Liar paradoxIn philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement.
Set theorySet theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.
Russell's paradoxIn mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen.
Epimenides paradoxThe Epimenides paradox reveals a problem with self-reference in logic. It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC) who is credited with the original statement. A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter: Epimenides was a Cretan who made the immortal statement: "All Cretans are liars." A paradox of self-reference arises when one considers whether it is possible for Epimenides to have spoken the truth.
TruthTruth is the property of being in accord with fact or reality. In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences. Truth is usually held to be the opposite of falsehood. The concept of truth is discussed and debated in various contexts, including philosophy, art, theology, law, and science. Most human activities depend upon the concept, where its nature as a concept is assumed rather than being a subject of discussion, including journalism and everyday life.
Thought experimentA thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. The ancient Greek deiknymi, "was the most ancient pattern of mathematical proof", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment. Johann Witt-Hansen established that Hans Christian Ørsted was the first to use the term Gedankenexperiment (from German: 'thought experiment') circa 1812.
Berry paradoxThe Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters). Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian Library. Russell called Berry "the only person in Oxford who understood mathematical logic". The paradox was called "Richard's paradox" by Jean-Yves Girard.
PhilosophyPhilosophy (love of wisdom in ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, values, mind, and language. It is a rational and critical inquiry that reflects on its own methods and assumptions. Historically, many of the individual sciences, like physics and psychology, formed part of philosophy. But they are considered separate academic disciplines in the modern sense of the term.
Paraconsistent logicA paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject the principle of explosion. Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beside the consistent") was first coined in 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias.
Set (mathematics)A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics.
Self-referenceSelf-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding.
Zeno's paradoxesZeno's paradoxes are a set of philosophical problems devised by the Eleatic Greek philosopher Zeno of Elea (c. 490–430 BC). The origins of the paradoxes are somewhat unclear, but they are generally thought to have been developed to support Parmenides' doctrine of monism, that all of reality is one, and that all change is impossible. Diogenes Laërtius, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise.
DialetheismDialetheism (from Greek δι- 'twice' and ἀλήθεια 'truth') is the view that there are statements that are both true and false. More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called "true contradictions", dialetheia, or nondualisms. Dialetheism is not a system of formal logic; instead, it is a thesis about truth that influences the construction of a formal logic, often based on pre-existing systems.
LogicLogic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
StoicismStoicism is one of the four major schools of thought established in the Hellenistic period. It was founded in the ancient Agora of Athens by Zeno of Citium around 300 BC. The Stoics believed that the practice of virtue is enough to achieve eudaimonia: a well-lived, flourishing life. The Stoics identified the path to achieving it with a life spent practicing certain virtues in everyday life such as courage or temperance and living in accordance with nature.
RationalityRationality 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.
Zeno of EleaZeno of Elea (ˈziːnoʊ...ˈɛliə; Ζήνων ὁ Ἐλεᾱ́της; 495-430 BC) was a pre-Socratic Greek philosopher of Magna Graecia (southern Italy) and a member of the Eleatic School founded by Parmenides. Plato and Aristotle called him the inventor of the dialectic. He is best known for his paradoxes. Little is known for certain about Zeno's life. The primary source of biographical information about Zeno is Plato's dialogue Parmenides, which recounts a fictionalized account of a visit that Zeno and Parmenides made to Ancient Athens in 450 BC, at a time when Parmenides is "about 65", Zeno is "nearly 40", and Socrates is "a very young man".
Truth valueIn logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false). In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true.
Quantum entanglementQuantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.
Circular reasoningCircular reasoning (circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect in an argument whereby the premises are just as much in need of proof or evidence as the conclusion, and as a consequence the argument fails to persuade. Other ways to express this are that there is no reason to accept the premises unless one already believes the conclusion, or that the premises provide no independent ground or evidence for the conclusion.
