Logical implication - Revision history

Jon Awbrey: update

2015-11-04T20:18:10Z

update

Jon Awbrey: unlink proposition for now

2012-05-16T17:00:18Z

unlink proposition for now

Jon Awbrey at 15:26, 16 May 2012

2012-05-16T15:26:23Z

Jon Awbrey: /* Discussion */

2012-05-15T11:24:42Z

Discussion

Jon Awbrey: /* Definition */

2012-05-15T11:24:01Z

Definition

Jon Awbrey at 02:14, 15 May 2012

2012-05-15T02:14:55Z

Jon Awbrey: /* Peer nodes */ update

2012-05-14T04:04:30Z

Peer nodes: update

Jon Awbrey: fix punctuation

2012-05-13T04:34:49Z

fix punctuation

Jon Awbrey: del extra comma

2012-05-13T04:32:07Z

del extra comma

Jon Awbrey: /* Discussion */ TeX

2012-05-13T03:49:55Z

Discussion: TeX

@@ Line 157: / Line 157: @@
 ===Focal nodes===
 * [[Inquiry Live]]
 * [[Logic Live]]
 ===Peer nodes===
-{{col-begin}}
+* [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://beta.wikiversity.org/wiki/Logical_implication Logical Implication @ Wikiversity Beta]
 ===Logical operators===
@@ Line 254: / Line 245: @@
 ===Related articles===
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
+{{col-begin}}
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
+* [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://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
+* [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://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
+{{col-break}}
-* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
+* [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://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
+* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
+{{col-end}}
 ==Document history==
@@ Line 274: / Line 264: @@
 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.
-{{col-begin}}
+* [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://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]]

@@ Line 48: / Line 48: @@
 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 [[proposition]]s, 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:
+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>

@@ Line 286: / Line 286: @@
 * [http://en.wikipedia.org/w/index.php?title=Logical_implication&oldid=77109738 Logical Implication], [http://en.wikipedia.org/ Wikipedia]
 {{col-end}}
 [[Category:Inquiry]]

@@ Line 48: / Line 48: @@
 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 [[proposition]]s, 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 is as follows:
+The main formal object under discussion is a logical operation on two logical values, typically the values of two [[proposition]]s, 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>

@@ Line 20: / Line 20: @@
 ==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 <math>\operatorname{false}</math> just in case 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.
-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> 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:
 <br>

@@ Line 169: / Line 169: @@
 {{col-break}}
 * [http://mywikibiz.com/Logical_implication Logical Implication @ MyWikiBiz]
-* [http://mathweb.org/wiki/Logical_implication Logical Implication @ MathWeb Wiki]
+* [http://intersci.ss.uci.edu/wiki/index.php/Logical_implication Logical Implication @ InterSci Wiki]
-* [http://netknowledge.org/wiki/Logical_implication Logical Implication @ NetKnowledge]
+* [http://wiki.oercommons.org/mediawiki/index.php/Logical_implication Logical Implication @ OER Commons]
 {{col-break}}
 * [http://p2pfoundation.net/Logical_Implication Logical Implication @ P2P Foundation]
-* [http://semanticweb.org/wiki/Logical_implication Logical Implication @ SemanticWeb]
+* [http://ref.subwiki.org/wiki/Logical_implication Logical Implication @ Subject Wikis]
 {{col-end}}

@@ Line 14: / Line 14: @@
 |}
-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.
+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.
@@ Line 70: / Line 70: @@
 <br>
-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.
+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%"

@@ Line 70: / Line 70: @@
 <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&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>\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%"
@@ Line 85: / Line 85: @@
 |}
-Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{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:
+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%"
@@ Line 99: / Line 99: @@
 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>\operatorname{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:
 {| align="center" cellspacing="10" width="90%"
@@ Line 119: / Line 119: @@
 The same object is achieved in the following way.  Begin with the binary operation:
-: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>
+{| align="center" cellspacing="10" width="90%"
-Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{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>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math>
+{| align="center" cellspacing="10" width="90%"
 This object is defined as follows:
-: <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond} (p, q) = \operatorname{T} \}.</math>
+{| align="center" cellspacing="10" width="90%"
-The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{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 \operatorname{Cond}^{-1}(T)\,.\!</math>
+{| align="center" cellspacing="10" width="90%"
-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.
+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==