[Inquiry] Re: Really Useful Logic

[Jon Awbrey]

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RUL.  Discussion Note 7

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JA = Jon Awbrey
PC = Party of Citizens

PC: Thank you for the opportunity to learn more about the ontogeny of
    the ontology of epistemology, aka 'logic fundamentals'.  I would
    like to ask a few questions below for clarification.

JA: Every now and then, somebody says something that reminds me of the work
    that I used to do in the real world -- I get the feeling that it's purely
    accidental on their part -- and of the hopes I still harbor that logic might
    actually oneday be applied to some good effect in that real world.

PC: I have never experienced people using long chains of complex
    logic in everyday life of the kind you find in logic texts.
    Even in courtrooms one doesn't find this.  However, it is
    obvious that long and complex sequences are used in the
    practical mathematics of a number of professions.
    These maths are used in their everyday work.

Yes, the particular "real world" that I had in mind was the world of applicable math,
applied science, empirical research (both basic and targeted), engineering, statistics,
and so on.  In those applications the knowledge that people have, if formalized at all,
is tied up in mathematical, quantitative, and statistical data, and the models thereof,
and what they need is good descriptions and better theories of what's going on with
their domain of practice, in order to explain that whole confused mass of data.

Notice how this is just the opposite of what many syntactile logicist types do,
starting out with some disconnected theory or detached set of axioms, and then
going looking for a formal semantics or a theory of models in order to anchor
their axiomatic dirigibles and keep them from drifting away in the breeze.

You might think that mathematics is different, but not if you look at its
actual practice, and especially if you look at its historical development.
When I think of some of my favorite formal objects -- groups and graphs --
that are currently defined by short lists of axioms, it's a fact that
that people were working with these things for a millennium of two
before they started to distinguish them under those precise names
and axiom sets, and then it has taken another century or two just
to work out a fraction of the major consequences of those axioms.

PC: First question:  Do you think all of these mathematical
    expressions could be translated into Standard English?

PC: If so, let's say we ask a rocket scientist how to make a succesful launch into
    orbit.  The answer, If ________ then __________ becomes  If ________ (a lot of
    complex mathematics and physics, etc.) then ___________(the rocket will go into
    orbit).  If you as a logician could (in co-operation with the rocket scientist)
    translate the first blank space into Standard English, the entire If-Then piece
    of logic would be Standard English.

But why?  Would it be useful in that form?  In the real world?
That just isn't what people do when they are really serious
about getting the best working models of complex realities.

One of the dead-ends that the likes of Bertrand Russell sent us down
was this idea of reducing mathematics and science to a very narrow
conception of logic, and maybe even to just abstract linguistics.
Not everybody who tries to build bridges between logic and
mathematics is trying to ford a conquering army of
one side over the other.

JA: So I will devote this thread to discussing some of the requirements that
    logical systems will need to have in order to make that so, at least, as
    I and many other folks I used to know found to be the constant lessons of
    their education and experience.

JA: Requirement 1.  Logic is qualitative statistics.

JA: This is a way of saying that logic must interface with statistics in such a way
    that it could almost be seen as a special case of the latter, satisfying a sort
    of "correspondence principle" where logic is the boolean case ...

PC: What are the "elements" of Boolean logic?  Will YES, NO, AND, OR
    (as connections) applied to EXPRESSIONS which can be evaluated
    as TRUE or FALSE suffice?  (Seven 'elements' in block letters).

It is a matter of what the expressions mean in the real world.
Look to the practical situations where you use these elements,
and abstract a bit, but not so far that you detach them from
all connection with reality.  Then you would start to notice
some analogies with the terms that refer to distributions in
statistics and the operations that combine these distributions.

JA: and statistics is the real case of the overarching general discipline.

PC: This is where you start to lose me so I'll ask some more questions.

JA: Requirement 2.  Functionality.

JA: In this view of things, a predicate P is always function P : X -> B, where X
    is some domain (population, sample space, universe of discourse) of interest
    and B = {0, 1}.

PC: Is this the same as saying that any truth-bearing
    expression can be evaluated as True (1) or False (0)?

I think, but true or false of each x in X.

It's just one way of providing a natural semantics
for propositional expressions or adjectival terms.
The important thing about this interpretation is
that it stays close to operational meanings in
practice, namely, the business of evalutaing
a predicate (say "green") with respect to
each object that comes your way, out of X,
and getting a decisive answer:  either
"yes, it applies" or "no, it does not".

The point is that a predicate name refers
to a whole pattern of conceivable results.
Patterns like that we call "functions".

Look, just think of "predicates" in the way that
McCulloch & Pitts or Minsky & Papert used them.

JA: Requirement 3.  Computability.

JA: This is optional, at least at the very beginning,
    but it is always highly desirable.  If you do not
    have a clue how to compute the value P(x) for almost ...

PC: Why "almost"?

Because of decidability issues -- the best that
we can expect in general is partial functions.

JA:  all of the x in X, then it is very probable that you
     do not have a clue what you mean by the predicate P.

PC: This is where you lose me for sure.  Let's say x is one ant from the
    population of ants, X, and we of course want to know all we can about
    ants because Proverbs tells us to observe the ant to be wise.  Can you
    apply your P : X -> B statement with probabilities of 0 or 1 to the ant?

I only said that logic is a limit case of statistics.
Predicates P : X -> B are normalized step functions.
Histograms H : X -> R are the more general case.
There is a level of analogy, not an identity.

JA: Requirement 4.  Semiotic Awareness.

JA: Defined this way, the predicate P : X -> B is a particular mathematical object,
    but that one object is the referent of many signs -- many names that denote it
    and many expressions that express it -- and a large part of the practical work
    of computation is busied with transforming among the referentially equivalent
    signs of the same object.  We tend to ignore the distinction between the sign
    and the object -- but we ignore it at our peril -- so let us follow a common
    good practice in computer science and refer to the names and expressions of
    predicates as "predicate names" and "predicate expressions", respectively.
    whenever we find it critical to make a point of this distinction.

PC: By sign you mean - and +?  Are these the same as
    NO, YES above re "elements" of Boolean logic?

No, I mean "sign" in the sense of "mark" or "signifier".
This is the generic term in semiotics from Augustine to
Peirce and beyond.  For example, the genus of signs is
divided into three species:  icons, indices, and symbols.

That whole paragraph boils down to the zeroth lesson of computing --
or all of mathematics, for that matter -- the distinction between
a number (the object) and a numeral (the sign).

JA: A related distinction is that between lattices and libraries.

PC: Matrix algebra = lattices?

No, "lattice" in the sense of a partially ordered set,
with LUB's and GLB's, Top and Bottom, and so on.

JA: Predicates, as mathematical objects, fit nicely in lattices.
    What's print to fit in libraries, though, are their codes.

That is, the set of functions {P : X -> B}
is isomorphic to the power set Pow(X),
the set of all subsets of X,
which is a lattice.

Jon Awbrey

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