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Topic: Contrapositive

	Contraposition - Wikipedia, the free encyclopedia
		In traditional logic, contraposition is a form of immediate inference in which from a given categorical proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality (affirmation or negation).
		Because nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it can be either the original subject, or its contradictory, resulting in a partial contraposition, or two contrapositives which are the obverts of one another.
		This is because the obverse of the "E" proposition is an "A" proposition which cannot be validly converted except by limitation, that is, contraposition plus a change in the quantity of the proposition from universal to particular.
	en.wikipedia.org /wiki/Contrapositive (534 words)

	Talk:Contrapositive - Wikipedia, the free encyclopedia
		Contraposition as an immediate inference is actually only one step in the method of deriving the tautology stated here and on the contraposition page.
		Contrapositive and contraposition are a terms of logic, math, and philosophy.
		The argument on the contrapositive and contraposition page is for the obversion (you can check the stub I expanded at obversion) of the contrapositive.
	en.wikipedia.org /wiki/Talk:Contrapositive (1064 words)

	PlanetMath: contrapositive
		An implication and its contrapositive are equivalent statements.
		Proof that contrapositive statement is true using logical equivalence
		This is version 5 of contrapositive, born on 2001-11-16, modified 2005-04-11.
	planetmath.org /encyclopedia/Contrapostive.html (73 words)

	Encyclopedia: Contrapositive
		In predicate logic, the contrapositive (or transposition) of the statement "p implies q" is "not-q implies not-p." A statement and its contrapositive are always logically equivalent, unlike a statement's inverse or its converse.
		Note that while a statement is logically equivalent to its contrapositive (where the two statements are both negated and "swapped"), it is not logically equivalent to its converse (with the two statements "swapped", but not negated).
		Contrapositive The contrapositive of a conditional statement is formed by interchanging the hypothesis and the conclusion, and negating both.
	www.nationmaster.com /encyclopedia/Contrapositive (1510 words)

	Contrapositive
		In predicate logic, the contrapositive (or transposition) of the statement "p implies q" is "not-q implies not-p." These are logically equivalent.
		The contrapositive of "All S is P" is "All P is S." (These are "A" propositions[?].)
		The contrapositive of "Some S is not P" is "Some P is not S." (These are "O" propositions[?].)
	www.ebroadcast.com.au /lookup/encyclopedia/co/Contrapositive.html (269 words)

	The Contrapositive and Equivalent Forms (Site not responding. Last check: 2007-10-20)
		Sometimes it is easier to prove a result in the contrapositive form rather than in the original conditional form.
		Proving the contrapositive might also be done as a direct proof, by induction or by contradiction.
		The contrapositive of the statement ``If x is odd and y is even then xy is even'' is ``If xy is odd then x is even or y is odd''.
	www.math.csusb.edu /notes/proofs/pfnot/node7.html (352 words)

	SparkNotes: Logic Statements: Variations on Conditional Statements
		The contrapositive of a statement is formed when the hypothesis and the conclusion are interchanged, and both are replaced by their negation.
		In other words, the contrapositive of a statement is the same as the inverse of that statement's converse, or the converse of its inverse.
		Its contrapositive, "A polygon with greater or less than three sides is not a triangle" is also true.
	www.sparknotes.com /math/geometry3/logicstatements/section3.rhtml (588 words)

	The independent bookstore on the web! (Site not responding. Last check: 2007-10-20)
		The contrapositive of a conditional assertion P-> Q is the statement (not Q) -> (not P).
		In mathematical arguments, the conditional assertion and its contrapositive are equivalent.
		Difficulties In contrast to the situation in mathematical reasoning, the contrapositive of a conditional sentence in ordinary English about everyday topics of conversation does not in general mean the same thing as the direct sentence.
	www.buybooksontheweb.com /peek.asp?ISBN=0-7414-1685-9 (945 words)

	PlanetMath: Proof that contrapositive statement is true using logical equivalence
		You can see that the contrapositive of an implication is true by considering the following:
		"Proof that contrapositive statement is true using logical equivalence" is owned by sprocketboy.
		This is version 4 of Proof that contrapositive statement is true using logical equivalence, born on 2003-06-19, modified 2005-03-26.
	planetmath.org /encyclopedia/SomethingRelatedToContrapositive.html (93 words)

