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	Vacuous truth - Wikipedia, the free encyclopedia
		Informally, a logical statement is vacuously true if it is true but doesn't say anything; examples are statements of the form "everything with property A also has property B", where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly.
		Vacuous truth is usually applied in classical logic, which in particular is two-valued, and most of the arguments in the next section will be based on this assumption.
		There are however vacuous truths that even most mathematicians will outright dismiss as "nonsense" and would never publish in a mathematical journal (even if grudgingly admitting that they are true).
	en.wikipedia.org /wiki/Vacuous_truth (2168 words)

	Vacuous truth -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		Informally, a (The branch of philosophy that analyzes inference) logical statement is vacuously true if it is true but doesn't say anything; examples are statements of the form "everything with property A also has property B", where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly.
		Vacuous truth should be compared to ((logic) a statement that is necessarily true) tautology, with which it is sometimes conflated.
		Vacuous truth is usually applied in (Click link for more info and facts about classical logic) classical logic, which in particular is two-valued, and most of the arguments in the next section will be based on this assumption.
	www.absoluteastronomy.com /encyclopedia/v/va/vacuous_truth.htm (1865 words)

	When is truth vacuous? Is infinity a bunch of nothing?
		Clearly, Dan's definition of vacuous truth--which is implied in his clever "(p and (p-->q is vacuously true) requires that q is true)--is the same as the definition I suggest.
		That is, in a vacuously true statement of this form it is irrelevant whether q is true or not.
		Vacuous truth in the sense that John mentions can be seen from tautologies such as all x e X(x e X) and E!x e X(x' e X) [where E is the existential quantifier and x' means a unique element].
	www.geocities.com /n_fold/vactruth.html (4095 words)

	Vacuous truth
		Informally, a logical statement is vacuously true if it is true but doesn't say anything; examples are statements of the form "everything with property A also has property B", where there is nothing with property A. Table of contents
		relevant logic) where "neither true nor false" is a possible truth value for a statement.
		Second, picking "true" as the truth value makes many mathematical propositions that people tend to think are true come out as true.
	www.brainyencyclopedia.com /encyclopedia/v/va/vacuous_truth.html (1560 words)

	Philosophy, et cetera: Fictional Worlds
		A sentence of the form "In fiction F, P" is non vacuously true iff some world where F is told as known fact and P is true differs less from our actual world, on balance, than does any world where F is told as known fact and P is not true.
		It is vacuously true iff there is no world where F is told as known fact.
		It is vacuously true iff there is no world where P is told as known fact.
	pixnaps.blogspot.com /2004/04/fictional-worlds.html (1086 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Notice that the statement being quantified, namely SH(c) -> TH(c), is an implication, and implications are false exactly when the antecedent is true and the consequent is false -- i.e., implications are true exactly when the antecedent is false, or the consequent is true, or both.
		Note that the consequent is immaterial for vacuously true statements.
		However, if you're unsure as to whether there is an x in D such that P(x) is true, then you could be left unsure as to whether the original statement is true or false.
	www.scar.utoronto.ca /~nick/cscA65/045/L04vacuous.txt (328 words)

	Philosophy — University of Northern Colorado - Philosophy of Language
		What he says is this: “It has often been maintained that certain special sentences are true solely in virtue of their meanings and/or the meanings the meanings of their component expressions, without regard to what the nonlinguistic world is like (‘No bachelor is married’; ‘If a thing is blue it is colored’).
		There’s a big difference between sentences that are true and sentences that are vacuously true.
		The sentence “Wayne Kimball is presently enrolled in PHIL 390” is true, but it’s not vacuously true: its truth has everything to do with what the nonlinguistic world is like.
	www.unco.edu /philosophy/current/forums/topic.asp?TOPIC_ID=80 (752 words)

	The Traditional Square of Opposition
		The O form is (vacuously) true if its subject term is empty, not false, and thus the logical interrelations of [SQUARE] are unobjectionable.
		So these cannot be true together, but their opposites may both be true with respect to the same thing, e.g.
		Thus we have passed from a true claim to a false one.
	plato.stanford.edu /entries/square (4451 words)

	Fake Barn Country: Truth in Fiction and Possible Worlds
		Recall Lewis’s theory: a proposition is true in the fiction if and only if it is true in all of the possible worlds that are maximally close to the actual world in which the fiction is told as known fact.
		It is true that we clearly do learn to interpret regular non-fictional speech, and that it is sometimes non-literal, and we typically don’t have a problem understanding such utterances.
		What Lewis' theory says (ought to say?) is that while it is true that Harry put one shoe on before the other, it's not true that he put a shoe on his right foot first, nor is it true that he put one on his left foor first.
	www.brown.edu /Departments/Philosophy/Blog/Archives/004702.html (5959 words)

