Jon Awbrey: /* Propositional forms on two variables */ test

2015-11-09T03:04:06Z

Propositional forms on two variables: test

Jon Awbrey: problems with MathJax

2015-11-08T16:02:10Z

problems with MathJax

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☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|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 <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.

 

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math>
|- style="background:ghostwhite"
|  
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
|  
|  
|  
|- style="background:ghostwhite"
|  
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
|  
|  
|  
|-
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
f_{0101}
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\texttt{(~)}
\\[4pt]
\texttt{(} x \texttt{)(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{)} ~ y ~
\\[4pt]
\texttt{(} x \texttt{)}
\\[4pt]
~ x ~ \texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} y \texttt{)}
\\[4pt]
\texttt{(} x \texttt{,} ~ y \texttt{)}
\\[4pt]
\texttt{(} x ~ y \texttt{)}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}\!</math>
|-
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\\[4pt]
f_{15}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\\[4pt]
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~ y
\\[4pt]
\texttt{((} x \texttt{,} ~ y \texttt{))}
\\[4pt]
y
\\[4pt]
\texttt{(} x ~ \texttt{(} y \texttt{))}
\\[4pt]
x
\\[4pt]
\texttt{((} x \texttt{)} ~ y \texttt{)}
\\[4pt]
\texttt{((} x \texttt{)(} y \texttt{))}
\\[4pt]
\texttt{((~))}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}\!</math>
|}

 

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

* Language '''L3''' describes each boolean function ''f'' : '''B'''2 → '''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 operator]]s'', the first few of which are given in the following variant notations:

: <math>\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}</math>

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

* 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==

* [http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 中文 : 零阶逻辑]

==Syllabus==

===Focal nodes===

* [[Inquiry Live]]
* [[Logic Live]]

===Peer nodes===

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic @ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic @ MyWikiBiz]
* [http://ref.subwiki.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Subject Wikis]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity]
* [http://beta.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic @ Wikiversity Beta]

===Logical operators===

{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Logical negation|Negation]]
{{col-end}}

===Related topics===

{{col-begin}}
{{col-break}}
* [[Ampheck]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean-valued function]]
* [[Differential logic]]
{{col-break}}
* [[Logical graph]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Parametric operator]]
* [[Peirce's law]]
{{col-break}}
* [[Propositional calculus]]
* [[Sole sufficient operator]]
* [[Truth table]]
* [[Universe of discourse]]
* [[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.

* [http://intersci.ss.uci.edu/wiki/index.php/Zeroth_order_logic Zeroth Order Logic], [http://intersci.ss.uci.edu/ InterSciWiki]
* [http://mywikibiz.com/Zeroth_order_logic Zeroth Order Logic], [http://mywikibiz.com/ MyWikiBiz]
* [http://planetmath.org/ZerothOrderLogic Zeroth Order Logic], [http://planetmath.org/ PlanetMath]
* [http://wikinfo.org/w/index.php/Zeroth_order_logic Zeroth Order Logic], [http://wikinfo.org/w/ Wikinfo]
* [http://en.wikiversity.org/wiki/Zeroth_order_logic Zeroth Order Logic], [http://en.wikiversity.org/ Wikiversity]
* [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]

[[Category:Inquiry]]
[[Category:Open Educational Resource]]
[[Category:Peer Educational Resource]]
[[Category:Computer Science]]
[[Category:Formal Languages]]
[[Category:Formal Sciences]]
[[Category:Formal Systems]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Normative Sciences]]
[[Category:Semiotics]]

@@ Line 5: / Line 5: @@
 ==Propositional forms on two variables==
-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.
+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.
 <br>

Talk:Zeroth order logic - Revision history

Jon Awbrey: /* Propositional forms on two variables */ test

Jon Awbrey: problems with MathJax