On 3/28/2013 6:11 AM, Hassan Aït-Kaci wrote:
> Haskell Curry and Robert Feys proposed Combinatory Logic and showed that
> all lambda-definable functions (in the sense of Alonzo Church), and thus
> all computable sets (in the sense of Turing) [the two being equivalent
> as shown by Stephen Kleene in his thesis on Recursion Theory done under
> Church's supervision in 1934], can be expressed as some combination of
> only two basic combinators S and K where: (01)
I agree, and I'll avoid mentioning any metaphysics. (02)
The fundamental principle can be seen when you draw diagrams that show
the interconnections: (03)
1. For each notation -- predicate calculus, existential graphs, lambda
calculus, combinators, Curried functions, or anything else you like
-- draw a graph that shows the interconnections. (04)
2. Each node of each graph should be labeled with one symbol from the
notation -- function, relation, individual, lambda, combinator, etc. (05)
3. Each arc of each graph should show how the data represented by each
node is referred to or transmitted to the other node by any means --
variables, functions, relations, combinators, direct connections... (06)
4. Finally, note that all the diagrams have similar connectivity.
They might have different numbers of nodes and arcs, but there
will be certain irreducible patterns that appear in all of them. (07)
5. If you look at the simplest of all the graphs, any cycles or any
triadic connections (one node with three attached arcs) that appear
in it will be reflected in the all the others: any triadic nodes
will be mapped to nodes with at least three connections and any
cycles will be mapped to other cycles. (08)
As just one simple example, see the following diagram, which shows
two conceptual graphs with different ontologies for representing
the sentence "Sue gives a child a book": (09)
http://www.jfsowa.com/figs/give.gif (010)
If you translate either CG to CGIF, CLIF, predicate calculus, or
any other notation, the graph for that notation will still have
at least one triadic connection. (011)
John (012)
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