	Truth, Prosentential Theory of [Internet Encyclopedia of Philosophy]
		To assert that a sentence is true is simply to assert or reassert that sentence; it is not to ascribe the property of truth to that sentence.
		In these examples, ‘that is true’ and ‘it is true’ serve as ‘prosentences of laziness.’ They inherit their content from antecedent statements, just as pronouns inherit their reference from antecedent singular terms.
		Similarly, according to Strawson, “It is true” (uttered after someone says that the sun is bright) and “It is true that the sun is bright” do not assert that some sentence or proposition has the property of being true.
	www.iep.utm.edu /t/truthpro.htm (6647 words)

	Final Exam Theory Quiz: Answers
		(a) is a vacuously true universal generalization Since nothing is both a tetrahedron and a cube, the wff in the antecedent of (a) comes out false of everything.
		If you thought (c) was vacuously true, it was probably because you know that in a Tarski world, no two blocks can be in the same column and same row simultaneously, because that would put them in the same square, which is impossible.
		This is true in every Tarski world, for of the two different size blocks that are both smaller than some other block, one must be medium, and to be larger than a medium block requires being large.
	faculty.washington.edu /smcohen/120/TheoryQuizFinalAnswers.htm (3684 words)

	Abstract (Site not responding. Last check: 2007-10-18)
		In model checking, a specification is vacuously true, if some subformula can be modified without affecting the truth value of the specification.
		In this paper, we argue that the common definition describes extreme cases of vacuity where the subformula indeed collapses to a constant truth value.
		We suggest a refined notion of vacuity (weak vacuity) which is parameterized by a user-defined class of vacuity causes.
	www.dbai.tuwien.ac.at /staff/samer/abstr/pv.htm (163 words)

	Laws of Nature
		Some laws are vacuously true: Newton's first law of motion — that all inertial bodies have no acceleration — is a law, even though there are no inertial bodies.
		Furthermore, it is reasonable to think that one goal of scientific theorizing is the formulation of true theories that are well balanced in terms of their simplicity and strength.
		It is standard to respond to such an example by arguing that this is not the pertinent notion of confirmation (that it is mere "content-cutting") and by suggesting that what does require lawlikeness is confirmation of the generalization's unexamined instances.
	plato.stanford.edu /entries/laws-of-nature (5171 words)

	Philosophy, et cetera: August 2005
		The problem with stipulating that empty universals are vacuously true was mentioned above: it implies the truth of apparent contradictions such as "all mermaids are not mermaids".
		His prediction is not a vacuous one, and it can be either true or false, no matter whether the conditions actually obtain.
		True beliefs are a "fuel for success": they form an information store about the world that advantages the animal in many different actions, but they are not tied to specific behaviors.
	pixnaps.blogspot.com /2005_08_01_pixnaps_archive.html (11152 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		Every statement P must be either true, or, if not true, false (principle of bivalence).
		So, it must already be true now that a sea battle will take place tomorrow or false now that a sea battle will take place tomorrow.
		If it is true now that a sea battle will take place tomorrow, then nobody can bring about that it might not; and if it is true now that a sea battle will not take place tomorrow, then nobody can bring about that it might.
	www.siue.edu /~evailat/i-fatalism.html (589 words)

	Articles - Mathematical induction (Site not responding. Last check: 2007-10-18)
		The proof that the statement is true for all natural numbers n proceeds as follows.
		Strictly speaking, it is not necessary in transfinite induction to prove the basis, because it is a vacuous special case of the proposition that if P is true of all n < m, then P is true of m.
		It is vacuously true precisely because there are no values of n < m that could serve as counterexamples.
	www.anfolk.com /articles/Mathematical_induction (771 words)

	Truth Table
		In this case p is true while q is false; this means that you did drink the Pepsi, but you weren't happy.
		In this case p is false while q is true; this means that you didn't drink the Pepsi but you were happy anyway.
		It shows that the only situation in which an "if...then" statement is FALSE is when the antecendant is true while the consequent is false (the second row of this table).
	www.math.fsu.edu /~wooland/hm/Unit1Module5/conditionalTable.html (359 words)

	Tuesday, May 2, 2000
		Suppose that (Ak E N.(Ax < k.Px)->Pk) is true.
		Basis step: Since (Ax < 0.Px) is vacuously true, P0 must be true by the hypothesis (Ak E N.(Ax < k.Px)->Pk), and this implies (Ax <= 0.Px), completing the proof of the basis step.
		Proof: Suppose for a fixed n that this statement is true for every smaller value of n.
	math.boisestate.edu /~holmes/M387syllabus/node63.html (690 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The lemma is vacuously true for graphs with one vertex or two vertices, since a degree of 2 is not possible in such graphs.
		(A vacuous basis is one that is true through the implication "false => true" -- the assumptions made in the statement of the theorem do not hold, so the implication is valid.
		(This is not a vacuous basis, because the hypothesis does apply, but you could certainly argue that it does not display the claimed properties very clearly.
	www.cs.unm.edu /~moret/computation/2/1.html (473 words)

