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===Cactus Language===
===Cactus Language===
'''Note.''' Need to find the original context of this fragment.
Table 13 illustrates the "existential interpretation"
of cactus graphs and cactus expressions by providing
Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
English translations for a few of the most basic and
commonly occurring forms.
Even though I do most of my thinking in the existential interpretation, I will continue to speak of these forms as ''logical graphs'', because I think it is an important fact about them that the formal validity of the axioms and theorems is not dependent on the choice between the entitative and the existential interpretations.
The first extension is the ''reflective extension of logical graphs'' (RefLog). It is obtained by generalizing the negation operator "<math>\texttt{(~)}</math>" in a certain way, calling "<math>\texttt{(~)}</math>" the ''controlled'', ''moderated'', or ''reflective'' negation operator of order 1, then adding another such operator for each finite <math>k = 2, 3, \ldots .</math>
of the axioms and theorems is not dependent on the choice between
In sum, these operators are symbolized by bracketed argument lists as follows: "<math>\texttt{(~)}</math>", "<math>\texttt{(~,~)}</math>", "<math>\texttt{(~,~,~)}</math>", …, where the number of slots is the order of the reflective negation operator in question.
In sum, these operators are symbolized by bracketed argument lists as follows:
"(_)", "(_,_)", "(_,_,_)", ..., where the number of slots is the order of the
reflective negation operator in question.
The cactus graph and the cactus expression
The cactus graph and the cactus expression shown here are both described as a ''spike''.
shown here are both described as a "spike".
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
o---------------------------------------o
o---------------------------------------o
| |
| |
| ( ) |
| ( ) |
o---------------------------------------o
o---------------------------------------o
</pre>
|}
The rule of reduction for a lobe is:
The rule of reduction for a lobe is:
{| align="center" cellspacing="6" width="90%"
| align="center" |
<pre>
x_1 x_2 ... x_k
x_1 x_2 ... x_k
o-----o--- ... ---o
o-----o--- ... ---o
\ /
\ /
@ = @
@ = @
</pre>
|}
if and only if exactly one of the x_j is a spike.
if and only if exactly one of the x_j is a spike.
<pre>
In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
