Difference between revisions of "Logical implication"

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In [[mathematics]] and [[mathematical logic]], the concept of '''logical implication''' encompasses, depending on the context of use, a specific logical [[function (mathematics)|function]], a specific logical [[relation (mathematics)|relation]], and the various symbols that are used to denote this function or this relation.  In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.
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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
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A close approximation to the concept of logical implication or material conditional is expressed in ordinary language by means of the following conditional form:
+
The concept of '''logical implication''' encompasses a specific logical [[function (mathematics)|function]], a specific logical [[relation (mathematics)|relation]], and the various symbols that are used to denote this function and this relation.  In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.
  
−
:* If ''p'' then ''q''.
+
Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:
  
−
Here ''p'' and ''q'' are propositional variables that stand for any propositions in a given language.  In a statement of the form "if ''p'' then ''q''", the first term, ''p'', is called the ''[[antecedent (logic)|antecedent]]'' and the second term, ''q'', is called the ''[[consequent]]'', while the statement as a whole is called either the ''conditional'' or the ''consequence''.  Assuming that the conditional is true, then 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.
+
{| align="center" cellspacing="10" width="90%"
 +
|
 +
<math>\begin{array}{l}
 +
p ~\text{implies}~ q.
 +
\\[6pt]
 +
\text{if}~ p ~\text{then}~ q.
 +
\end{array}</math>
 +
|}
 +
 
 +
Here <math>p\!</math> and <math>q\!</math> are propositional variables that stand for any propositions in a given language.  In a statement of the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},</math> the first term, <math>p,\!</math> is called the ''antecedent'' and the second term, <math>q,\!</math> is called the ''consequent'', while the statement as a whole is called either the ''conditional'' or the ''consequence''.  Assuming that the conditional statement is true, then 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.
 +
 
 +
'''Note.'''  Many writers draw a technical distinction between the form <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> and the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.</math>  In this usage, writing <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q,\!</math> whereas writing <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}</math> merely forms a compound statement whose logical value is a function of the logical values of <math>p\!</math> and <math>q.\!</math>  This will be discussed in detail below.
  
 
==Definition==
 
==Definition==
  
−
The '''material conditional''' and '''logical implication''' are both associated with an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if the first operand is true and the second operand is false.
+
The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of ''false'' just in case the first operand is true and the second operand is false.
  
−
The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;→&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;⇒&nbsp;q''') is as follows:
+
In the interpretation where <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}</math>, the truth table associated with the statement <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},</math> symbolized as <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},</math> appears below:
  
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
<br>
−
|+ '''Logical Implication'''
+
 
−
|- style="background:paleturquoise"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
−
! style="width:15%" | p
+
|+ style="height:30px" | <math>\text{Logical Implication}\!</math>
−
! style="width:15%" | q
+
|- style="height:40px; background:#f0f0ff"
−
! style="width:15%" | p ⇒ q
+
| style="width:33%" | <math>p\!</math>
 +
| style="width:33%" | <math>q\!</math>
 +
| style="width:33%" | <math>p \Rightarrow q\!</math>
 
|-
 
|-
−
| F || F || T
+
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
|-
 
|-
−
| F || T || T
+
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 
|-
 
|-
−
| T || F || F
+
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
 
|-
 
|-
−
| T || T || T
+
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
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The usage of the terms '''''logical implication''''' and '''''material conditional''''' varies from field to field and even across different contexts of discussion.  One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.
 
The usage of the terms '''''logical implication''''' and '''''material conditional''''' varies from field to field and even across different contexts of discussion.  One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.
  
