Re: Linear Logic Maybe?

From: Jan Burse (janburse_at_fastmail.fm)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 11:43:32 +0100
To: tmp123 <tmp123@menta.net>

Hi

tmp123 wrote:
> Probably I do not understand something, because if
> "!" means "->" ("then")
> "&" means "/\" ("and")
> "v" means "\/" ("or")

With ! I have more a composition/application
in mind then an implikation.

Let @ denote the application. It takes an
argument f and g, where f is a function A->B
and g is an element of A. Then f@g is an element
of B. Namely:

    f:A->B, g:A
    -----------
      f@g:B

Then ! can denote the inverse composition as
follows. It takes argument h and f, where h
is a function B->C and f is a function A->B.
Then h!f is a function A->C. Namely:

    h:B->C, f:A->B
    --------------
       h!f:A->C

We can model ! by @. Namely h!f = \x.h@(f@x)
where \x. is the lambda abstraction.

Now we can assume that Nouns are "Merkmale",
thus they map Objects to Subobjects. Namely
a Noun is a function O->O, where is O is
the set of a objects, an infinite set.

Further we can assume that Adjects are
"Ausprägungen", thus they map Objects to
Truthvalues. Namely an Adjective is a
function O->2. Where 2 ist the set {0,1},
and 2 and O are disjoint.

In the example we had apples and bananas.
Lets assume apples and bananas refers not
to apples and bananas in general, but we
have a situation which is modeled as a
record {apple:x,banna:y}.

    apple:O->O
    apple{apple:x,..}=x
    banana:O->O
    banana{banna:y,..}=y

In the example we had also red and green.
Lets assume that the x and y are futher
records of the form {color:z,...}. Then
we can define:

   red:O->T
   red{color:"red",..}=true
   red{color:"green",..}=false
   green:O->T
   green{color:"red",..}=false
   gren{color:"green",..}=true

Now red!apple means that the apple in the
current situation is red. The expression
is of the type O->T:

   red:O->T apple:O->O
   --------------------
     red!apple:O->T

Now (red v green) can be easily interpreted
as the truth function which results from
the truth function of red and green by
the following rule:

    p:O->T, q:O->T
    --------------------------------------
    (pvq)(x) := p(x) v q(x), (pvq):O->T

But what happens if we make a disjunction
or conjunction of Nouns?

    p:O->O, q:O->O
    -----------------
       pvq ?? q&p ??

What disjunction and conjunction should
be defined. How could a calculus look like?

Best Regards 

---
Spuntik III

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