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

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