Difference between revisions of "Implication" - New World Encyclopedia

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:''This article is about logical implication. For the historical legal practice of restricting the descent of property, see [[fee tail]].''
 
  
 
'''Implication''' or '''entailment''' is used in [[propositional calculus|propositional logic]] and [[predicate logic]] to describe a relationship between two sentences or sets of sentences, in which one sentence or set of sentences is said to "lead to" or "imply" or "entail" the other sentence or set of sentences, and the other is said to "follow from" or be "derived from" or be "entailed by" or be "implied by" the former.
 
'''Implication''' or '''entailment''' is used in [[propositional calculus|propositional logic]] and [[predicate logic]] to describe a relationship between two sentences or sets of sentences, in which one sentence or set of sentences is said to "lead to" or "imply" or "entail" the other sentence or set of sentences, and the other is said to "follow from" or be "derived from" or be "entailed by" or be "implied by" the former.
 
+
{{toc}}
 
==Logical Implication==
 
==Logical Implication==
 
<math>A \vdash B</math>
 
<math>A \vdash B</math>
  
states that the set '''A''' of sentences logically entails the set '''B''' of sentences. It can be read as "B can be proven from A".
+
states that the set '''A''' of sentences logically entails the set '''B''' of sentences. It can be read as "B can be proven from A."
  
 
Definition: '''A''' logically entails '''B''' if, by assuming all sentences in '''A''' are true, and applying a finite sequence of inference rules to them (for example, those from [[propositional calculus]]), one can derive all sentences in '''B'''.
 
Definition: '''A''' logically entails '''B''' if, by assuming all sentences in '''A''' are true, and applying a finite sequence of inference rules to them (for example, those from [[propositional calculus]]), one can derive all sentences in '''B'''.
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<math>A \models B</math>
 
<math>A \models B</math>
  
states that the set '''A''' of sentences semantically entails the set '''B''' of sentences.
+
states that the set '''A''' of sentences semantically entails the set '''B''' of sentences.  
  
Formal definition: the set '''A''' entails the set '''B''' [[if and only if]], in every model in which all sentences in '''A''' are true, all sentences in '''B''' are also true. In diagram form, it looks like this:
+
Formal definition: the set '''A''' entails the set '''B''' if and only if, in every model in which all sentences in '''A''' are true, all sentences in '''B''' are also true. In diagram form, it looks like this:
 
 
[[image:Venn_A_subset_B.png|A entails B]]
 
  
 
We need the definition of entailment to demand that ''every'' model of '''A''' must also be a model of '''B''' because a formal system like a knowledge base can't possibly know the interpretations which a user might have in mind when they ask whether a set of facts ('''A''') entails a proposition ('''B''').
 
We need the definition of entailment to demand that ''every'' model of '''A''' must also be a model of '''B''' because a formal system like a knowledge base can't possibly know the interpretations which a user might have in mind when they ask whether a set of facts ('''A''') entails a proposition ('''B''').
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In [[pragmatics]] ([[linguistics]]), [[entailment (pragmatics)|entailment]] has a different, but closely related, meaning.
 
In [[pragmatics]] ([[linguistics]]), [[entailment (pragmatics)|entailment]] has a different, but closely related, meaning.
  
If <math>\varnothing \models X</math> for a formula '''X''' then '''X''' is said to be "valid" or "[[tautology|tautological]]".
+
If <math>\varnothing \models X</math> for a formula '''X''' then '''X''' is said to be "valid" or "[[tautology|tautological]]."
  
 
==Relationship between Semantic and Logical Implication==
 
==Relationship between Semantic and Logical Implication==
Ideally, semantic implication and logical implication would be [[equivalent]]. However, this may not always be feasible. (See [[Gödel's incompleteness theorem]], which states that some languages (such as [[arithmetic]]) contain true but unprovable sentences.) In such a case, it is useful to break the equivalence down into its two parts:
+
Ideally, semantic implication and logical implication would be [[equivalent]]. However, this may not always be feasible. (See [[Gödel's incompleteness theorem]], which states that some languages (such as [[arithmetic]]) contain true but unprovable sentences.) In such a case, it is useful to break the equivalence down into its two parts:
  
 
A deductive system '''S''' is [[completeness|complete]] for a language '''L''' if and only if <math>A \models_L X</math> implies <math>A \vdash_S X</math>: that is, if all [[valid]] arguments are provable.
 
