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
