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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 15:14, 30 November 2015
, 15:14, 30 November 2015modify "~~~", "~~~~", "~~~~~" strings for compatibility with MathJax sites
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| The first kind of propositional expression is a parenthesized sequence of propositional expressions, written as <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\!</math> and read to say that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> is false, in other words, that their [[minimal negation]] is true. A clause of this form maps into a PARC structure called a ''lobe'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.
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| The second kind of propositional expression is a concatenated sequence of propositional expressions, written as <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k\!</math> and read to say that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> are true, in other words, that their [[logical conjunction]] is true. A clause of this form maps into a PARC structure called a ''node'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k\!</math> as shown below.
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<br>
<br>
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
|+ <math>\text{Table 1.}~~\text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
|- style="height:40px; background:ghostwhite"
|- style="background:#f0f0ff"
| <math>\text{Graph}\!</math>
| <math>\text{Expression}~\!</math>
| <math>\text{Expression}~\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
| <math>\text{Other Notations}\!</math>
|-
|-
|
| height="100px" | [[Image:Rooted Node.jpg|20px]]
| <math>\text{True}\!</math>
| <math>~\!</math>
| <math>\mathrm{true}\!</math>
| <math>1\!</math>
| <math>1\!</math>
|-
|-
| height="100px" | [[Image:Rooted Edge.jpg|20px]]
| <math>\texttt{(~)}\!</math>
| <math>\texttt{(~)}\!</math>
| <math>\text{False}\!</math>
| <math>\mathrm{false}\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| <math>x\!</math>
| height="100px" | [[Image:Cactus A Big.jpg|20px]]
| <math>x\!</math>
| <math>a\!</math>
| <math>x\!</math>
| <math>a\!</math>
| <math>a\!</math>
|-
|-
| <math>\texttt{(} x \texttt{)}\!</math>
| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]
| <math>\text{Not}~ x\!</math>
| <math>\texttt{(} a \texttt{)}\!</math>
|
| <math>\mathrm{not}~ a\!</math>
<math>\begin{matrix}
| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime~\!</math>
\\
\tilde{x}
\\
|-
|-
| <math>x~y~z\!</math>
| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math>
| <math>a ~ b ~ c\!</math>
| <math>x \land y \land z\!</math>
| <math>a ~\mathrm{and}~ b ~\mathrm{and}~ c\!</math>
| <math>a \land b \land c\!</math>
|-
|-
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math>
| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math>
| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\!</math>
| <math>x \lor y \lor z\!</math>
| <math>a ~\mathrm{or}~ b ~\mathrm{or}~ c\!</math>
| <math>a \lor b \lor c\!</math>
|-
|-
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math>
| height="120px" | [[Image:Cactus (A(B)) Big.jpg|60px]]
| <math>\texttt{(} a \texttt{(} b \texttt{))}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a ~\mathrm{implies}~ b
\\
\\[6pt]
\mathrm{If}~ x ~\text{then}~ y
\mathrm{if}~ a ~\mathrm{then}~ b
\end{matrix}</math>
\end{matrix}\!</math>
| <math>x \Rightarrow y\!</math>
| <math>a \Rightarrow b\!</math>
|-
|-
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>
| height="120px" | [[Image:Cactus (A,B) Big ISW.jpg|65px]]
| <math>\texttt{(} a \texttt{,} b \texttt{)}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a ~\mathrm{not~equal~to}~ b
\\
\\[6pt]
a ~\mathrm{exclusive~or}~ b
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a \neq b
\\
\\[6pt]
a + b
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>
| height="160px" | [[Image:Cactus ((A,B)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{,} b \texttt{))}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a ~\mathrm{is~equal~to}~ b
\\
\\[6pt]
a ~\mathrm{if~and~only~if}~ b
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
a = b
\\
\\[6pt]
a \Leftrightarrow b
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math>
| height="120px" | [[Image:Cactus (A,B,C) Big.jpg|65px]]
| <math>\texttt{(} a \texttt{,} b \texttt{,} c \texttt{)}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\text{Just one of}
\mathrm{just~one~of}
\\
\\
a, b, c
\\
\\
\text{is false}.
