Logical consequence

{{Short description|Relationship where one statement follows from another}} {{Redirect|Entailment||Entail (disambiguation)}} {{Redirect|Therefore|the therefore symbol ∴|Therefore sign}} {{Redirect|Logical implication|the binary connective|Material conditional}} {{Redirect|⊧|the symbol|Double turnstile}}

'''Logical consequence''' (also '''entailment''' or '''logical implication''') is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the following questions: ''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 is meant to provide accounts of the nature of logical consequence and the nature of logical truth.<ref>Quine, Willard Van Orman, ''Philosophy of Logic''.</ref>

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.<ref name="sep"/> A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e., without regard to any ''personal'' interpretations of the sentences) the sentence must be true if every sentence in the set is true.<ref name="iep">McKeon, Matthew, ''[http://www.iep.utm.edu/logcon/ Logical Consequence]'' Internet Encyclopedia of Philosophy.</ref>

Logicians make precise accounts of logical consequence regarding a given language <math>\mathcal{L}</math>, either by constructing a deductive system for <math>\mathcal{L}</math> or by formal intended semantics for language <math>\mathcal{L}</math>. The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the logical form of the sentences: (2) The relation is a priori, i.e., it can be determined with or without regard to empirical evidence (sense experience); and (3) The logical consequence relation has a modal component.<ref name="iep" />

== Formal accounts == The most widely prevailing view on how best to 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 schemes using inference rules. For instance, we can express the logical form of a valid argument as:

: All ''X'' are ''Y'' : All ''Y'' are ''Z'' : Therefore, all ''X'' are ''Z''.

This argument is formally valid, because every instance of arguments constructed using this scheme is 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 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 "''P'' is ''Q''{{'}}s brother's son, therefore ''P'' is ''Q''{{'}}s nephew" is valid in all cases, but is not a ''formal'' argument.<ref name="sep" />

== A priori property ==

If it is known that <math>Q</math> follows logically from <math>P</math>, then 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 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|editor1=Maria Luisa Dalla Chiara |editor1-link= Maria Luisa Dalla Chiara |editor2=Kees Doets |editor3=Daniele Mundici |editor4=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|chapter-url=https://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 === {{See also|Therefore_sign|label 1= ∴|Turnstile_(symbol)|label 2= ⊢}}

A formula <math>A</math> is a '''syntactic consequence'''<ref>Dummett, Michael (1993) [https://books.google.com/books?id=EYP7uCZIRQYC&q=syntactic+consequence%27%27Frege%3A&pg=PA82 ''philosophy of language''] Harvard University Press, p.82ff</ref><ref>Lear, Jonathan (1986) [https://books.google.com/books?id=lXI7AAAAIAAJ&q=syntactic+consequence%27%27Aristotle&pg=PA1 ''and Logical Theory''] Cambridge University Press, 136p.</ref><ref>Creath, Richard, and Friedman, Michael (2007) [https://books.google.com/books?id=87BcFLgJmxMC&q=syntactic+consequence%27%27The&pg=PA189 ''Cambridge companion to Carnap''] Cambridge University Press, 371p.</ref><ref>[http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence FOLDOC: "syntactic consequence"] {{webarchive|url=https://web.archive.org/web/20130403201417/http://www.swif.uniba.it/lei/foldop/foldoc.cgi?syntactic+consequence |date=2013-04-03 }}</ref><ref name="Kleene52">S. C. Kleene, ''[https://www.worldcat.org/oclc/523942 Introduction to Metamathematics]'' (1952), Van Nostrand Publishing. p.88.</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>. This is denoted <math>\Gamma \vdash_{\mathcal {FS} } A</math>. The turnstile symbol <math>\vdash</math> was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935).<ref name="Kleene52" />

Syntactic consequence does not depend on any interpretation of the formal system.<ref>{{Hunter 1996|p=101}}</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> 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>Etchemendy, John, ''Logical consequence'', The Cambridge Dictionary of Philosophy</ref> This is denoted <math>\Gamma \models_{\mathcal {FS} } A</math>. 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 necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantifier over possible worlds, 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 not 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 "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 intuitionists.<!-- missing reference to citation. Putting his name here doesn't serves the argument, I would rather change it to a reference to his work. -->

=== Non-monotonic logical consequence === {{See also|Non-monotonic logic|Belief revision#Non-monotonic inference relation}}

The accounts discussed above all yield 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}.

