"Therefore" redirects here. For the therefore symbol ∴, see
Therefore sign.
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 , either by constructing a deductive system for or by formal intended semantics for language . 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]
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.[1]
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]
Beall, JC and Restall, Greg, Logical Consequence The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971, p. 75.
- Anderson, A.R.; Belnap, N.D. Jr. (1975), Entailment, vol. 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, ed. (1965), The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions, New York: Raven Press, ISBN 9780486432281. Papers include those by Gödel, Church, Rosser, Kleene, and Post.
- Dummett, Michael (1991), The Logical Basis of Metaphysics, Harvard University Press, ISBN 9780674537866.
- Edgington, Dorothy (2001), Conditionals, Blackwell in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic.
- Edgington, Dorothy (2006), "Indicative Conditionals", Conditionals, Metaphysics Research Lab, Stanford University 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, Blackwell.
- Hanson, William H (1997), "The concept of logical consequence", The Philosophical Review, 106 (3): 365–409, doi:10.2307/2998398, JSTOR 2998398 365–409.
- Hendricks, Vincent F. (2005), Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, ISBN 978-87-991013-7-5
- 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.
- Ryszard Wójcicki (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations. Springer. ISBN 978-90-277-2785-5.
- A paper on 'implication' from math.niu.edu, Implication Archived 2014-10-21 at the Wayback Machine
- A definition of 'implicant' AllWords