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Rational Functions: How to Graph and Solve Them



Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in rational animal, to a psychological process, like reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either arational, if it is outside the domain of rational evaluation, or irrational, if it belongs to this domain but does not fulfill its standards.


There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accounts, which define rationality as internal coherence among the agent's mental states. Many rules of coherence have been suggested in this regard, for example, that one should not hold contradictory beliefs or that one should intend to do something if one believes that one should do it. Goal-based accounts characterize rationality in relation to goals, such as acquiring truth in the case of theoretical rationality. Internalists believe that rationality depends only on the person's mind. Externalists contend that external factors may also be relevant. Debates about the normativity of rationality concern the question of whether one should always be rational. A further discussion is whether rationality requires that all beliefs are reviewed from scratch rather than trusting pre-existing beliefs.




rational



Various types of rationality are discussed in the academic literature. The most influential distinction is between theoretical and practical rationality. Theoretical rationality concerns the rationality of beliefs. Rational beliefs are based on evidence that supports them. Practical rationality pertains primarily to actions. This includes certain mental states and events preceding actions, like intentions and decisions. In some cases, the two can conflict, as when practical rationality requires that one adopts an irrational belief. Another distinction is between ideal rationality, which demands that rational agents obey all the laws and implications of logic, and bounded rationality, which takes into account that this is not always possible since the computational power of the human mind is too limited. Most academic discussions focus on the rationality of individuals. This contrasts with social or collective rationality, which pertains to collectives and their group beliefs and decisions.


Rationality is important for solving all kinds of problems in order to efficiently reach one's goal. It is relevant for and discussed in many disciplines. In ethics, one question is whether one can be rational without being moral at the same time. Psychology is interested in how psychological processes implement rationality. This also includes the study of failures to do so, as in the case of cognitive biases. Cognitive and behavioral sciences usually assume that people are rational to predict how they think and act. Logic studies the laws of correct arguments. These laws are highly relevant for the rationality of beliefs. A very influential conception of practical rationality is given in decision theory: it states that a decision is rational if the chosen option has the highest expected utility. Other relevant fields include game theory, Bayesianism, economics, and artificial intelligence.


In its most common sense, rationality is the quality of being guided by reasons or being reasonable.[1][2][3][4] For example, a person who acts rationally has good reasons for what they do. This usually implies that they reflected on the possible consequences of their action and the goal it is supposed to realize. In the case of beliefs, it is rational to believe something if the agent has good evidence for it and it is coherent with the agent's other beliefs.[5][6] While actions and beliefs are the most paradigmatic forms of rationality, the term is used both in ordinary language and in many academic disciplines to describe a wide variety of things, such as persons, desires, intentions, decisions, policies, and institutions.[7][8] Because of this variety in different contexts, it has proven difficult to give a unified definition covering all these fields and usages. In this regard, different fields often focus their investigation on one specific conception, type, or aspect of rationality without trying to cover it in its most general sense.[9]


These different forms of rationality are sometimes divided into abilities, processes, mental states, and persons.[7][2][1][9][10] For example, when it is claimed that humans are rational animals, this usually refers to the ability to think and act in reasonable ways. It does not imply that all humans are rational all the time: this ability is exercised in some cases but not in others.[7][9][10] On the other hand, the term can also refer to the process of reasoning that results from exercising this ability. Often many additional activities of the higher cognitive faculties are included as well, such as acquiring concepts, judging, deliberating, planning, and deciding as well as the formation of desires and intentions. These processes usually affect some kind of change in the thinker's mental states. In this regard, one can also talk of the rationality of mental states, like beliefs and intentions.[7] A person who possesses these forms of rationality to a sufficiently high degree may themselves be called rational.[1] In some cases, also non-mental results of rational processes may qualify as rational. For example, the arrangement of products in a supermarket can be rational if it is based on a rational plan.[7][2]


