mergefrom Contraposition traditional logic date March 2012 For contraposition in the field of traditional logic, see Contraposition traditional logic . For contraposition in the field of symbolic logic, see Transposition logic . In logic, contraposition is a logical relationship between two conditional statements. For example, take the following proposition All bats are mammals , which can be stated equivalently as the conditional statement If something is a bat, then it is a mammal . The contrapositive of this statement is If something is not a mammal, then it is not a bat . The contrapositive of a conditional statement is true if the original statement is true, and false if the original statement is false. The contrapositive can be compared with three other relationships between conditional statements Inverse logic Inversion the inverse If something is not a bat, then it is not a mammal . Unlike the contrapositive, the inverse s truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true. Conversion logic Conversion the converse If something is a mammal, then it is a bat . The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation There exists a bat that is not a mammal . If the negation is true, the original proposition and by extension the contrapositive is untrue. Here, of course, the negation is untrue. Simple proof using Venn diagrams File Venn A subset B.svg thumb right Consider the Venn diagram on the right. It appears clear that if something is in A, it must ... true , the other is also true. Likewise with falsity. Strictly speaking, a contraposition can only exist in two simple conditionals. However, a contraposition may also exist in two complex conditionals ... tool for proving mathematical theorems is the Proof by contrapositive proof by contraposition ... more details
mergeto Contraposition date March 2012 In traditional logic , contraposition is a form of immediate inference ... in modern logic see the Transposition logic rule of transposition . Contraposition also has distinctive ... types. Traditional logic In Term logic traditional logic the process of contraposition is a schema ... where in quantification existence is instantiated existential instantiation . Conversion by contraposition ... full contraposition. Since in the process of contraposition the obversion obverse can be obtained in all ... predicate, contraposition is first obtained by converting the obvert of the original proposition. Thus, partial contraposition can be obtained conditionally in an E type proposition with a change in quantity. Because nothing is said in the definition of contraposition with regard to the predicate ... proposition, All non voters are non residents . The schema of contraposition ref Stebbing ... th th th th Contraposition th th Obverted Contraposition th tr tr style background color DDD td ... non P is S td td O Some non P is not non S td tr table div Notice that contraposition is a valid form ..., where the obverse is an O proposition which has no conversion logic converse . The contraposition ... proposition is an A proposition which cannot be validly converted except by limitation, that is, contraposition ... to particular . Also, notice that contraposition is a method of inference which may require the use of other rules of inference. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial. The successive applications of conversion and obversion within the process of contraposition may be given by a variety ... transposition , or the law of contraposition. In its technical usage within the field of philosophic logic, the term contraposition may be limited by logicians e.g. Irving Copi , Susan Stebbing to traditional logic and categorical propositions. In this sense the use the term contraposition is usually ... more details
Merge contraposition date February 2010 In logic , the contrapositive of a indicative conditional conditional statement of the form if A then B is formed by negating both terms and reversing the direction of inference. Thus, the contrapositive of the statement if A, then B is if not B, then not A. A statement and its contrapositive are logically equivalent if the statement is true, then its contrapositive is true, and vice versa. ref Regents Exam Prep, http www.regentsprep.org Regents math geometry GP2 Lcontrap.htm contrapositive definition ref In logic, proof by contrapositive is a form of Mathematical proof proof ref http www.jimloy.com math proof.htm ref that establishes the Truth Formal theories truth or validity of a proposition by demonstrating the truth or validity of the converse of its negated parts. ref http zimmer.csufresno.edu larryc proofs proofs.contrapositive.html ref In other words, to prove by contraposition that math P Rightarrow Q math , prove that math lnot Q Rightarrow lnot P math . Any proof by contrapositive can also be trivially formulated in terms of a Proof by contradiction To prove the proposition math P Rightarrow Q