logical consequence
   HOME

TheInfoList



OR:

Logical consequence (also entailment or logical implication) is a fundamental
concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...
in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
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 series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
is one in which the conclusion is entailed by the
premise A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...
s, because the conclusion is the consequence of the premises. The
philosophical analysis Philosophical analysis is any of various 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 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 philosophic ...
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 whic ...
. 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 atti ...
, by way of examples that explain with
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to 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 that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
,
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
, 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 that consists of grammar and vocabulary. It is the primary means by which humans convey meaning, both in spoken and signed language, signed forms, and may also be conveyed through writing syste ...
\mathcal, either by constructing a
deductive system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
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 ...
identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the
logical form In logic, the 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, unamb ...
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 (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
, i.e., it can be determined with or without regard to
empirical evidence Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how the ...
(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, the 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, unamb ...
of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using
inference rule Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the co ...
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. 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 , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
whereas the study of (its) semantic consequence is called (its)
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
.


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 and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
\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 (known as well-formed formulas when relating to 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", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by e ...
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 met ...
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 not 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.


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 ord ...
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, 200 ...
*
Ampheck In Boolean logic, logical NOR, non-disjunction, 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 precis ...
*
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
*
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 functi ...
*
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
* Causality *
Deductive reasoning Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, t ...
*
Logic gate A logic gate is a device that performs 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 gate, one that has, for ...
*
Logical graph An existential graph is a type of diagrammatic or visual notation for logical expressions, created by Charles Sanders Peirce, who wrote on graphical logic as early as 1882, and continued to develop the method until his death in 1914. They include ...
*
Peirce's law In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an Axiom#Mathematics, axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written ...
* Probabilistic logic *
Propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
* Sole sufficient operator *
Strawson entailment In formal semantics, Strawson entailment is a variant of the concept of entailment which is insensitive to presupposition failures. Formally, a sentence ''P'' Strawson-entails a sentence ''Q'' iff ''Q'' is always true when ''P'' is true and ''Q'' ...
*
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 ) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ...
* 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. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". ...
*
Double turnstile In logic, the symbol ⊨, ⊧ or \models is called the double turnstile. It is often read as " entails", " models", "is a semantic consequence of" or "is stronger than". It is closely related to the turnstile symbol \vdash, which has a single bar ...
* Validity


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 place/building for Christian religious activities and praying * Church (congregation), a local congregation of a Christian denomination * Church service, a formalized period of Christian comm ...
, 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 of ...
, 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 publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
. Originally published in Polish and German. * * A paper on 'implication' from math.niu.edu
Implication
* A definition of 'implicant


External links

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