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Logical consequence
 
 
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"Therefore" redirects here. For the symbol, see therefore sign
Therefore sign

In a mathematical proof, the therefore sign is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism....
.


Logical consequence is a fundamental concept in logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
. It is the relation
Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
 that holds between a set of sentences
Sentence (linguistics)

In linguistics, a sentence is a grammatical unit of one or more words, bearing minimal syntactic relation to the words that precede or follow it, often preceded and followed in speech by pauses, having one of a small number of characteristic intonation patterns, and typically expressing an independent statement, question, request, command, et...
 (or proposition
Proposition

This article is about the term proposition in logic and philosophy; for other uses see PropositionIn logic and philosophy, proposition refers to either the "content" or Meaning of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence....
s) and a sentence (proposition) when the former "entails"
Entailment

In logic and mathematics, entailment or logical implication is a logical relation that holds between a set T of propositions and a proposition B when every Model theory of T is also a model of B....
 the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory" to affirm the latter and deny the former.






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"Therefore" redirects here. For the symbol, see therefore sign
Therefore sign

In a mathematical proof, the therefore sign is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism....
.


Logical consequence is a fundamental concept in logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
. It is the relation
Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
 that holds between a set of sentences
Sentence (linguistics)

In linguistics, a sentence is a grammatical unit of one or more words, bearing minimal syntactic relation to the words that precede or follow it, often preceded and followed in speech by pauses, having one of a small number of characteristic intonation patterns, and typically expressing an independent statement, question, request, command, et...
 (or proposition
Proposition

This article is about the term proposition in logic and philosophy; for other uses see PropositionIn logic and philosophy, proposition refers to either the "content" or Meaning of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence....
s) and a sentence (proposition) when the former "entails"
Entailment

In logic and mathematics, entailment or logical implication is a logical relation that holds between a set T of propositions and a proposition B when every Model theory of T is also a model of B....
 the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory" to affirm the latter and deny the former. Logical consequence is the relationship between the premises and the conclusion of a valid argument. These explanations and definitions tend to be circular; the provision of a satisfactory account of logical consequence and entailment is an important topic of philosophy of logic
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 be termed either philosophy of logic or philosophical logic if no longer simply logic....
.

The truth of the above consequence depends on both the truth of the antecedents and the relationship of logical consequence between the antecedents and the consequence. The consequence might NOT be true if not all frogs were green. Logical consequences or inferences by deductive reasoning
Deductive reasoning

Deductive reasoning, sometimes called deductive logic, is reasoning which constructs or evaluates deductive Argument s.In logic, an argument is said to be deductive when the truth of the conclusion is purported to follow necessarily or be a logical consequence of the premises and its corresponding conditional is a necessary truth....
 are a major aspect of epistemology
Epistemology

Epistemology or theory of knowledge is the branch of philosophy concerned with the nature and scope of knowledge. It addresses the questions:...
 that communicates to the general public hypotheses about causality
Causality

Causality denotes a necessary relationship between one event and another event which is the direct consequence of the first.While this informal understanding suffices in everyday use, the Philosophy analysis of how best to characterize causality extends over millennia....
 of risk factors
Risk factors

A risk factor is a concept in finance theory such as the Capital Asset Pricing Model, Arbitrage Pricing Theory and other theories that use pricing kernels....
.

A formally specified logical consequence relation may be characterized model-theoretically
Model theory

In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
 or proof-theoretically
Proof theory

Proof theory is a branch of mathematical logic that represents Mathematical proofs as formal mathematical objects, facilitating their analysis by mathematical techniques....
 (or both).

Logical consequence can also be expressed as a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 from sets of sentences to sets of sentences (Tarski
Alfred Tarski

Alfred Tarski was a Poles logician and mathematician. Educated in the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and did research in mathematics at the University of California, Berkeley, from 1942 until his death....
's preferred formulation), or as a relation between two sets of sentences (multiple-conclusion logic
Multiple-conclusion logic

A multiple-conclusion logic is one in which logical consequence is a relation , , between two Set of sentence . is typically interpreted as meaning that whenever each element of is true, some element of is true; and whenever each element of is false, some element of is false....
).

Accounts of logical consequence


This section describes some common accounts of logical consequence.

