For contraposition in the field of traditional logic, see Contraposition traditional logic . For contraposition in the field of symbolic logic, see Transposition logic . Contraposition is a logical relationship between two proposition s, or statements. For example, take the following true proposition All bats are mammals . We can restate that as If something is a bat, then it is a mammal . The contrapositive is, If something is not a mammal, then it is not a bat . In mathematics and in logic , the contrapositive is always guaranteed to be true, as long as the original proposition is true. If the original proposition is false, the contrapositive will always be false, as well. Compare that to the other common relationships between propositions Inverse logic Inversion If something is not a bat, then it is not a mammal . Unlike the contrapositive, the inversion s truth value is not dependent upon whether the original proposition is true, as evidenced here. The inverse, here, is clearly not true. Conversion logic Conversion If something is a mammal, then it is a bat . The conversion is actually the contrapositive of the inversion and always has the same truth value as the inversion, which is not necessarily the same as that of the original proposition. Contradiction There exists a bat that is not a mammal . If the contradiction is true, then the original proposition and, by extension, the contrapositive are untrue. Here, of course, the contradiction 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 within A, then it must also be within B. We can rephrase that as math A to B ... , 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 ... by contrapositive proof by contraposition . Category Mathematical logic Category Inference ca Contraposici ... more details
In traditional logic , contraposition is a form of immediate inference in which from a given proposition ... logic rule of transposition . Contraposition also has distinctive applications in its philosophical ... logic traditional logic the process of contraposition is a schema composed of several steps of inference ... is instantiated exististential instantiation . Conversion by contraposition is the simultaneous ... with limitations and changes in quantity. This is considered full contraposition. Since in the process of contraposition the obversion obverse can be obtained in all four types of traditional propositions, yielding propositions with the contradictory of the original predicate, contraposition is first obtained by converting the obvert of the original proposition. Thus, partial contraposition can ... in the definition of contraposition with regard to the predicate of the inferred proposition, it can ... are non residents . The schema of contraposition ref Stebbing, L. Susan. A Modern Introduction to Logic ... style background color CCF th Original Proposition th th Obversion th th th th Contraposition th th Obverted Contraposition th tr tr style background color DDD td A All S is P td td E No S is non P ... 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 ... , or the law of contraposition. In its technical usage within the field of philosophic ... more details
An immediate inference is an inference which can be made from only one statement or proposition . For instance, from the statement All toads are green. we can make the immediate inference that No toads are not green. This new statement is known as the contrapositive of the original statement. There are a number of logical operations which can validly be made as an immediate inference. See also Contraposition traditional logic Conversion logic Obversion Transposition logic Inverse logic Square of opposition Superaltern Category Traditional logic Category Inference zh ... more details
men from All unmarried men are bachelors . Transposition and the method of contraposition In traditional ... propositions 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 ... All non P is non S . Since nothing is said in the definition of contraposition with regard to the predicate ... between transposition and contraposition Note that the method of transposition and contraposition should not be confused. Contraposition is a type of immediate inference in which from a given ... of the original predicate. Since nothing is said in the definition of contraposition with regard ... application to categorical propositions the result of contraposition is two contrapositives, each ..., MacMillan, 1979, fifth edition. ref of contraposition and is also referred to as the law of contraposition ... Improper Transposition Fallacy Files See also col begin col break Contraposition traditional logic Contraposition Mathematics Conversion logic Inference col break Obversion Propositional ... 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 . It has similarities to the Proof by contradiction , where math P wedge lnot Q Rightarrow perp math is proved in order to show math P Rightarrow Q math . See main article on contrapositionContraposition traditional logic . Example To prove For all whole numbers y , if y sup 2 sup is even then y 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. So by contrapositive if the square of x is even, x is even. ref cite book first J. last Franklin authorlink James Franklin philosopher date 2011 title http www.maths.unsw.edu.au jim proofs.html Proof in Mathematics An Introduction publisher Kew Books location Sydney isbn 0646545094 coauthors A. Daoud p. 50 . ref References Reflist See also Proof by contradiction Reductio ... more details
