The notion of *possible worlds* has proved incredibly frutiful in providing formal semantics for various systems of quantified modal logic. Perhaps so fruitful that philosophers interested in related issues such as semantics for the terms ‘necessarily’ and ‘possibly’ of natural language or in the metaphysical nature of necessity and possibility often make use of the notion of possible worlds to frame their discussions. In terms of a formal system of quantified modal logic, possible worlds serve only as a kind of *index* in the mathematical structure which provides semantics for the system. If possible worlds are employed in more properly philosophical discussions of natural language semantics or modality, one might wonder (with good motivation!) about the ontological status of these possible worlds. Are they *concrete* just as the *actual* universe is as David Lewis suggests? Or are they in some sense or other *abstract*?

Michael Jubien has suggested in seminar(s?) at UF that a specific approach for understanding possible worlds as *abstracta* leads to contradiction. Specifically, he considers the proposal on which a possible world is taken to be a maximally consistent set of propositions which are true in that world. For example, if a possible world contains the proposition expressed by the sentence ‘the cat is on the mat’ then in that possible world, the cat *is* on the mat. Jubien claims to demonstrate that this understanding of possible worlds is untenable because it leads to contradiction on the assumption of just a few straightforward, unexceptionable claims. Here’s his argument:

**Assumption 1**: for any proposition, p, there is a set of all propositions entailed by p.

**Assumption 2** (the proposal for understanding possible worlds): Let W be a possible world understood as a maximal, consistent set of all propositions which are true in that world. Assume W exits.

**Step 1**: Let E be the set of all propositions entailed by each member of W. By maximality of W, for any proposition q, either W entails q or W entails ~q (the negation of q).

**Step 2**: Let E^{–} be the set of all negations of members of E.

**Step 3**: Let **P** be the set whose members are all members of E ∪ E^{–}.

**Step 4**: Then **P** is the set of all propositions.

**Step A**: Consider 2** ^{P}**, the power set of all propositions (the set of all set subsets of

**P**). Each member of 2

**is a (possibly infinite) set of propositions.**

^{P}**Step B**: For each member s of 2** ^{P}**, consider the proposition expressed by the sentence ‘some members of s are true’. Clearly, if s ≠ s’, where both s and s’ are members of 2

**, then the corresponding propositions are distinct. So there are at least as many propositions as there are member of 2**

^{P}**.**

^{P}**Step C**: By Cantor’s Theorem cardinality(2** ^{P}**) > cardinality(

**P**). So there are more propositions than there are members of the assumed set of all propositions. This is absurd so W doesn’t exist – that is, we give up Assumption 2.

So far, so good. But I believe that the involvement of W is a sort of red herring. I think we can show that the same sort of contradiction follows on only the assumption that there are infinitely many propositions. Specifically, the argument goes like this:

**Assumption 1′**: There is an infinite set of propositions.

**Assumption 2′**: For an arbitrary set, s, of propositions (possibly infinite) a proposition can be such that it asserts that some of the members of s are true.

(Now Steps A’, B’ and C’ are analogs of Steps A, B and C from Jubien’s argument)

**Step A’**: Let **P’ **be an infinite set of propositions, and so 2** ^{P’}** is the set of all subsets of

**P’**.

**Step B’**: For each s ∈ 2** ^{P’}**, let p

_{s}be a proposition which asserts that some of s are true. (Recall that the members of s are propositions.)

Step C’: Since for each s ∈ 2** ^{P’}**, there is a corresponding p

_{s}, and cardinality(2

**)> cardinality(**

^{P’}**P’**), there must be more of {p

_{s}} (the set of all p

_{s}‘s) than there are propositions, but each member of {p

_{s}}

*is*

*itself*a proposition. Contradiction.

The only assumption that is suspect is Assumption 2′. And so it seems that the *expressive power of propositions* given that there are an infinite number of propositions is what causes this inconsistency. Interestingly, the same problem does not arise if we speak of *sentences* rather than *propositions* in Assumption 1′ and Assumption 2′. I’ll address this issue in a later post or in a comment on this one.

