SUO: RE: Re: Montague's type system
John,
I see what's happened now; I've been using IE, and the
symbol that looks like "necessary" is coming up when you
use the "element of" and "union" symbols as well as
when you use the "necessary" symbols.
Pardon my comments. Now with Netscape I can read the
paper the way you intended it to be read.
Thanks,
Rich
John F. Sowa wrote:
> Rich,
>
> I paraphrased every symbolic statement in English. You can
> read the English text and get the major points -- if you
> already know modal logic and at least something about
> Kripke's model theory
> (which is a prerequisite for Montague's).
>
> The phrase "deductive closure of a set of axioms" is familiar to
> anybody who knows logic: it is the set of all propositions that
> can be deduced from those axioms. The "turnstile" symbol, |-,
> is used to mean provability: A|-L means that everything in the
> set L is provable from A.
>
> The phrase "an ordered pair (M,L)" is typical math terminology.
> It means just that: two things, M and L with M coming before L.
>
> In the paper, those symbols are introduced together with their
> English equivalents:
>
> > 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. For this article, the following
> > conventions are assumed:
>
> Note that the four points below are assumed to be familiar
> to any reader who knows modal logic. I was just giving the
> English words that are used for the symbols:
>
> * 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.
>
> Note the English phrase is "maximally consistent". It is a
> common term in logic, but for convenience, I included a
> definition: any proposition p or its negation ~p is in M,
> but not both.
>
> * Contingent facts. The set M−L of all facts that
> are not
> laws is called the contingent facts.
>
> This is just a definition in English of something the readers
> are expected to know. And it gives a notation M-L for it.
>
> * Closure. The facts are the deductive closure of any
> axiom set A and the contingent facts:
> A ∪ (M − L) |− M.
>
> This gives the English statement first. Then it gives the
> notation. The notation uses nothing but set notation plus
> the turnstile for provability.
>
> That is a lot more hand-holding than you'll find in most
> papers on math and logic.
>
> Section 2 of the paper is a summary of Dunn's paper for the
> benefit of readers who don't have a copy of his 1973 paper.
> It assumes some background in modal logic, because that is
> the topic of the paper. The terms necessary, possible,
> and contingent are as old as Aristotle, and they are covered
> in any introduction to modal logic.
>
> I could write a tutorial on modal logic, but that would be
> out of place in a paper written for people who wouldn't go
> to a conference on modal logic unless they already knew
> quite a lot about modal logic.
>
> John
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