Confusing date December 2006 Context date February 2009 Rules of inference In logic , a destructive dilemma is any logical argument of the following form math P rightarrow Q math math R rightarrow S math math neg Q lor neg S math math vdash neg P lor neg R math where math vdash math represents the logical assertion . The argument makes use of contraposition , a property which states that math P rightarrow Q math math neg Q rightarrow neg P math The destructive dilemma is the disjunctive version of modus tollens . The disjunctive version of modus ponens is the constructive dilemma . Here is an example of the destructive dilemma in English If it rains, we will stay inside. If it is sunny, we will go for a walk. Either we will not stay inside, or we will not go for a walk. Therefore, either it will not rain, or it will not be sunny. References Reflist 1. http en.wikipedia.org wiki List of logic symbols External links http mathworld.wolfram.com DestructiveDilemma.html DEFAULTSORT Destructive Dilemma Category Rules of inference Logic stub ... 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 postiive 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
The inverse is a type of conditional sentence in formal logic. Any conditional sentence has an inverse the contrapositive of the 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 Conversion logic Obversion Transposition logic Contraposition DEFAULTSORT Inverse Logic Category Traditional logic Category Inference logic stub ja ... more details
Converse may refer to wiktionary converse A logical implication with the propositions reversed see conversion logic . Also see examples at Contraposition . Converse accident , a type of logical fallacy Conversion linguistics , a kind of word formation Conversion logic , a concept in traditional logic Religious conversion , the adaption of a new religious belief Converse shoe company , an American shoe company Seroconversion , the development of antibodies in the blood serum People Florence Converse 1871&ndash 1967 , American Frank Converse 1938&ndash , American actor Frederick Converse 1871&ndash 1940 , American composer of classical music Julius Converse 1798&ndash 1885 , American politician George A. Converse 1844&ndash 1909 , officer of the United States Navy George L. Converse 1827&ndash 1897, American politician Philip Converse , 1879&ndash 1928 , American political scientist Ric Converse 1979&ndash , American professional wrestler Emma M. Converse 19th century American science writer Places In the United States Converse, Indiana , in Miami County Converse, Blackford County, Indiana Converse, Louisiana Converse, Texas Converse County, Wyoming Converse College , a women s college in Spartanburg, South Carolina Converse Island , Maine Vessels Named after George A. Converse USS Converse DD 291 USS Converse DD 291 , U.S. Navy destroyer USS Converse DD 509 USS Converse DD 509 , U.S. Navy destroyer disambig de Converse Begriffskl rung es Converse desambiguaci n fr Converse it Converse disambigua nl Converse pl Converse pt Converse ru sk Converse vo Converse ... 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 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 inference ... more details
proposition. Contraposition The statements are logically equivalent . File Carl Sagan Viking.JPG ... Contraposition and Transposition Contraposition traditional logic Contraposition is a logically valid ... of the leading proposition that supports it. See Contraposition and Transposition logic Transposition ... 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
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. Definitionally, if B is a mean preserving spread of A then A is said to be a mean preserving contraction of B. 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 , w ... more details
Confusing section date October 2010 Rules of inference In logic , a transformation rule or rule of inference is a Syntax logic syntactic rule or function which takes premises and returns a conclusion or in multiple conclusion logic , conclusion s . For example, the rule of inference modus ponens takes two premises, one of the form If p then q and another of the form p and returns the conclusion q. The rule is sound 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 the semantic property of truth or designationhood more generally see many valued logic . But taken purely syntactically, a rule of inference need not 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 of interest 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 date 2007 publisher Cambridge University Press location Cambridge isbn 0 521 87752 0 page 364 ref Well known rules of inference include, besides the already mentioned 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 br ... br u Premise n u br ... 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
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