−
The main formal object under discussion is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' just in case the first operand is true and the second operand is false.  By way of a temporary name, the logical operation in question may be written as Cond&nbsp;(''p'', ''q''), where ''p'' and ''q'' are logical values.  The [[truth table]] associated with this operation is as follows:
+
The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of <math>\operatorname{false}</math> just in case the first operand is true and the second operand is false.  By way of a temporary name, the logical operation in question may be written as <math>\operatorname{Cond}(p, q),</math> where <math>p\!</math> and <math>q\!</math> are logical values.  The [[truth table]] associated with this operation appears below:
 +
 
 +
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:60%"
−
|+ '''Conditional Operation : B<sup>2</sup> → B'''
+
|+ style="height:30px" | <math>\text{Conditional Operation} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math>
−
|- style="background:paleturquoise"
+
|- style="height:40px; background:#f0f0ff"
−
! style="width:15%" | p
+
| style="width:33%" | <math>p\!</math>
−
! style="width:15%" | q
+
| style="width:33%" | <math>q\!</math>
−
! style="width:15%" | Cond (p, q)
+
| style="width:33%" | <math>\operatorname{Cond}(p, q)</math>
 
|-
 
|-
−
| F || F || T
+
| <math>\operatorname{F}</math> || <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
 
|-
 
|-
−
| F || T || T
+
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
 
|-
 
|-
−
| T || F || F
+
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math> || <math>\operatorname{F}</math>
 
|-
 
|-
−
| T || T || T
+
| <math>\operatorname{T}</math> || <math>\operatorname{T}</math> || <math>\operatorname{T}</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
Some logicians draw a firm distinction between the conditional connective (the syntactic sign "<math>\rightarrow</math>"), and the implication relation (the formal object denoted by the sign "<math>\Rightarrow</math>"). These logicians use the phrase ''if–then'' for the conditional connective and the term ''implies'' for the implication relation.  Some explain the difference by saying that the conditional is the ''contemplated'' relation while the implication is the ''asserted'' relation.  In most fields of mathematics, it is treated as a variation in the usage of the single sign "<math>\Rightarrow</math>", not requiring two separate signs.  Not all of those who use the sign "<math>\rightarrow</math>" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called ''[[syncategorematic sign]]'', that is, a sign with a purely syntactic function.  For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "<math>\rightarrow</math>" to denote the [[boolean function]] that is associated with the [[truth table]] of the material conditional.  These considerations result in the following scheme of notation.
+
Some logicians draw a firm distinction between the conditional connective, the symbol <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime},</math> and the implication relation, the object denoted by the symbol <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}.</math>  These logicians use the phrase ''if&ndash;then'' for the conditional connective and the term ''implies'' for the implication relation.  Some explain the difference by saying that the conditional is the ''contemplated'' relation while the implication is the ''asserted'' relation.  In most fields of mathematics, it is treated as a variation in the usage of the single sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime},</math> not requiring two separate signs.  Not all of those who use the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called ''syncategorematic sign'', that is, a sign with a purely syntactic function.  For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> to denote the [[boolean function]] that is associated with the [[truth table]] of the material conditional.  These considerations result in the following scheme of notation.
  
−
 
+
{| align="center" cellspacing="10" width="90%"
−
: <math>\begin{matrix}
+
|
−
p \rightarrow q & \quad & \quad & p \Rightarrow q \\
+
<math>\begin{matrix}
−
\mbox{if}\ p \ \mbox{then}\ q & \quad & \quad & p \ \mbox{implies}\ q
+
p \rightarrow q
 +
& \quad & \quad &
 +
p \Rightarrow q
 +
\\
 +
\text{if}~ p ~\text{then}~ q
 +
& \quad & \quad &
 +
p ~\text{implies}~ q
 
\end{matrix}</math>
 
\end{matrix}</math>
 +
|}
  
 +
Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' consisting of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows:
  
−
Let <math>\mathbb{B} = \{\mathbf{F},\ \mathbf{T}\}</math> be the ''[[boolean domain]]'' of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}</math> that is defined as follows:
+
{| align="center" cellspacing="10" width="90%"
−
 
+
| <math>L = \{ (p, q, r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = r \}.</math>
−
: <math>L = \{(p,\ q,\ r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q)\ = r \}\,.</math>
+
|}
  
 
Regarded as a set, this triadic relation is the same thing as the binary operation:
 
Regarded as a set, this triadic relation is the same thing as the binary operation:
  
−
: <math>Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>
 +
|}
  
−
The relationship between <math>Cond\!</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.
+
The relationship between <math>\operatorname{Cond}</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.
  
