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Possible Worlds Semantics

Thus far, three of our episodes (12, 25 and 28) have contained some discussion of possible worlds semantics.  Most memorably, we learned in our last episode that John Searle is rather critical of the enterprise.  But what is possible worlds semantics?  Let’s take a look.

This possible worlds business originally stems from the work of 17th-century philosopher and polymath, Gottfried Wilhelm Leibniz.  Leibniz got the ball rolling on possible worlds by putting forth the idea that a statement is necessarily true just in case it is true in all possible worlds, and possibly true just in case it is true in some possible world.  The idea also plays a role in his controversial view that we live in the best of all possible worlds.  Leibniz was resolutely mocked for this idea in Voltaire’s Candide.

Ok, fine.  All well and good, but we still haven’t said what a possible world is.  Well, the idea is intuitive enough.  Here are two ways to think about possible worlds.  First, you can think of a possible world as a way the world might be.  In actuality, Barack Obama was elected president in 2008.  However, he might not have been elected president in 2008–John McCain might have won the election instead.  So we can say that in addition to the possible world where Obama was elected president, there is another possible world in which McCain, rather than Obama, was elected president.  What does that mean?  Well, if you think about it in terms of this ‘way’ talk, it means that there’s a way the world actually is: i.e. in actuality, Obama is president.  And there’s also another way the world might have been: i.e. the world might very well have turned out such that McCain was president.

Second, perhaps a bit more fancifully, you might think of a possible world as a coordinate in logical space.  The idea of logical space comes from 20th-century philosopher Ludwig Wittgenstein.  To get a feel for what logical space is, consider an analogy to physical, 3D space.  Let’s label the point in the middle of the top of my desk <0, 0, 0>, and think about what makes this point the point that it is.  Part of what makes it that point is its relation to all the other points–that it’s to the right of <-1, 0, 0>, that it’s to the left of <1, 0, 0>, that it’s below <0, 0, 1>, that it’s twice as far away from <0, 0, 2> as it is from <0,0, 1>, and so on and so forth.  Draw up a list of all the points (it will be infinitely long, but don’t worry about that for now), say something about how they’re all related to one another, and you’ve got yourself a description of 3D space.

We can describe other kinds of space in more or less the same way.  For instance, take the color red.  If my mittens are red, then they aren’t yellow and they aren’t blue–their color is at a particular ‘location’ on the visible light spectrum, as opposed to a bunch of other locations on that spectrum.  So write down all the colors of the rainbow, rank them by wavelength, and you’ve got a description of color space.  Wittgenstein’s idea was that you can even think of claims as being ‘located’ in a kind of space.  Our claims, after all, are related to one another–they aren’t all independent.  If I say that I am at home and my home is at 400 Park Avenue, then it follows that I am not at 200 Madison Avenue.  So note down all the different ways the world might be, say which of them are incompatible (a person can’t simultaneously be over here and over there, nor simultaneously hungry and full, etc.), and you’ve got a description of logical space.

Possible worlds semantics is based around this second idea.  Claims, or what philosophers like to call propositions, can be thought of as sets of possible worlds.  For instance, in possible worlds semantics, the proposition ‘Matt has a dog’ would be thought of as the set of worlds in which Matt has a dog.  So there’s a world in which Matt has a dog and he lives in a blue house, a world in which Matt has a dog and he lives in a red house, a world in which Matt has a dog, lives in a blue house, and has three sisters, a world like the last but in which he has two sisters, and so on ad infinitum.  Furthermore, there will also be infinitely many worlds that aren’t contained in the proposition ‘Matt has a dog': the world where he only has a parakeet, the world where he only has an iguana, the world where he has no pets, etc.  Since each possible world can be thought of as a point in logical space, and every claim you make can be thought of as a set of possible worlds, that means that every claim you make can be thought of as a region in logical space.  ‘Matt has a dog’ draws a dividing line between all the ways in which Matt could have a dog and all the ways in which he could fail to have a dog.

This notion of what a claim is has all sorts of uses in philosophy and linguistics, though we won’t go into most of them here.  (As you may remember, Robert Stalnaker discusses some applications of possible worlds semantics in our conversation with him.)  For now, we’ll just mention one.  Possible worlds semantics gives us a neat way to describe what happens when one claim logically follows from another.  In general, B logically follows from A just in case it is impossible for A to be true and B false.  Using the apparatus of possible worlds semantics, we can rephrase this idea by saying that B logically follows from A just in case A is a subset of B–just in case all the worlds contained in A are also contained in B.

Consider the following example.  If I own a black puppy, then it logically follows from that claim that I own a puppy.  (Not the most exciting claim in the world, but it will suffice!)  How come?  Well, there’s simply no way I could own a black puppy without also owning a puppy.  If the proposition that I own a black puppy is the set of all the ways I could own a black puppy, and the proposition that I own a puppy is the set of all the ways I could own a puppy, then clearly the one is contained in the other.  Furthermore, recall that if we think of a possible world as a point in logical space and a proposition as a set of possible worlds, then we’re thinking of a proposition as a region in logical space.  This gives us a nice way to visualize logical consequence: B logically follows from A just in case the region that A occupies in logical space is contained in the region that B occupies in logical space:

Another way to put the idea is that possible worlds semantics gives us a way to define the relative strength of a claim.  Here, ‘I own a black puppy’ is the stronger claim, because it’s saying more than ‘I own a puppy.’  It’s giving more information.  Having a formal definition of the strength of claims allows us to do all sorts of fun stuff.  For example, it lets us imagine every claim we could ever make arranged in a ranking according to its strength, then to try to figure out what the mathematical properties of that ranking are.  (If we’re so inclined.)

But the true power of possible worlds semantics lies in its ability to give a precise characterization of possibility and necessity, or what philosophers like to call modality.  This is a huge topic, which we’ll have to save for a later post.  Hopefully, though, we’ve managed to give you at least a glimpse of what philosophers are on about when they allude to other possible worlds.

Matt Teichman

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2 Responses

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  1. Mark Balaz says

    This whole idea of included and not included reminds me of basic computer programming books with all those tree diagrams and such.
    Quite particular subjects indeed!!!

    • Matt Teichman says

      Indeed! Yeah; that’s set theory for you. Immensely valuable for both computer programming and semantics. And much more! If you’re curious about the origins of set theory, this is how it used to look back when it was invented in the 19th century, before we had either computers or semantics. (pp. 14-58.)

      http://www.gutenberg.org/files/21016/21016-pdf.pdf



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