math , we consider the opposite, math lnot P Rightarrow Q equiv lnot lnot P vee Q equiv P wedge lnot Q math . Since we have a proof that math lnot Q Rightarrow lnot P math , we have math P wedge lnot Q Rightarrow P wedge lnot P equiv bot math which arrives at the contradiction we want. So proof by contrapositive is in some sense at least as hard to formulate as proof by contradiction. See main article on contrapositionContraposition traditional logic . Example To prove For all whole numbers x , if x sup 2 sup is even then x is even. A direct proof is difficult, so a proof by contrapositive is preferable. Suppose x is not even that is, x is odd. Then x 2k 1 , for some whole number k . So x sup 2 sup 2k 1 sup 2 sup 4k sup 2 sup 4k 1 2 2k sup 2 sup 2k 1 which is odd. Thus we have proved if x is not even, then the square of x is not even ... more details
wiktionarypar highlight Highlight may refer to In photography, any of the brightest parts of a subject, in contraposition to shadow An especially significant or interesting detail or phenomenon . Highlight is also used in the following expressions Specular highlight , Spot of light that appears on shiny objects when illuminated Hair highlighting , Changing a person s hair at different levels Syntax highlighting , Displaying of text in different fonts or colors, often by category Highlighter , Marker like tool which adds translucent color to paper Highlights album Highlights album , Album by Tom Hingley and the Lovers Highlights for Children , Children s magazine Highlights , the school newspaper of Beverly Hills High School Highlights, public page in Russian social network VKontakte disambig de Highlight ko ja ... more details
An immediate inference is an inference which can be made from only one wikt statement statement or proposition . For instance, from the statement All toads are green. we can make the immediate inference that No toads are not green. There are a number of immediate inferences which can Validity validly be made using logical operations, the result of which is a Logical equivalence logically equivalent statement form to the given statement. There are also invalid immediate inferences which are syllogistic fallacy syllogistic fallacies . Valid immediate inferences Conversion main Conversion logic Given a type E statement, from the traditional square of opposition , No S are P . , one can make the immediate inference that No P are S which is the converse of the given statement. Given a type I statement, Some S are P . , one can make the immediate inference that Some P are S which is the converse of the given statement. Obversion main Obversion Given a type A statement, All S are P . , one can make the immediate inference that No S are non P which is the obverse of the given statement. Given a type E statement, No S are P . , one can make the immediate inference that All S are non P which is the obverse of the given statement. Given a type I statement, Some S are P . , one can make the immediate inference that Some S are not non P which is the obverse of the given statement. Given a type O statement, Some S are not P . , one can make the immediate inference that Some S are non P which is the obverse of the given statement. Contraposition main Contraposition traditional logic Given a type A statement, All S are P . , one can make the immediate inference that All non P are non S which is the contraposition of the given statement. Given a type O statement, Some S are not P . , one can make the immediate inference that Some non P are not non S which is the contraposition of the given statement. Invalid immediate inferences Cases of the incorrect application of the contrary, su ... more details
All unmarried men are bachelors . Transposition and the method of contraposition In traditional logic ... through contraposition and obversion , ref Stebbing, 1961, p. 65 66. For reference to the initial step of contraposition as obversion and conversion, see Copi, 1953, p. 141. ref a series of immediate ... in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible ..., and obversion again, see Copi, 1953, p. 141. ref Differences between transposition and contraposition Note that the method of transposition and contraposition should not be confused. Contraposition ... nothing is said in the definition of contraposition with regard to the predicate of the inferred ... the result of contraposition is two contrapositives, each being the obvert of the other, ref See ... inferences see Copi, Irving. Symbolic Logic . pp. 171 174, MacMillan, 1979, fifth edition. ref of contraposition and is also referred to as the law of contraposition . ref Prior, A.N. Logic, Traditional ... Contraposition traditional logic col break Syllogism Term logic col end References reflist Further ... more details
