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Topic: Conditional proof

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	Conditional proof - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-07)
		Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion.
		Proving this requires assuming the premise and deriving, from that assumption, the consequent of the conditional.
		By proving the connection between the antecedent and the consequent, the assumption of the antecedent is justified post hoc.
	www.arikah.com /encyclopedia/Conditional_proof (197 words)

	Methods of mathematics proof
		It is called Conditional Proof, because we have not proved the truth of r; we have only proved that if q is true then r is true.
		We recommend that a Proof by Contradiction be one that begins with p and ~q and ends up obtaining the negation of the premise, and that a Reductio Ad Absurdum Proof be one that ends up obtaining any contradiction of a known truth.
		Although RAA proofs are often easier and more convenient, a direct proof is preferred for the reason that RAA depends for its validity on the assumption that the unprovability of the negation of p is tantamount to the provability of the negation of the negation of p.
	www.mathpath.org /proof/proof.methods.htm (2455 words)

	Maggie Johnson
		Conditional Proof: A technique where we prove a conditional (A®B) by showing that the consequent B must be true when the antecedent A is true.
		Conditional Elimination: This is the formal counterpart of modus ponens.
		Conditional Introduction: This is the formal counterpart of a conditional proof.
	cse.stanford.edu /class/cs103a/h14Conditionals.htm (934 words)

	Conditional Proofs
		Given an argument with a conditional conclusion, assume the antecedent of the conclusion and derive the consequent:
		The general effect of conditional proof is to make proofs a lot shorter and a lot easier to see.
		Note that the conclusion is not a conditional.
	www.hu.mtu.edu /~wsewell/hu250/cond_proof.htm (985 words)

	Logical conditional - Enpsychlopedia (Site not responding. Last check: 2007-10-07)
		In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent).
		The hypothesis is sometimes also called sufficient condition for the conclusion, while the conclusion may be called necessary condition for the hypothesis.
		The logical conditional, and particularly the material conditional, is closely related to inclusion (for sets), subsumption (for concepts), or implication (for propositions).
	www.grohol.com /psypsych/Logical_conditional (1018 words)

	Conditionals
		If we knew the truth conditions of conditionals, we would handle uncertainty about conditionals in terms of a general theory of what it is to be uncertain of the truth of a proposition.
		Conditional desires appear to be like conditional beliefs: to desire that B is to prefer B to ~B; to desire that B if A is to prefer AandB to Aand~B; there is no proposition X such that one prefers X to ~X just to the extent that one prefers AandB to Aand~B.
		Extending Jackson's account to conditional commands, the doctor said "Make it the case that either the patient is not alive in the morning, or you change the dressing", and indicated that she would still command this if she knew that the patient would be alive.
	plato.stanford.edu /entries/conditionals (12732 words)

	Philosophy 222:Class VII
		The way we capture this in a proof is as follows: (1) We will assume the antecedent of the conclusion, D, as a premise (an assumption for the purposes of conditional proof, ACP).
		Conditional proof can also be used more than once, either in sequence, or with one embedded in another.
		In the next proof with discovery numbers you'll notice gaps where I wasn't after anything in particular--I was just looking for two contradictory statements wherever they occurred.
	mywebpages.comcast.net /reasoning/symlogic/classes/class7.htm (1200 words)

	Logic: Web Lecture: Continued Techniques and Rules for Proofs (Site not responding. Last check: 2007-10-07)
		Although we have examined proofs in their basic form, and we have nineteen different rules, we still occasionally find ourselves forced to build unwieldy and seemingly meaningless proofs because of the complexity of the rules and of the arguments.
		Indeed, this is the shortest possible proof outside of using the Conditional Proof.
		Second, wile the conditional proof need not be the last premise in an argument, you may not use any of the premises from the conditional proof outside of the conditional proof.
	www.cstone.net /~lbrannon/Logic/weblecture012403.htm (1259 words)

	Tutorial- Reductio ad Absurdum
		Conditional proof (CP) can be used to prove a conditional statement by making an assumption that is used to derive a consequent which, in turn, justifies a conditional conclusion.
		The Latin phrase means a reduction to absurdity or contradiction; the technique is to show that if we accept the premises of an argument, but deny the conclusion, we contradict ourselves.
		The reductio ad absurdum technique is to take the negation of the conclusion, add it as an assumption (just as in conditional proof), and then from the premises and that assumption, derive a contradiction--a statement of the form p
	www.wwnorton.com /college/phil/logic3/ch13/reductio.htm (403 words)

	CONDITIONAL PROOF
		CP allows you derive a conditional (hence the name) that you need in a proof, either as the conclusion or as an intermediate step.
		The conditional proof must be bracketed from the assumed premise to the conclusion with the last line outside the bracket always a material implication.
		But that is all that the conditional says: If P then Q. When using conditional proof, you are allowed to make multiple assumptions.
	cstl-cla.semo.edu /hill/pl120/notes/conditional%20proof.htm (655 words)