	Proofs by Contrapositive
		Proof by contrapositive takes advantage of the logical equivalence between "P implies Q" and "Not Q implies Not P".
		The contrapositive version of this theorem is "If x and y are two integers with opposite parity, then their sum must be odd." So we assume x and y have opposite parity.
		Method of Contrapositive: Assume Not Q and prove Not P. The method of Contrapositive has the advantage that your goal is clear: Prove Not P. In the method of Contradiction, your goal is to prove a contradiction, but it is not always clear what the contradiction is going to be at the start.
	zimmer.csufresno.edu /~larryc/proofs/proofs.contrapositive.html (706 words)

	Immediate Inferences
		The contrapositive of any categorical proposition is the new categorical proposition that results from putting the complement of the predicate term of the original proposition in the subject place of the new proposition and the complement of the subject term of the original in the predicate place of the new.
		Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true.
		(And since conversion and contraposition "by limitation" derive from subalternation, they too must be forsworn.) From now on, therefore, we will rely only upon the immediate inferences in the table at the end of the previous section of this lesson and suppose that A and O propositions and E and I propositions are genuinely contradictory.
	www.philosophypages.com /lg/e07b.htm (1313 words)

	Logic - Contrapositive
		Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.
		An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statement.
		The contrapositive is the mixing of the inverse and the converse.
	regentsprep.org /regents/math/relcond/lcontrap.htm (153 words)

	On Truth and Lies in a Nonmoral Sense
		Despite their shared tradition of contrapositivity that developed from the work of Hume, Wittgenstein is working within the philosophy of language while Popper is studying the philosophy of science.
		Both rely on a notion of the contrapositive in the tradition of Humean scepticism, as outlined by Kripke in the case of Wittgenstein.
		This contrapositivity is paired with a pragmatic view of philosophy which leads both thinkers to reject absolutist or foundational views of either science (primary sense experience) or language (formal logic).
	www.littlesputnik.net /trpearce/wittgenstein.htm (3995 words)

	MATH220 Formal Methods -- L. E. Rogers, Pepperdine University ... (Site not responding. Last check: 2007-10-20)
		For any implication if A then B, the contrapositive if not B, then not A is logically equivalent to the original implication — i.e., if one is true then so is the other; and if one is false then so is the other.
		Therefore, if we prove that contrapositive, then we know that the original implication was also true.
		Whenever it is easier to prove the contrapositive than the original implication, we would use this approach.
	faculty.pepperdine.edu /lrogers/ma220/ch1/1-3.htm (735 words)

	n_solns_4
		The contrapositive of the contrapositive is the original implication.
		Notice that the contrapositive is equivalent to the original; it is not the same as the opposite implication
		Using the contrapositive is so frequent that in simple situations you can use it without calling attention to it.
	www.math.ucla.edu /~baker/115ah.1.01f/handouts/n_solns_4/node1.html (461 words)

	[No title]
		The converse of a proposition is not necessarily logically equivalent to it, that is they may or may not take the same truth value at the same time.
		On the other hand, the contrapositive of a proposition is always logically equivalent to the proposition.
		That is, they take the same truth value regardless of the values of their constituent variables.
	grail.cba.csuohio.edu /~sanchita/cis505/classnotes/Logic2.ppt (801 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-20)
		What you are allowed to use in a logic proof is the contrapositive.
		The contrapositive of p --> q is ~q --> ~p.
		In our example, the contrapositive of "If X is 2 then X is an even number" would read, "If X is NOT an even number then X is NOT 2." We can see that this is also true.
	mathforum.org /library/drmath/view/55706.html (321 words)

	[No title]
		Contrapositives In the propositional calculus of elementary symbolic logic, p -> q, material implication, has the following truth table: p q p->q -------------- 1.
		In case you still are not convinced of the logical equivalence of a sentence and its contrapositive, select a conditional sentence and try to imagine a world in which it is true, but its contrapositive is false (or vice versa).
		You are awarded the Nobel Prize for Avian Trivia and retire to Woodshole, where you spend your declining years wiring flowers to the grave of Burt Lancaster to commemorate his portrayal of the Bird Man of Alcatraz.
	linuxmafia.com /skeptic/files-to-classify/hodes.txt (977 words)