In logic , a conditional quantifier is a kind of Lindstr m quantifier or generalized quantifier math Q A math that, relative to a classical model math A math , satisfies some or all of the following conditions X and Y range over arbitrary formulas in one free variable math Q AXX math reflexivity math Q AXY Rightarrow Q AX Y land X math right conservativity math Q AX Y land X Rightarrow Q AXY math left conservativity math Q AXY Rightarrow Q AX Y lor Z math positive confirmation math Q AX Y land Z Rightarrow Q A X land Y Z math math Q AXY Rightarrow Q A X lor Z Y lor Z math positive and negative confirmation math Q AXY Rightarrow Q A lnot X lnot Y math contraposition math Q AXY land Q AYZ Rightarrow Q AXZ math transitivity math Q AXY Rightarrow Q A X land Z Y math weakening math Q AXY land Q AXZ Rightarrow Q AX Y land Z math conjunction math Q AXZ land Q AYZ Rightarrow Q A X lor Y Z math disjunction math Q AXY Rightarrow Q AYX math symmetry . The implication arrow denotes material implication in the metalanguage. The minimal conditional logic M is characterized by the first six properties, and stronger conditional logics include some of the other ones. For example, the quantifier math forall A math , which can be viewed as set theoretic inclusion, satisfies all of the above except symmetry . Clearly symmetry holds for math exists A math while e.g. contraposition fails. A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure i.e. a relation between properties defined on the structure. Some of the details can be found in the article Lindstr m quantifier . Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first order language as they relate to other connectives, such as conjunction or disjunction. While they can cover nested conditionals, the greater complexity of the formula, ... more details
In traditional logic , an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. Any conditional sentence has an inverse the contrapositive of the Conversion logic converse . The inverse of math P rightarrow Q math is thus math neg P rightarrow neg Q math . For example, substituting propositions in natural language for logical variables, the inverse of the conditional proposition, If it s raining, then Sam will meet Jack at the movies is If it s not raining, then Sam will not meet Jack at the movies. The inverse of the inverse, that is, the inverse of math neg P rightarrow neg Q math , is math neg neg P rightarrow neg neg Q math . Since a double negation has no logical effect, the inverse of the inverse is logically equivalent to the original conditional math P rightarrow Q math . Thus it is permissible to say that math neg P rightarrow neg Q math and math P rightarrow Q math are inverses of each other. Likewise, we may say that math P rightarrow neg Q math and math neg P rightarrow Q math are inverses of each other. The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. But the inverse of a conditional is not inferable from the conditional. For example, If it s not raining, then Sam will not meet Jack at the movies cannot be inferred from If it s raining, then Sam will meet Jack at the movies. It could easily be the case that Sam and Jack are attending the movies no matter the weather. See also Converse logic Obversion Transposition logic Contraposition DEFAULTSORT Inverse Logic Category Immediate inference logic stub am ja pl Twierdzenie przeciwne ... more details
Unreferenced date December 2009 In logic , statements var p var and var q var are logically equivalent if they have the same logical content. Syntax logic Syntactically , var p var and var q var are equivalent if each can be proof logic proved from the other. Semantic ally, var p var and var q var are equivalent if they have the same truth value in every model logic model . The logical equivalence of var p var and var q var is sometimes expressed as math p equiv q math , E pq , or math p Leftrightarrow q math . However, these symbols are also used for material equivalence the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related. Example The following statements are logically equivalent If Lisa is in France , then she is in Europe . In symbols, math f rightarrow e math . If Lisa is not in Europe, then she is not in France. In symbols, math neg e rightarrow neg f math . Syntactically, 1 and 2 are derivable from each other via the rules of contraposition and double negation . Semantically, 1 and 2 are true in exactly the same models interpretations, valuations namely, those in which either Lisa is in France is false or Lisa is in Europe is true. Note that in this example classical logic is assumed. Some non classical logic s do not deem 1 and 2 logically equivalent. Relation to material equivalence Logical equivalence is different from material equivalence . The material equivalence of p and q often written p q is itself another statement in same formal system object language as p and q , which expresses the idea p if and only if q . In particular, the truth value of p q can change from one model to another. The claim that two formulas are logically equivalent is a statement in the metalanguage , expressing a relationship between two statements p and q . The claim that p and q are semantically equivalent does not depend on any particular model it says that in every possib ... more details
