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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:

*>
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*> In one little sentence: modal logic is a tool for refining truth
*

*> by making it relative to context, situations, etc. Those last
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*> notions are in general captured by some abstract mathematical
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*> spaces, like set + binary (accessibility) relations with Kripke,
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*> 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

Date: Mon, 9 Sep 2002 10:53:23 -0700

On Monday, September 9, 2002, at 01:39 AM, Bruno Marchal wrote:

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

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