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In logic, the law of non-contradiction (LNC) states that contradictory propositions cannot both be true in the same sense at the same time, e. g. the two propositions "p is the case" and "p is not the case" are mutually exclusive. Formally, this is expressed as the tautology ¬(p ∧ ¬p). The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case", holds.
Causality is an influence by which one event, process, state, or object (acause) contributes to the production of another event, process, state, or object (an effect) where the cause is at least partly responsible for the effect, and the effect is at least partly dependent on the cause. In general, a process can have multiple causes, which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.
The unexpected hanging paradox or surprise test paradox is a paradox about a person's expectations about the timing of a future event which they are told will occur at an unexpected time. The paradox is variously applied to a prisoner's hanging or a surprise school test. It was first introduced to the public in Martin Gardner's March 1963 Mathematical Games column in Scientific American magazine.
The omnipotence paradox is a family of paradoxes that arise with some understandings of the term omnipotent. The paradox arises, for example, if one assumes that an omnipotent being has no limits and is capable of realizing any outcome, even a logically contradictory one such as creating a square circle. Atheological arguments based on the omnipotence paradox are sometimes described as evidence for countering theism. Other possible resolutions to the paradox hinge on the definition of omnipotence applied and the nature of God regarding this application and whether omnipotence is directed toward God Himself or outward toward his external surroundings.
Simpson's paradox is a phenomenon in probability and statistics in which a trend appears in several groups of data but disappears or reverses when the groups are combined. This result is often encountered in social-science and medical-science statistics, and is particularly problematic when frequency data are unduly given causal interpretations. The paradox can be resolved when confounding variables and causal relations are appropriately addressed in the statistical modeling.
In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future.
Will, within philosophy, is a faculty of the mind. Will is important as one of the parts of the mind, along with reason and understanding. It is considered central to the field of ethics because of its role in enabling deliberate action.
The argument from free will, also called the paradox of free will or theological fatalism, contends that omniscience and free will are incompatible and that any conception of God that incorporates both properties is therefore inconceivable. See the various controversies over claims of God's omniscience, in particular the critical notion of foreknowledge. These arguments are deeply concerned with the implications of predestination.
Decision theory or the theory of rational choice is a branch of probability, economics, and analytic philosophy that uses the tools of expected utility and probability to model how individuals should behave rationally under uncertainty. It differs from the cognitive and behavioral sciences in that it is prescriptive and concerned with identifying optimal decisions for a rational agent, rather than describing how people really do make decisions. Despite this, the field is extremely important to the study of real human behavior by social scientists, as it lays the foundations for the rational agent models used to mathematically model and analyze individuals in fields such as sociology, economics, criminology, cognitive science, and political science.
Meno is a Socratic dialogue written by Plato. Meno begins the dialogue by asking Socrates whether virtue is taught, acquired by practice, or comes by nature. In order to determine whether virtue is teachable or not, Socrates tells Meno that they first need to determine what virtue is. When the characters speak of virtue, or rather arete, they refer to virtue in general, rather than particular virtues, such as justice or temperance. The first part of the work showcases Socratic dialectical style; Meno, unable to adequately define virtue, is reduced to confusion or aporia. Socrates suggests that they seek an adequate definition for virtue together. In response, Meno suggests that it is impossible to seek what one does not know, because one will be unable to determine whether one has found it.
Judea Pearl is an Israeli-American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks. He is also credited for developing a theory of causal and counterfactual inference based on structural models. In 2011, the Association for Computing Machinery (ACM) awarded Pearl with the Turing Award, the highest distinction in computer science, "for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning". He is the author of several books, including the technical Causality: Models, Reasoning and Inference, and The Book of Why, a book on causality aimed at the general public.