Unreferenced date December 2009 Zeroth order logic is first order logic without quantifier s. A finitely Axiomatic system axiomatizable zeroth order logic is isomorphic to a propositional logic . Zeroth order logic with axiom schema is a more expressive system than propositional logic. An example is given by the system Primitive recursive arithmetic , or PRA. Example The well known syllogism All men are mortal Socrates is a man Therefore, Socrates is mortal cannot be formalized in propositional logic, because of the use of predicate grammar predicate s like is a man and is mortal . The obvious formalization in first order logic uses universal quantification to model the use of All . The following weak version of the syllogism can be formalized in propositional logic If Socrates is a man, then Socrates is mortal Socrates is a man Therefore, Socrates is mortal This can be done by introducing propositional constants SMN for Socrates is a man and SML for Socrates is mortal , and the two axioms SMN SML , and SMN . Together with the usual rule of modus ponens the conclusion, SML , follows. In this weak version most of the essence of the original syllogism has been lost. In predicate logic one can instead introduce predicates Man for is a man , Mortal for is mortal , constants A for Aristotle , S for Socrates , Z for Zeus , and so on, and use a multitude of axioms, one for each individual Man A Mortal A Man S Mortal S Man Z Mortal Z ... Man S Mortal Z Again, modus ponens allows to conclude Mortal S . If the axioms for contraposition are added, also Man Z becomes a theorem. By using an axiom schema , the above can be collapsed into Man x Mortal x Man S Mortal Z The first line uses the variable x , which can be instantiated by any constant for an individual, such as S . The axioms are then the substitution instance s of the schema. An equivalent approach is to declare the schema to be a plain axiom and to make First order logic Substitution variable substitution a special in ... more details
Italic title For the mathematical form of proof by contradiction, see Proof by contradiction . Reductio ad absurdum Latin reduction to the absurd is a form of argument in which a proposition is disproven by following its implications logically to an absurd consequence. ref name IEP cite web url http www.utm.edu research iep r reductio.htm work The Internet Encyclopedia of Philosophy title Reductio ad absurdum author Nicholas Rescher accessdate 21 July 09 ref A common type of reductio ad absurdum is proof by contradiction also called indirect proof , where a proposition is proved true by proving that it is impossible for it to be false. That is to say, if A being false implies that B must also be false and it is known that B is true, then A cannot be false and therefore A is true. Where such an argument is premised on a false dichotomy , the ostensible proof is a logical fallacy . Two simple examples of reductio ad absurdum are blockquote Proposition Raising taxation rates always results in increased tax revenue. br Proposition Lowering taxation rates always results in increased tax revenue. br blockquote These can both be disproved using reductio ad absurdum as follows blockquote If taxes were raised to 100 of income, individuals would not work, and companies would not operate, resulting in zero income, and thus zero tax. That is less than current tax income, thus the proposition is false. br If taxes were lowered to 0 , no taxes at all would be collected. Zero will always be less revenue than even the lowest non zero tax rate would produce, thus the proposition is false. blockquote Reductio ad absurdum should be contrasted from a similar, but irrational, argument known as a straw man . A straw man tactic relies on constructing an argument against an inaccurate representation of the original proposition. See also Contraposition Slippery slope fallacy Reductio ad ridiculum Reductio ad Hitlerum References Reflist 2 DEFAULTSORT Reductio Ad Absurdum Category Latin logic ... more details
as true, or perhaps exactly as untrue, as the original proposition. The statements are contraposition ... address issues of philosophic burden of proof . Related terms Contraposition and Transposition Contraposition traditional logic Contraposition is a logically valid rule of inference that allows ... Contraposition and Transposition logic Transposition in the Related terms section in this article ... more details
