logical consequence


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 bod ...

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=none, lit ...

, 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 statement that an argument claims will induce or justify a Logical consequence, conclusion. It is an assumption that something is true. Explanation In logic, an argument requires a Set (mathematics), set of (at least) ...
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 ** An action somebody may feel they must do ** An important task or essential thing to do at a particular time or by a particular moment * Necessary and sufficient condition, in logic, something that is ...
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 and interpretation (logic), models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given Formal language, 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.Matthew W. McKeon, McKeon, Matthew,
Logical Consequence
' Internet Encyclopedia of Philosophy.
Logicians make precise accounts of logical consequence regarding a given formal language, language \mathcal, either by constructing a deductive system for \mathcal or by formal Intended interpretation, 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 and a posteriori, 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 logic, 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 schema (logic), 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 Substitution (logic), 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 conditional, 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 A priori and a posteriori, 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 consequence within some formal system \mathcal of a set \Gamma of formulas if there is a formal proof in \mathcal of A from the set \Gamma. :\Gamma \vdash_ A Syntactic consequence does not depend on any interpretation (logic), 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 :\Gamma \models_ A, if and only if there is no model \mathcal in which all members of \Gamma are true and A is false.John Etchemendy, Etchemendy, John, ''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 logic, 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 truth, logical necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantification, 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 "Theory of justification, 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.

Non-monotonic logical consequence

The accounts discussed above all yield Monotonicity of entailment, monotonic 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 * Ampheck * Boolean algebra (logic) * Boolean domain * Boolean function * Boolean logic * Causality * Deductive reasoning * Logic gate * Logical graph * Peirce's law * Probabilistic logic * Propositional calculus * Sole sufficient operator * Strict conditional * Tautology (logic) * Tautological consequence * Therefore sign * Turnstile (symbol) * Double turnstile * Validity (logic), Validity



* . * 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, Alonzo Church, Church, J. Barkley Rosser, Rosser, Kleene, and Emil Leon Post, 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. Originally published in Polish language, Polish and German language, German. * * A paper on 'implication' from math.niu.edu
* A definition of 'implicant

External links

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