−
The conditional sign "<math>\rightarrow</math>" denotes the same formal object as the function name "<math>Cond\mbox{ }</math>", the only difference being that the first is written infix while the second is written prefix.  Thus we have the following equation:
+
The conditional sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the function name <math>{}^{\backprime\backprime} \operatorname{Cond} {}^{\prime\prime},</math> the only difference being that the first is written infix while the second is written prefix.  Thus we have the following equation:
  
−
: <math>(p \rightarrow q) = Cond (p,\ q)\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>(p \rightarrow q) = \operatorname{Cond}(p, q).</math>
 +
|}
  
−
Consider once again the triadic relation <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}</math> that is defined in the following equivalent fashion:
+
Consider once again the triadic relation <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined in the following equivalent fashion:
  
−
: <math>L = \{(p,\ q,\ Cond (p,\ q))\ :\ (p,\ q) \in \mathbb{B} \times \mathbb{B} \}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>L = \{ (p, q, \operatorname{Cond}(p, q) ) : (p, q) \in \mathbb{B} \times \mathbb{B} \}.</math>
 +
|}
  
−
Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{..T} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''[[image (mathematics)|fiber]]'' of <math>L\!</math> with <math>T\!</math> in the third place.  This object is defined as follows:
+
Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place.  This object is defined as follows:
  
−
: <math>L_{..T} = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ (p,\ q,\ T) \in L \}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math>
 +
|}
  
 
The same object is achieved in the following way.  Begin with the binary operation:
 
The same object is achieved in the following way.  Begin with the binary operation:
  
−
: <math>Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>
 +
|}
  
−
Form the binary relation that is called the ''fiber'' of <math>Cond\!</math> at <math>T\!</math>, notated as follows:
+
Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T},</math> notated as follows:
  
−
: <math>Cond^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math>
 +
|}
  
 
This object is defined as follows:
 
This object is defined as follows:
  
−
: <math>Cond^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q) = T \}\,.</math>
+
{| align="center" cellspacing="10" width="90%"
 +
| <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = \operatorname{T} \}.</math>
 +
|}
  
−
The implication sign "<math>\Rightarrow</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }</math>" and "<math>Cond^{-1}(T)\mbox{ }</math>", the only differences being purely syntactic.  Thus we have the following logical equivalence:
+
The implication sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the relation names <math>{}^{\backprime\backprime} L_{\underline{~} \, \underline{~} \, \operatorname{T}} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \operatorname{Cond}^{-1}(T) {}^{\prime\prime},</math> the only differences being purely syntactic.  Thus we have the following logical equivalence:
  
−
: <math>(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in Cond^{-1}(T)\,.</math>
+
{| align="center" cellspacing="10" width="90%"
−
 
+
| <math>(p \Rightarrow q) \iff (p, q) \in L_{\underline{~} \, \underline{~} \, \operatorname{T}} \iff (p, q) \in \operatorname{Cond}^{-1}(T).</math>
−
This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow</math>" and "<math>\Rightarrow</math>" in this discussion.  It needs to be remembered, though, that not all writers observe this distinction in every context.  Especially in mathematics, where the single arrow sign "<math>\rightarrow</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow</math>" being used for both concepts.
+
|}
−
 
 
−
==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'' <br>
 
−
''A''—Kermit is a frog.<br>
 
−
''B''—Muppets are animals.
 
−
 
 
−
Using the horseshoe "⊃" symbol for implication is falling out favor due to its conflict with the superset symbol <math>\supset</math> used by the [[Algebra of sets]].  A set interpretation of "<math> A \to B</math>" is "{''x''| ''A''(''x'') is true} <math>\subseteq</math> {''x''| ''B''(''x'') is true}".
 
−
 
 
−
==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.
+
This completes the derivation of the mathematical objects that are denoted by the signs <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> in this discussion.  It needs to be remembered, though, that not all writers observe this distinction in every context.  Especially in mathematics, where the single arrow sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> is reserved for function notation, it is common to see the double arrow sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> being used for both concepts.
−
 
 
−
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 → 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.)
 
