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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)
Revision as of 20:16, 19 June 2009
, 20:16, 19 June 2009→Note 1: convert graphics
Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
| [[Image:Cactus Graph Existential P And Q.jpg|500px]]
|}
|}
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph Existential True.jpg|500px]]
| [[Image:Cactus Graph Existential True.jpg|500px]]
|}
|}
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph Existential False.jpg|500px]]
| [[Image:Cactus Graph Existential False.jpg|500px]]
|}
|}
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
Back to the proposition <math>pq.\!</math> Imagine yourself standing in a fixed cell the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Venn Diagram P And Q.jpg|500px]]
| [[Image:Venn Diagram P And Q.jpg|500px]]
|}
|}
Don't think about it — just compute:
Don't think about it — just compute:
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
| [[Image:Cactus Graph (P,dP)(Q,dQ).jpg|500px]]
|}
|}
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph (P,dP).jpg|500px]]
| [[Image:Cactus Graph (P,dP).jpg|500px]]
|}
|}
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
proposition <math>pq\!</math> over there, at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| [[Image:Cactus Graph ((P,dP)(Q,dQ),PQ).jpg|500px]]
| o=o-----------o |
|}
|}