# Possible worlds in logic

In modern logic there is often talked about possible worlds. I understand this idea in this way:

Every proposition may be true or false. Possible world W is a set of all propositions where there is not logical contradiction among them. And the set P of all W is the possible worlds set.

Could you describe a few core ideas behind this concept and why it's useful to incorporate possible worlds into logical reasoning and semantics? Probably the basic ideas originated from Gottfried Leibniz.

What was the flaw (bug) in logical reasoning or semantics without possible worlds, so that ancestors (e.g. Gottfried Leibniz) came with possible worlds idea to correct this flaw? Could you show this contrast?

• I'm not sure, but I guess if you have many worlds, and in each world every proposition is either true or false, then the propositions (probably) form a Boolean algebra. Most important, the distributive law will still be valid (as will be all other laws of "classical" logic). I recently read an essay by Hilary Putnam about the logic of quantum mechanics, where he tried to explain suggestions by Birkhoff, von Neumann and later supporters of their suggestions. He explained how the subspaces of a Hilbert space replace "classical" logic propositions, and that only the distributive law is violated. Commented Mar 6, 2012 at 18:28
• @ThomasKlimpel Thank you for tip. This one? In quantum physics there is an interesting problem about deciding if electrons are particles or wave (both are possible). Commented Mar 6, 2012 at 18:48
• It actually was this one socsci.uci.edu/~dmalamen/courses/prob-determ/Putnam.pdf from 1968. I'm not sure whether it is actually so interesting. But it wasn't too complicated either, and the logic is close enough to common logic. Commented Mar 7, 2012 at 0:56
• I don't know anyone who has used possible world semantics to explain quantum mechanics, but I know David Deutsche has advocated a 'multiverse' of infinite actual worlds all of which realise the probabilities encoded in the Schrödinger equation. But QM Is off topic I guess! Commented Mar 25, 2012 at 11:55

Possible worlds have been used as a way to understand modalities like necessity and possibility. For instance something is necessary if and only if it is true in all possible worlds. This is a way of trying to understand what it means to say that X is necessary.

More basically, you can use possible worlds to cash out the idea of an inference's being valid: the inference from X to Y is valid if and only if Y is true in every possible world where X is true.

Possible worlds have also been used to give a semantics for counterfactual conditionals, though this is more controversial. How do you evaluate conditionals like:

• "If I were to drop this glass, it would smash"
• "If I were to drop this glass, it would turn into a porcupine"

Now, as a matter of fact, I don't drop the glass. Thus the antecedent of both conditionals is false, so if they were material conditionals, both would be true conditionals. But intuitively, the first is true, the second false. So we need a better understanding of conditionals like this. David Lewis offered an account of what makes the first true and the second false in terms of possible worlds. The idea is that in the closest possible world where I drop the glass the glass smashes, and it does not turn into a porcupine. Thus the first conditional is true, the second is not. Obviously, the devil is in the detail and much ink has been spilled trying to get a handle on closeness of possible worlds. But the intuitive idea should be clear.

I don't think possible worlds fix a flaw or bug in logic: they are a neat conceptual tool to make understanding things easier.

• So, there are two basic usages? 1) inferences and 2) counterfactual conditionals? Is this all? The first thing seems to me more interesting. Is there some reading material about this? The work of David Lewis I have found already but I'm not sure if this is the right thing what he wants to achieve "something that is empirical (depending on possible world) he wants to make truth because of intuition", but maybe interesting reading. Commented Mar 8, 2012 at 14:09
• No I think both are more or less the same kind of use. It's about interpreting a aspects of a formal language in an intuitive way. In both cases you're trying to interpret sentences of a formal language in an intuitive way by giving "meaning" to the symbols by associating them with aspects of possible worlds. Hmm. That probably isn't very clear. Commented Mar 8, 2012 at 14:33
• I will wait for a while if somebody else wants something to say. It's not clear enough for me yet. Commented Mar 8, 2012 at 17:23
• What isn't clear? The stack exchange format works best if you ask specific questions with determinate answers… Commented Mar 8, 2012 at 22:49
• The format is for more answers. Your is the best so far. David Lewis work is from 1973, Gottfried Leibniz dies at 1716. There is a gap. It seems suspicious for me that 200 hundred years it was used for nothing. So, I want to wait for some time yet. There is no need to hurry with closing the question. Commented Mar 9, 2012 at 8:15

I think it's just used to help people understand validity and truth with regards to necessity and possibility, first by having people realize that one thing needn't be true on this world for the statement to be possibly true. But also, through this use of this "possible worlds" idea, it is easier to also demonstrate necessity, such that something that is necessary must exist on all possible worlds (or none) as well.

While Gottfried Leibniz may have been credited with it (which seems somewhat odd to me), the notion certainly has existed for a very long time, perhaps not under the "possible worlds" name though; but really any discussion of necessity or possibility invokes exactly what the concept of "possible worlds" is supposed to help illustrate: in which instances ("on what worlds?") is X proposition true?

To answer your final question, it was indeed quite likely used to help highlight to people the scope of their truth (or falsity) claims, such that they are either necessary or contingent, for example.

• I understand that this is related to necessity and possibility (contingency). That `7 is a prime` is true in all possible worlds and `Barack Obama is a president of the USA in 2012` is true only in some worlds. But what is this good for? When we reason about something we work with our actual world, not with the possibility of another world. This is what I don't understand. Commented Mar 6, 2012 at 16:38
• I have an idea that this can be useful for looking for contradiction. We can say "Proposition A could not be true in this world because propositions B, C and D certainly hold.". But I haven't seen some utilization of this concept yet. Maybe we work with "possibility" when we don't know the actual world or know only part of it (actually we always know only part, I think). Commented Mar 6, 2012 at 16:42

As has already been mentioned, Possible Worlds are used to do quite a lot of work in Metaphysics and the Philosophy of Language. For example:

• In Philosophy of Language, one usually tries to understand the semantics of the things we say. What makes it true that

• (1) "Had I eaten less breakfast this morning, my stomach wouldn't have hurt so much",

but false that

• (2) "Had I eaten less breakfast, I would be President of the United States"?

The answer, according to most people in the field, has something to do with some notion of possible worlds. The simple story usually goes like this: In all the closest possible worlds where I ate less breakfast, I also don't have a tummy ache; but in none of those worlds I become president. Therefore (1) is True, (2) is False. The details are crazy complicated and literally fill books (e.g., Bennett's A Philosophical Guide to Conditionals)

• In Metaphysics, one of the many uses of possible worlds has been in the theory of Causation. For C to be a cause of E would mean that: "Had C not occurred, E would not have occurred", and that is true if (in some interpretations) the closest possible world in which C did not occur, E did not either. (This is an equally hotly debated question.)

• There is also a whole branch of logic called modal logic in which operators such as Possibly, Necessarily, etc. are studied. From what I gather (don't quote me on it), lots of work there involves possible worlds.

Leibniz was indeed one of the first to bring up possible worlds, although he used them in much more restricted fashion. (For much more on that, start with http://plato.stanford.edu/entries/leibniz-modal/). From what I gather, his interest was partly just to explain modal operators (possibly, necessarily) and partly to give an account of freedom within determinism.

Lastly, possible worlds as a tool in philosophy indeed seem to have only had a resurgence in the latter half of the 20th century. I'll leave it to someone with more knowledge of the history of philosophy to explain that one. Of course, most people after Kant seemed to have somewhat given up on Leibnizian thoughts.