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Intensional Logic
Intensional logic is an approach to predicate logic that extends first-order logic, which has quantifiers that range over the individuals of a universe (''extensions''), by additional quantifiers that range over terms that may have such individuals as their value (''intensions''). The distinction between intensional and extensional entities is parallel to the distinction between sense and reference. Overview Logic is the study of proof and deduction as manifested in language (abstracting from any underlying psychological or biological processes). Logic is not a closed, completed science, and presumably, it will never stop developing: the logical analysis can penetrate into varying depths of the language (sentences regarded as atomic, or splitting them to predicates applied to individual terms, or even revealing such fine logical structures like modal, temporal, dynamic, epistemic ones). In order to achieve its special goal, logic was forced to develop its own formal tools ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predicate 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truth Value
In 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''). Truth values are used in computing as well as various types of logic. Computing 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 are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called falsy and truthy. For example, in Lisp, nil, the empty list, is treated as false, and all other values are treated as true. In C, the number 0 or 0.0 is false, and all other values are treated as true. In JavaScript, the empty string (""), null, undefined, NaN, +0, −0 and false are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Referential Transparency
In analytic philosophy and computer science, referential transparency and referential opacity are properties of linguistic constructions, and by extension of languages. A linguistic construction is called ''referentially transparent'' when for any expression built from it, Rewriting, replacing a subexpression with another one that Denotation, denotes the same value does not change the value of the expression. Also: Otherwise, it is called ''referentially opaque''. Each expression built from a referentially opaque linguistic construction states something about a subexpression, whereas each expression built from a referentially transparent linguistic construction states something not about a subexpression, meaning that the subexpressions are ‘transparent’ to the expression, acting merely as ‘references’ to something else. For example, the linguistic construction ‘_ was wise’ is referentially transparent (e.g., ''Socrates was wise'' is equivalent to ''The founder of Weste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Morgan's Laws
In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of Logical conjunction, conjunctions and Logical disjunction, disjunctions purely in terms of each other via logical negation, negation. The rules can be expressed in English as: * The negation of "A and B" is the same as "not A or not B". * The negation of "A or B" is the same as "not A and not B". or * The Complement (set theory), complement of the union of two sets is the same as the intersection of their complements * The complement of the intersection of two sets is the same as the union of their complements or * not (A or B) = (not A) and (not B) * not (A and B) = (not A) or (not B) where "A or B" is an "inclusive or" meaning ''at least' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meta (prefix)
''Meta'' (from the , ''wikt:meta-, meta'', meaning 'after' or 'beyond') is an adjective meaning 'more comprehensive' or 'transcending'. In modern nomenclature, the prefix meta can also serve as a prefix meaning self-referential, as a field of study or endeavor (metatheory: theory about a theory; metamathematics: mathematical theories about mathematics; meta-axiomatics or meta-axiomaticity: axioms about axiomatic systems; metahumor: joking about the ways humor is expressed; etc.). Original Greek meaning In Ancient Greek, Greek, the prefix ''meta-'' is generally less esoteric than in English language, English; Greek ''meta-'' is equivalent to the Latin language, Latin words ''post-'' or ''ad-''. The use of the prefix in this sense occurs occasionally in English language, scientific English terms derived from Greek (language), Greek. For example, the term ''Metatheria'' (the name for the clade of marsupial mammals) uses the prefix ''meta-'' in the sense that the ''Metatheria'' occur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alethic Logic
Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula \Box P can be used to represent the statement that P is known. In deontic modal logic, that same formula can represent that P is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula \Box P \rightarrow P as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operators such as \Diamond and \Box, representing possibility and necessity respectively. For instance the modal formula \Diamond P can be read as "possibly P" while ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transparent Intensional Logic
Transparent intensional logic (frequently abbreviated as TIL) is a logical system created by Pavel Tichý. Due to its rich ''procedural semantics'' TIL is in particular apt for the logical analysis of natural language. From the formal point of view, TIL is a hyperintensional, partial, typed lambda calculus. TIL applications cover a wide range of topics from formal semantics, philosophy of language, epistemic logic, philosophical, and formal logic. TIL provides an overarching semantic framework for all sorts of discourse, whether colloquial, scientific, mathematical or logical. The semantic theory is a procedural one, according to which sense is an abstract, pre-linguistic procedure detailing what operations to apply to what procedural constituents to arrive at the product (if any) of the procedure. TIL procedures, known as ''constructions'', are hyperintensionally individuated. Construction is the single most important notion of transparent intensional logic, being a philosophic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Calculus
In mathematical logic, a proof calculus or a proof system is built to prove statements. Overview A proof system includes the components: * Formal language: The set ''L'' of formulas admitted by the system, for example, propositional logic or first-order logic. * Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems. * Axioms: Formulas in ''L'' assumed to be valid. All theorems are derived from axioms. A formal proof of a well-formed formula in a proof system is a set of axioms and rules of inference of proof system that infers that the well-formed formula is a theorem of proof system. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sense And Reference
In the philosophy of language, the distinction between sense and reference was an idea of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the two ways he believed a singular term may have meaning. The reference (or "referent"; ''Bedeutung'') of a ''proper name'' is the object it means or indicates (''bedeuten''), whereas its sense (''Sinn'') is what the name expresses. The reference of a ''sentence'' is its truth value, whereas its sense is the thought that it expresses."On Sense and Reference" Über Sinn und Bedeutung" '' Zeitschrift für Philosophie und philosophische Kritik'', vol. 100 (1892), pp. 25–50, esp. p. 31. Frege justified the distinction in a number of ways. #Sense is something possessed by a name, whether or not it has a reference. For example, the name "Odysseus" is intelligible, and therefore has a sense, even though there is no individual object (its refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-dimensional Semantics
Two-dimensionalism is an approach to semantics in analytic philosophy. It is a theory of how to determine the sense and reference of a word and the truth-value of a sentence. It is intended to resolve the puzzle: How is it possible to discover empirically that a necessary truth is true? Two-dimensionalism provides an analysis of the semantics of words and sentences that makes sense of this possibility. The theory was first developed by Robert Stalnaker, but it has been advocated by numerous philosophers since, including David Chalmers. Two-dimensional semantic analysis According to two-dimensionalism, any statement, for example "Water is H2O", is taken to express two distinct propositions, often referred to as a ''primary intension'' and a ''secondary intension'', which together compose its meaning. The ''primary intension'' of a word or sentence is its sense, i.e., is the idea or method by which we find its referent. In other words, it's how we identify something in any possibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, philosophy of logic, logic, and Philosophy of mathematics, mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the History of logic#Rise of modern logic, development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