The term "rational" has two opposites: irrational and arational. Arational things are outside the domain of rational evaluation, like digestive processes or the weather. Things within the domain of rationality are either rational or irrational depending on whether they fulfill the standards of rationality.[11][8] For example, beliefs, actions, or general policies are rational if there is a good reason for them and irrational otherwise. It is not clear in all cases what belongs to the domain of rational assessment. For example, there are disagreements about whether desires and emotions can be evaluated as rational and irrational rather than arational.[7] The term "irrational" is sometimes used in a wide sense to include cases of arationality.[12]


The meaning of the terms "rational" and "irrational" in academic discourse often differs from how they are used in everyday language. Examples of behaviors considered irrational in ordinary discourse are giving into temptations, going out late even though one has to get up early in the morning, smoking despite being aware of the health risks, or believing in astrology.[13][14] In the academic discourse, on the other hand, rationality is usually identified with being guided by reasons or following norms of internal coherence. Some of the earlier examples may qualify as rational in the academic sense depending on the circumstances. Examples of irrationality in this sense include cognitive biases, making arithmetic mistakes, and violating the laws of probability theory when assessing the likelihood of future events.[13] This article focuses mainly on irrationality in the academic sense.


The terms "rationality", "reason", and "reasoning" are frequently used as synonyms. But in technical contexts, their meanings are often distinguished.[8][13][1] Reason is usually understood as the faculty responsible for the process of reasoning.[8][15] This process aims at improving mental states. Reasoning tries to ensure that the norms of rationality obtain. It differs from rationality nonetheless since other psychological processes besides reasoning may have the same effect.[8] Rationality derives etymologically from the Latin term rationalitas.[7]


There are many disputes about the essential characteristics of rationality. It is often understood in relational terms: something, like a belief or an intention, is rational because of how it is related to something else.[7][1] But there are disagreements as to what it has to be related to and in what way. For reason-based accounts, the relation to a reason that justifies or explains the rational state is central. For coherence-based accounts, the relation of coherence between mental states matters. There is a lively discussion in the contemporary literature on whether reason-based accounts or coherence-based accounts are superior.[16][6] Some theorists also try to understand rationality in relation to the goals it tries to realize.[1][17]


Other disputes in this field concern whether rationality depends only on the agent's mind or also on external factors, whether rationality requires a review of all one's beliefs from scratch, and whether we should always be rational.[7][1][13]


A common idea of many theories of rationality is that it can be defined in terms of reasons. On this view, to be rational means to respond correctly to reasons.[2][1][16] For example, the fact that a food is healthy is a reason to eat it. So this reason makes it rational for the agent to eat the food.[16] An important aspect of this interpretation is that it is not sufficient to merely act accidentally in accordance with reasons. Instead, responding to reasons implies that one acts intentionally because of these reasons.[2]


Some theorists understand reasons as external facts. This view has been criticized based on the claim that, in order to respond to reasons, people have to be aware of them, i.e. if have some form of epistemic access.[16][6] But lacking this access is not automatically irrational. In one example by John Broome, the agent eats a fish contaminated with salmonella, which is a strong reason against eating the fish. But since the agent could not have known this fact, eating the fish is rational for them.[18][19] Because of such problems, many theorists have opted for an internalist version of this account. This means that the agent does not need to respond to reasons in general, but only to reasons they have or possess.[2][16][6][20] The success of such approaches depends a lot on what it means to have a reason and there are various disagreements on this issue.[8][16] A common approach is to hold that this access is given through the possession of evidence in the form of cognitive mental states, like perceptions and knowledge. A similar version states that "rationality consists in responding correctly to beliefs about reasons". So it is rational to bring an umbrella if the agent has strong evidence that it is going to rain. But without this evidence, it would be rational to leave the umbrella at home, even if, unbeknownst to the agent, it is going to rain.[2][20] These versions avoid the previous objection since rationality no longer requires the agent to respond to external factors of which they could not have been aware.[2] 2ff7e9595c


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