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Logical consequence (also entailment) is a fundamental
concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

concept
in
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
, which describes the relationship between
statement Statement or statements may refer to: Common uses *Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a Sentence_(lin ...
s that hold true when one statement logically ''follows from'' one or more statements. A valid logical
argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
is one in which the conclusion is entailed by the
premise A premise or premiss is a true or false statement that helps form the body of an argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, apply ...
s, because the conclusion is the consequence of the premises. The
philosophical analysis Philosophical analysis refers to a set of techniques, typically used by philosophers in the analytic tradition, in order to "break down" (i.e. analyze) philosophical issues. Arguably the most prominent of these techniques is the analysis of conce ...
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 Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophi ...
is meant to provide accounts of the nature of logical consequence and the nature of
logical truth Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and arg ...
. Logical consequence is
necessary Necessary or necessity may refer to: * Need A need is something that is necessary Necessary or necessity may refer to: * Need ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by ...
and
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set theory, set of requirements (substantial form, forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for forma ...
, by way of examples that explain with
formal proof In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label= ...
and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given
language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...
,
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
, 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 A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the ...
\mathcal, either by constructing a
deductive system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentiall ...
for \mathcal or by formal intended semantics for language \mathcal. The Polish logician
Alfred Tarski Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. was a Polish-American logician ...
identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the
logical form In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
of the sentences: (2) The relation is
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
, i.e., it can be determined with or without regard to
empirical evidence Empirical evidence for a proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is s ...
(sense experience); and (3) The logical consequence relation has a modal component.


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 In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using
inference rule In the philosophy of logic Following the developments in formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to ...
s. 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.


A priori property of logical consequence

If it is known that Q follows logically from P, then 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 Empirical evidence is the information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of its characteristics. The concept of ' ...
. 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 Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Ma ...
whereas the study of (its) semantic consequence is called (its)
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
.


Syntactic consequence

A formula A is a syntactic consequence within some
formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essen ...
\mathcal of a set \Gamma of formulas if there is a
formal proof In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label= ...
in \mathcal of A from the set \Gamma. :\Gamma \vdash_ A Syntactic consequence does not depend on any
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event i ...
of the formal system.


Semantic consequence

A formula A is a semantic consequence within some formal system \mathcal of a set of statements \Gamma :\Gamma \models_ A, if and only if there is no model \mathcal in which all members of \Gamma are true and A is false.
Etchemendy, John John W. Etchemendy (born 1952 in Reno, Nevada) is an American logician and philosopher who served as Stanford University's twelfth Provost (education), Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on J ...
, ''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 Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a Statement (logic), statement which is truth, true regardless of the truth or falsity of its constituent propositions. In other words, a logical tr ...
and
logical possibility Logical possibility refers to a logical proposition that cannot be disproved, using the axioms and rules of a given system of logic. The logical possibility of a proposition will depend upon the system of logic being considered, rather than on the ...
. 'It is necessary that' is often expressed as a
universal quantifier In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
over
possible world A possible world is a complete and consistent way the world is or could have been. They are widely used as a formal device in logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously ...
s, 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
intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathe ...
s such as
Michael Dummett Sir Michael Anthony Eardley Dummett (1925–2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham ...
.


Non-monotonic logical consequence

The accounts discussed above all yield
monotonic Figure 3. A function that is ''not'' monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
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 : but not of :.


See also

*
Abstract algebraic logic In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 2003 ...
*
Ampheck In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical disjunction, logical or. That is, a sentence of the form (''p'' NOR ''q'') is true precisely when neither ''p'' ...
*
Boolean algebra (logic) In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary a ...
*
Boolean domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Boolean function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
Boolean logic In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary ...
*
Causality Causality (also referred to as causation, or cause and effect) is influence by which one Event (relativity), event, process, state or object (a ''cause'') contributes to the production of another event, process, state or object (an ''effect'') ...
*
Deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ...
*
Logic gate A logic gate is an idealized model of computation A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list of steps with details of timing and resourc ...
* Logical graph * Peirce's law *
Probabilistic logicThe aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal proof, formal argument. ...
*
Propositional calculus Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...
*
Sole sufficient operatorIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
*
Strict conditionalIn logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, acc ...
*
Tautology (logic) In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical pr ...
* Tautological consequence *
Therefore sign In logical argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thou ...
*
Turnstile (symbol) In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical pr ...
*
Double turnstile In logic, the symbol (formal), symbol ⊨, ⊧ or \models is called the double turnstile. It is often read as "logical consequence, entails", "Model theory, models", "is a semantic consequence of" or "is stronger than". It is closely related to the ...
*
Validity Validity or Valid may refer to: Science/mathematics/statistics: * Validity (logic), a property of a logical argument * Scientific: ** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments ** ...


Notes


Resources

* . * London: College Publications. Series
Mathematical logic and foundations
* . * 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003. * . Papers include those by Gödel,
Church Church may refer to: Religion * Church (building) A church building, church house, or simply church, is a building used for Christian worship services and other Christian religious activities. The term is usually used to refer to the p ...
, Rosser,
Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an United States, American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of t ...

Kleene
, and
Post Post or POST commonly refers to: *Mail The mail or post is a system for physically transporting postcards, letters, and parcels. A postal service can be private or public, though many governments place restrictions on private systems. Since ...

Post
. * . * in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic''. * in Edward N. Zalta (ed.), ''The Stanford Encyclopedia of Philosophy''. * . * . * 365–409. * * in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell. * (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), (4th edition, 1982). * in D. Jacquette, ed., ''A Companion to Philosophical Logic''. Blackwell. * Reprinted in Tarski, A., 1983. ''Logic, Semantics, Metamathematics'', 2nd ed.
Oxford University Press Oxford University Press (OUP) is the university press A university press is an academic publishing Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for fre ...

Oxford University Press
. Originally published in
Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a country located in Central Europe. It is divided into 16 Voivodeships of Pol ...
and
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

German
. * * A paper on 'implication' from math.niu.edu
Implication
* A definition of 'implicant


External links

* * * * * {{Authority control Philosophical logic Metalogic Propositional calculus Deductive reasoning Concepts in logic Syntax (logic) Binary operations