ONT Re: Differential Logic
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DLOG. Note D1
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Differential Logic and Dynamic Systems
Author: Jon Awbrey
Created: 16 Dec 1993
Relayed: 31 Oct 1994
Revised: 06 May 2003
| Stand and unfold yourself.
|
| Hamlet, 1.1.2
Purpose
This series of reports develops a differential extension of
propositional calculus and applies it to a context of problems
arising in dynamic systems. The work pursued here is coordinated
with a parallel application that focuses on neural network systems,
but the dependencies are arranged to make the present series the
main and the more self-contained work, to serve as a conceptual
frame and a technical background for the network project.
Review and Transition
This note continues a previous discussion on the problem of dealing
with change and diversity in logic-based intelligent systems. For
ease of reference, I begin by summarizing essential material from
previous reports.
Table 1 outlines the notation that I use for propositional calculus.
Explained as briefly as possible, I am using only two basic kinds
of truth-functional connectives, both of variable k-ary scope.
1. For the first, I use concatenation as a connective
to signify the logical conjunctions of k arguments.
2. For the other, I use a bracket of the form ( , , , )
as a connective which says that exactly one of its k
arguments is false.
All other truth-functional connectives can be obtained in a very
efficient style of representation through combinations of these
two forms.
This treatment of propositional logic is derived from the work of
C.S. Peirce [P1, P2], who gave this approach an extensive development
in his graphical systems of predicate, relational, and modal logic [Rob].
More recently, these ideas were revived and supplemented in an alternative
interpretation by G. Spencer-Brown [SpB]. Both of these authors used other
forms of enclosure where I use parentheses, but the structural topologies of
expression and the functional varieties of interpretation are fundamentally
the same.
While working with expressions solely in propositional calculus, the use
of plain parentheses to represent logical connectives is simplest to write
and easiest to read for both human and machine parsers. In the present text
I preserve this form of expression in tables and set-off displays, but in
contexts where parentheses are needed for functional notation I will use
a distinctive font for logical operators. [Not available in Ascii.]
The briefest expression for logical truth is the empty word, usually denoted by
epsilon or lambda in formal languages, where it forms the identity element for
concatenation. To make it visible in this text, I denote it by the equivalent
expression "(())", or, especially if operating in an algebraic context, by a
simple "1". Also when working in an algebraic mode, I use the plus sign "+"
for exclusive disjunction. Thus, we may express the following paraphrases
of algebraic forms:
A + B = (A, B)
A + B + C = ((A, B), C) = (A, (B, C))
One should be careful to note that this last pair of
expressions are not equivalent to the form (A, B, C).
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
| Expression | Interpretation | Other Notations |
o-------------------o-------------------o-------------------o
| " " | True. | 1 |
o-------------------o-------------------o-------------------o
| () | False. | 0 |
o-------------------o-------------------o-------------------o
| A | A. | A |
o-------------------o-------------------o-------------------o
| (A) | Not A. | A' |
| | | ~A |
o-------------------o-------------------o-------------------o
| A B C | A and B and C. | A & B & C |
o-------------------o-------------------o-------------------o
| ((A)(B)(C)) | A or B or C. | A v B v C |
o-------------------o-------------------o-------------------o
| (A (B)) | A implies B. | A => B |
| | If A then B. | |
o-------------------o-------------------o-------------------o
| (A, B) | A not equal to B. | A =/= B |
| | A exclusive-or B. | A + B |
o-------------------o-------------------o-------------------o
| ((A, B)) | A equals B. | A = B |
| | A if & only if B. | A <=> B |
o-------------------o-------------------o-------------------o
| (A, B, C) | Just one of | A'B C v |
| | A, B, C | A B'C v |
| | is false. | A B C' |
o-------------------o-------------------o-------------------o
| ((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. | |
o-------------------o-------------------o-------------------o
| ((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 |
| | | A'B C' v |
| | | A'B'C |
o-------------------o-------------------o-------------------o
| (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v |
| | into A, B, C. | Q A B'C' v |
| | | Q A'B C' v |
| | Genus Q comprises | Q A'B'C |
| | species A, B, C. | |
o-------------------o-------------------o-------------------o
NB. The usage that one often sees, of a plus sign "+"
to represent inclusive disjunction, and the reference
to this operation as "boolean addition", is a misnomer
on at least two counts. Boole used the plus sign to
represent exclusive disjunction (at any rate, an
operation of aggregation restricted in its logical
interpretation to cases where the represented sets
are disjoint [Boo, 32]), as any mathematician with
a sensitivity to the ring and field properties of
algebra would do:
| The expression x + y seems indeed uninterpretable,
| unless it be assumed that the things represented
| by x and the things represented by y are entirely
| separate; that they embrace no individuals in
| common. [Boo, 66].
It was only later that Peirce and Jevons treated inclusive
disjunction as a fundamental operation, but these authors,
with a respect for the algebraic properties that were already
associated with the plus sign, used a variety of other symbols
for inclusive disjunction [Sty, 177, 189]. It seems to have
been Schroeder who later reassigned the plus sign to inclusive
disjunction [Sty, 208]. Additional information, discussion,
and references can be found in [Boo] and [Sty, 177-263].
Aside from these historical points, which never really
count against a current practice that has gained a life
of its own, this usage does have a further disadvantage
of cutting or confounding the lines of communication
between algebra and logic. For this reason, I am
forced to avoid it here.
Jon Awbrey
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