# The Thesis of Modal Realism

A few post ago I was talking about formal systems.  As an example I talked a little about Hofstadter’s example of the MIU system and promised to talk about decision procedures.  I would hate to be a liar and so I do plan on returning to this matter.  However, it seems appropriate to me to take a diversion into the philosophical foundations of Tegmark’s view before delving into the more mathematical aspects.  For this reason, and in keeping with the announcement of my previous post, I am going to interact a bit with David Lewis’ work On the Plurality of Worlds.

So, what is Modal Realism?  From what I gather, the name was actually coined by Lewis, though he expresses certain regrets for calling it this based on some ambiguity in the term ‘realism’ and what it has come to mean in academia.  As one who is less familiar with the larger body of literature on realism, I find the name perfectly appropriate and satisfying.

The term ‘modal’ comes from modal logic which analyses the notions of necessity and possibility.  G.E. Hughes and M.J. Cresswell summarize this discipline similarly:

Modal logic can be described briefly as the logic of necessity and possibility, of ‘must be’ and ‘may be’.

‘Realism’ as Lewis intends to use it deals with ontology.  Put together, modal realism is

… the thesis that the world we are part of is but one of a plurality of worlds, and that we who inhabit this world are only a few out of all the inhabitants of all the worlds.

In other words, modal realism takes possible worlds semantics quite seriously as well as literally.  Upon reading this description, one may be tempted to think of modal realism as a multi-verse theory and indeed it is (we’ll see that it corresponds to Tegmark’s Level IV multiverse), but not in the sense often portrayed in popular literature.  The worlds of modal realism are not some distant universes located some vast, albeit finite, distance away.  Nor are they located in some alternate dimension of this reality.  As Lewis puts it,

They are isolated; there are no spatiotemporal relations at all between things that belong to different worlds.  Nor does anything that happens at one world cause anything to happen at another.

One should liken this to the way in which the vector space $\mathbb{R}$ is isolated from the finite field $\mathbb{Z}_{7}$, to sneak in some mathematics.

Lewis goes on to say,

The difference between this and the other worlds is not a categorial difference.  Nor does this world differ from the others in its manner of existing.  I do not have the slightest idea what a difference in manner of existing is supposed to be. [emphasis mine]

Thank you!!!  I expressed this very sentiment in A Philosophical Interlude.  It always feels amazing when one’s own thoughts are legitimized by world renown experts.  Okay, enough tooting my own horn.  The idea, here, is that some things exist, say, on Earth, while other things exist elsewhere and this is a difference in location of existing things (not a difference in any mode of existence).  Similarly, though somewhat more controversial, some things exist in the past, others now and still others in the future.  This is a temporal difference (we will deal with the nature of time in another post).  To extend this further, it seems reasonable that the same could be said for worlds.  Some things exist in this world and other things exist in other worlds and this is still a difference between existing things as opposed to a difference in any mode of existing.

For those not familiar with the language of possible worlds, I should hasten to point out that a world need not be a synonym for a universe (though it might be).  Rather, a world is a complete state of affairs, an entire reality.  Thus, in our world, if there is a God who created the universe, then clearly our world is more than just our universe.  Alvin Plantinga in his book The Nature of Necessity defines a possible world to be a maximal state of affairs and a maximal state of affairs $S$ is one such that for every state of affairs $S'$, $S$ includes $S'$ or precludes $S'$.  Note the similarity of this definition with that of completeness for a mathematical system:

A system is complete if, for every statement $p$, we can find a proof of $p$, or a proof of not-$p$.

Here “proof” refers to a demonstration within the system in question using the axioms and the rules of inference. In other words, a “proof”, here, of a statement $p$ within a system $S$ “consists of a sequence of statements, each of which is either an axiom or a logical consequence of certain preceding statements in the list, such that the last statement in the list is $p$.”

At this point, we needn’t get bogged down in technicalities.  The basic idea is that modality is best understood by actually taking possible worlds semantics as more than just semantics.  According to modal realism, all possible worlds are actual worlds, where actuality is taken to be indexical.