SUO: RE: Re: Montague's type system
John F. Sowa wrote:
> Rich and Pierluigi,
>
> Before saying anything about Montague's system, let me say
> that I admire it very much -- as an abstract mathematical
> construction. I have learned a great deal from it. One of
> the things I learned is that there has to be a better way.
>
> You can only make sense of Montague's system when you study
> the mathematical Lego blocks that he put together in order
> to construct it. Following are the pieces:
>
> 1. Leibniz's idea that a necessary statement is true in all
> possible worlds.
>
> 2. Carnap's idea that a sentence is a function from possible
> worlds to truth values -- i.e., given a world w and a
> sentence S, the result of applying S to w, written S(w),
> is true if S makes a true statement about w, and S(w) is
> false if S makes a false statement about w.
>
> 3. Kripke's semantics of modal logic, which combines the
> ideas of Leibniz, Carnap, and Tarski to form a model
> theory for modal logic.
>
> 4. Ajdukiewicz's categorial grammar, in which the grammar
> rules combine the types of the constituents to derive
> the type of any combination of constituents.
>
> 5. Church's lambda calculus, which is designed to create
> functions by designating one or more constituents of
> an expression as formal parameters. For example, the
> following sentence is a function from possible worlds
> to truth values:
>
> John walks.
>
> This function maps worlds in which John walks to truth,
> and worlds in which John does not walk to false. When
> you replace "John" by a parameter "x", you get
>
> (lambda x)(x walks)
>
> This lambda expression is a function that corresponds
> to the verb "walks" by itslef. It is a function from
> proper names (or noun phrases in general) to sentences.
>
> In general, if X and Y are any two types, the notation <X,Y>
> represents the type of functions from X to Y.
>
> The only two primitive types in Montague's ontology are
> e for entities and t for truth values. Every other type is
> a function from some previously defined type to some other
> previously defined type.
>
> Given those two assumptions, a noun such as "dog" has the
> type <e,t>, which implies that the noun "dog" is a function
> that applies to entities. If some entity is a dog, then
> the value of the function "dog" is _true_. If some entity is
> not a dog, then the value of the function "dog" is _false_.
>
> When you take a noun such as "dog" and put a quantifier
> such as "a" in front of it, you get the noun phrase "a dog".
> Montague assigned the type <<e,t>,t> to noun phrases. That
> means that "a dog" is a function from (functions from entities
> to truth values) to truth values. Any noun phrase, including
> proper names such as "Fido" or "John" has the same type.
>
> I am not even going to try to make that sound intuitively
> plausible by itself. But if you puzzle over Montague's paper
> long enough, you can begin to manipulate his symbols in the
> same way that he did. You will then have entered the weird
> and wonderful world that constituted Montague's mind. But I
> don't recommend that you stay there.
>
> If you really want to understand natural language semantics,
> I recommend an equally elegant, but more realistic theory,
> namely, conceptual graphs. See my paper about how to replace
> possible worlds with a model theory based on graphs:
>
> http://www.jfsowa.com/pubs/laws.htm
> Laws, Facts, and Contexts
>
> John Sowa
John, near the beginning of this paper, I ran into some
terminology issues. When I sit in a math class and listen
to a professor read off the symbols he uses on the whiteboard,
I get some kind of interpretation that is deeper rooted
in experience than just the symbolic notation itself. When
I read a paper that jumps into symbolic notation without
clear English rephrasing of the symbols, I get confused.
From your description of Dunn's mathematics, I quote:
--------------------
For every Kripke world w, Dunn assumed an ordered pair (M,L),
where M is a Hintikka-style model set called the facts of
w and L is a subset of M called the laws of w.
--------------------
I'm assuming that a Kripke world is a specific situation or
set of situations that can be expressed in the usual FOL
list of facts and rules, while Dunn simply partitioned the
Kripke world into facts and rules that are specific to the
situation, and laws are those that are general enough to
apply to many possible worlds, all drawn from the same
context.
The word "facts" in this article seems to apply equally
well to both FOL assertions and FOL implications.
In particular, M includes both facts and rules (in the usual
expert systems sense of those words) and so does L; it is
just in their applicability to a context that members of
L are distinguished from those of M.
Is this a correct interpretation?
Then the paper establishes some symbolic groundwork:
---------------------
For this article,
the following conventions are assumed:
Axioms. Any subset A of L whose deductive closure is the
laws (A |? L) is called an axiom set for (M,L).
Facts. The set of all facts M is maximally consistent:
for any proposition p, either p?M or ~p?M, but not both.
---------------------
Q:How do you read p?M. Is it "if p then necessarily M"?
How do you read ~p?M. Is it "if not p then necessarily M"?
---------------------
Contingent facts. The set M?L of all facts that are not
laws is called the contingent facts.
---------------------
Q: I guess this means contingent facts are those that happen
to be true of this possible world w, but might not be true
of other possible worlds in which M may also hold.
---------------------
Closure. The facts are the deductive closure of any axiom
set A and the contingent facts:
A ? (M ? L) |? M.
---------------------
Q: Whoa! Does this mean that any fact implied by the axioms
in A is necessarily able to produce the facts of M, but may
also produce some Ls as well. So the Ls have to be taken
out of A's consequences to get the kernel of Ms.
Is that a correct English interpretation?
Thanks,
Rich