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Identity Of Indiscernibles
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' is also possessed by ''y'' and vice versa. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below. A form of the principle is attributed to the German philosopher Gottfried Wilhelm Leibniz. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontology
Ontology is the philosophical study of existence, being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every entity within it. To articulate the basic structure of being, ontology examines the commonalities among all things and investigates their classification into basic types, such as the Theory of categories, categories of particulars and Universal (metaphysics), universals. Particulars are unique, non-repeatable entities, such as the person Socrates, whereas universals are general, repeatable entities, like the color ''green''. Another distinction exists between Abstract and concrete, concrete objects existing in space and time, such as a tree, and abstract objects existing outside space and time, like the number 7. Systems of categories aim to provide a comprehensive inventory of reality by employing categories such as Substance t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Relation
A symmetric relation is a type of binary relation. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: : \forall a, b \in X(a R b \Leftrightarrow b R a) , where the notation ''aRb'' means that . An example is the relation "is equal to", because if is true then is also true. If ''R''T represents the converse of ''R'', then ''R'' is symmetric if and only if . Symmetry, along with reflexivity and transitivity, are the three defining properties of an equivalence relation. Examples In mathematics * "is equal to" ( equality) (whereas "is less than" is not symmetric) * "is comparable to", for elements of a partially ordered set * "... and ... are odd": :::::: Outside mathematics * "is married to" (in most legal systems) * "is a fully biological sibling of" * "is a homophone of" * "is a co-worker of" * "is a teammate of" Relationship to asymmetric and antisymmetric relations By definition, a nonempty relation cannot be both symmetric and asymmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Etymology The word ''reflexive'' is originally derived from the Medieval Latin ''reflexivus'' ('recoiling' reflex.html" ;"title="f. ''reflex">f. ''reflex'' or 'directed upon itself') (c. 1250 AD) from the classical Latin ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. ''Reflexive verb'' and ''Reflexive pronoun''). The first e ... [...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. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). 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". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from ) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equals Sign
The equals sign (British English) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality. In an equation it is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. In Unicode and ASCII it has the code point U+003D. It was invented in 1557 by the Welsh mathematician Robert Recorde. History Prior to 16th century there was no common symbol for equality, and equality was usually expressed with a word, such as ''aequales, aequantur, esgale, faciunt, ghelijck'' or ''gleich,'' and sometimes by the abbreviated form ''aeq'', or simply and . Diophantus's use of , short for ( 'equals'), in '' Arithmetica'' () is considered one of the first uses of an equals sign. The symbol, now universally accepted in mathematics for equality, was first recorded by the Welsh mathematician Robert Recorde in '' The Whetstone of Witte'' (1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. Biography Carnap's father rose from being a poor ribbon-weaver to be the owner of a ribbon-making factory. His mother came from an academic family; her father was an educational reformer and her oldest brother was the archaeologist Wilhelm Dörpfeld. As a ten-year-old, Carnap accompanied Wilhelm Dörpfeld on an expedition to Greece. Carnap was raised in a profoundly religious Protestant family, but later became an atheist. He began his formal education at the Barmen Gymnasium (school), Gymnasium and the Gymnasium in Jena. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. He also intently studied Immanuel Kant's ''Critique of Pure Reason'' during a course taught by Bruno Bauch, and was one of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, type theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school, Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.#FefA, Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equality (mathematics)
In mathematics, equality is a relationship between two quantities or Expression (mathematics), expressions, stating that they have the same value, or represent the same mathematical object. Equality between and is written , and read " equals ". In this equality, and are distinguished by calling them ''sides of an equation, left-hand side'' (''LHS''), and ''right-hand side'' (''RHS''). Two objects that are not equal are said to be distinct. Equality is often considered a primitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality like Reflexive relation, reflexivity, Symmetric relation, symmetry, and Transitive relation, transitivity have been understood intuitively since at least the ancient Greeks, but were not symboli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity (philosophy)
In metaphysics, identity (from , "sameness") is the relation each thing bears only to itself. The notion of identity gives rise to many philosophical problems, including the identity of indiscernibles (if ''x'' and ''y'' share all their properties, are they one and the same thing?), and questions about change and personal identity over time (what has to be the case for a person ''x'' at one time and a person ''y'' at a later time to be one and the same person?). It is important to distinguish between ''qualitative identity'' and ''numerical identity''. For example, consider two children with identical bicycles engaged in a race while their mother is watching. The two children have the ''same'' bicycle in one sense (''qualitative identity'') and the ''same'' mother in another sense (''numerical identity''). This article is mainly concerned with ''numerical identity'', which is the stricter notion. The philosophical concept of identity is distinct from the better-known notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