	Physics Help and Math Help - Physics Forums - Quantifiers, emptiness, and vacuous truth
		Though if we assume that the statements are "vacuously true", then every possible objection raised in that article seems to vanish.(This particular thing is also quoted in the article)
		But i believe that assuming these statements to be vacuously true was a problem to many logicians.
		as meaning that all statements whose subjects are empty are false by default (which isn't true since O statements were denied existential import and considered true when their subjects were empty (at least in some versions)).
	www.physicsforums.com /printthread.php?t=62945 (1398 words)

	Tarski's World: More Information (3)
		It is what we called a vacuously true generalization, since there are no objects that satisfy the antecedent.
		Confirm that all of sentences 1--3 are vacuously true in the current world.
		Two more vacuously true sentences are given in sentences 4 and 5.
	www-csli.stanford.edu /hp/Tarski3.html (913 words)

	CmSc180 Conditional statements (Site not responding. Last check: 2007-10-18)
		The only case when P → Q is false, is when P is true however Q is false (in red).
		We say that in this case the implication is true by default or vacuously true.
		is also true - vacuously true, given that I am not the president of the U.S.A. Semantic relation between the variables in a logical expression - there may be no relation.
	www.simpson.edu /~sinapova/cmsc180-02/L03-Implication.htm (919 words)

	[No title]
		As it is well known, the Lewis truth conditions require that for a counterfactual p ((q to be true, q has to be true in all p-worlds which are closest to the actual one with respect to their similarity.
		However, SS point out that according to Stapp’s semantics of counterfactuals both (1) and (2) may be true even if there is a world which is very much accessible from the actual one, and in which R2 is true (this is exactly what they call “the anomaly”).
		Under this reading, the counterfactual p ((q would be true in w if q were true in all p-worlds w(that have the same past light-cone of region R as w. With this interpretation, changes in regions space-like separated from R cannot in any way influence the truth of the counterfactual.
	philsci-archive.pitt.edu /archive/00001771/01/Article2.doc (7022 words)

	The Java Language Specification Definite Assignment
		Except for the special treatment of certain boolean operators and of boolean-valued constant expressions, the values of expressions are not taken into account in the flow analysis.
		V is definitely assigned after a when true and V is definitely assigned after b when true.
		V is definitely assigned after a when true and V is definitely assigned after b when false.
	tinf2.vub.ac.be /~dvermeir/java/draft5.2-html/16.doc.html (3469 words)

	CSE 3315: Notes on Propositional Logic and Predicate Calculus
		The implication statement as a whole is considered vacuously true.
		A WFF is a tautology (theorem) if it is valid (true) for all interpretations of its individual propositions (as is the case in the example above).
		P(x) means "for all objects x (in universe), relation P on x is true (or, said another way, all x has property P)".
	home.flash.net /~kennieg/cse3315/logic.htm (848 words)

	[No title]
		If P is false before executing the program text, then the correctness condition is, "If false before then Q after," which is vacuously true regardless of what the program does.
		In some cases we could remove the comments around P and Q, and test that are assertions are true (in some example input) by running the program.
		loop invariant: a predicate that is true on each iteration of the loop body.
	www.cs.berkeley.edu /~yelick/61bf97/lectures/09/09.inst~ (875 words)

	[No title]
		Foot note 3_3 and equally clear that there are many timelines that could satisfy the antecedent of (2) and (3).
		The real task we face, of course, is explicating the middle case -- subjunctive conditionals such as (2) and (3) which have non-vacuous truth values -- and to this the remainder of this paper is devoted.
		It is also now clear why (3), represented as ~([[forall]]x)(Mx->Px), is non-vacuously true: It is simply the negation of a proposition that is false, with «might not» clueing us in to its proper representation.
	www.sorites.org /Issue_05/item4.htm (938 words)

	Theorem 4.7 (Site not responding. Last check: 2007-10-18)
		Assume that the result is true for n = k and consider n = k + 1.
		If n = 0, then n has no proper subsets, so the result is vacuously true.
		Assume that the result is true if n = k and we seek to prove that it is true if n = k + 1.
	www.sonoma.edu /users/w/wilsonst/Papers/finite/4/t4-7.html (123 words)

	Project Objectives (Site not responding. Last check: 2007-10-18)
		is true in the interval's final state or is vacuously true if the interval does not terminate.
		However, when reasoning about a system built out of sequential parts, it is advantageous to consider certain kinds of assumptions and commitments which readily lend themselves to suitable proof rules.
		is true on an interval, it is also true in all subintervals.
	www.cse.dmu.ac.uk /~cau/pcsihomepage/node4.html (328 words)

	Brainstorms: Karl D. Stephan: Tegmark’s Parallel Universes: A Challenge to Intelligent Design?
		We have vacuous truths which are logical statements that dont really say anything for example "all elephants inside a loaf of bread are pink".
		The statement is vacuously true but there really is no consequence of it being true.
		Jacob says that "a statement can be logical but that in itself does not confer any veracity to the claims it makes." This is true on one level, but ceases to be true with respect to scientific applications of model theory (which involve semantic correspondences).
	www.iscid.org /boards/ubb-get_topic-f-6-t-000351-p-3.html (4658 words)

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