Difference between revisions of "Zeroth order logic"

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'''''Zeroth order logic''''' is a term in popular use among practitioners for the subject matter otherwise known as [[boolean function]]s, [[monadic logic|monadic predicate logic]], [[propositional calculus]], or sentential calculus.  One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent [[isomorphism]]s.
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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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By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type ''X''&nbsp;&times;&nbsp;''Y''&nbsp;&rarr;&nbsp;'''B''' and abstract type '''B'''&nbsp;&times;&nbsp;'''B'''&nbsp;&rarr;&nbsp;'''B''' in a number of different languages for zeroth order logic.
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'''''Zeroth order logic''''' is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, [[boolean functions]], logical connectives, monadic predicate calculus, [[propositional calculus]], and sentential logic.&nbsp; The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.
  
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
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==Propositional forms on two variables==
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By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> in a number of different languages for zeroth order logic.
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<br>
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{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:90%"
 
|+ '''Table 1.  Propositional Forms on Two Variables'''
 
|+ '''Table 1.  Propositional Forms on Two Variables'''
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! style="width:15%" | L<sub>6</sub>
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| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
 
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
 
|}
 
|}
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<br>
 
<br>
  
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* Language '''L<sub>6</sub>''' expresses the sixteen functions in one of several notations that are commonly used in formal logic.
 
* Language '''L<sub>6</sub>''' expresses the sixteen functions in one of several notations that are commonly used in formal logic.
  
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==See also==
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==Translations==
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* [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 &#20013;&#25991; : &#38646;&#38454;&#36923;&#36753;]
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==Syllabus==
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===Focal nodes===
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* [[Inquiry Live]]
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* [[Logic Live]]
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===Peer nodes===
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* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
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* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
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* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
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* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]
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===Logical operators===
 
===Logical operators===
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{|
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| valign=top |
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{{col-begin}}
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{{col-break}}
 
* [[Exclusive disjunction]]
 
* [[Exclusive disjunction]]
 
* [[Logical conjunction]]
 
* [[Logical conjunction]]
 
* [[Logical disjunction]]
 
* [[Logical disjunction]]
 
* [[Logical equality]]
 
* [[Logical equality]]
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| valign=top |
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{{col-break}}
 
* [[Logical implication]]
 
* [[Logical implication]]
 
* [[Logical NAND]]
 
* [[Logical NAND]]
 
* [[Logical NNOR]]
 
* [[Logical NNOR]]
 
* [[Logical negation|Negation]]
 
* [[Logical negation|Negation]]
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|}
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{{col-end}}
  
 
===Related topics===
 
===Related topics===
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{|
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| valign=top |
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{{col-begin}}
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{{col-break}}
 
* [[Ampheck]]
 
* [[Ampheck]]
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* [[Boolean algebra]]
 
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* [[Boolean algebra topics]]
 
 
* [[Boolean domain]]
 
* [[Boolean domain]]
 
* [[Boolean function]]
 
* [[Boolean function]]
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* [[Boolean logic]]
 
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| valign=top |
 
 
* [[Boolean-valued function]]
 
* [[Boolean-valued function]]
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* [[Entitative graph]]
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* [[Differential logic]]
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* [[Existential graph]]
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{{col-break}}
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* [[First order logic]]
 
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* [[Indicator function]]
 
 
* [[Logical graph]]
 
* [[Logical graph]]
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| valign=top |
 
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* [[Logical value]]
 
 
* [[Minimal negation operator]]
 
* [[Minimal negation operator]]
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* [[Monadic predicate calculus]]
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* [[Multigrade operator]]
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* [[Operation (mathematics)|Operation]]
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* [[Parametric operator]]
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* [[Peirce's law]]
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{{col-break}}
 
* [[Propositional calculus]]
 
* [[Propositional calculus]]
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* [[Sole sufficient operator]]
 
* [[Truth table]]
 