A deductive system '''S''' is [[completeness|complete]] for a language '''L''' if and only if <math>A \models_L X</math> implies <math>A \vdash_S X</math>: that is, if all [[valid]] arguments are provable.
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==Material Conditional==
 
==Material Conditional==
In [[propositional calculus]], or logical calculus in [[mathematics]], the '''material conditional''' or the '''implies operator''' is a [[binary_relation|binary]] [[truth-functional]] [[logical operator]] yielding the form  
+
In [[propositional calculus]], or logical calculus in [[mathematics]], the ''material conditional'' or the ''implies operator'' is a [[binary_relation|binary]] [[truth-functional]] [[logical operator]] yielding the form  
  
 
''If'' a ''then'' c,  
 
''If'' a ''then'' c,  
  
where ''a'' and ''c'' are statement variables (to be replaced by any meaningful indicative sentence of the language). In a statement of this form, the first term, in this case ''a'', is called the ''[[antecedent]]'' and the second term, in this case ''c'', is called the ''[[consequent]]''. The truth of the antecedent is a [[sufficient condition]] for the truth of the consequent, while the truth of the consequent is a [[necessary condition]] for the truth of the antecedent.
+
where ''a'' and ''c'' are statement variables (to be replaced by any meaningful indicative sentence of the language). In a statement of this form, the first term, in this case ''a'', is called the ''[[antecedent]]'' and the second term, in this case ''c'', is called the ''[[consequent]]''. The truth of the antecedent is a [[sufficient condition]] for the truth of the consequent, while the truth of the consequent is a [[necessary condition]] for the truth of the antecedent.
  
The operator is symbolized using a right-arrow "&#8594;" (or sometimes a horseshoe "&sup;"). "If A then B" is written like this:
+
The operator is symbolized using a right-arrow "&#8594;" (or sometimes a horseshoe "&sup;"). "If A then B" is written like this:
  
 
<math> A \to B</math>
 
<math> A \to B</math>
Line 46: Line 43:
  
 
==Relationship with Material Implication==
 
==Relationship with Material Implication==
In many cases, entailment corresponds to [[material implication]]: that is, <math>A, X \models Y</math> if and only if <math>A \models X \to Y</math> . However, this is not true in some [[many-valued logic]]s.
+
In many cases, entailment corresponds to [[material implication]]: that is, <math>A, X \models Y</math> if and only if <math>A \models X \to Y</math> . However, this is not true in some [[many-valued logic]]s.
  
 
Standard logic is two-valued, meaning that statements can be only true or false, and every statement is either true or false. So if a statement is not false it is true, and if it is not true it is false. In many-valued logics those conditions do not necessarily hold.
 
Standard logic is two-valued, meaning that statements can be only true or false, and every statement is either true or false. So if a statement is not false it is true, and if it is not true it is false. In many-valued logics those conditions do not necessarily hold.
 
[[zh:蕴涵]]
 
  
 
==Symbolization==
 
==Symbolization==
  
A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, [[disjunction]], [[conjunction]], [[negation]], and (frequently) [[biconditional]]. More advanced logic books and later chapters of introductory volumes often add [[identity]], [[Existential quantification]], and [[Universal quantification]].
+
A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, [[disjunction]], [[conjunction]], [[negation]], and (frequently) [[biconditional]]. More advanced logic books and later chapters of introductory volumes often add [[identity]], [[Existential quantification]], and [[Universal quantification]].
  
Different phrases used to identify the material conditional in ordinary language include ''if'', ''only if'', ''given that'', ''provided that'', ''supposing that'', ''implies'', ''even if'', and ''in case''. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "A only if B" is captured by the statement  
+
Different phrases used to identify the material conditional in ordinary language include ''if'', ''only if'', ''given that'', ''provided that'', ''supposing that'', ''implies'', ''even if'', and ''in case''. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "A only if B" is captured by the statement  
  
 
A &#8594; B,
 
A &#8594; B,
Line 64: Line 59:
 
B &#8594; A
 
B &#8594; A
  
When doing symbolization exercises, it is often required that the student give a [[scheme of abbreviation]] that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:
+
When doing symbolization exercises, it is often required that the student give a [[scheme of abbreviation]] that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:
  
 
A &#8594; B,
 
A &#8594; B,
Line 87: Line 82:
  
 
==Comparison with other conditional statements==
 
==Comparison with other conditional statements==
The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.
+
The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.
  
