SUO: Re: Montague's type system
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