ONT Re: Propositional Equation Reasoning Systems
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PERS. Note 12
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Analysis of Contingent Expressions (cont.)
Let us now examine the propositional equation:
o-----------------------------------------------------------o
| Equation E1 |
o-----------------------------------------------------------o
| |
| q r |
| q o o r o |
| | | | |
| p o o p p o |
| \ / | |
| @ = @ |
| |
| (p (q)) (p (r)) = (p (q r)) |
| |
| [p=>q] & [p=>r] = [p=>[q&r]] |
| |
o-----------------------------------------------------------o
There are three distinct ways that I can think of right off as to
how we might go about formally proving or systematically checking
the proposed equivalence, the evidence of whose truth we already
have before us clearly enough, and in a visually intuitive form,
from the venn diagrams that we examined above.
While we go through each of these ways let us keep one eye out
for the character and the conduct of each type of proceeding as
a semiotic process, that is, as an orbit, in this case discrete,
through a locus of signs, in this case propositional expressions,
and as it happens in this case, a sequence of transformations that
perseveres in the denotative objective of each expression, that is,
in the abstract proposition that it expresses, while it preserves
the informed constraint on the universe of discourse that gives
us one viable candidate for the informational content of each
expression in the interpretive chain of sign metamorphoses.
A "sign relation" is a subset L of a cartesian product OxSxI, where
O, S, I are sets known as the "object", "sign", "interpretant sign"
domains, respectively. One writes "L c OxSxI", where the symbol "c"
is Ascii for the subset relation "contained as a subset of". Thus,
a sign relation L consists of ordered triples of the form <o, s, i>,
where o, s, i belong to the relational domains O, S, I, respectively.
The "syntactic domain" of a sign relation L c OxSxI is just the
union S |_| I of its sign domain S and its interpretant domain I.
It is not unusual, especially in artificial and formal examples,
to have S = I, in other words, to have the sign domain and the
interpretant domain being equal as sets.
Elsewhere I have discussed examples of sign relations that consist
of a finite set of triples of the form <o, s, i>, where o, s, i are
the "object", "sign", "interpretant sign", respectively, of what is
called the "sign triple" or the "elementary sign relation" <o, s, i>.
We will be taking a bit of a jump up from the finite case now,
since most of the examples of sign relations that interest us
in logic will have S and I being infinite sets, and usually O
will be infinite, too, in the long run, at least, although we
will frequently work up to the infinite object domains by way
of various series of finite approximations and gradual stages.
With that preamble behind us, then, let us turn to consider
the case of semiosis, or sign transforming process, that is
constituted by our first proof of propositional equation E1.
o-----------------------------------------------------------o
| Equation E1. Proof 1. |
o-----------------------------------------------------------o
| |
| |
| q o o r |
| | | |
| p o o p |
| \ / |
| @ |
| |
| (p (q)) (p (r)) |
| |
o=============================< Double Negation >===========o
| |
| q o o r |
| | | |
| p o o p |
| \ / |
| o |
| | |
| o |
| | |
| @ |
| |
| (( (p (q)) (p (r)) )) |
| |
o=============================< Collection >================o
| |
| q o o r |
| | | |
| o o |
| \ / |
| o |
| | |
| p o |
| | |
| @ |
| |
| (p ( ((q)) ((r)) )) |
| |
o=============================< Double Negation >===========o
| |
| q r |
| o |
| | |
| p o |
| | |
| @ |
| |
| (p (q r)) |
| |
o=============================< QED >=======================o
For some reason I always think of this
as the way that our DNA would prove it.
Jon Awbrey
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