vacuous

WordNet

devoid of matter; "a vacuous space"
indicative of or marked by mental vacuity and an absence of ideas; "the vacuousness of her face belied her feelings"
in a vacuous manner

PrepTutorEJDIC

からの,何も入っていない / 頭がからっぽの,魂の抜けた / 目的のない,無意味な

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/08/22 00:11:19」(JST)

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A vacuous truth is a statement that asserts that all members of the empty set have a certain property. For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on and turned off".

More formally, a relatively well-defined usage refers to a conditional statement with a false antecedent. One example of such a statement is "if Ayers Rock is in France, then the Eiffel Tower is in Bolivia".^{[note 1]} Such statements are considered vacuous because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. They are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true).

In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction.^[1] This notion has relevance as well as in any other field which uses classical logic.

Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might tell his or her parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with.

Scope of the concept

A statement is "vacuously true" if it resembles the statement , where is known to be false.

Statements that can be reduced (with suitable transformations) to this basic form include the following:

, where it is the case that .
, where the set is empty.
, where the symbol is restricted to a type that has no representatives.

Vacuous truth is usually applied in classical logic, which in particular is two-valued. However, vacuous truth also appears in, for example, intuitionistic logic in the same situations given above. Indeed, the first two forms above will yield vacuous truth in any logic that uses material conditional, but there are other logics which do not.

Notes

^ Such statements can also be used sarcastically, as in the line "If she's a lady, then I'm a Vermicious Knid!" spoken by the character Grandpa Joe from the film Willy Wonka & the Chocolate Factory.

References

^ Baldwin, Douglas L.; Scragg, Greg W. (2011), Algorithms and Data Structures: The Science of Computing, Cengage Learning, p. 261, ISBN 9781285225128 .

Bibliography

Blackburn, Simon (1994). "vacuous," The Oxford Dictionary of Philosophy. Oxford: Oxford University Press, p. 388.
David H. Sanford (1999). "implication." The Cambridge Dictionary of Philosophy, 2nd. ed., p. 420.
Beer, Ilan; Ben-David, Shoham; Eisner, Cindy; Rodeh, Yoav (1997). "Efficient Detection of Vacuity in ACTL Formulas". Lecture Notes in Computer Science 1254: 279–290. doi:10.1007/3-540-63166-6_28.

External links

Conditional Assertions: Vacuous truth

English Journal

Effect of ethanolic extract of Coriandrum sativum L. on tacrine induced orofacial dyskinesia.

Mohan M, Yarlagadda S, Chintala S.
Indian journal of experimental biology.Indian J Exp Biol.2015 May;53(5):292-6.
The effect of ethanolic extract of Coriandrum sativum L. seeds (100, 200 mg/kg) was studied on tacrine induced orofacial dyskinesia. Tacrine (2.5 mg/kg, i.p.) treated animals were observed for vacuous chewing movements (VCM), tongue protrusions (TP) and orofacial bursts (OB) for 1 h followed by obse
PMID 26040026

Evaluating whether nature's intrinsic value is an axiom of or anathema to conservation.

Vucetich JA1, Bruskotter JT, Nelson MP.
Conservation biology : the journal of the Society for Conservation Biology.Conserv Biol.2015 Apr;29(2):321-32. doi: 10.1111/cobi.12464. Epub 2015 Feb 19.
That at least some aspects of nature possess intrinsic value is considered by some an axiom of conservation. Others consider nature's intrinsic value superfluous or anathema. This range of views among mainstream conservation professionals potentially threatens the foundation of conservation. One cha
PMID 25704250

Analysis of Vacuous Pulse and Replete Pulse Using a Clip-type Pulsimeter Equipped with a Hall Sensor.

Lee NK1, Rhee JK2, Lee SS3.
Journal of acupuncture and meridian studies.J Acupunct Meridian Stud.2015 Apr;8(2):83-93. doi: 10.1016/j.jams.2014.12.002. Epub 2014 Dec 6.
A logistic regression equation for the vacuous pulse and the replete pulse was determined based on data obtained using a clip-type pulsimeter equipped with a Hall device that sensed the change in the magnetic field due to the minute movement of a radial artery. To evaluate the efficacy of the two di
PMID 25952125

Japanese Journal

空洞性腫瘤を呈し肺真菌感染を伴った肺腺癌の1切除例

杉浦真貴子,斉藤春洋,伊藤宏之,横瀬智之,中山治彦,山田耕三
肺癌 52(4), 414-419, 2012
背景．肺癌と感染症の合併例は，これまでにしばしば報告されているが，肺癌と真菌感染の同部位での合併例は稀である．今回我々は，肺癌の空洞内部にアスペルギルスと考えられる真菌の感染を合併した1例を経験し，これまでの報告と併せて検討を行った．症例．76歳男性．胸部X線で左下葉に長径約10 cmの空洞性陰影を指摘された．thin-section CT画像では不均一に肥厚する空洞壁および空洞内に隔壁の所見を認 …
NAID 130002060263

空移動仮説と主語の獲得

福田稔,古川武史
宮崎公立大学人文学部紀要 18(1), 197-210, 2011-03-04
本稿では、線形語順に影響を与えない移動は生じないという空移動仮説について、言語獲得の側面から問題点を指摘する。空移動仮説のもとでは主語whは顕在的に移動していないと予測する。言語獲得の過程で主語が顕在的に移動している資料に子どもが接する機会がないためである、というのがその論拠の1つである。事実、英語におけるwh疑問文の獲得についての研究からは、主語whの統語位置について決定的な証拠を得ることが難し …
NAID 110008434510

CONTEMPORARY ENVIRONMENTAL AESTHETICS AND THE REQUIREMENTS OF ENVIRONMENT ALISM

CARLSON Allen
JTLA : Journal of the Faculty of Letters, the University of Tokyo, Aesthetics 34, 1-21, 2010-03-24
… Some environmentalists have found fault with the modes of aesthetic appreciation of nature that are associated with these two views, charging that they are anthropocentric, scenery-obsessed, superficial, subjective, and morally vacuous. …
NAID 120003874417

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