	[No title]
		So "not P" is written ~P. Contrapositive -------------- The "contrapositive" of P -> Q is ~Q -> ~P. In other words, contrapositive of "P implies Q" is "all non-Qs are non-Ps".
		The contrapositive of P -> Q is ~Q -> ~P. The contrapositive of ~Q -> ~P is ~~P -> ~~Q, i.e., P -> Q! Converse -------- The "converse" of P -> Q is Q -> P. In other words, the converse of "P implies Q" is "Q implies P".
		The contrapositive of (4) is xy != 0 -> x != 0 It is equivalent to (4).
	www.cs.toronto.edu /~fpitt/CSC165/20039/lectures/L02.txt (771 words)

	hw2ans
		Contrapositive: If you need a purpose in life, then you do not look good or do not dress well.
		Contrapositive: If California is not a fine place to live, then you don't happen to be an orange.
		Contrapositive: If I am not free, then you cannot be free.
	www.cs.umd.edu /class/spring2004/cmsc250/hw/hw2ans (715 words)

	Conditional Statements
		The contrapositive form is frequently used in mathematics and computer science to make solutions simpler or to offer another approach to a tough problem.
		This logical equivalence is the basis for one of the most important laws of deduction, modus tollens, and for the contrapositive method of proof.
		Verify the conditional and its contrapositive are equivalent by evaluating their truth tables.
	people.uncw.edu /tompkinsj/133/logic/condition.htm (931 words)

	CSC 432 Notes (Site not responding. Last check: 2007-10-20)
		use proof by contradiction (or contrapositive) to show how negating the statement to be proved would lead to undesirable and untenable consequences (e.g.
		with contrapositive we prove "if P then Q" by first negating Q and then using it to prove "not P" with contradiction we prove "if P then Q" by first negating Q and then showing how "not Q" and P together lead to something that cannot be true.
		The goal is not always as specific with contradiction as it is with contrapositive.
	web.presby.edu /~wasmith/courses/432/notes/chap0Proofs.htm (475 words)

	Re: inverse and contrapositive (Site not responding. Last check: 2007-10-20)
		In Reply to: inverse and contrapositive posted by jalitza on May 21, 2002 at 14:58:47:
		Contrapositive: if Jane is not well, then she is not here.
		: Contrapositive: if Jane is not well, then she is not here.
	www.mathguide.com /bbs/Messages2002/1572.html (214 words)

	tlex1prep
		State the converse, the obverse, and the contrapositive of each of the following statements.
		Contrapositive: non-D()C - but this is not equivalent to the original.
		Contrapositive: B()non-A, but this is not equivalent to the original.
	pegasus.cc.ucf.edu /~stanlick/tlproblems.html (495 words)

	Taxonomy of proof: contrapositive (Site not responding. Last check: 2007-10-20)
		Instead of going from the assumptions and trying to derive the result, you start by assuming the result is false and show that this violates one of the assumptions.
		Since we are assuming that the result is not true and end up by contradicting our assumptions, this is a kind of proof by contradiction known as proof by contrapositive.
		By applying the law of contrapositive, it is fairly easy to see that all three forms are in fact equivalent.
	www-cs-students.stanford.edu /~csilvers/proof/node6.html (163 words)

	Tutorial- Contraposition (Site not responding. Last check: 2007-10-20)
		Contraposition is not a legitimate operation for I and E propositions.
		The E proposition, "No primate is an aquatic animal," is clearly not equivalent to its contrapositive, "No nonaquatic animal is a nonprimate," because the first is true and the second false (cows are nonaquatic animals but they are nonprimates).
		The O proposition is the only one, besides the A, that is equivalent to its contrapositive.
	www.wwnorton.com /college/phil/logic3/ch8/contpos.htm (203 words)

	Contrapositive Encyclopedia Article, Information, History and Biography @ OneOnly.com (Site not responding. Last check: 2007-10-20)
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	oneonly.com /encyclopedia/Contrapositive (675 words)

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