G will represent an arbitrary set of premise
Premise

Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument* Premises, land and buildings together considered as a property...
s and A an arbitrary conclusion. G/A will denote the logical argument having G as its (set of) premises and A as its conclusion. G A will mean that A is a logical consequence of G.

Syntactic and semantic consequence


A formula A is a syntactic consequence within some formal system FS of a set ? of formulas iff there is a formal proof
Derivation

Derivation may refer to:* Derivation , a function on an algebra which generalizes certain features of the derivative operator* Derivation * Derivation in differential algebra, a unary function satisfying the Leibniz product law...
 in formal system
Formal system

In logic, a formal system consists of a formal language together with a deductive system which consists of a set of inference rules and/or axioms....
 FS of A from the set ?.

Syntactic consequence does not depend on any interpretation of the formal system.

A formula A is a semantic consequence of a set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 of statements
Proposition

This article is about the term proposition in logic and philosophy; for other uses see PropositionIn logic and philosophy, proposition refers to either the "content" or Meaning of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence....
 ?

? A,


if and only if no interpretation makes all members of ? true and A false.

Modal accounts


Modal accounts of logical consequence are variations on the following basic idea:

  • G A just in case it is necessary that if all of the elements of G are true, then A is true.


Alternatively (and, most would say, equivalently):

  • G A just in case it is impossible for all of the elements of G to be true and A false.


Such accounts are called "modal" because they appeal to the modal notions of necessity and (im)possibility
Logical possibility

A logically possible proposition is one that can be asserted without implying a logical contradiction. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true....
. 'It is necessary that' is often cashed out as a universal quantifier over possible worlds
Possible Worlds

Possible Worlds may refer to:* Possible worlds, a concept in philosophy* Possible Worlds , by John Mighton** Possible Worlds , by Robert Lepage, based on the Mighton play...
, so that the accounts above translate as:

  • G A just in case there is no possible world at which all of the elements of G 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.

Formal accounts


Formal accounts of logical consequence are variations on the following basic idea:

  • G A just in case no argument with the same logical form
    Logical form

    The form or logical form of an argument is the representation of its sentences using the formal grammar and symbolism of a logical system to display its similarity with all other arguments of the same type....
     as G/A has true premises and a false conclusion.


Two common variations on this basic idea are:

  1. G A just in case no uniform substitution of the nonlogical terms in G/A yields an argument with true premises and a false conclusion.
  2. G A just in case there is no way of interpreting
    Interpretation (logic)

    In logic an interpretation gives meaning to an artificial or formal language or to a Sentence of such a language by assigning a denotation to each non-logical symbol in that language or in that sentence....
     the nonlogical terms in G/A that yields an argument with true premises and a false conclusion.


Let us again consider the argument:

All frogs are green.
Kermit is a frog.
Therefore, Kermit is green.


Formal account (1) says that the conclusion is a logical consequence of the premises because no matter how we uniformly replace the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Consider for example:

All skyscrapers are tall.
The Empire State Building is a skyscraper.
Therefore, the Empire State Building is tall.


All squares are rectangles.
All rectangles are quadrilaterals.
Therefore, a square is a quadrilateral.


All matter has mass.
Coffee tables are matter.
Therefore, coffee tables have mass.


All birds have feathers.
Penguins are birds.
Therefore, penguins have feathers.


We can make up arguments of this form all day, but we will never come up with one that has true premises and a false conclusion. The argument is deductively valid
Validity

The term Validity in logic applies to Argument or statements....
 by virtue of its logical form, which might be characterized with the following template (in which F, G, and a are meaningless placeholders):

All Fs are Gs.
a is an F.
Therefore, a is a G.


Formal account (2) says that the conclusion of the "Kermit" argument is a logical consequence of the premises because no matter how we interpret the nonlogical terms (frog, green, Kermit) in the argument, we do not get true premises and a false conclusion. Suppose, for example, we interpret frog to mean plumber, green to mean shy, and Kermit to mean Madonna (the singer). Then the argument has two false premises (for not all plumbers are shy, and Madonna is not a plumber) and a false conclusion (for Madonna is not shy). We can come up with as many interpretations of frog, green, and Kermit as we like, but this will never result in an argument with true premises and a false conclusion.

Modal-formal accounts


Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:

  • G A just in case it is impossible for an argument with the same logical form as G/A to have true premises and a false conclusion.