* [[Truth table]]
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|}
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* [[Universe of discourse]]
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* [[Zeroth order logic]]
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{{col-end}}
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===Relational concepts===
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{{col-begin}}
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{{col-break}}
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* [[Continuous predicate]]
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* [[Hypostatic abstraction]]
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* [[Logic of relatives]]
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* [[Logical matrix]]
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{{col-break}}
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* [[Relation (mathematics)|Relation]]
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* [[Relation composition]]
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* [[Relation construction]]
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* [[Relation reduction]]
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{{col-break}}
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* [[Relation theory]]
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* [[Relative term]]
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* [[Sign relation]]
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* [[Triadic relation]]
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{{col-end}}
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===Information, Inquiry===
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{{col-begin}}
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{{col-break}}
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* [[Inquiry]]
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* [[Dynamics of inquiry]]
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{{col-break}}
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* [[Semeiotic]]
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* [[Logic of information]]
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{{col-break}}
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* [[Descriptive science]]
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* [[Normative science]]
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{{col-break}}
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* [[Pragmatic maxim]]
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* [[Truth theory]]
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{{col-end}}
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===Related articles===
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{{col-begin}}
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{{col-break}}
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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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{{col-break}}
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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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{{col-break}}
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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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{{col-end}}
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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/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
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* [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
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* [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo]
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* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity]
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* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://beta.wikiversity.org/ Wikiversity Beta]
 +
* [http://en.wikipedia.org/w/index.php?title=Zeroth-order_logic&oldid=77109225 Zeroth Order Logic], [http://en.wikipedia.org/ Wikipedia]
 +
* [http://web.archive.org/web/20050323065233/http://www.altheim.com/cs/zol.html Zeroth Order Logic], [http://web.archive.org/web/20070305032442/http://www.altheim.com/cs/ Altheim.com]
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[[Category:Inquiry]]
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[[Category:Open Educational Resource]]
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[[Category:Peer Educational Resource]]
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[[Category:Computer Science]]
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[[Category:Formal Languages]]
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[[Category:Formal Sciences]]
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[[Category:Formal Systems]]
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[[Category:Logic]]
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[[Category:Mathematics]]
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[[Category:Normative Sciences]]
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[[Category:Semiotics]]

Latest revision as of 16:04, 8 November 2015

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

Zeroth order logic is an informal term that is sometimes used to indicate the common principles underlying the algebra of sets, boolean algebra, boolean functions, logical connectives, monadic predicate calculus, propositional calculus, and sentential logic.  The term serves to mark a level of abstraction in which the more inessential differences among these subjects can be subsumed under the appropriate isomorphisms.

Propositional forms on two variables

By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type \(X \times Y \to \mathbb{B}\) and abstract type \(\mathbb{B} \times \mathbb{B} \to \mathbb{B}\) in a number of different languages for zeroth order logic.


Table 1. Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
  x : 1 1 0 0      
  y : 1 0 1 0      
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1


These six languages for the sixteen boolean functions are conveniently described in the following order:

  • Language L3 describes each boolean function f : B2 → B by means of the sequence of four boolean values (f(1,1), f(1,0), f(0,1), f(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values F and T instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a truth table.
  • Language L2 lists the sixteen functions in the form fi, where the index i is a bit string formed from the sequence of boolean values in L3.
  • Language L1 notates the boolean functions fi with an index i that is the decimal equivalent of the binary numeral index in L2.
  • Language L4 expresses the sixteen functions in terms of logical conjunction, indicated by concatenating function names or proposition expressions in the manner of products, plus the family of minimal negation operators, the first few of which are given in the following variant notations:

\[\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}\]

It may also be noted that \((x, y)\!\) is the same function as \(x + y\!\) and \(x \ne y\), and that the inclusive disjunctions indicated for \((x, y)\!\) and for \((x, y, z)\!\) may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function \((x, y, z)\!\) is not the same thing as the function \(x + y + z\!\).

  • Language L5 lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents.
  • Language L6 expresses the sixteen functions in one of several notations that are commonly used in formal logic.

Translations

Syllabus

Focal nodes

Peer nodes

Logical operators

Template:Col-breakTemplate:Col-breakTemplate:Col-end

Related topics

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Relational concepts

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Information, Inquiry

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate:Col-end

Related articles

Template:Col-breakTemplate:Col-breakTemplate:Col-breakTemplate: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.

Retrieved from "https://mywikibiz.com/index.php?title=Zeroth_order_logic&oldid=466849"