ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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It is recognized as something of a persistent phenomenon
in pragmatic semiotics to see the semiotics overwhelm the
pragmatics, that is, to find the rush of signs submerging
any sense of an object. So let us remind ourselves why we
are fiddling about with such rudimentary logical systems as
ZOL(k), for small k -- not so much for their own sake, surely,
but to arrive at initial insights into how a logical language
can serve a logical object, and further, as a second purpose,
to observe in a non-distracting but not too trivial case how
this whole business of logic looks from a sign-theoretic POV.
I have recently drawn your attention to one such "logical object" (LO),
going about it in the clumsiest way possible, through all manner of
gesticulation, verbal clatter, and visual clutter that is hardly
a cut above what all the world calls "ostentation", and I have
done this deliberately -- I might as well say "deliberate",
since I could not find any other way -- all in order to
illustrate the character of the informal process
of communication that is our first recourse
and our last resort.
But now we have need of a more formal language for discussing these objects
and their kin, and so I shall draw on two such calculi, one that the reader
will already know, and then, for the sake of illustrating what it means to
make a critical comparison among languages with regard to their objects --
since that is another of our interests here -- one that I have designed
to the purpose.
Here is a Table that will serve as an introduction to this tale of two calculi.
The language that I have come to use more and more for propositional calculus
is exemplified by an initial sampling in the first column, an indication of
one of its intended interpretations is shown in the second column, while
a motley crew of familiar expressions is exhibited in the third column:
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| Document History:
|
| Subject: Differential Logic & Dynamic Systems
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: 2.0
| Created: 16-Dec-1993
| Revised: 31-Oct-1994
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Table 1 (Syntax & Semantics of a Calculus for Propositional Logic)
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
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| Expression | Interpretation | Other Notations |
o-------------------o-------------------o-------------------o
| | | |
| " " | True. | 1 |
| | | |
| () | False. | 0 |
| | | |
| A | A. | A |
| | | |
| (A) | Not A. | A' |
| | | ~A |
| | | |
| A B C | A and B and C. | A & B & C |
| | | A · B · C |
| | | |
| ((A)(B)(C)) | A or B or C. | A v B v C |
| | | |
| (A (B)) | A implies B. | A => B |
| | If A then B. | |
| | | |
| (A, B) | A not equal to B. | A =/= B |
| | A exclusive-or B. | A + B |
| | | |
| ((A, B)) | A equals B. | A = B |
| | A if & only if B. | A <=> B |
| | | |
| (A, B, C) | Just one of | A'B·C v |
| | A, B, C | A·B'C v |
| | is false. | A·B·C' |
| | | |
| ((A),(B),(C)) | Just one of | A·B'C' v |
| | A, B, C | A'B·C' v |
| | is true. | A'B'C |
| | | |
| | Partition all | |
| | into A, B, C. | |
| | | |
| ((A, B), C) | Oddly many of | A + B + C |
| (A, (B, C)) | A, B, C | |
| | are true. | A·B·C v |
| | | A·B'C' v |
| | One or all of | A'B·C' v |
| | A, B, C | A'B'C |
| | are true. | |
| | | |
| (X, (A),(B),(C)) | Partition X | X'A'B'C' v |
| | into A, B, C. | X·A·B'C' v |
| | | X·A'B·C' v |
| | Genus X comprises | X·A'B'C |
| | species A, B, C. | |
| | | |
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Have a good weekend,
Jon Awbrey
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