  
 
==References==
 
==References==
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* [[W.V. Quine|Quine, W.V.]] (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
 
* [[W.V. Quine|Quine, W.V.]] (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
  
−
==See also==
+
==Syllabus==
 +
 
 +
===Focal nodes===
 +
 
 +
* [[Inquiry Live]]
 +
* [[Logic Live]]
 +
 
 +
===Peer nodes===
 +
 
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication @ InterSciWiki]
 +
* [http://mywikibiz.com/Logical_implication Logical Implication @ MyWikiBiz]
 +
* [http://ref.subwiki.org/wiki/Logical_implication Logical Implication @ Subject Wikis]
 +
* [http://en.wikiversity.org/wiki/Logical_implication Logical Implication @ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Logical_implication Logical Implication @ Wikiversity Beta]
  
 
===Logical operators===
 
===Logical operators===
 +
 
{{col-begin}}
 
{{col-begin}}
 
{{col-break}}
 
{{col-break}}
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* [[Logical implication]]
 
* [[Logical implication]]
 
* [[Logical NAND]]
 
* [[Logical NAND]]
−
* [[Logical NOR]]
+
* [[Logical NNOR]]
−
* [[Negation]]
+
* [[Logical negation|Negation]]
 
{{col-end}}
 
{{col-end}}
  
 
===Related topics===
 
===Related topics===
 +
 
{{col-begin}}
 
{{col-begin}}
 
{{col-break}}
 
{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
−
* [[Boolean algebra]]
 
 
* [[Boolean domain]]
 
* [[Boolean domain]]
 
* [[Boolean function]]
 
* [[Boolean function]]
 +
* [[Boolean-valued function]]
 +
* [[Differential logic]]
 
{{col-break}}
 
{{col-break}}
−
* [[Boolean logic]]
 
−
* [[Laws of Form]]
 
−
* [[Logic gate]]
 
 
* [[Logical graph]]
 
* [[Logical graph]]
 +
* [[Minimal negation operator]]
 +
* [[Multigrade operator]]
 +
* [[Parametric operator]]
 +
* [[Peirce's law]]
 
{{col-break}}
 
{{col-break}}
−
* [[Peirce's law]]
+
* [[Propositional calculus]]
−
* [[Propositional logic]]
 
 
* [[Sole sufficient operator]]
 
* [[Sole sufficient operator]]
 +
* [[Truth table]]
 +
* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
 
{{col-end}}
 
{{col-end}}
  
−
{{aficionados}}<sharethis />
+
===Relational concepts===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [[Continuous predicate]]
 +
* [[Hypostatic abstraction]]
 +
* [[Logic of relatives]]
 +
* [[Logical matrix]]
 +
{{col-break}}
 +
* [[Relation (mathematics)|Relation]]
 +
* [[Relation composition]]
 +
* [[Relation construction]]
 +
* [[Relation reduction]]
 +
{{col-break}}
 +
* [[Relation theory]]
 +
* [[Relative term]]
 +
* [[Sign relation]]
 +
* [[Triadic relation]]
 +
{{col-end}}
 +
 
 +
===Information, Inquiry===
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* [[Inquiry]]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
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==Document history==
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Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
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* [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [http://mywikibiz.com/Logical_implication Logical Implication], [http://mywikibiz.com/ MyWikiBiz]
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* [http://wikinfo.org/w/index.php/Logical_implication Logical Implication], [http://wikinfo.org/w/ Wikinfo]
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* [http://beta.wikiversity.org/wiki/Logical_implication Logical Implication], [http://beta.wikiversity.org/ Wikiversity Beta]
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* [http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738 Logical Implication], [http://en.wikipedia.org/ Wikipedia]
  
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Latest revision as of 20:18, 4 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

The concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

\(\begin{array}{l} p ~\text{implies}~ q. \\[6pt] \text{if}~ p ~\text{then}~ q. \end{array}\)

Here \(p\!\) and \(q\!\) are propositional variables that stand for any propositions in a given language. In a statement of the form \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},\) the first term, \(p,\!\) is called the antecedent and the second term, \(q,\!\) is called the consequent, while the statement as a whole is called either the conditional or the consequence. Assuming that the conditional statement is true, then 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.

