en>Cydebot: Robot - Speedily moving category Units of measure to :Category:Units of measurement per CFDS.

2012-12-19T08:26:38Z

Robot - Speedily moving category Units of measure to Category:Units of measurement per CFDS.

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{{redirect|Entailment||Entail (disambiguation)}}
{{redirect|Therefore|the therefore symbol (∴)|therefore sign}}
{{redirect|Logical implication|the binary connective|material conditional}}

'''Logical consequence''' (also '''entailment''') is one of the most fundamental [[concept]]s in [[logic]]. It is the relationship between [[statement (logic)|statement]]s that holds true when one logically "follows from" one or more others. [[Validity|Valid]] logical [[argument]]s are ones in which the [[Consequent|conclusions]] follow from its [[premise]]s, and its conclusions are consequences of its premises. The [[philosophical analysis]] of logical consequence involves asking, 'in what sense does a conclusion follow from its premises?' and 'what does it mean for a conclusion to be a consequence of premises?'<ref name="sep" >Beall, JC and Restall, Greg, ''[http://plato.stanford.edu/archives/fall2009/entries/logical-consequence/ Logical Consequence]'' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.).</ref> All of [[philosophical logic]] can be thought of as providing accounts of the nature of logical consequence, as well as [[logical truth]].<ref>[[Willard Van Orman Quine|Quine, Willard Van Orman]], ''Philosophy of logic''</ref>

Logical consequence is taken to be both [[logical truth|necessary]] and [[Formalism (mathematics)|formal]] with examples explicated using [[interpretation (logic)|models]] and [[Formal proof|proofs]].<ref name="sep" /> A sentence is said to be a logical consequence of a set of sentences, for a given [[Formal language|language]], [[if and only if]], using logic alone (i.e. without regard to any interpretations of the sentences) the sentence must be true if every sentence in the set were to be true.<ref name="iep">[[Matthew W. McKeon|McKeon, Matthew]], ''[http://www.iep.utm.edu/logcon/ Logical Consequence]'' Internet Encyclopedia of Philosophy.</ref>

Logicians make precise accounts of logical consequence with respect to a given [[formal language|language]] <math>\mathcal{L}</math> by constructing a [[deductive system]] for <math>\mathcal{L}</math>, or by formalizing the [[Intended interpretation|intended semantics]] for <math>\mathcal{L}</math>. [[Alfred Tarski]] highlighted three salient features for which any adequate characterization of logical consequence needs to account: 1) that the logical consequence relation relies on the [[logical form]] of the sentences involved, 2) that the relation is [[a priori and a posteriori|a priori]], i.e. it can be determined whether or not it holds without regard to [[empirical evidence|sense experience]], and 3) that the relation has a [[modal logic|modal]] component.<ref name="iep" />

== Formal accounts of logical consequence ==
The most widely prevailing view on how to best account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on the structure or [[logical form]] of the statements without regard to the contents of that form.

Syntactic accounts of logical consequence rely on [[schema (logic)|schemes]] using [[inference rule]]s. For instance, we can express the logical form of a valid argument as "All <math>A</math> are <math>B</math>. All <math>C</math> are <math>A</math>. Therefore, All <math>C</math> are <math>B</math>." This argument is formally valid, because every [[Substitution (logic)|instance]] of arguments constructed using this scheme are valid.

This is in contrast to an argument like "Fred is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called [[material conditional|material consequence]] of "Fred is Mike's brother's son," not a formal consequence. A formal consequence must be true ''in all cases'', however this is an incomplete definition of formal consequence, since even the argument "<math>P</math> is <math>Q</math>'s brother's son, therefore <math>P</math> is <math>Q</math>'s nephew" is valid in all cases, but is not a ''formal'' argument.<ref name="sep" />

== A priori property of logical consequence ==

If you know that <math>Q</math> follows logically from <math>P</math> no information about the possible interpretations of <math>P</math> or <math>Q</math> will affect that knowledge. Our knowledge that <math>Q</math> is a logical consequence of <math>P</math> cannot be influenced by [[A priori and a posteriori|empirical knowledge]].<ref name="sep" /> Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.<ref name="sep" /> However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.<ref name="sep" />

== Proofs and models ==
The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of ''proofs'' and via ''models''. The study of the syntactic consequence (of a logic) is called (its) [[proof theory]] whereas the study of (its) semantic consequence is called (its) [[model theory]].<ref name="ChiaraDoets1996">{{cite book|editor=Maria Luisa Dalla Chiara, Kees Doets, Daniele Mundici, Johan van Benthem|title=Logic and Scientific Methods: Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995|url=http://books.google.com/books?id=TCthvF8xLIAC&pg=PA292|year=1996|publisher=Springer|isbn=978-0-7923-4383-7|page=292|chapter=Logical consequence: a turn in style|author=Kosta Dosen}}</ref>