Most logicians would probably agree that logical consequence, as we intuitively understand it, has both a modal and a formal aspect, and that some version of the modal/formal account is therefore closest to being correct.

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
Theory of justification

Theory of justification is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and probability....
-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
Michael Dummett

Knight Bachelor Michael Anthony Eardley Dummett Fellow of the British Academy Doctor of Letters is a leading British philosopher. He has both written on the history of analytic philosophy, and made original contributions to the subject, particularly in the areas of philosophy of mathematics, philosophy of logic, philosophy of language and me...
.

Non-monotonic logical consequence


The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if A is a consequence of G, then A is a consequence of any superset of G. 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

.


For more on this, see the article on non-monotonic logic
Non-monotonic logic

A non-monotonic logic is a formal logic whose Logical consequence Relation is not Monotonic#Monotonic_logic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences....
.

See also

  • Deductive reasoning
    Deductive reasoning

    Deductive reasoning, sometimes called deductive logic, is reasoning which constructs or evaluates deductive Argument s.In logic, an argument is said to be deductive when the truth of the conclusion is purported to follow necessarily or be a logical consequence of the premises and its corresponding conditional is a necessary truth....
  • Tautology (logic)
    Tautology (logic)

    In propositional logic, a tautology is a propositional formula that is true under any possible Valuation of its propositional variables. For example, the propositional formula is a tautology, because the statement is true for any valuation of A....
  • Validity
    Validity

    The term Validity in logic applies to Argument or statements....
  • Semantic consequence


Resources

  • Michael Dummett
    Michael Dummett

    Knight Bachelor Michael Anthony Eardley Dummett Fellow of the British Academy Doctor of Letters is a leading British philosopher. He has both written on the history of analytic philosophy, and made original contributions to the subject, particularly in the areas of philosophy of mathematics, philosophy of logic, philosophy of language and me...
    , 1991. The Logical Basis of Metaphysics. Harvard University Press.
  • John Etchemendy
    John Etchemendy

    John W. Etchemendy is Stanford University's twelfth and current Provost . He succeeded John L. Hennessy to the post on September 1, 2000.John Etchemendy received his bachelor's and master's degrees at the University of Nevada, Reno before earning his PhD in philosophy at Stanford in 1982....
    , 1990. The Concept of Logical Consequence. Harvard University Press.
  • Hanson, William H., 1997, "The concept of logical consequence," The Philosophical Review 106: 365-409.
  • Vincent F. Hendricks
    Vincent F. Hendricks

    Vincent F. Hendricks , is a distinguished philosopher and logician. He holds two doctoral degrees in philosophy and is Professor of Formal Philosophy at Roskilde University in Denmark and member of IIP, the Institut Internationale de Philosophie....
    , 2005. Thought 2 Talk: A Crash Course in Reflection and Expression. New York: Automatic Press / VIP. ISBN 87-991013-7-8
  • Planchette, P. A., 2001, "Logical Consequence," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
  • Stewart Shapiro
    Stewart Shapiro

    Stewart Shapiro is Professor of Philosophy at the Ohio State University and a regular visiting professor at the University of St Andrews in Scotland....
    , 2002, "Necessity, meaning, and rationality: the notion of logical consequence" In D. Jacquette, ed., A Companion to Philosophical Logic. Blackwell.
  • Alfred Tarski
    Alfred Tarski

    Alfred Tarski was a Poles logician and mathematician. Educated in the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and did research in mathematics at the University of California, Berkeley, from 1942 until his death....
    , 1936, "On the concept of logical consequence." Reprinted in Tarski, A., 1983. Logic, Semantics, Metamathematics, 2nd ed. Oxford University Press
    Oxford University Press

    Oxford University Press is a publisher and a department of the University of Oxford in England. It is the largest university press in the world, being larger than all the American university presses combined with Cambridge University Press....
    . Originally published in Polish
    Polish language

    Polish , an official language of Poland, has the largest number of speakers of any West Slavic languages. Polish-speakers use the language in a uniform manner through most of Poland, and it has a regular orthography....
     and German
    German language

    German is a West Germanic languages, thus related to and classified alongside English language and Dutch language. It is one of the world's world language and the most widely spoken mother tongue in the European Union....
    .


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