Buridan's ass is an illustration of a paradox in philosophy in the conception of free will. It refers to a hypothetical situation wherein an ass (donkey) that is equally hungry and thirsty is placed precisely midway between a stack of hay and a pail of water. Since the paradox assumes the donkey will always go to whichever is closer, it dies of both hunger and thirst since it cannot make any rational decision between the hay and water. A common variant of the paradox substitutes the hay and water for two identical piles of hay; the ass, unable to choose between the two, dies of hunger.
A temporal paradox, time paradox, or time travel paradox, is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time travel or other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistic time dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.
David Harold Blackwell was an American statistician and mathematician who made significant contributions to game theory, probability theory, information theory, and statistics. He is one of the eponyms of the Rao–Blackwell theorem. He was the first African American inducted into the National Academy of Sciences, the first African American full professor at the University of California, Berkeley, and the seventh African American to receive a Ph.D. in mathematics. In 2012, President Barack Obama posthumously awarded Blackwell the National Medal of Science.
The private language argument argues that a language understandable by only a single individual is incoherent, and was introduced by Ludwig Wittgenstein in his later work, especially in the Philosophical Investigations. The argument was central to philosophical discussion in the second half of the 20th century.
The 19th-century philosopher Friedrich Nietzsche is known as a critic of Judeo-Christian morality and religions in general. One of the arguments he raised against the truthfulness of these doctrines is that they are based upon the concept of free will, which, in his opinion, does not exist.
Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree", which introduced the set theoretic description of common knowledge. The theorem concerns agents who share a common prior and update their probabilistic beliefs by Bayes' rule. It states that if the probabilistic beliefs of such agents, regarding a fixed event, are common knowledge then these probabilities must coincide. Thus, agents cannot agree to disagree, that is have common knowledge of a disagreement over the posterior probability of a given event.
In statistics, Lord's paradox raises the issue of when it is appropriate to control for baseline status. In three papers, Frederic M. Lord gave examples when statisticians could reach different conclusions depending on whether they adjust for pre-existing differences. Holland & Rubin (1983) use these examples to illustrate how there may be multiple valid descriptive comparisons in the data, but causal conclusions require an underlying (untestable) causal model. Pearl used these examples to illustrate how graphical causal models resolve the issue of when control for baseline status is appropriate.
Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based on the idea that beliefs can be interpreted as subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesians have applied these fundamental principles to various epistemological topics but Bayesianism does not cover all topics of traditional epistemology. The problem of confirmation in the philosophy of science, for example, can be approached through the Bayesian principle of conditionalization by holding that a piece of evidence confirms a theory if it raises the likelihood that this theory is true. Various proposals have been made to define the concept of coherence in terms of probability, usually in the sense that two propositions cohere if the probability of their conjunction is higher than if they were neutrally related to each other. The Bayesian approach has also been fruitful in the field of social epistemology, for example, concerning the problem of testimony or the problem of group belief. Bayesianism still faces various theoretical objections that have not been fully solved.
References
- ↑ Savage, L. J. (1954), The foundations of statistics. John Wiley & Sons Inc., New York.
- 1 2 3 Jeffrey, Richard (1982). "The Sure Thing Principle". Proceedings of the Biennial Meeting of the Philosophy of Science Association. 1982 (2): 719–730. doi:10.1086/psaprocbienmeetp.1982.2.192456. JSTOR 192456. S2CID 124506828.
- 1 2 Pearl, Judea (2009). Causality: Models, Reasoning, and Inference (2nd ed.). New York: Cambridge University Press.
- ↑ Blyth, C. (1972). "On Simpson's paradox and the sure-thing principle". Journal of the American Statistical Association. 67 (338): 364–366. doi:10.2307/2284382. JSTOR 2284382.
- ↑ Pearl, Judea (December 2015). "The sure-thing principle" (PDF). UCLA Cognitive Systems Laboratory, Technical Report R-466.
- ↑ Samet, Dov (2022). "The impossibility of agreeing to disagree: An extension of the sure-thing principle". Games and Economic Behavior (338): 390–399. doi:10.1016/j.geb.2022.01.016.