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S5 (modal Logic)
In logic and philosophy, S5 is one of five systems of modal logic proposed by Clarence Irving Lewis and Cooper Harold Langford in their 1932 book ''Symbolic Logic''. It is a normal modal logic, and one of the oldest systems of modal logic of any kind. It is formed with propositional calculus formulas and tautologies, and inference apparatus with substitution and modus ponens, but extending the syntax with the modal operator ''necessarily'' \Box and its dual ''possibly'' \Diamond. The axioms of S5 The following makes use of the modal operators \Box ("necessarily") and \Diamond ("possibly"). S5 is characterized by the axioms: *K: \Box(A\to B)\to(\Box A\to\Box B); *T: \Box A \to A, and either: * 5: \Diamond A\to \Box\Diamond A; * or both of the following: :* 4: \Box A\to\Box\Box A, and :* B: A\to\Box\Diamond A. The (5) axiom restricts the accessibility relation R of the Kripke frame to be Euclidean, i.e. (wRv \land wRu) \implies vRu . Kripke semantics In terms of Kripke seman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Five Ways (Aquinas)
The ''Quinque viæ'' (Latin for "Five Ways") (sometimes called "five proofs") are five logical arguments for the existence of God summarized by the 13th-century Catholic philosopher and theologian Thomas Aquinas in his book ''Summa Theologica''. They are: #the argument from "first mover"; #the argument from universal causation; #the argument from contingency; #the argument from degree; #the argument from final cause or ends ("teleological argument"). Aquinas expands the first of these – God as the "unmoved mover" – in his ''Summa Contra Gentiles''. Background Need for demonstration of the existence of God Aquinas did not think the finite human mind could know what God is directly, therefore God's existence is not self-evident to us.''ST'', I, Q 2, A 1Latinan So instead the proposition ''God exists'' must be "demonstrated" from God's effects, which are more known to us.''ST'', I, Q 2, A 2an However, Aquinas did not hold that what could be demonstrated philosoph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Aquinas
Thomas Aquinas, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican friar and priest who was an influential philosopher, theologian and jurist in the tradition of scholasticism; he is known within the tradition as the , the , and the . The name ''Aquinas'' identifies his ancestral origins in the county of Aquino in present-day Lazio, Italy. Among other things, he was a prominent proponent of natural theology and the father of a school of thought (encompassing both theology and philosophy) known as Thomism. He argued that God is the source of both the light of natural reason and the light of faith. He has been described as "the most influential thinker of the medieval period" and "the greatest of the medieval philosopher-theologians". His influence on Western thought is considerable, and much of modern philosophy is derived from his ideas, particularly in the areas of ethics, natural law, metaphysics, and political theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects X in \mathcal and Y in \mathcal, a bijection between the respective morphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontological Argument
An ontological argument is a philosophical argument, made from an ontological basis, that is advanced in support of the existence of God. Such arguments tend to refer to the state of being or existing. More specifically, ontological arguments are commonly conceived ''a priori'' in regard to the organization of the universe, whereby, if such organizational structure is true, God must exist. The first ontological argument in Western Christian traditionSzatkowski, Miroslaw, ed. 2012. ''Ontological Proofs Today''. Ontos Verlag. "There are three main periods in the history of ontological arguments. The first was in 11th century, when St. Anselm of Canterbury came up with the first ontological argument" (p. 22). was proposed by Saint Anselm of Canterbury in his 1078 work, ''Proslogion'' (), in which he defines God as "a being than which no greater can be conceived," and argues that such being must exist in the mind, even in that of the person who denies the existence of God. Oppy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontological Argument
An ontological argument is a philosophical argument, made from an ontological basis, that is advanced in support of the existence of God. Such arguments tend to refer to the state of being or existing. More specifically, ontological arguments are commonly conceived ''a priori'' in regard to the organization of the universe, whereby, if such organizational structure is true, God must exist. The first ontological argument in Western Christian traditionSzatkowski, Miroslaw, ed. 2012. ''Ontological Proofs Today''. Ontos Verlag. "There are three main periods in the history of ontological arguments. The first was in 11th century, when St. Anselm of Canterbury came up with the first ontological argument" (p. 22). was proposed by Saint Anselm of Canterbury in his 1078 work, ''Proslogion'' (), in which he defines God as "a being than which no greater can be conceived," and argues that such being must exist in the mind, even in that of the person who denies the existence of God. Oppy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Possible World
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional logic, intensional and modal logic. Their metaphysics, metaphysical status has been a subject of controversy in philosophy, with Modal realism, modal realists such as David Lewis (philosopher), David Lewis arguing that they are literally existing alternate realities, and others such as Robert Stalnaker arguing that they are not. Logic Possible worlds are one of the foundational concepts in modal logic, modal and intensional logics. Formulas in these logics are used to represent statements about what ''might'' be true, what ''should'' be true, what one ''believes'' to be true and so forth. To give these statements a formal interpretation, logicians use structures containing possible worlds. For instance, in the relational semantics for classical propositional mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alvin Plantinga
Alvin Carl Plantinga (born November 15, 1932) is an American analytic philosopher who works primarily in the fields of philosophy of religion, epistemology (particularly on issues involving epistemic justification), and logic. From 1963 to 1982, Plantinga taught at Calvin University before accepting an appointment as the John A. O'Brien Professor of Philosophy at the University of Notre Dame. He later returned to Calvin University to become the inaugural holder of the Jellema Chair in Philosophy. A prominent Christian philosopher, Plantinga served as president of the Society of Christian Philosophers from 1983 to 1986. He has delivered the Gifford Lectures two times and was described by ''Time'' magazine as "America's leading orthodox Protestant philosopher of God". In 2014, Plantinga was the 30th most-cited contemporary author in the Stanford Encyclopedia of Philosophy. A fellow of the American Academy of Arts and Sciences, he was awarded the Templeton Prize in 2017. Some of Pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