Multiple issues disputed March 2008 POV March 2008 The hypothetico deductive model or method , first so named by William Whewell , ref William Whewell 1837 History of the Inductive Sciences ref ref William Whewell 1840 , Philosophy of the Inductive Sciences ref is a proposed description of scientific method . According to it, scientific inquiry proceeds by formulating a hypothesis in a form that could conceivably be falsified by a test on observable data. A test that could and does run contrary to predictions of the hypothesis is taken as a Falsifiability falsification of the hypothesis. A test that could but does not run contrary to the hypothesis corroborates the theory. It is then proposed to compare the explanatory value of competing hypotheses by testing how stringently they are corroborated by their predictions. Quotation2 From the long tradition of empiricism we have inherited the hypothetico deductive model of scientific research . p.86 Brody, Thomas A. 1993 , The Philosophy Behind Physics , Springer Verlag, ISBN 0 387 55914 0 . Luis De La Pe a and Peter E. Hodgson, eds. Qualification of corroborating evidence is sometimes raised as philosophically problematic. The raven paradox is a famous example. The hypothesis that all ravens are black would appear to be corroborated by observations of only black ravens. However, all ravens are black is Logical equivalence logically equivalent to all non black things are non ravens this is the contraposition form of the original implication . This is a green tree is an observation of a non black thing that is a non raven and therefore corroborates all non black things are non ravens . It appears to follow that the observation this is a green tree is corroborating evidence for the hypothesis all ravens are black . Attempted resolutions may distinguish non falsifying observations as to strong, moderate, or weak corroborations investigations that do or do not provide a potentially falsifying test of the hypothesis. ref John ... more details
In probability and statistics , a mean preserving spread MPS ref Michael Rothschild Rothschild, Michael , and Joseph Stiglitz Stiglitz, Joseph , Increasing risk I A definition, Journal of Economic Theory , 1970, 225&ndash 243. ref is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A s probability density function while leaving the mean the expected value unchanged. As such, the concept of mean preserving spreads provides a stochastic ordering of equal mean gambles probability distributions according to their degree of risk this ordering is partial, meaning that of two equal mean gambles, it is not necessarily true that either is a mean preserving spread of the other. A is said to be a mean preserving contraction of B if B is a mean preserving spread of A. Ranking gambles by mean preserving spreads is a special case of ranking gambles by second order stochastic dominance &ndash namely, the special case of equal means If B is a mean preserving spread of A, then A is second order stochastically dominant over B and the contraposition Examples converse holds if A and B have equal means. If B is a mean preserving spread of A, then B has a higher variance than A but the converse is not in general true, because the variance is a complete ordering while ordering by mean preserving spreads is only partial. Example This example from ref Landsberger, M., and Meilijson, I., Mean preserving portfolio dominance, Review of Economic Studies 60, April 1993, 479&ndash 485. ref shows that to have a mean preserving spread does not require that all or most of the probability mass move away from the mean. Let A have equal probabilities math 1 100 math on each outcome math x Ai math , with math x Ai 198 math for math i 1, dots, 50 math and math x Ai 202 math for math i 51, dots,100 math and let B have equal probabilities math 1 100 math on each outcome math x Bi math , with math x B1 100 mat ... more details
Confusing section date October 2010 Transformation rules In logic , a rule of inference , inference rule , or transformation rule is the act of drawing a conclusion based on the Logical form form of premise s interpreted as a function which takes premises, analyses their Syntax logic syntax , and returns a conclusion or multiple conclusion logic conclusions . For example, the rule of inference modus ponens takes two premises, one in the form of If p then q and another in the form of p and returns the conclusion q . The rule is valid with respect to the semantics of classical logic as well as the semantics of many other non classical logic s , in the sense that if the premises are true under an interpretation then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many valued logic , it preserves a general designation. But a rule of inference s action is purely syntactic, and does not need to preserve any semantic property any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are Recursion recursive are important i.e. rules such that there is an effective procedure for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary rule . ref Cite book last1 Boolos first1 George last2 Burgess first2 John last3 Jeffrey first3 Richard C. title Computability and logic year 2007 publisher Cambridge University Press location Cambridge isbn 0 521 87752 0 page 364 ref Popular rules of inference include modus ponens, modus tollens from propositional logic and contraposition . First order predicate logic uses rules of inference to deal with logical quantifier s. See List of rules of inference for examples. Overview In formal logic and many related areas , rules of inference are usually given in the following standard form Premise 1 br Premise 2 b ... more details