These unexpected truths arise because speakers of English (and other natural languages) are tempted to [[equivocation|equivocate]] between the material conditional and the [[indicative conditional]], or other conditional statements, like the [[counterfactual conditional]] and the [[logical biconditional |material biconditional]]. This temptation can be lessened by reading conditional statements without using the words "if" and "then"The most common way to do this is to read ''A &#8594; B'' as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true"(This [[equivalence|equivalent]] statement is captured in logical notation by <math>\neg A \vee B</math>, using negation and disjunction.)
+
These unexpected truths arise because speakers of English (and other natural languages) are tempted to [[equivocation|equivocate]] between the material conditional and the [[indicative conditional]], or other conditional statements, like the [[counterfactual conditional]] and the [[logical biconditional |material biconditional]]. This temptation can be lessened by reading conditional statements without using the words "if" and "then." The most common way to do this is to read ''A &#8594; B'' as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true." (This [[equivalence|equivalent]] statement is captured in logical notation by <math>\neg A \vee B</math>, using negation and disjunction.)
 
 
[[Category:Logic]]
 
[[Category:Binary operations]]
 
[[Category:Philosophy and religion]]
 
  
 
==References==
 
==References==
 
 
Most logic texts have sections dealing with implication and/or material implication.
 
Most logic texts have sections dealing with implication and/or material implication.
  
Here are some such representative tests:
+
Here are some such representative texts:
  
 
*Copi, Irving M., and Carl Cohen. ''Introduction to Logic''. Prentice Hall. (Many editions; the latest, from 2004, is the 12th.)
 
*Copi, Irving M., and Carl Cohen. ''Introduction to Logic''. Prentice Hall. (Many editions; the latest, from 2004, is the 12th.)
Line 111: Line 101:
 
*C. Lewis and C. Langford, ''Symbolic Logic''. 1932. Dover reprint, 1960.
 
*C. Lewis and C. Langford, ''Symbolic Logic''. 1932. Dover reprint, 1960.
 
*Sandford, David H, ''If P, then Q: Conditionals and the foundations of reasoning''. London and New York: Routledge, 1989, 1992, 2nd ed. 2003.
 
*Sandford, David H, ''If P, then Q: Conditionals and the foundations of reasoning''. London and New York: Routledge, 1989, 1992, 2nd ed. 2003.
 +
 +
== External links ==
 +
All links retrieved February 27, 2018.
 +
 +
===General Philosophy Sources===
 +
*[http://plato.stanford.edu/ Stanford Encyclopedia of Philosophy]
 +
*[http://www.iep.utm.edu/ The Internet Encyclopedia of Philosophy]
 +
*[http://www.bu.edu/wcp/PaidArch.html Paideia Project Online]
 +
*[http://www.gutenberg.org/ Project Gutenberg]
  
  
[[Category:Logic]]
 
[[Category:Binary operations]]
 
 
[[Category:Philosophy and religion]]
 
[[Category:Philosophy and religion]]
 
+
[[Category:Philosophy]]
  
 
{{Credit2|Entailment|43729795|Material_conditional|48054407}}
 
{{Credit2|Entailment|43729795|Material_conditional|48054407}}

Latest revision as of 00:41, 28 February 2018


Implication or entailment is used in propositional logic and predicate logic to describe a relationship between two sentences or sets of sentences, in which one sentence or set of sentences is said to "lead to" or "imply" or "entail" the other sentence or set of sentences, and the other is said to "follow from" or be "derived from" or be "entailed by" or be "implied by" the former.

Logical Implication

states that the set A of sentences logically entails the set B of sentences. It can be read as "B can be proven from A."

Definition: A logically entails B if, by assuming all sentences in A are true, and applying a finite sequence of inference rules to them (for example, those from propositional calculus), one can derive all sentences in B.

Semantic Implication

states that the set A of sentences semantically entails the set B of sentences.

Formal definition: the set A entails the set B if and only if, in every model in which all sentences in A are true, all sentences in B are also true. In diagram form, it looks like this:

We need the definition of entailment to demand that every model of A must also be a model of B because a formal system like a knowledge base can't possibly know the interpretations which a user might have in mind when they ask whether a set of facts (A) entails a proposition (B).