	List of rules of inference - Wikipedia, the free encyclopedia
		Rules can also be either conditional or biconditional.
		Conditional rules, sometimes known simply as rules of inference, are rules which one can use to infer the first type of statement from the second, but where the second can not be inferred from the first.
		Biconditional rules, also known as rules of replacement, are rules which one can use to infer the first type of statement from the second, or vice versa.
	en.wikipedia.org /wiki/List_of_rules_of_inference (285 words)

	[No title]
		That is, the deduction of any formula via conditional proof is always dependent upon the formula assumed.
		When using conditional proof, any formula derived within the conditional proof lines, that is, between the assumption and the conclusion, is dependent on the assumption made.
		You can construct more than one conditional proof in a deduction, but, when you do so, they either need to be (1) nested or (2) separate entirely.
	www.csus.edu /indiv/n/nogalesp/SymbolicLogicSPG05/SymbolicLogicOverheads/Phil60GusCh3DeductiveMethods/ConditionalProof/ConditionalproofSteps.doc (257 words)

	Logical Consequence, Deductive-Theoretic Conceptions [Internet Encyclopedia of Philosophy]
		A proof of X from K that appeals exclusively to the inference rules of N is a formal deduction or formal proof.
		The structure of this proof is that of a conditional proof: a deduction of a conditional from a set of basis sentence which starts with the assumption of the antecedent, then a derivation of the consequent, and concludes with the conditional.
		To assert P is to have a proof of P, and to assert not-P is to have a refutation of P.
	www.iep.utm.edu /l/logcon-d.htm (9103 words)

	Conditional proof (Site not responding. Last check: 2007-10-07)
		Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditionalnecessarily leads to the conclusion.
		Proving this requires assuming the premise and deriving, from that assumption, theconsequent of the conditional.
		Note that I haven't proved that you'll be late for work: I've only proven the conditional, that the consequent followsnecessarily from the antecedent.
	www.therfcc.org /conditional-proof-44508.html (155 words)

	Using Conditional Proof: Pointers on Strategy
		If so, it is very often helpful to assume the antecedent of that conditional schema and then derive its consequent—thus proving the conditional schema to be true given the original premises; or perhaps it will be easier to assume the negation of the consequent and then derive the negation of the antecedent (remembering Transposition).
		After you make your initial assumption, you may discover some further conditional schema whose proof would be useful: in that case, make a second assumption of the antecedent of that second schema, and work to derive its consequent.
		When the desired conclusion is found in the premises as the consequent in a conditional premise, you might try to prove (not: assume) the antecedent by Conditional Proof.
	www.lawrence.edu /fast/boardmaw/cond_proof_strat.html (876 words)

	MATH3181 - Hyperbolic Geometry
		Often the rule of conditional proof is used to prove the contrapositive.
		A proof by contradiction of a statement P is a proof that assumes ~P and yields a sentence of the type
		It is helpful-nay, it is essential in starting a proof to examine all previously proved theorems for results which might be relevant to the proof.
	www.math.uncc.edu /~droyster/math3181/proof.html (1350 words)

	conproof (Site not responding. Last check: 2007-10-07)
		CP's can be regarded as mini subproofs within the context of a longer proof; they can be thought of as subroutines needed to deduce some statement which is then used by the full proof to derive the required conclusion.
		In case a) above where the implication is itself the required conclusion of the whole proof, the CP in effect provides the "guts" of the proof, it does all the work, and the whole proof in which it is set is nothing more than setting up and terminating the CP.
		In the case where the required conclusion is a conjunction of implications (b ii), use two separate CP's each to derive one of the required conjuncts in the conjunction.
	www.loyno.edu /~folse/conproof.html (983 words)

	Peter Suber, "Mathematical Induction" (Site not responding. Last check: 2007-10-07)
		Prove that the property of complying with the theorem is "hereditary" and extends to all the successors of the minimal case.
		The induction step is the proof of a conditional statement, namely, "if the theorem is true of the ancestor case, then it is true of the descendant cases." The if-clause of this conditional statement, asserting that the theorem is true of the ancestor case, is called the induction hypothesis.
		It is assumed for the sake of a conditional proof; we don't have to prove it.
	www.earlham.edu /~peters/courses/logsys/math-ind.htm (1191 words)

	[No title]
		Conditional Proof (CP) This rule could also be called ‘conditional introduction’ because it allows the introduction of a conditional.
		We clearly state that the rule CP is applied to line 3 where the antecedent of the conditional we want to prove is introduced as an assumption, and to line 6 where the consequent of the conditional is derived from p and the other assumptions.
		The conditional in line 7 then depends only on the assumptions in lines 1 and 2, hence we have proved the sequent.
	www.bris.ac.uk /depts/Philosophy/UG/ugunits9900/lecture8.doc (277 words)