=== Syntactic consequence ===

A formula <math>A</math> is a '''syntactic consequence'''<ref>[[Michael Dummett|Dummett, Michael]] (1993)[http://books.google.com/books?id=EYP7uCZIRQYC&pg=PA82&lpg=PA82&dq=syntactic+consequence&source=bl&ots=Ms58438B6w&sig=FE38FCaZpRpAr18gtG7INX4wieM&hl=en&ei=qOy7SoLlEI7KsQPgnYG7BA&sa=X&oi=book_result&ct=result&resnum=6#v=onepage&q=syntactic%20consequence&f=false''Frege: philosophy of language''] Harvard University Press, p.82ff</ref><ref>[[Jonathan Lear|Lear, Jonathan]] (1986)[http://books.google.com/books?id=lXI7AAAAIAAJ&pg=PA1&lpg=PA1&dq=syntactic+consequence&source=bl&ots=8IYWyFYTN-&sig=wrOg75cFxQwn1Uq-8LShBNXf9w0&hl=en&ei=I-y7SpHtLZLotgOsnLHcBQ&sa=X&oi=book_result&ct=result&resnum=10#v=onepage&q=syntactic%20consequence&f=false''Aristotle and Logical Theory''] Cambridge University Press, 136p.</ref><ref>Creath, Richard, and [[Michael Friedman (philosopher)|Friedman, Michael]] (2007) [http://books.google.com/books?id=87BcFLgJmxMC&pg=PA189&lpg=PA189&dq=syntactic+consequence&source=bl&ots=Fn2zomcMZP&sig=8hnJWsJFysNhmWLskICo4IQDYAc&hl=en&ei=I-y7SpHtLZLotgOsnLHcBQ&sa=X&oi=book_result&ct=result&resnum=6#v=onepage&q=syntactic%20consequence&f=false''The Cambridge companion to Carnap''] Cambridge University Press, 371p.</ref><ref>[http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence FOLDOC: "syntactic consequence"]</ref> within some [[formal system]] <math>\mathcal{FS}</math> of a set <math>\Gamma</math> of formulas if there is a [[formal proof]] in <math>\mathcal{FS}</math> of <math>A</math> from the set <math>\Gamma</math>.

:<math>\Gamma \vdash_{\mathcal {FS} } A</math>

Syntactic consequence does not depend on any [[interpretation (logic)|interpretation]] of the formal system.<ref>[[Geoffrey Hunter (logician)|Hunter, Geoffrey]], Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971, p. 75.</ref>

=== Semantic consequence ===
{{See also|Double turnstile|label 1= ⊨}}

A formula <math>A</math> is a '''semantic consequence''' within some formal system <math>\mathcal{FS}</math> of a set of statements <math>\Gamma</math>

:<math>\Gamma \models_{\mathcal {FS} } A,</math>

if and only if there is no model <math>\mathcal{I}</math> in which all members of <math>\Gamma</math> are true and <math>A</math> is false.<ref>[[John Etchemendy|Etchemendy, John]], ''Logical consequence'', The Cambridge Dictionary of Philosophy</ref> Or, in other words, the set of the interpretations that make all members of <math>\Gamma</math> true is a subset of the set of the interpretations that make <math>A</math> true.

== Modal accounts ==

Modal accounts of logical consequence are variations on the following basic idea:

:<math>\Gamma</math> <math>\vdash</math> <math>A</math> is true if and only if it is ''necessary'' that if all of the elements of <math>\Gamma</math> are true, then <math>A</math> is true.

Alternatively (and, most would say, equivalently):

:<math>\Gamma</math> <math>\vdash</math> <math>A</math> is true if and only if it is ''impossible'' for all of the elements of <math>\Gamma</math> to be true and <math>A</math> false.

Such accounts are called "modal" because they appeal to the modal notions of [[Logical truth|logical necessity]] and [[logical possibility]]. 'It is necessary that' is often expressed as a [[universal quantification|universal quantifier]] over [[possible world]]s, so that the accounts above translate as:

:<math>\Gamma</math> <math>\vdash</math> <math>A</math> is true if and only if there is no possible world at which all of the elements of <math>\Gamma</math> are true and <math>A</math> is false (untrue).

Consider the modal account in terms of the argument given as an example above:

:All frogs are green.
:Kermit is a frog.
:Therefore, Kermit is green.

The conclusion is a logical consequence of the premises because we can't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green.

=== Modal-formal accounts ===

Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:

:<math>\Gamma</math> <math>\vdash</math> <math>A</math> if and only if it is impossible for an argument with the same logical form as <math>\Gamma</math>/<math>A</math> to have true premises and a false conclusion.

=== Warrant-based accounts ===

The accounts considered above are all "truth-preservational," in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed "[[Theory of justification|warrant]]-preservational" accounts, according to which the characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored by [[intuitionist]]s such as [[Michael Dummett]].

=== Non-monotonic logical consequence ===
{{main|Non-monotonic logic}}

The accounts discussed above all yield [[Monotonicity of entailment|monotonic]] consequence relations, i.e. ones such that if <math>A</math> is a consequence of <math>\Gamma</math>, then <math>A</math> is a consequence of any superset of <math>\Gamma</math>. It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of

:{Birds can typically fly, Tweety is a bird}

but not of

:{Birds can typically fly, Tweety is a bird, Tweety is a penguin}.