\mathrm{is~false}.
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
& \bar{a} ~ b ~ c
\\
\\
\lor & a ~ \bar{b} ~ c
\\
\\
\lor & a ~ b ~ \bar{c}
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>
| height="160px" | [[Image:Cactus ((A),(B),(C)) Big.jpg|65px]]
| <math>\texttt{((} a \texttt{),(} b \texttt{),(} c \texttt{))}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\text{Just one of}
\mathrm{just~one~of}
\\
\\
a, b, c
\\
\\
\text{is true}.
\mathrm{is~true}.
\\[6pt]
\mathrm{partition~all}
\\
\\
\mathrm{into}~ a, b, c.
\end{matrix}\!</math>
\text{into}~ x, y, z.
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
& a ~ \bar{b} ~ \bar{c}
\\
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| height="160px" | [[Image:Cactus (A,(B,C)) Big.jpg|90px]]
| <math>\texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}
\mathrm{oddly~many~of}
\\
\\
a, b, c
\\
\\
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}
\mathrm{are~true}.
\end{matrix}\!</math>
\end{matrix}\!</math>
|
|
<p><math>a + b + c\!</math></p>
<p><math>x + y + z\!</math></p>
<br>
<br>
<p><math>\begin{matrix}
<p><math>\begin{matrix}
& a ~ b ~ c
\\
\\
\lor & a ~ \bar{b} ~ \bar{c}
\\
\\
\lor & \bar{a} ~ b ~ \bar{c}
\\
\\
\lor & \bar{a} ~ \bar{b} ~ c
\end{matrix}\!</math></p>
\end{matrix}\!</math></p>
|-
|-
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>
| height="160px" | [[Image:Cactus (X,(A),(B),(C)) Big.jpg|90px]]
| <math>\texttt{(} x \texttt{,(} a \texttt{),(} b \texttt{),(} c \texttt{))}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\text{Partition}~ w
\mathrm{partition}~ x
\\
\\
\text{into}~ x, y, z.
\mathrm{into}~ a, b, c.
\\[6pt]
\mathrm{genus}~ x ~\mathrm{comprises}
\\
\\
\mathrm{species}~ a, b, c.
\\
\end{matrix}\!</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
& \bar{x} ~ \bar{a} ~ \bar{b} ~ \bar{c}
\\
\\
\lor & x ~ a ~ \bar{b} ~ \bar{c}
\\
\\
\lor & x ~ \bar{a} ~ b ~ \bar{c}
\\
\\
\lor & x ~ \bar{a} ~ \bar{b} ~ c
\end{matrix}</math>
\end{matrix}~\!</math>
|}
|}
(p)~q~
(p)~q~
\\[4pt]
\\[4pt]
(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
(p)~ ~
\\[4pt]
\\[4pt]
~p~(q)
~p~(q)
\\[4pt]
\\[4pt]
~ ~(q)
\\[4pt]
\\[4pt]
(p,~q)
(p,~q)
((p,~q))
((p,~q))
\\[4pt]
\\[4pt]
~ ~ ~q~~
\\[4pt]
\\[4pt]
~(p~(q))
~(p~(q))
\\[4pt]
\\[4pt]
~~p16:16, 29 November 2015 (UTC)
~~p~ ~ ~
\\[4pt]
\\[4pt]
((p)~q)~
((p)~q)~
|}
|}
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 16:16, 29 November 2015 (UTC)}\, {}^{\prime\prime}\!</math> and the associated 2-adic relation <math>M \subseteq X \times X,\!</math> the general pattern of whose common structure is represented by the following matrix:
For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}\!</math> and the associated 2-adic relation <math>M \subseteq X \times X,\!</math> the general pattern of whose common structure is represented by the following matrix:
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Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 16:16, 29 November 2015 (UTC)}\, {}^{\prime\prime}\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c\!</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~ ~ ~}\, {}^{\prime\prime}\!</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a\!</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call “multiplying on the left”.
Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j\!</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,\!</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <math>j.\!</math> This is the mode of reading that we call “multiplying on the left”.