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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 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. Logical consequence is necessary and formal, 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 seq ...
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 bic ...
, 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 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 ("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 ex ...
, 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 ...
(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 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.

# 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. 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.

## 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 seq ...
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 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 ...
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 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 whic ...
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 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 order ...
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, 200 ...
* Ampheck * Boolean algebra (logic) * Boolean domain * Boolean function * Boolean logic * Causality * Deductive reasoning *
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 A logical graph is a special type of diagrammatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. In his papers on ''qualitative logic'', ''entitative graphs'', and '' existential grap ...
*
Peirce's law In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that inv ...
* Probabilistic logic *
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 b ...
*
Sole sufficient operator In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression.. ("Complete set of logical connectives").. ( ...
* Strict conditional * Tautology (logic) *
Tautological consequence In propositional logic, tautological consequence is a strict form of logical consequenceBarwise and Etchemendy 1999, p. 110 in which the tautologousness of a proposition is preserved from one line of a proof to the next. Not all logical consequence ...
* Therefore sign * Turnstile (symbol) * Double turnstile * Validity

# 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 * Chri ...
, Rosser, Kleene, and 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 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 book ...
. 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