NationMaster - Encyclopedia: Deductive system

A deductive system (also called a deductive apparatus of a formal system) consists of the axioms (or axiom schemata) and rules of inference that can be used to derive the theorems of the system.^[1] In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... This article is about a logical statement. ... In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... ... A mathematical picture paints a thousand words: the Pythagorean theorem. ...

Such a deductive system is intended to preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead. Look up deduction in Wiktionary, the free dictionary. ... In mathematical logic, a formula is a formal syntactic object that expresses a proposition. ... Time Saving Truth from Falsehood and Envy, FranÃ§ois Lemoyne, 1737 For other uses, see Truth (disambiguation). ... In formal logic, a modal logic is any logic for handling modalities: concepts like possibility, existence, and necessity. ... Justification can mean: justification (jurisprudence) justification (typesetting) justification (theology) In epistemology, justification of a belief is what renders it worth believing in terms of its probable truth. ... For other uses, see Believe. ...

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a syntactic consequence of the lines that precede it. There should be no element of any interpretation of the language that gets involved with the deductive nature of the system. In logic, given some formal language L and a set Form(L) (the set of well-formed formulas of L) closed under the rules of the grammar (or syntax) for L, there may be an interpretation intended to apply to some privileged subset Con(L) of the connectives of L... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

See also

In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language â€” that is, of a set of strings. ... In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

