SUO: *Date 10 Apr 2002 -- Dispossessed Logic
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| Guarded by Bhishma, the strength
| of our army is without limit;
| but the strength of their army,
| guarded by Bhima, is limited.
|
| Duryodhana -- Bhagavad Gita, 1.10.
|
|'The Bhagavad-Gita: Krishna’s Counsel in Time of War'.
| B.S. Miller (trans.), Bantam Books, New York, NY, 1986.
Dispossessed Logic
I am going to go back to the four themes that I raised last time
and try to stay with them until I have made them moderately clear.
I will need to begin by unpacking their more opaque compressions,
as these are the sorts of codes that developed over many years
of not writing things down but just laying down layer on layer
of heuristic mnemonics that I kept in my head as I worked away.
| 1. All boolean operators are created equal,
| and need to be regarded as equal B-ings.
You can read this as the normal sort of design requirement
to treat all propositional forms as "first class citizens".
| Be wary of any form of syntaxation that
| that warps the ballot of representation.
Here we just have to get real. No matter how much we might wish
to deny it, syntax really does have a differential effect on the
alacrity and clarity with which we are permitted to think by its
means, and by extension, the facility and flexibility with which
real programs can run on real machines. It is like those other
physical symbol systems, the catalysts and the inhibitors that
determine the reaction rates of metabolic kinetics, and thus
make the difference between life and death, whatever their
meanings may be when it comes to that kind of semantics.
| A freefolk's "philosophy of logic" (POL)
| must never be rent subject to a POL tax.
Now, the issue of syntactic catalysis comes up in the matter
of equal representation precisely because of the effect that
syntax can have on the preservation of syntactic equilibrium.
| 2. The equal B-ings that they are find their most
| fruitful interpretations as functions B^k -> B.
| If logic is to comport with the communities of
| effective description and formal icon building
| in mathematics, programming, statistics, along
| with the full spectrum of systems and sciences,
| it shall be obligated to keep this option open,
| bordering nigh unto a correspondence principle.
|
| 3. The distincture that a thinker may lay on these B-ings,
| between the asserted and the merely contemplated, does
| not constitute any brand of logical essence but purely
| a hermeneutic, interpretive, pragmatic color or nuance.
| It amounts to no more than the transit of orientations
| between the modes of operation that we commonly depict
| as "computing a function" versus "solving an equation".
| A proposition q : B^k -> B may be regarded as a charge
| to evaluate q on more or less points of the domain B^k,
| or else a manner of indicating the subset of points of
| type B^k on which q evalues to true, which is all that
| any assertion is really meant to do.
Time for a concrete example.
Consider the sixteen functions of the type B^2 -> B, along with
their associated connectives or corresponding logical operators.
We would like to consider the set of functions in its invariant
aspect as an object domain, freely choosing whatever notational
system of signs seems to serve best for denoting its members in
a given context of application. And we would like to treat all
of these functions and their chosen notations on an equal basis,
granting to all of them whatever freedom of interpretation that
we accord to any one of them.
One degree of interpretive freedom that comes to mind in this
connection is that between the forward and the inverse use of
the function, between computing values and computing "fibers",
as this reflects in functional terms the pragmatic difference
between asserting a proposition and merely contemplating it.
But if we write a couple of different boolean functions
in one of the usual notations, we find that whole hosts
of cultural, historical, and psychological associations
are borne as riders and rein in the desired flexibility.
For instance, if we write "x & y", then it seems very
easy for us to regard this in either one of two ways:
(1) as the sort of function of the variables x, y that
would lead us to compute or to contemplate its value;
(2) as the sort of assertion that is associated with
the imposition of a constraint on the values of x, y,
namely, the constraint that selects the combinations
of values for x, y that makes the function x & y true.
However, if we write "x = y", then it is very unlikely
that we will find it easy to keep in mind the functional
interpretation of this expression, feeling almost forced
to read it as an assertion, a constraint, or an imposition.
That is one of the sorts of thing that I mean by the "bias",
the "tax", or the "warp" of the syntactic representation.
Jon Awbrey
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