# OpenMath Content Dictionary: logic1

Canonical URL:
http://www.openmath.org/cd/logic1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
logic1.ocd
CD as XML Encoded OpenMath:
logic1.omcd
Defines:
and, equivalent, false, implies, nand, nor, not, or, true, xnor, xor
Date:
2004-03-30
Version:
4
Review Date:
2006-03-30
Status:
official


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document is stated prominently in the derived work.  Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
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Dictionary Group whose name is, for example, math' containing
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not name a derived Content Dictionary mathN' where N is an integer.
However you are free to name it private_mathN' or some such.  This
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  Author: OpenMath Consortium
SourceURL: https://github.com/OpenMath/CDs


This CD holds the basic logic functions.

## equivalent

Role:
application
Description:

This symbol is used to show that two boolean expressions are logically equivalent, that is have the same boolean value for any inputs.

Commented Mathematical property (CMP):
The condition (A is equivalent to B) is equivalent to the condition that (A implies B and B implies A)
Formal Mathematical property (FMP):
$A\equiv B\equiv \left(\left(A⇒B\right)\wedge \left(B⇒A\right)\right)$
Signatures:
sts

 [Next: not] [Last: false] [Top]

## not

Role:
application
Description:

This symbol represents the logical not function which takes one boolean argument, and returns the opposite boolean value.

Commented Mathematical property (CMP):
for all x | not(not(x))=x
Formal Mathematical property (FMP):
$\forall x.¬¬x=x$
Signatures:
sts

 [Next: and] [Previous: equivalent] [Top]

## and

Role:
application
Description:

This symbol represents the logical and function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if all arguments are true or false otherwise.

Commented Mathematical property (CMP):
for all x | x and not(x) = false
Formal Mathematical property (FMP):
$\forall x.x\wedge ¬x=F$
Signatures:
sts

 [Next: nand] [Previous: not] [Top]

## nand

Role:
application
Description:

This symbol represents the logical nand function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if all arguments are true or true otherwise.

Commented Mathematical property (CMP):
for all x | x nand not(x) = true
Formal Mathematical property (FMP):
$\forall x.\mathrm{nand}\left(x,¬x\right)=T$
Formal Mathematical property (FMP):
$\forall x,y.\mathrm{nand}\left(x,y\right)=¬\left(x\wedge y\right)$
Signatures:
sts

 [Next: xor] [Previous: and] [Top]

## xor

Role:
application
Description:

This symbol represents the logical xor function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if there are an odd number of true arguments or false otherwise.

Commented Mathematical property (CMP):
for all x | x xor x = false
Formal Mathematical property (FMP):
$\forall x.xxorx=F$
Commented Mathematical property (CMP):
for all x | x xor not(x) = true
Formal Mathematical property (FMP):
$\forall x.xxor¬x=T$
Signatures:
sts

 [Next: xnor] [Previous: nand] [Top]

## xnor

Role:
application
Description:

This symbol represents the logical xnor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if there are an odd number of true arguments or true otherwise.

Commented Mathematical property (CMP):
for all x | x xnor x = true
Formal Mathematical property (FMP):
$\forall x.\mathrm{xnor}\left(x,x\right)=T$
Commented Mathematical property (CMP):
for all x | x xnor not(x) = false
Formal Mathematical property (FMP):
$\forall x.\mathrm{xnor}\left(x,¬x\right)=F$
Formal Mathematical property (FMP):
$\forall x,y.\mathrm{xnor}\left(x,y\right)=¬\left(xxory\right)$
Signatures:
sts

 [Next: or] [Previous: xor] [Top]

## or

Role:
application
Description:

This symbol represents the logical or function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if any of the arguments are true or false otherwise.

Commented Mathematical property (CMP):
for all x | x or not(x) = true
Formal Mathematical property (FMP):
$\forall x.x\vee ¬x=T$
Commented Mathematical property (CMP):
for all a,b | not(a and b)= (not(a) or not(b))
Formal Mathematical property (FMP):
$\forall a,b.¬\left(a\wedge b\right)=¬a\vee ¬b$
Signatures:
sts

 [Next: nor] [Previous: xnor] [Top]

## nor

Role:
application
Description:

This symbol represents the logical nor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if any of the arguments are true or true otherwise.

Commented Mathematical property (CMP):
for all x | x nor not(x) = false
Formal Mathematical property (FMP):
$\forall x.\mathrm{nor}\left(x,¬x\right)=F$
Commented Mathematical property (CMP):
for all a,b | a and b = (not(a) nor not(b))
Formal Mathematical property (FMP):
$\forall a,b.a\wedge b=\mathrm{nor}\left(¬a,¬b\right)$
Formal Mathematical property (FMP):
$\forall x,y.\mathrm{nor}\left(x,y\right)=¬\left(x\vee y\right)$
Signatures:
sts

 [Next: implies] [Previous: or] [Top]

## implies

Role:
application
Description:

This symbol represents the logical implies function which takes two boolean expressions as arguments. It evaluates to false if the first argument is true and the second argument is false, otherwise it evaluates to true.

Commented Mathematical property (CMP):
for all x | false implies x
Formal Mathematical property (FMP):
$\forall x.F⇒x$
Signatures:
sts

 [Next: true] [Previous: nor] [Top]

## true

Role:
constant
Description:

This symbol represents the boolean value true.

Commented Mathematical property (CMP):
not true = false
Formal Mathematical property (FMP):
$¬T=F$
Signatures:
sts

 [Next: false] [Previous: implies] [Top]

## false

Role:
constant
Description:

This symbol represents the boolean value false.

Commented Mathematical property (CMP):
not false = true
Formal Mathematical property (FMP):
$¬F=T$
Signatures:
sts

 [First: equivalent] [Previous: true] [Top]