# ONT Re: Differential Logic A -- Discussion

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DLOG A.  Discussion Note 1

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GR = Gary Richmond
JA = Jon Awbrey
JS = John Sowa

Re: DLOG A10.  http://suo.ieee.org/ontology/msg05373.html

Gary,

In Texas they tell a variety of joke that goes a bit like this:

| Q.  What do you do when your 100-dollar 10-gallon hat
|     blows off in a dust-storm?
|
| A.  Reach up in the air and pull down another one.

The story that I called a "genealogy" not a "history"
has to do with the sorts of ideas that are always in
the air and only occasionally get seized on in novel
fashions.  Or call it a Hegelian history if you like.

It is customary to give Bentham first billing for
the "paraphrasis" idea, so I went along with that.
Quine's comment in Van Heijenoort gives additional
snippets about Schonfinkel's "Bausteine".  A very
enjoyable way to learn about combinator calculus
is provided by the second half of Ray Smullyan's
'To Mock a Mockingbird'.  Folks usually give Curry
and Church credit for independently rediscovering
what are more or less computationally equivalent
ideas in the various lambda calculi.  I tend to be
suspicious of how independent anybody can be from
their collective unconscious background/culture,
but that's just me.

B. and/or C. Peirce still get credit, I haven't been
able to sort out which deserves the lion's share yet,
for a very general form of algebraic representation
principle that's been a blowin' in more or less
the same wind since about the days of Galois.

Somewhere in the mid 1970's I figured out the relationship between
the pragmatic maxim and these representation principles, and that
has been the important thing to me since then.  Mathematics and
physics are still just about the only places where something
like the prag-maxim gets applied on a routine basis, most
often by people who never heard of it under that name.

I see John's note covers most of the other questions.

Jon Awbrey

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CSP: | Consider what effects that might 'conceivably'
| have practical bearings you 'conceive' the
| objects of your 'conception' to have.  Then,
| your 'conception' of those effects is the
| whole of your 'conception' of the object.
|
| Peirce, "Maxim of Pragmaticism",
| 'Collected Papers', CP 5.438.

JA: The genealogy of this conception of pragmatic representation is very intricate.
I will delineate some details that I presently fancy I remember clearly enough,
subject to later correction.  Without checking historical accounts, I will not
be able to pin down anything like a real chronology, but most of these notions
were standard furnishings of the 19th Century mathematical study, and only the
last few items date as late as the 1920's.

JA: The idea about the regular representations of a group is universally known
as "Cayley's Theorem", usually in the form:  "Every group is isomorphic to
a subgroup of Aut(X), the group of automorphisms of an appropriate set X".
There is a considerable generalization of these regular representations to
a broad class of relational algebraic systems in Peirce's earliest papers.
The crux of the whole idea is this:

JA: Contemplate the effects of the symbol
whose meaning you wish to investigate
as they play out on all the stages of
conduct on which you have the ability
to imagine that symbol playing a role.

JA: This idea of contextual definition is basically the same as Jeremy Bentham's
notion of "paraphrasis", a "method of accounting for fictions by explaining
various purported terms away" (Quine, in Van Heijenoort, page 216).  Today
we'd call these constructions "term models".  This, again, is the big idea
behind Schönfinkel's combinators {S, K, I}, and hence of lambda calculus,
and I reckon you know where that leads.

GR: I managed to follow this discussion pretty well,

JA: This, again, is the big idea behind Schönfinkel's combinators {S, K, I},
and hence of lambda calculus, and I reckon you know where that leads.

GR: Confronted with *lambda calculus*, I think first of what John Sowa has
written (see, especially, 'Conceptual Structures' [1984], 162-3, 373-4).
A few matters I'm quite unclear about:

GR: 1. Sowa describes the lambda calculus in this way (I'm starting
in media res, with comments involving a shipping example):

GR, quoting JS:

JS: | The denotations operation [delta] would search the database to find
| some part x, date y, and shipment z that would make the predicates
| in the body of the formula true.  If the denotation is true, then
| the answer to the question is yes. If the denotation is false,
| then the answer is no.
|
| For a wh-type type question *What suppliers shipped parts to Dept. 85?*,
| the answer is a set of suppliers.  In symbolic logic, a yes-no question
| corresponds to a proposition where every variable is *bound* by a quantifier.
| A wh-question, however, is mapped to a *lambda expressions* with one or more
| parameters ... 162 [the text diagrams this situation through symbolic logic
| and a conceptual graph] 162-3

GR: The upshot of this is:

JS: | The denotation of the lambda expression is not a truth value like
| true or false, but rather the set of all instances of SUPPLIERs that
| could be substituted for w to make the body of the expression true.
| Lambda calculus combined with symbolic logic makes a powerful database
| query language. 163

GR: What do you think of this description, which really constitutes a kind
of definition?  Is it yours?  Is it pointing to what your last, perhaps
rhetorical, question is meant to point to?  (Collateral knowledge needed,

GR: 2. In the second passage discussing the lambda calculus there
is no mention made of Schönfinkel in Sowa's discussion of
the history of the lambda calculus.

GR, quoting JS:

JS: | A definition by extension is only possible when the domain A is finite.
| In all other cases, the function must be defined by a rule, which is
| called the *intension* of f.  (One could, of course, define the
| extension of a function as an infinite set, but the set itself
| would have to be defined by some rule or intension.)
|
| Defining a function by a rule is more natural  or intuitive than defining
| it as a set of ordered pairs.  But a problem can occur when two or more
| different rules or intensions lead to the same sets of ordered pairs or
| extensions.  Are two functions considered the same if they have the same
| sets of ordered pairs, but different mapping rules?  To distinguish the
| intension and extension of functions and to formalize the rules for
| defining them, Alonzo Church (1941) developed a system called the
| *lambda calculus* which uses the Greek letter [lambda] to indicate
| the parameters of a function 373]

GR: So what's the exact intellectual history here?  I it connected to Peirce
in any way?  John's earlier discussion hints at this being a triadic logic
in the Peircean sense (yes/no being insufficient for the particular purpose
at hand -- see the shipping example above) though I don't believe he explicitly
states it as such.  Later he comments:

JS: | An important advantage of the lambda notation is that it defines
| a mapping independently of the act of naming it.  As a result, an
| unnamed lambda expression can be used anywhere that a function name
| could be used.  This feature is especially useful for applications
| that create new functions dynamically and then immediately pass them
| as arguments to another function. [e.g., some database systems]

GR: John concludes this analysis thus:

JS: An important result of the lambda calculus is the Church-Rosser theorem:
when more than one function in an expression is expandable, the order of
expansion is irrelevant because the same canonical form would be obtained
with any sequence of expansion.  [He then points to Sect. 3.6 of his text
relating this to graphs] 374

GR: What do you think about all this (perhaps even beyond databases)?
What are the pragmatic import of the lambda calculus?  [Btw, I have
not yet read much of John's most recent textbook, so perhaps he's
commented on some of these matters there.]

GR: If my questions appear naive, I hope you will realize that I am a mere
beginning student of logic hoping to get some light thrown on matters
that look to me to be of potential (pragmatic) value.

GR: Can Peirce's Gamma graphs be related to any of this, btw?

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http://www.cs.bsu.edu/homepages/mighty/history.html
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