References

Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. ... In Islamic philosophy, logic played an important role. ... For other uses, see Reason (disambiguation). ... Philosophical logic is the application of formal logical techniques to problems that concern philosophers. ... Philosophy of logic is the branch of philosophy that is concerned with the nature and justification of systems of logic. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... The metalogic of a system of logic is the formal proof supporting its soundness. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Reasoning is the mental (cognitive) process of looking for reasons to support beliefs, conclusions, actions or feelings. ... Deductive reasoning is reasoning whose conclusions are intended to necessarily follow from its premises. ... Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ... Abduction, or inference to the best explanation, is a method of reasoning in which one chooses the hypothesis that would, if true, best explain the relevant evidence. ... Informal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial (technical) or formal language (see formal logic). ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... Look up argument in Wiktionary, the free dictionary. ... In logic, the form of an argument is valid precisely if it cannot lead from true premises to a false conclusion. ... An argument is cogent if and only if the truth of the arguments premises would render the truth of the conclusion probable (i. ... Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... are you kiddin ? i was lookin for it for hours ... Look up fallacy in Wiktionary, the free dictionary. ... A syllogism (Greek: â€” conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... Argumentation theory, or argumentation, embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. ... Philosophy of logic is the branch of philosophy that is concerned with the nature and justification of systems of logic. ... Platonic realism is a philosophical term usually used to refer to the idea of realism regarding the existence of universals after the Greek philosopher Plato who lived between c. ... Logical atomism is a philosophical belief that originated in the early 20th century with the development of analytic philosophy. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... In philosophy, nominalism is the theory that abstract terms, general terms, or universals do not represent objective real existents, but are merely names, words, or vocal utterances (flatus vocis). ... Fictionalism is a doctrine in philosophy that suggests that statements of a certain sort should not be taken to be literally true, but merely a useful fiction. ... Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... In mathematics, logic, and computer science, a formal language is a language that is defined by precise mathematical or machine processable formulas. ... In computer science and linguistics, a formal grammar, or sometimes simply grammar, is a precise description of a formal language â€” that is, of a set of strings. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... ... In theoretical computer science formal semantics is the field concerned with the rigorous mathematical study of the meaning of programming languages and models of computation. ... In mathematical logic, a formula is a formal syntactic object that expresses a proposition. ... In logic, WFF is an abbreviation for well-formed formula. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... This article is about a logical statement. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... A mathematical picture paints a thousand words: the Pythagorean theorem. ... Logical consequence is the relation that holds between a set of sentences and a sentence when the latter follows from the former. ... Look up Consistency in Wiktionary, the free dictionary. ... (This article discusses the soundess notion of informal logic. ... Look up completeness in Wiktionary, the free dictionary. ... A logical system or theory is decidable if the set of all well-formed formulas valid in the system is decidable. ... 3SAT redirects here. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. ... Syntax in logic is a systematic statement of the rules governing the properly formed formulas (WFFs) of a logical system. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. ... In logic, the monadic predicate calculus is the fragment of predicate calculus in which all predicate letters are monadic (that is, they take only one argument), and there are no function letters. ... In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules allows to establish that certain formulas are theorems of the formal system. ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... In formal logic, a modal logic is any logic for handling modalities: concepts like possibility, existence, and necessity. ... Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. ... Michaels the greatest boyfriend in the whole wide world, and Id love to call him in a phonebooth sometime. ... In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. ... doxastic logic is a modal logic that is concerned with reasoning about beliefs. ... Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. ... Introduced by Giorgi Japaridze in 2003, Computability logic is a research programme and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. ... For the Super Furry Animals album, see Fuzzy Logic (album). ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. ... A non-monotonic logic is a formal logic whose consequence relation is not monotonic. ... A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Look up paradox in Wiktionary, the free dictionary. ... Antinomy (Greek anti-, against, plus nomos, law) is a term used in logic and epistemology, which, loosely, means a paradox or unresolvable contradiction. ... Is logic empirical? is the title of two articles that discuss the idea that the algebraic properties of logic may, or should, be empirically determined; in particular, they deal with the question of whether empirical facts about quantum phenomena may provide grounds for revising classical logic as a consistent logical... Al Farabi (870-950) was born of a Turkish family and educated by a Christian physician in Baghdad, and was himself later considered a teacher on par with Aristotle. ... Abu HÄmed Mohammad ibn Mohammad al-GhazzÄlÄ« (1058-1111) (Persian: ), known as Algazel to the western medieval world, born and died in Tus, in the Khorasan province of Persia (modern day Iran). ... For the Christian theologian, see Abd al-Masih ibn Ishaq al-Kindi. ... Fakhr al-Din al-Razi (1149â€“1209) was a well-known Persian theologian and philosopher from Ray. ... For other uses, see Aristotle (disambiguation). ... Ibn Rushd, known as Averroes (1126 â€“ December 10, 1198), was an Andalusian-Arab philosopher and physician, a master of philosophy and Islamic law, mathematics, and medicine. ... (Persian: Ø§Ø¨Ù† Ø³ÙŠÙ†Ø§) (c. ... Not to be confused with George Boolos. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] â€“ January 6, 1918) was a German mathematician. ... Rudolf Carnap (May 18, 1891, Ronsdorf, Germany â€“ September 14, 1970, Santa Monica, California) was an influential philosopher who was active in central Europe before 1935 and in the United States thereafter. ... â€¹ The template below (Expand) is being considered for deletion. ... Dharmakirti (circa 7th century), was an Indian scholar and one of the Buddhist founders of Indian philosophical logic. ... DignÄga (5th century AD), was an Indian scholar and one of the Buddhist founders of Indian philosophical logic. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar â€“ 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Gerhard Karl Erich Gentzen (November 24, 1909 â€“ August 4, 1945) was a German mathematician and logician. ... Kanada (also transliterated as Kanad and in other ways; Sanskrit à¤•à¤£à¤¾à¤¦) was a Hindu sage who founded the philosophical school of Vaisheshika. ... Kurt GÃ¶del (IPA: ) (April 28, 1906 BrÃ¼nn, Austria-Hungary (now Brno, Czech Republic) â€“ January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... The NyÄya SÅ«tras is an ancient Indian text on of philosophy composed by (also Gotama; c. ... | name = David Hilbert | image = Hilbert1912. ... Ala-al-din abu Al-Hassan Ali ibn Abi-Hazm al-Qarshi al-Dimashqi (Arabic: Ø¹Ù„Ø§Ø¡ Ø§Ù„Ø¯ÙŠÙ† Ø£Ø¨Ùˆ Ø§Ù„ØØ³Ù† Ø¹Ù„ÙŠÙ‘ Ø¨Ù† Ø£Ø¨ÙŠ ØØ²Ù… Ø§Ù„Ù‚Ø±Ø´ÙŠ Ø§Ù„Ø¯Ù…Ø´Ù‚ÙŠ ) known as ibn Al-Nafis (Arabic: Ø§Ø¨Ù† Ø§Ù„Ù†ÙÙŠØ³ ), was an Arab physician who is mostly famous for being the first to describe the pulmonary circulation of the blood. ... Abu Muhammad Ali ibn Ahmad ibn Sa`id ibn Hazm (Ø£Ø¨Ùˆ Ù…ØÙ…Ø¯ Ø¹Ù„ÙŠ Ø¨Ù† Ø§ØÙ…Ø¯ Ø¨Ù† Ø³Ø¹ÙŠØ¯ Ø¨Ù† ØØ²Ù…) (November 7, 994 â€“ August 15, 1069) was an Andalusian Muslim philosopher and theologian of Persian descent [1] born in CÃ³rdoba, present day Spain. ... Taqi al-Din Ahmad Ibn Taymiyyah (Arabic: )(January 22, 1263 - 1328), was a Sunni Islamic scholar born in Harran, located in what is now Turkey, close to the Syrian border. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ... Mozi (Chinese: ; pinyin: ; Wade-Giles: Mo Tzu, Lat. ... For other uses, see Nagarjuna (disambiguation). ... Indian postage stamp depicting (2004), with the implication that he used (à¤ªà¤¾à¤£à¤¿à¤¨à¤¿; IPA ) was an ancient Indian grammarian from Gandhara (traditionally 520â€“460 BC, but estimates range from the 7th to 4th centuries BC). ... Giuseppe Peano Giuseppe Peano (August 27, 1858 â€“ April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... Charles Sanders Peirce (IPA: /pÉs/), (September 10, 1839 â€“ April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... For people named Quine, see Quine (surname). ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... Shahab al-Din Yahya as-Suhrawardi (from the ArabicØ´Ù‡Ø§Ø¨ Ø§Ù„Ø¯ÙŠÙ† ÙŠØÙŠÙ‰ Ø³Ù‡Ø±ÙˆØ±Ø¯Ù‰, also known as Sohrevardi) (born 1153 in North-West-Iran; died 1191 in Aleppo) was a persian philosopher and Sufi, founder of School of Illumination, one of the most important islamic doctrine in Philosophy. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland â€“ October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Alan Mathison Turing, OBE, FRS (23 June 1912 â€“ 7 June 1954) was an English mathematician, logician, and cryptographer. ... Alfred North Whitehead, OM (February 15, 1861, Ramsgate, Kent, England â€“ December 30, 1947, Cambridge, Massachusetts, U.S.) was an English-born mathematician who became a philosopher. ... Lotfali Askar Zadeh (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California, Berkeley. ... This is a list of topics in logic. ... For a more comprehensive list, see the List of logic topics. ... This is a list of mathematical logic topics, by Wikipedia page. ... Algebra of sets George Boole Boolean algebra Boolean function Boolean logic Boolean homomorphism Boolean Implicant Boolean prime ideal theorem Boolean-valued model Boolean satisfiability problem Booles syllogistic canonical form (Boolean algebra) compactness theorem Complete Boolean algebra connective -- see logical operator de Morgans laws Augustus De Morgan duality (order... Set theory Axiomatic set theory Naive set theory Zermelo set theory Zermelo-Fraenkel set theory Kripke-Platek set theory with urelements Simple theorems in the algebra of sets Axiom of choice Zorns lemma Empty set Cardinality Cardinal number Aleph number Aleph null Aleph one Beth number Ordinal number Well... A logician is a person, such as a philosopher or mathematician, whose topic of scholarly study is logic. ... This is a list of rules of inference. ... This is a list of paradoxes, grouped thematically. ... This is a list of fallacies. ... In logic, a set of symbols is frequently used to express logical constructs. ...

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