that a given feature of one object also characterizes the other object. Another factor concerns not just the degree of similarity but also its relevance. For example, an artificial strawberry made of plastic may be similar to a real strawberry in many respects, including its shape, color, and surface structure. But these similarities are irrelevant to whether the artificial strawberry tastes as sweet as the real one.[91]

Analogical reasoning plays a central role in

problem-solving, decision-making, and learning. It can be used both for simple physical characteristics and complex abstract ideas.[92][93] In science, analogies are often used in models to understand complex phenomena in a simple way. For example, the Bohr model explains the interactions of sub-atomic particles in analogy to how planets revolve around the sun.[94][95]

Fallacies

Main article: Fallacy

A fallacy is an incorrect argument or a faulty form of reasoning. This means that the premises provide no or not sufficient support for the conclusion. Fallacies often appear to be correct on the first impression and thereby seduce people into accepting and using them. In logic, the term "fallacy" does not mean that the conclusion is false. Instead, it only means that some kind of error was committed on the way to reaching the conclusion. An argument can be a fallacy even if, by a fortuitous accident, the conclusion is true. Outside the field of logic, the term "fallacy" is sometimes used in a slightly different sense for a false belief or theory and not for an argument.[32][96][97]

Fallacies are usually divided into

denying a conjunct, and the fallacy of the undistributed middle.[32][96][101]

Informal fallacies are expressed in natural language. Their main fault usually lies not in the form of the argument but has other sources, like its content or context.

strawman fallacies, even involve correct deductive reasoning on the formal level.[97] The content of an argument is the idea that is expressed in it. For example, a false dilemma is an informal fallacy that is based on an error in one of the premises. The faulty premise oversimplifies reality: it states that things are either one way or another way but ignore many other viable alternatives.[102][103] False dilemmas are often used by politicians when they claim that either their proposal is accepted or there will be dire consequences. Such claims usually ignore that various alternatives exist to avoid those consequences, i.e. that their proposal is not the only viable solution.[104]

The strawman fallacy is another informal fallacy. Its error happens on the level of the context. It consists in misrepresenting the view of an opponent and then refuting this view. The refutation itself is often correct but the error lies in the false assumption that the opponent actually defends this view. For example, an alcohol lobbyist may respond to the suggestion to ban alcohol advertisements on television by claiming that it is impossible to make people give up drinking alcohol. This is a strawman fallacy since the suggestion was merely to ban advertisements and not to stop all alcohol consumption.[105][96][106]

Ambiguous and vague expressions in natural language are often responsible for the faulty reasoning in informal fallacies. For example, this is the case for fallacies of ambiguity, like the argument "(1) feathers are light; (2) light is opposed to darkness; (3) therefore feathers are opposed to darkness". The error is found in the ambiguous term "light", which has one meaning in the first premise ("not heavy") and a different meaning in the second premise ("visible electromagnetic radiation").[107][108][109]

As a skill

Some theorists discuss logical reasoning in a very wide sense that includes its role as a broad skill responsible for high-quality thinking. In this sense, it is roughly equivalent to critical thinking and includes the capacity to select and apply the appropriate rules of logic to specific situations.[110] It encompasses a great variety of abilities besides drawing conclusions from premises. Examples are to understand a position, to generate and evaluate reasons for and against it as well as to critically assess whether to accept or reject certain information. It is about making judgments and drawing conclusions after careful evaluation and contrasts in this regard with uncritical snap judgments and gut feelings.[17] Other core skills linked to logical reasoning are to assess reasons before accepting a claim and to search for new information if more is needed to reach a reliable conclusion. It also includes the ability to consider different courses of action and compare the advantages and disadvantages of their consequences, to use common sense, and to avoid inconsistencies.[111][112] The skills responsible for logical reasoning can be learned, trained, and improved.[17][113]

Logical reasoning is relevant both on the

skeptical and open-minded at the same time.[120]

On the practical level, logical reasoning concerns the issue of making rational and effective decisions.[114][115] For many real-life decisions, various courses of action are available to the agent. For each possible action, there can be conflicting reasons, some in favor of it and others opposed to it. In such cases, logical reasoning includes weighing the potential benefits and drawbacks as well as considering their likelihood in order to arrive at a balanced all-things-considered decision.[121][122] For example, when a person runs out of drinking water in the middle of a hiking trip, they could employ the skills associated with logical reasoning to decide whether to boil and drink water from a stream that might contain dangerous microorganisms rather than break off the trip and hike back to the parking lot. This could include considering factors like assessing how dangerous the microorganisms are and the likelihood that they survive the boiling procedure. It may also involve gathering relevant information to make these assessments, for example, by asking other hikers.[123]

Time also plays a central role in logical reasoning.[124] If one lacks important information, it is often better to delay a decision and look for new information before coming to a conclusion.[111] If the decision is time-sensitive, on the other hand, logical reasoning may imply making a fast decision based on the currently available evidence even if it is very limited. For example, if a friend yells "Duck!" during a baseball game the most logical response may be to blindly trust them and duck instead of demanding an explanation or investigating what might have prompted their exclamation.[124][125] Generally speaking, the less time there is, the more significant it is to trust intuitions and gut feelings. If there is more time, on the other hand, it becomes important to examine ambiguities and assess contradictory information.[126]

See also

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Sources

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