Note. Many writers draw a technical distinction between the form \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}\) and the form \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.\) In this usage, writing \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}\) asserts the existence of a certain relation between the logical value of \(p\!\) and the logical value of \(q,\!\) whereas writing \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}\) merely forms a compound statement whose logical value is a function of the logical values of \(p\!\) and \(q.\!\) This will be discussed in detail below.

Definition

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.

In the interpretation where \(0 = \operatorname{false}\) and \(1 = \operatorname{true}\), the truth table associated with the statement \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},\) symbolized as \({}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},\) appears below:


\(\text{Logical Implication}\!\)
\(p\!\) \(q\!\) \(p \Rightarrow q\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(0\!\) \(0\!\)
\(1\!\) \(1\!\) \(1\!\)


Discussion

The usage of the terms logical implication and material conditional varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.

The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of \(\operatorname{false}\) just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as \(\operatorname{Cond}(p, q),\) where \(p\!\) and \(q\!\) are logical values. The truth table associated with this operation appears below:


\(\text{Conditional Operation} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!\)
\(p\!\) \(q\!\) \(\operatorname{Cond}(p, q)\)
\(\operatorname{F}\) \(\operatorname{F}\) \(\operatorname{T}\)
\(\operatorname{F}\) \(\operatorname{T}\) \(\operatorname{T}\)
\(\operatorname{T}\) \(\operatorname{F}\) \(\operatorname{F}\)
\(\operatorname{T}\) \(\operatorname{T}\) \(\operatorname{T}\)


Some logicians draw a firm distinction between the conditional connective, the symbol \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime},\) and the implication relation, the object denoted by the symbol \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}.\) These logicians use the phrase if–then for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime},\) not requiring two separate signs. Not all of those who use the sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) to denote the boolean function that is associated with the truth table of the material conditional. These considerations result in the following scheme of notation.

\(\begin{matrix} p \rightarrow q & \quad & \quad & p \Rightarrow q \\ \text{if}~ p ~\text{then}~ q & \quad & \quad & p ~\text{implies}~ q \end{matrix}\)

Let \(\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}\) be the boolean domain consisting of two logical values. The truth table shows the ordered triples of a triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!\) that is defined as follows:

\(L = \{ (p, q, r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = r \}.\)

Regarded as a set, this triadic relation is the same thing as the binary operation:

\(\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\)

The relationship between \(\operatorname{Cond}\) and \(L\!\) exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The conditional sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) denotes the same formal object as the function name \({}^{\backprime\backprime} \operatorname{Cond} {}^{\prime\prime},\) the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation:

\((p \rightarrow q) = \operatorname{Cond}(p, q).\)

Consider once again the triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!\) that is defined in the following equivalent fashion:

\(L = \{ (p, q, \operatorname{Cond}(p, q) ) : (p, q) \in \mathbb{B} \times \mathbb{B} \}.\)

Associated with the triadic relation \(L\!\) is a binary relation \(L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}\) that is called the fiber of \(L\!\) with \(\operatorname{T}\) in the third place. This object is defined as follows:

\(L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.\)

The same object is achieved in the following way. Begin with the binary operation:

\(\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\)

Form the binary relation that is called the fiber of \(\operatorname{Cond}\) at \(\operatorname{T},\) notated as follows:

\(\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.\)

This object is defined as follows:

\(\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = \operatorname{T} \}.\)

The implication sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) denotes the same formal object as the relation names \({}^{\backprime\backprime} L_{\underline{~} \, \underline{~} \, \operatorname{T}} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \operatorname{Cond}^{-1}(T) {}^{\prime\prime},\) the only differences being purely syntactic. Thus we have the following logical equivalence:

\((p \Rightarrow q) \iff (p, q) \in L_{\underline{~} \, \underline{~} \, \operatorname{T}} \iff (p, q) \in \operatorname{Cond}^{-1}(T).\)

This completes the derivation of the mathematical objects that are denoted by the signs \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) and \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}\) in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) is reserved for function notation, it is common to see the double arrow sign \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}\) being used for both concepts.

References

  • Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
  • Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
  • Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.
  • Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.

Syllabus

Focal nodes

Peer nodes

Logical operators

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Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

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