	Making Mathematics: Mathematics Tools: Conditional Statements
		A conditional statement is one that can be put in the form if A, then B where A is called the premise (or antecedent) and B is called the conclusion (or consequent).
		Note that the inverse and converse of the same statement are each other’s contrapositive and are, therefore, both true or both false conditional statements.
		The equivalence of a statement and its contrapositive is at the heart of the method of proof by contradiction, which proves that the contrapositive of a conjecture is true and, therefore, that the original conjecture is true.
	www2.edc.org /makingmath/mathtools/conditional/conditional.asp (578 words)

	SingaporeMoms - Parenting Encyclopedia - Logical conditional
		This table needn't be taken as "the definition of →", however, because its contents can also be derived from the axioms of the propositional calculus.
		Another trouble is that the material conditional is such that P AND ¬P → Q, regardless of what Q is taken to mean.
		From this it may be seen at once that the propositional interpretation is more homogeneous than the conceptual, since it alone makes it possible to give the same meaning to the copula in both primary and secondary propositions.
	www.singaporemoms.com /parenting/Logical_conditional (999 words)

	Courses: Gabbay / Barringer
		Conditional logic and non-monotonic logic are central areas in philosophy, computer science and language.
		Moreover, the connection between non-monotonic consequence "A entails B" and the conditional "A>B" is well known, so too are the formal similarities between the conditional and substuctural implications.
		The semantic modelling (possible worlds, probabilistic, translational, etc) of the conditional and non-monotonic consequence seems to be relatively well developed but not much work has been done on the proof-theoretic aspects.
	www.let.uu.nl /esslli/Courses/gabbay-barringer.html (247 words)

	LESSON TEN -- Repetition and Conditional Proofs (Site not responding. Last check: 2007-10-07)
		This rule is the rule of conditional proof.
		When you assume something in a proof, you indicate that with the word "Assumption." Thus, our proof should look like this so far: 1.
		You cannot use any conclusions you draw inside your sub-proof outside of it in the rest of the proof.
	www.ling.rochester.edu /~duniho/harmonics/102E.html (460 words)

	Intelligent Book Maths Tutor
		The student is given the first three statements as three assumptions in a Conditional Proof.
		The student is permitted to enter "trivial" as the proof.
		The student is shown a Conditional proof, with all of the previous statements as assumptions.
	www.cl.cam.ac.uk /~whb21/research/mathsTutor/scenario_num_induction_2.html (217 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The purpose of presenting proofs is to demonstrate unequivocally that a given set of premises entails a particular conclusion.
		Given a sentence (at line n), conclude a conditional having it as the consequent and whose antecedent appears in the proof as an assumption (at line m).
		Given a conditional sentence (at line m) and another sentence that is its antecedent (at line n), conclude the consequent of the conditional.
	logic.tamu.edu /Primer/Edition1/1.4.html (729 words)

	An Elementary Introduction to Logic and Set Theory: Methods of Proof
		A Formal Proof is a derivation of a theorem that consists of a finite sequence of well-formed formulas.
		A special case of Conditional Proof is to assume p and then reach as a contradiction the conjunction of q and ~ q for some sentence q.
		The method of Indirect Proof is related to the reasoning used in Hypotheses Testing in statistics (an application of Inductive Logic), where one assumes the Null Hypothesis and then tries to show that it can't be supported by the available empirical evidence.
	matcmadison.edu /alehnen/weblogic/logproof.htm (2447 words)

	What Our Students Really Know About Proof and Reasoning in Geometry
		A proof of this statement might show me why this statement is true.
		You are given the statement "The base angles of an isosceles triangle are congruent." You may use the fact that the base angles are congruent as given information in your proof.
		Consider the conditional statement and the accompanying diagram.
	www.math.ilstu.edu /tsmartin/NCTM2000.html (1736 words)

	Insolubles: Supplementary Document: A Supplement to Insolubles
		Here is a proof that on Bradwardine's theory, every proposition signifies that it is true.
		Spade's reason for attributing CBP to Bradwardine is that "it is presupposed in some of his reasoning" [see Spade 1981, p.
		If he had, he would not have given a long and complex proof of the much weaker claim that every insoluble signifies its own truth, or at least, he would have followed it by that stronger claim.
	www.science.uva.nl /~seop/entries/insolubles/supplement.html (765 words)

	OTTER Encapsulated
		This is the sort of simple proof that students of logic learn at the outset of their education.
		Normally, many such rules are added to the arsenal of the human who learns to do proofs in first-order logic.
		Lines 4 and 5 in the OTTER proof are applications of the rules of inference `binary resolution' and `hyperresolution,' respectively.
	www.rpi.edu /~faheyj2/SB/SELPAP/MBR/mbr1/node13.html (475 words)

	Untitled Normal Page
		Notice how this proof is related to the proof we did earlier for sequent 10.
		Whenever you see that you are trying to derive a conditional sentence, it is a good idea to consider using the CP strategy.
		And (b) construct a proof to show that the reasoning is valid in each case.
	personal.bgsu.edu /~mbelzer/insidelogic8.html (1251 words)

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