What Can We Say About Isomorphisms Between Canonical Frames?

Timothy J. Surendonk

April 12, 1996

Abstract

This short paper will look at a number of straightforward conclusions we can draw about the number and nature of automorphisms of and isomorphisms between canonical (relational) frames for normal monomodal logics.

1 Introduction

Often when looking at a particular modal logic, we will have a relational frame in mind. This will be the ?typical? or ?representative? frame which we hope our logic will capture and will be the frame we immediately visualize when we approach problems of that logic. In most cases our `frame of mind' will be reasonably rigid?we have such a strong and restricted idea of the structure of the frame that it won't permit any internal adjustment or automorphisms (e.g. (!; <))?or it will be a frame that is pretty loose?a frame which, while well de?ned and strongly present within our mind, has no particularly distinguished elements and allows all sorts of automorphisms. However, the modal logician has to carry out completeness proofs and so (mostly) relys on the canonical frame construction to bring these proofs to a speedy conclusion. Unfortunately the canonical frame is not a `frame of mind,' with even the simplest logic K having an unimaginably bizarre canonical frame?the reason being that it contains the canonical frames for all normal modal logics. Since these canonical objects are a basic component of most studies of modal logic it should certainly be of interest to analyse their behaviour and try to get a better idea of their structure.
The question we will take up in this paper is ?What kind of isomorphisms are there between canonical frames?? Of course, this question is easily answered if the canonical frames are for different canonical logics,1 however the question does become engagingly interesting if we start asking about canonical frames for the same logic which are built up from different starting blocks. Recall that a canonical frame is made up of maximal consistent sets of formulae within some language over a set of propositional variables. If we were to vary that set of variables we would get different canonical frames. Of course, if we hold the cardinality of the set of variables ?xed, we would have no dif?culty in concluding that the different frames so produced were isomorphic. But what if we use sets of propositional variables of different cardinalities?

1Logics where the canonical frame is a frame for the logic.

Now, the size of a canonical frame is easy to compute (we will assume that all sets of propositional variables are in?nite): For a set of propositional letters P of cardinality >=, the size of the language, S (P ), would then be >=. Then how many maximal consistent sets could there be? Well, at least 2>= many because each subset of P gives rise to a distinct maximal consistent set, and at most 2>= because each maximal consistent set is a subset of S (P ). So we have

Proposition 1.1. FLP = (XLP ; RLP ), the canonical frame for a logic L formed using propositional variables P , has size 2>=.

Now, it is a fact [1, p. 427], that 2! = 2!1 is relatively consistent with the axioms of ZFC, and so we can reasonably ask ?Is FL! ?= FL!1 when the axiom 2! = 2!1 is added to ZFC??
Without thinking about it long we can see that if such an isomorphism were to exist, then it would have to look a little strange; It wouldn't be a map which is just the result of a bijection between the underlying set of propositional variables?the sets are of different cardinalities so no such bijection could exist. Such a map is called standard.

De?nition 1.2. A map f : FLP ?! FLQ is called standard if it

satis?es2

(8p 2 P ) (9' 2 S (Q)) [f (kpk) = k'k] :

So, in answering our question we will, necessarily, be delving into questions about non-standard maps. Even the most ardent supporter of the continuum hypothesis (a set theoretic assumption which would totally demolish our current basis of investigation) will agree that any such investigation will shed light on how canonical frames of dfferent cardinality can relate to each other even when we are not restricting our attention to isomorphisms. After all, there is the open problem of GOLD- BLATT:
Does FL! ffl L =) FL!1 ffl L?

This problem has failed to yield to investigation and any attempt to analyse the relationships between canonical frames of different cardinalities will certainly help in efforts to solve this problem.
If we did want to do away with set theoretic assumptions totally, however, we can simply ask how many (non-standard) automorphisms a canonical frame may have.

2k'k, for ' a formula, is the set ?x 2 XLP j ' 2 x?.