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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?^[1] All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.^[2]

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.^[1] 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.^[3]

Logicians make precise accounts of logical consequence regarding a given language Ldisplaystyle mathcal L, either by constructing a deductive system for Ldisplaystyle mathcal L or by formal intended semantics for language Ldisplaystyle 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.^[3]

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 Q's brother's son, therefore P is Q's nephew" is valid in all cases, but is not a formal argument.^[1]

A priori property of logical consequence

If you know that Qdisplaystyle Q follows logically from Pdisplaystyle P no information about the possible interpretations of Pdisplaystyle P or Qdisplaystyle Q will affect that knowledge. Our knowledge that Qdisplaystyle Q is a logical consequence of Pdisplaystyle P cannot be influenced by empirical knowledge.^[1] Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori.^[1] 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.^[1]

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.^[4]

Syntactic consequence

A formula Adisplaystyle A is a syntactic consequence^[5][6][7][8] within some formal system FSdisplaystyle mathcal FS of a set Γdisplaystyle Gamma of formulas if there is a formal proof in FSdisplaystyle mathcal FS of Adisplaystyle A from the set Γdisplaystyle Gamma .

Γ⊢FSAdisplaystyle Gamma vdash _mathcal FSA

Syntactic consequence does not depend on any interpretation of the formal system.^[9]

Semantic consequence

A formula Adisplaystyle A is a semantic consequence within some formal system FSdisplaystyle mathcal FS of a set of statements Γdisplaystyle Gamma

Γ⊨FSA,displaystyle Gamma models _mathcal FSA,

if and only if there is no model Idisplaystyle mathcal I in which all members of Γdisplaystyle Gamma are true and Adisplaystyle A is false.^[10] Or, in other words, the set of the interpretations that make all members of Γdisplaystyle Gamma true is a subset of the set of the interpretations that make Adisplaystyle A true.

Modal accounts

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

Γdisplaystyle Gamma ⊢displaystyle vdash Adisplaystyle A is true if and only if it is necessary that if all of the elements of Γdisplaystyle Gamma are true, then Adisplaystyle A is true.

Alternatively (and, most would say, equivalently):

Γdisplaystyle Gamma ⊢displaystyle vdash Adisplaystyle A is true if and only if it is impossible for all of the elements of Γdisplaystyle Gamma to be true and Adisplaystyle 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:

Γdisplaystyle Gamma ⊢displaystyle vdash Adisplaystyle A is true if and only if there is no possible world at which all of the elements of Γdisplaystyle Gamma are true and Adisplaystyle 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:

Γdisplaystyle Gamma ⊢displaystyle vdash Adisplaystyle A if and only if it is impossible for an argument with the same logical form as Γdisplaystyle Gamma /Adisplaystyle 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 Adisplaystyle A is a consequence of Γdisplaystyle Gamma , then Adisplaystyle A is a consequence of any superset of Γdisplaystyle 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.

See also

Notes

Resources

• 
  Anderson, A.R.; Belnap, N.D., Jr. (1975), Entailment, 1, Princeton, NJ: Princeton.


• 
  Augusto, Luis M. (2017), Logical consequences. Theory and applications: An introduction. London: College Publications. Series: Mathematical logic and foundations.


• 
  Barwise, Jon; Etchemendy, John (2008), Language, Proof and Logic, Stanford: CSLI Publications.


• 
  Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.


• 
  Davis, Martin, (editor) (1965), The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven PressCS1 maint: Extra text: authors list (link) . Papers include those by Gödel, Church, Rosser, Kleene, and Post.


• 
  Dummett, Michael (1991), The Logical Basis of Metaphysics, Harvard University Press.


• 
  Edgington, Dorothy (2001), Conditionals, Blackwell in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic.


• 
  Edgington, Dorothy (2006), Conditionals in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.


• 
  Etchemendy, John (1990), The Concept of Logical Consequence, Harvard University Press.


• 
  Goble, Lou, ed. (2001), The Blackwell Guide to Philosophical Logic, BlackwellCS1 maint: Extra text: authors list (link) .


• 
  Hanson, William H (1997), "The concept of logical consequence", The Philosophical Review, 106 365–409.


• 
  Hendricks, Vincent F. (2005), Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, ISBN 87-991013-7-8
  


• 
  Planchette, P. A. (2001), Logical Consequence in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.


• 
  Quine, W.V. (1982), Methods of Logic, Cambridge, MA: Harvard University Press (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), (4th edition, 1982).


• 
  Shapiro, Stewart (2002), Necessity, meaning, and rationality: the notion of logical consequence in D. Jacquette, ed., A Companion to Philosophical Logic. Blackwell.


• 
  Tarski, Alfred (1936), 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.


• A paper on 'implication' from math.niu.edu, Implication
  


• A definition of 'implicant' AllWords
  


• 
  Ryszard Wójcicki (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations. Springer. ISBN 978-90-277-2785-5.
  

External links

Wikimedia Commons has media related to Logical consequence.

• 
  Zalta, Edward N. (ed.). "Logical Consequence". Stanford Encyclopedia of Philosophy.
  


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  Logical consequence at the Indiana Philosophy Ontology Project
  


• 
  "Logical consequence". Internet Encyclopedia of Philosophy.
  


• 
  Logical consequence at PhilPapers
  


• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Implication", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4