In pragmatics (linguistics), entailment has a different, but closely related, meaning.

If for a formula X then X is said to be "valid" or "tautological."

Relationship between Semantic and Logical Implication

Ideally, semantic implication and logical implication would be equivalent. However, this may not always be feasible. (See Gödel's incompleteness theorem, which states that some languages (such as arithmetic) contain true but unprovable sentences.) In such a case, it is useful to break the equivalence down into its two parts:

A deductive system S is complete for a language L if and only if implies : that is, if all valid arguments are provable.

A deductive system S is sound for a language L if and only if implies : that is, if no invalid arguments are provable.

Material Conditional

In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form

If a then c,

where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). In a statement of this form, the first term, in this case a, is called the antecedent and the second term, in this case c, is called the consequent. The truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.

The operator is symbolized using a right-arrow "→" (or sometimes a horseshoe "⊃"). "If A then B" is written like this:


Relationship with Material Implication

In many cases, entailment corresponds to material implication: that is, if and only if . However, this is not true in some many-valued logics.

Standard logic is two-valued, meaning that statements can be only true or false, and every statement is either true or false. So if a statement is not false it is true, and if it is not true it is false. In many-valued logics those conditions do not necessarily hold.

Symbolization

A common exercise for an introductory logic text to include is symbolizations. These exercises give a student a sentence or paragraph of text in ordinary language which the student has to translate into the symbolic language. This is done by recognizing the ordinary language equivalents of the logical terms, which usually include the material conditional, disjunction, conjunction, negation, and (frequently) biconditional. More advanced logic books and later chapters of introductory volumes often add identity, Existential quantification, and Universal quantification.

Different phrases used to identify the material conditional in ordinary language include if, only if, given that, provided that, supposing that, implies, even if, and in case. Many of these phrases are indicators of the antecedent, but others indicate the consequent. It is important to identify the "direction of implication" correctly. For example, "A only if B" is captured by the statement

A → B,

but "A, if B" is correctly captured by the statement

B → A

When doing symbolization exercises, it is often required that the student give a scheme of abbreviation that shows which sentences are replaced by which statement letters. For example, an exercise reading "Kermit is a frog only if muppets are animals" yields the solution:

A → B, A - Kermit is a frog. B - Muppets are animals.

Truth table

The truth value of expressions involving the material conditional is defined by the following truth table:

p q pq
F F T
F T T
T F F
T T T

Comparison with other conditional statements

The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement "2 is odd implies 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "If pigs fly, then Paris is in France" is true.

These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. This temptation can be lessened by reading conditional statements without using the words "if" and "then." The most common way to do this is to read A → B as "it is not the case that A and/or it is the case that B" or, more simply, "A is false and/or B is true." (This equivalent statement is captured in logical notation by , using negation and disjunction.)

References
ISBN links support NWE through referral fees

Most logic texts have sections dealing with implication and/or material implication.

Here are some such representative texts:

  • Copi, Irving M., and Carl Cohen. Introduction to Logic. Prentice Hall. (Many editions; the latest, from 2004, is the 12th.)
  • Hurley, Patrick J. A Concise Introduction to Logic. Belmont, CA: Wadsworth/Thompson Learning. (Many editions; the latest is the 9th.)
  • Johnson, Robert M. Fundamentals of Reasoning: A Logic Book. Belmont, CA: Wadsworth. (Latest is the 4th edition.)

Also:

  • Reese, William L. "Implication," in Dictionary of Philosophy and Religion, New and Enlarged Edition. Atlantic Highlands, NJ: Humanities Press, 1996.
  • "Implication," in Ted Hondereich, ed. The Oxford Companion to Philosophy. Oxford and New York: Oxford University Press, 1995.

Other valuable texts:

  • A. Anderson and Nuel Belnap, Entailments.
  • C. Lewis and C. Langford, Symbolic Logic. 1932. Dover reprint, 1960.
  • Sandford, David H, If P, then Q: Conditionals and the foundations of reasoning. London and New York: Routledge, 1989, 1992, 2nd ed. 2003.

External links

All links retrieved February 27, 2018.

General Philosophy Sources

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