==See also== {{div col begin}} * Abstract algebraic logic * Ampheck * Boolean algebra (logic) * Boolean domain * Boolean function * Boolean logic * Causality * Deductive reasoning * Logic gate * Logical graph * Peirce's law * Probabilistic logic * Propositional calculus * Sole sufficient operator * Strawson entailment * Strict conditional * Tautology (logic) * Tautological consequence * Therefore sign * Turnstile (symbol) * Double turnstile * Validity {{div 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|last=Augusto|first=Luis M.|year=2017|title=Logical consequences. Theory and applications: An introduction.}} London: College Publications. Series: [http://www.collegepublications.co.uk/logic/mlf/?00029 Mathematical logic and foundations]. * {{citation|last1=Barwise|first1=Jon|author1-link=Jon Barwise|last2=Etchemendy|first2=John|author2-link=John Etchemendy|year=2008|title=Language, Proof and Logic|publisher=CSLI Publications|location=Stanford}}. * {{citation|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|author-link=Martin Davis (mathematician)|editor-last=Davis|editor-first= Martin|title=The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions|publisher=Raven Press|location=New York|year=1965|url=https://books.google.com/books?id=qW8x7sQ4JXgC&q=consequence|isbn=9780486432281}}. Papers include those by Gödel, Church, Rosser, Kleene, and Post. * {{citation |first=Michael |last=Dummett |year=1991 |title=The Logical Basis of Metaphysics |publisher=Harvard University Press|url=https://books.google.com/books?id=lvsVFxK3BPcC&q=consequence|isbn=9780674537866 }}. * {{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|chapter-url=http://plato.stanford.edu/entries/conditionals| chapter=Indicative Conditionals| publisher=Metaphysics Research Lab, Stanford University}} 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 |editor-last=Goble |editor-first=Lou|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|issue= 3|pages= 365–409|jstor= 2998398|doi= 10.2307/2998398}} 365–409. * {{citation |author-link=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= 978-87-991013-7-5}} * {{citation |last=Planchette |first=P. A. |year=2001 |title=Logical Consequence}} in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell. * {{citation|author-link=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 |author-link=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 |author-link=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 and German. * {{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|url-access=registration|url=https://archive.org/details/theoryoflogicalc0000wojc}} * A paper on 'implication' from math.niu.edu, [http://www.math.niu.edu/~richard/Math101/implies.pdf Implication] {{Webarchive|url=https://web.archive.org/web/20141021082239/http://www.math.niu.edu/~richard/Math101/implies.pdf |date=2014-10-21 }} * A definition of 'implicant' [http://www.allwords.com/word-implicant.html AllWords]

==External links== {{Commons category}} *{{cite SEP |url-id=logical-consequence |title=Logical Consequence|date=2013-11-19|edition=Winter 2016|last=Beall|first=Jc|last2=Restall|first2=Greg|author-link=Jc Beall|author2-link=Greg Restall}} *{{cite IEP |url-id=logcon/ |title=Logical consequence}} *{{InPho|taxonomy|2409}} *{{PhilPapers|category|logical-consequence-and-entailment}} *{{springer|title=Implication|id=p/i050280}}

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Category:Logical consequence Category:Philosophical logic Category:Metalogic Category:Propositional calculus Category:Semantic units Category:Deductive reasoning Category:Concepts in logic Category:Syntax (logic) Category:Binary operations