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{{short description|Fundamental concept in logic}}{{redirect|Entailment||Entail (disambiguation)}}{{redirect|Therefore|the therefore symbol âˆ´|Therefore sign}}{{redirect|Logical implication|the binary connective|Material conditional}}{{redirect|âŠ§|the symbol|Double turnstile}}Logical consequence (also entailment) 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 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?Beall, JC and Restall, Greg, Logical Consequence The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.Quine, Willard Van Orman, Philosophy of Logic.Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. 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.McKeon, Matthew, Logical Consequence Internet Encyclopedia of Philosophy.Logicians make precise accounts of logical consequence regarding a given language mathcal{L}, either by constructing a deductive system for mathcal{L} or by formal intended semantics for language mathcal{L}. 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.

Formal accounts

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 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 Qs brother's son, therefore P is Qs nephew" is valid in all cases, but is not a formal argument.

A priori property of logical consequence

If you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a logical consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. 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.

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.BOOK, Maria Luisa Dalla Chiara, Kees Doets, Daniele Mundici, Johan van Benthem, Logic and Scientific Methods: Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995,weblink 1996, Springer, 978-0-7923-4383-7, 292, Logical consequence: a turn in style, Kosta Dosen,

Syntactic consequence

{{See also|Therefore_sign|label 1= âˆ´|Turnstile_(symbol)|label 2= âŠ¢}}A formula A is a syntactic consequenceDummett, Michael (1993) philosophy of language Harvard University Press, p.82ffLear, Jonathan (1986) and Logical Theory Cambridge University Press, 136p.Creath, Richard, and Friedman, Michael (2007) Cambridge companion to Carnap Cambridge University Press, 371p.FOLDOC: "syntactic consequence" {{webarchive|url=https://web.archive.org/web/20130403201417weblink |date=2013-04-03 }} within some formal system mathcal{FS} of a set Gamma of formulas if there is a formal proof in mathcal{FS} of A from the set Gamma.
Gamma vdash_{mathcal {FS} } A
Syntactic consequence does not depend on any interpretation of the formal system.Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971, p. 75.

Semantic consequence

{{See also|Double turnstile|label 1= âŠ¨}}A formula A is a semantic consequence within some formal system mathcal{FS} of a set of statements Gamma
Gamma models_{mathcal {FS} } A,
if and only if there is no model mathcal{I} in which all members of Gamma are true and A is false.Etchemendy, John, Logical consequence, The Cambridge Dictionary of Philosophy Or, in other words, the set of the interpretations that make all members of Gamma true is a subset of the set of the interpretations that make A true.

Modal accounts

Modal accounts of logical consequence are variations on the following basic idea:
Gamma vdash A is true if and only if it is necessary that if all of the elements of Gamma are true, then A is true.
Alternatively (and, most would say, equivalently):
Gamma vdash A is true if and only if it is impossible for all of the elements of Gamma to be true and A 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:
Gamma vdash A is true if and only if there is no possible world at which all of the elements of Gamma are true and A 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:
Gamma vdash A if and only if it is impossible for an argument with the same logical form as Gamma/A 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 such as Michael Dummett.

Non-monotonic logical consequence

The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if A is a consequence of Gamma, then A is a consequence of any superset of Gamma. 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 Non-monotonic inference relation.

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Resources

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• {{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 and German.
• BOOK, Ryszard WÃ³jcicki, Theory of Logical Calculi: Basic Theory of Consequence Operations, 1988, Springer, 978-90-277-2785-5,
• A paper on 'implication' from math.niu.edu, Implication
• A definition of 'implicant' AllWords

• SEP, logical-consequence, Logical Consequence, 2013-11-19, Winter 2016, Beall, Jc, Restall, Greg, Jc Beall, Greg Restall,
• IEP, logcon/, Logical consequence,
• {{InPho|taxonomy|2409}}
• {{PhilPapers|category|logical-consequence-and-entailment}}
• {{springer|title=Implication|id=p/i050280}}
{{Logic}}{{Mathematical logic}}{{Logical connectives}}

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