Difference between revisions of "Logical implication"

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The '''logical implication''' and the '''material conditional''' 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.
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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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The [[truth table]] associated with the material conditional '''if p then q''' (symbolized as '''p&nbsp;&rarr;&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;&rArr;&nbsp;q''') is as follows:
+
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.
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+
Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:
−
|+ '''Logical Implication'''
+
 
−
|- style="background:paleturquoise"
+
{| align="center" cellspacing="10" width="90%"
−
! style="width:15%" | p
+
|
−
! style="width:15%" | q
+
<math>\begin{array}{l}
−
! style="width:15%" | p &rArr; q
+
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==
 +
 
 +
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 <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:
 +
 
 +
<br>
 +
 
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
 +
|+ style="height:30px" | <math>\text{Logical Implication}\!</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| style="width:33%" | <math>p\!</math>
 +
| style="width:33%" | <math>q\!</math>
 +
| style="width:33%" | <math>p \Rightarrow q\!</math>
 +
|-
 +
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 +
|-
 +
| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
 +
|-
 +
| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
 +
|-
 +
| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
 +
|}
 +
 
 +
<br>
 +
 
 +
==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 <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="text-align:center; width:60%"
 +
|+ style="height:30px" | <math>\text{Conditional Operation} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math>
 +
|- style="height:40px; background:#f0f0ff"
 +
| style="width:33%" | <math>p\!</math>
 +
| style="width:33%" | <math>q\!</math>
 +
| 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>
  
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==See also==
+
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
 +
\\
 +
\text{if}~ p ~\text{then}~ q
 +
& \quad & \quad &
 +
p ~\text{implies}~ q
 +
\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:
 +
 
 +
{| 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>
 +
|}
 +
 
 +
Regarded as a set, this triadic relation is the same thing as the binary operation:
 +
 
 +
{| align="center" cellspacing="10" width="90%"
 +
| <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>
 +
|}
 +
 
 +
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>{}^{\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:
 +
 
 +
{| 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:
 +
 
 +
{| 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_{\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:
 +
 
 +
{| 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:
 +
 
 +
{| 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>\operatorname{Cond}</math> at <math>\operatorname{T},</math> notated as follows:
 +
 
 +
{| 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:
 +
 
 +
{| 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>{}^{\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:
 +
 
 +
{| 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>{}^{\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.
 +
 
 +
==References==
 +
 
 +
* [[Frank Markham Brown|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'',  [http://plato.stanford.edu/entries/conditionals/ Eprint].
 +
 
 +
* [[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.
 +
 
 +
==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===
−
{|
+
 
−
| valign=top |
+
{{col-begin}}
 +
{{col-break}}
 
* [[Exclusive disjunction]]
 
* [[Exclusive disjunction]]
 
* [[Logical conjunction]]
 
* [[Logical conjunction]]
 
* [[Logical disjunction]]
 
* [[Logical disjunction]]
 
* [[Logical equality]]
 
* [[Logical equality]]
−
| valign=top |
+
{{col-break}}
 
* [[Logical implication]]
 
* [[Logical implication]]
 
* [[Logical NAND]]
 
* [[Logical NAND]]
 
* [[Logical NNOR]]
 
* [[Logical NNOR]]
 
* [[Logical negation|Negation]]
 
* [[Logical negation|Negation]]
−
|}
+
{{col-end}}
 +
 
 
===Related topics===
 
===Related topics===
−
{|
+
 
−
| valign=top |
+
{{col-begin}}
 +
{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
−
* [[Boolean algebra]]
 
 
* [[Boolean domain]]
 
* [[Boolean domain]]
 
* [[Boolean function]]
 
* [[Boolean function]]
−
| valign=top |
+
* [[Boolean-valued function]]
−
* [[Boolean logic]]
+
* [[Differential logic]]
−
* [[Laws of Form]]
+
{{col-break}}
−
* [[Logic gate]]
 
 
* [[Logical graph]]
 
* [[Logical graph]]
−
| valign=top |
+
* [[Minimal negation operator]]
 +
* [[Multigrade operator]]
 +
* [[Parametric operator]]
 
* [[Peirce's law]]
 
* [[Peirce's law]]
 +
{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
 
* [[Sole sufficient operator]]
 
* [[Sole sufficient operator]]
 +
* [[Truth table]]
 +
* [[Universe of discourse]]
 
* [[Zeroth order logic]]
 
* [[Zeroth order logic]]
−
|}
+
{{col-end}}
 +
 
 +
===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===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [[Inquiry]]
 +
* [[Dynamics of inquiry]]
 +
{{col-break}}
 +
* [[Semeiotic]]
 +
* [[Logic of information]]
 +
{{col-break}}
 +
* [[Descriptive science]]
 +
* [[Normative science]]
 +
{{col-break}}
 +
* [[Pragmatic maxim]]
 +
* [[Truth theory]]
 +
{{col-end}}
 +
 
 +
===Related articles===
 +
 
 +
{{col-begin}}
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 +
{{col-break}}
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
 +
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 +
{{col-end}}
 +
 
 +
==Document history==
 +
 
 +
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.
  
−
{{aficionados}}<sharethis />
+
* [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication], [http://intersci.ss.uci.edu/ InterSciWiki]
 +
* [http://mywikibiz.com/Logical_implication Logical Implication], [http://mywikibiz.com/ MyWikiBiz]
 +
* [http://wikinfo.org/w/index.php/Logical_implication Logical Implication], [http://wikinfo.org/w/ Wikinfo]
 +
* [http://en.wikiversity.org/wiki/Logical_implication Logical Implication], [http://en.wikiversity.org/ Wikiversity]
 +
* [http://beta.wikiversity.org/wiki/Logical_implication Logical Implication], [http://beta.wikiversity.org/ Wikiversity Beta]
 +
* [http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738 Logical Implication], [http://en.wikipedia.org/ Wikipedia]
  
 +
[[Category:Inquiry]]
 +
[[Category:Open Educational Resource]]
 +
[[Category:Peer Educational Resource]]
 +
[[Category:Charles Sanders Peirce]]
 
[[Category:Computer Science]]
 
[[Category:Computer Science]]
 
[[Category:Formal Languages]]
 
[[Category:Formal Languages]]

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.

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