For more on this, see [[Belief revision#Non-monotonic inference relation]].

==See also==
{{col-begin}}
{{col-begin}}
{{col-break}}
* [[Abstract algebraic logic]]
* [[Ampheck]]
* [[Boolean algebra (logic)]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean logic]]
{{col-break}}
* [[Deductive reasoning]]
* [[Logic gate]]
* [[Logical graph]]
{{col-break}}
* [[Peirce's law]]
* [[Probabilistic logic]]
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Strict conditional]]
* [[Tautology (logic)]]
{{col-break}}
* [[Tautological consequence]]
* [[Therefore sign]]
* [[Turnstile (symbol)]]
* [[Double turnstile]]
* [[Validity]]
{{col-end}}

== Notes ==
{{reflist}}

== Resources ==
* {{citation|last1=Anderson|first1=A.R.|last2=Belnap|first2=N.D., Jr.|title=Entailment|year=1975|publisher=Princeton|location=Princeton, NJ|volume=1}}.
* {{citation|last1=Barwise|first1=Jon|last2=Etchemendy|first2=John|year=2008|title=Language, Proof and Logic|publisher=CSLI Publications|location=Stanford}}.
* {{citation|authorlink=Frank Markham Brown|last=Brown | first=Frank Markham | year=2003 |title=Boolean Reasoning: The Logic of Boolean Equations}} 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
* {{citation|authorlink=Martin Davis|last=Davis|first= Martin, (editor)|title=The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions|publisher=Raven Press|location=New York|year=1965}}. Papers include those by [[Gödel]], [[Alonzo Church|Church]], [[J. Barkley Rosser|Rosser]], [[Kleene]], and Post.
* {{citation |first=Michael |last=Dummett |year=1991 |title=The Logical Basis of Metaphysics |publisher=Harvard University Press}}.
* {{citation|last=Edgington| first=Dorothy|year=2001|title=Conditionals|publisher=Blackwell}} in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic''.
* {{citation|last=Edgington| first=Dorothy|year=2006|title=Conditionals|url=http://plato.stanford.edu/entries/conditionals}} in Edward N. Zalta (ed.), ''The Stanford Encyclopedia of Philosophy''.
* {{citation |first=John |last= Etchemendy |year= 1990 |title=The Concept of Logical Consequence |publisher= Harvard University Press}}.
* {{citation |last=Goble |first=Lou, ed.|year=2001 |title=The Blackwell Guide to Philosophical Logic |publisher= Blackwell}}.
* {{citation |last=Hanson |first= William H|year= 1997 |title=The concept of logical consequence| journal=The Philosophical Review| volume=106}} 365–409.
* {{citation |authorlink=Vincent F. Hendricks|last=Hendricks |first=Vincent F. |year=2005 |title=Thought 2 Talk: A Crash Course in Reflection and Expression |location=New York |publisher=Automatic Press / VIP |isbn= 87-991013-7-8}}
* {{citation |last=Planchette |first=P. A. |year=2001 |title=Logical Consequence}} in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.
* {{citation|authorlink=W.V. Quine|last=Quine|first=W.V.| year=1982| title=Methods of Logic|location=Cambridge, MA|publisher=Harvard University Press}} (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), (4th edition, 1982).
*{{citation |authorlink=Stewart Shapiro |last=Shapiro |first=Stewart |year=2002 |title=Necessity, meaning, and rationality: the notion of logical consequence}} in D. Jacquette, ed., ''A Companion to Philosophical Logic''. Blackwell.
*{{citation |authorlink=Alfred Tarski |last=Tarski |first=Alfred |year= 1936 |title=On the concept of logical consequence}} Reprinted in Tarski, A., 1983. ''Logic, Semantics, Metamathematics'', 2nd ed. [[Oxford University Press]]. Originally published in [[Polish language|Polish]] and [[German language|German]].
* A paper on 'implication' from math.niu.edu, [http://www.math.niu.edu/~richard/Math101/implies.pdf Implication]
* A definition of 'implicant' [http://www.allwords.com/word-implicant.html AllWords]
* {{cite book|author=Ryszard Wójcicki|title=Theory of Logical Calculi: Basic Theory of Consequence Operations|year=1988|publisher=Springer|isbn=978-90-277-2785-5}}

==External links==
* {{SEP|logical-consequence}}
* {{InPho|taxonomy|2409}}
* {{IEP|logcon/}}
* {{PhilPapers|category|logical-consequence-and-entailment}}
* {{springer|title=Implication|id=p/i050280}}

{{logic}}
{{Logical connectives}}

[[Category:Philosophical logic]]
[[Category:Metalogic]]
[[Category:Propositional calculus]]
[[Category:Deductive reasoning]]
[[Category:Concepts in logic]]
[[Category:Concepts in logic#]]
[[Category:Syntax (logic)]]
[[Category:Logical consequence]]
[[Category:Binary operations]]

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[[ja:論理的帰結]]
[[pl:Konkluzja]]
[[pt:Acarretamento]]
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Snowden (physics) - Revision history

en>Cydebot: Robot - Speedily moving category Units of measure to :Category:Units of measurement per CFDS.