Mar a Luisa Anido Isabel Mar a Luisa Anido Gonz lez was a Spain Spanish classical guitarist . She was born 26 January 1907 in Mor n, in the province of Buenos Aires , Argentina she died 4 June 1996 in Tarragona , Spain , and was buried there. Biography She was the fourth daughter of Juan Carlos Anido and Betilda Gonz lez Rigaud. Her family moved to Buenos Aires when she was very young. Mar a Luisa Anido s compositions for guitar Composing is a wonderful task because of the sincerity it carries within, because of the act of creation itself... because it reveals the greatest depths of the human soul. Mar a Luisa Anido Some excellent guitar performers of the 20th century turned to composition as an additional outlet for expressing their artistry. They include Miguel Llobet, Andr s Segovia, Agust n Barrios, Emilio Pujol, Abel Carlevaro, Nikita Koshkin, Stefan Rak, Carlo Domeniconi, Andrew York, and Du an Bogdanovi . Two women, Mar a Luisa Anido in Argentina and the Austrian Luise Walker, also left us products of their inspiration. Possibly out of modesty, Mar a Luisa Anido did not record all her works. They are miniatures that reflect, with her characteristic honesty, several aspects of her personality. Aire Norte o , her most popular piece, is a Bailecito , a little dance present in all festivities in north western Argentina which is generally accompanied by charangos, quenas and cajas. Anido frequently plays the bass notes pizzicato to emphasise the contraposition of 3 4 time in the bass and 6 8 in the melody, a characteristic that is frequently found in Argentine folklore. In 1927 Mar a Luisa Anido composed her first piece, Barcarola . Miguel Llobet, the Catalan guitarist, wrote to her shortly after that I have read and played your Barcarola the voices are carried magnificently with admirable taste of their natural characteristics the tone colours are perfect. Bravo, very well done. I think you should continue writing your excellent inspirations. In Canci n del Yucat ... more details
File Il ponte degli angeli by Scipione 1930.jpg thumb Il ponte degli angeli The Bridge of Angels,1930 , work by Scipione Gino Bonichi Scuola romana or Scuola di via Cavour was a 20th century art movement defined by a group of painters within Expressionism and active in Rome between 1928 and 1945, and with a second phase in the mid 1950s. Birth of the Movement In November 1927, artists Antonietta Rapha l and Mario Mafai ref See also it Wiki for it Antonietta Rapha l Antonietta Rapha l and it Mario Mafai Mario Mafai . ref move to No. 325 of Roman street Via Cavour, Rome via Cavour , in a Savoyan palace subsequently demolished in 1930 in order to allow the fascist construction of the New Empire Way currently the via dei Fori Imperiali . The apartment s larger room is transformed into a studio . Within a short time, this studio becomes a meeting point for intellectual literati such as it Enrico Falqui scrittore Enrico Falqui , Giuseppe Ungaretti , it Libero de Libero Libero de Libero , Leonardo Sinisgalli , as well as young artists Scipione Gino Bonichi Scipione , Renato Marino Mazzacurati , ref On Mazzacurati, see also his biographical it note at http www.scuolaromana.it artisti mazzacur.htm Scuola Romana.it ref and Corrado Cagli . Contraposition to the sensitivity of the Return to Order Movement From start, this spontaneous confluence of artists at the via Cavour studio does not appear to be led by true and proper programmes or manifesto s, but rather by friendship, cultural syntheses and a singular pictorial cohesion. With their firm approach to European expressionism , they formally contrapose the solid and orderly painting of Neoclassicism neoclassic character, promoted by the socalled Return to order current of the 1920s, particularly strong in the Italian sensitivity of post World War II . The first identification of this artistic group should be attributed to Roberto Longhi , who writes ref name Longhi in L Italia Letteraria Literary Italy of 7 April 1929. ref q ... more details
System NARS and Ben Goertzel s OpenCog system. See also col begin col break Aristotle ContrapositionContraposition traditional logic Conversion logic De Interpretatione Obversion col break Portal Logic ... more details
of a randomized balanced experiment fail to have constant variance, then by contraposition the assumption ..., by contraposition , a necessary condition for unit treatment additivity is that the variance ... more details
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