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Logical consequence (also entailment) is a fundamental
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by ...
in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
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, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agr ...
s, 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 Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophica ...
is meant to provide accounts of the nature of logical consequence and the nature of
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whi ...
. Logical consequence is necessary and
formal Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal attir ...
, by way of examples that explain with
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the s ...
and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given
language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...
,
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
, 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 Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...
$\mathcal$, either by constructing a deductive system for $\mathcal$ or by formal intended semantics for language $\mathcal$. The Polish logician
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the
logical form In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambigu ...
of the sentences: (2) The relation is
a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
, i.e., it can be determined with or without regard to
empirical evidence Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences a ...
(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, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambigu ...
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, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
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 for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
. 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, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding par ...
whereas the study of (its) semantic consequence is called (its)
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
.

## Syntactic consequence

A formula $A$ is a syntactic consequenceS. C. Kleene,
Introduction to Metamathematics
' (1952), Van Nostrand Publishing. p.88.
within some
formal system A formal system is an abstract structure 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 fo ...
$\mathcal$ of a set $\Gamma$ of formulas if there is a
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the s ...
in $\mathcal$ of $A$ from the set $\Gamma$. This is denoted $\Gamma \vdash_ A$. The turnstile symbol $\vdash$ was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934--1935). Syntactic consequence does not depend on any interpretation of the formal system.

## Semantic consequence

A formula $A$ is a semantic consequence within some formal system $\mathcal$ of a set of statements $\Gamma$ if and only if there is no model $\mathcal$ in which all members of $\Gamma$ are true and $A$ is false. Etchemendy, John, ''Logical consequence'', The Cambridge Dictionary of Philosophy This is denoted $\Gamma \models_ A,$. 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 In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In oth ...
over
possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their me ...
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 is considered to be purely the result of the constructive mental activity of humans rather than the discovery of ...
s such as
Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 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 ...
.

## Non-monotonic logical consequence

The accounts discussed above all yield
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
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 :.

*
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, 2 ...
* Ampheck *
Boolean algebra (logic) In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
*
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
*
Boolean function In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...
*
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
*
Deductive reasoning Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
*
Logic gate A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic ga ...
* Logical graph * Peirce's law *
Probabilistic logic Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A diffic ...
*
Propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
* Sole sufficient operator *
Strict conditional In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necess ...
*
Tautology (logic) In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always ...
* Tautological consequence * Therefore sign *
Turnstile (symbol) In mathematical logic and computer science the symbol \vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies ...
* Double turnstile *
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 * ...

# 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 building for Christian religious activities * Church (congregation), a local congregation of a Christian denomination * Church service, a formalized period of Christian communal worship * Chr ...
, Rosser,
Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an 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 the branch ...
, and
Post Post or POST commonly refers to: *Mail, the postal system, especially in Commonwealth of Nations countries **An Post, the Irish national postal service **Canada Post, Canadian postal service **Deutsche Post, German postal service ** Iraqi Post, Ir ...
. * . * 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 of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
. Originally published in Polish and
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
. * * A paper on 'implication' from math.niu.edu
Implication
* A definition of 'implicant