Time-varying sets and modal logic

From: Tim May <tcmay.domain.name.hidden>
Date: Mon, 9 Sep 2002 10:53:23 -0700

On Monday, September 9, 2002, at 01:39 AM, Bruno Marchal wrote:
>
> In one little sentence: modal logic is a tool for refining truth
> by making it relative to context, situations, etc. Those last
> notions are in general captured by some abstract mathematical
> spaces, like set + binary (accessibility) relations with Kripke,
> quasi topological space with Scott and Montague, etc.


Or, seen naturally all around us in the world, with time-varying sets.

A time-varying set, informally, is one whose set of members varies with
time. (Time is just about the most important kind of _context_
mentioned above by Bruno.) The set of nations in the U.N. varies with
time, the set of air molecules in a room varies with time, the set of
descendants of a person varies with time, and so on.

The logic and algebra associated with such variable sets are Heyting
logic and Heyting algebra, not the more commonly studied Boolean logic
and Boolean algebra. I outlined this in some earlier posts. (And there
are synonyms for Heyting: intuitionistic logic, Brouwerian lattices,
forms of modal logic, etc.)

The connection between time-varying sets and time-varying logic is of
course straightforward. Propositions within a logical system can be
translated into set inclusion relationships.

The connection with branching forks of a universe, where different
forks are BY DEFINITION contradictory (and hence are not analyzable
with Boolean logic), is clear...to me at least.

Two outcomes of the flip of a coin, for example, form a fork which is
part of a poset. The outcomes, H or T, do not obey the usual law of
trichotomy, hence the set is a poset. I outlined this in earlier posts
as well.

--Tim May
Received on Mon Sep 09 2002 - 10:53:59 PDT