– Jesse Butler

on January 25, 2007 at 4:23 am |Craig EwertYou don’t need propositions, either.

That result is one of the basics of set theory. Russell spent a lot of work on it.

Usually, it isn’t assumption 2 you throw out, but assumption 1. When you gather together all of the propositions, or all of the sets, that thing isn’t a set. It’s less well behaved.

At least, when I’m faced with these paradoxes, it’s the first assumption I discard.

on January 25, 2007 at 9:42 am |Andrew BaconI don’t think the result will work for sentences. It is relatively easy to show that the number of sentences is countable (that is, if the alphabet is countable – in our case its finite).

If you could derive a contradiction for sentences then we would not even have the option of appealing to proper classes since no proper class bijects with the naturals. However I do not think the result follows. Suppose you could form the sentence ‘some member of s is true’ for each set of sentences s, then you would need a way of constructing names for each set of sentences. But this is impossible since there are only countably many strings but uncountably many sets of sentences.

The disanalogy with sentences comes from the fact that sentences aren’t as expressive as propositions. The idea is that for any truth condition there is a proposition that captures it so the construction ‘some member of s is true’ should represent a proposition for each set of propositions. With sentences this just isn’t the case – for us to grasp a language the sentences over that language must be recursively enumerable and hence countable. A similar result can be found in computer science – you can construct domains which are isomorphic to their own function space (D =~ D^D) – *but* only if we consider only recursive functions – if we considered all functions we’d fall into problems with cantor’s theorem again.

on January 25, 2007 at 10:15 am |Andrew BaconWhoops, I seem to have missed a crucial ‘not’ when I read (misread) your post. I thought you where claiming that there was an analogous result for sentences. My apologies.

Also just out of interest, Jubien’s original argument was aimed directly against propositions I think (see Jubien, Michael. 1988. “Problems with possible worlds” – David F. Austin (ed.), Philosophical Analysis: A Defence by Example. Kluwer Academic Publishers, Dordrecht.) There are also variants of the argument aimed directly at possible worlds – e.g. Kaplan’s paradox – that for a given person and time, for any proposition (set of worlds) it is possible that that person is thinking that proposition at that time (and no others). This gives an injective map between worlds and sets of worlds and is problematic for the same reasons.

on January 25, 2007 at 4:59 pm |Rico VitzThanks to Brit Brogaard , of Lemmings, for providing a link to this discussion.

I encourage those who are interested in discussions of Language, Epistemology, Metaphysics, and Mind (hence ‘LEMMings’) to check out her blog and become a reader, if you aren’t already.

on January 27, 2007 at 12:52 am |Jared WarrenYou need to assume that there is a set of all propositions, not just an infinite set, in order to generate the paradox.

The result is essentially Cantor’s paradox. Generally, whenever we have a set S of all Fs, with cardinality k, if we can generate an F for each and every member of the power set of S, we have a paradox. This can be done for propositions, sets, and possible worlds. See Lewis’s discussion in ‘On the Plurality of Worlds’ section 2.3 where he discusses Kaplan’s paradox (mentioned by Andrew above).

As Craig pointed out; the common response in such cases is to deny that there is a set of all Fs.

It’s a bit of a tangent, but I also want to point out that Lewis hedges a bit about calling worlds ‘concrete.’ See section 1.7 of OtPoW where he has a nice discussion of the abstract/concrete distinction.

on January 27, 2007 at 4:05 pm |Lee WaltersJared is correct, infinity is not doing any work here.

I think there is something fishy about this puzzle for possible worlds or else the thesis under attack is loosely formulated. Presumably, and intuitvely, possible worlds are constituted by atomic propositions and the negations of atomic sentences, so that for every atomic proposition a pw contains it or its negation (there may be some complexities to-do with truth-value gaps but these are orthogonal) and quantificational propositions (and conjunctive and disjunctive) follow from those but are not contained in the set of props that constitute the pw. If so then quantificational props like “some member of s is true” does not generate a problem.

Again intutively such metalinguistic truths as “p is true” are not what we want to include in the original constitution of pw and so such truths need not generate a problem.

The pw theorist can plausibly claim there is a set of all atomic propositions. What Jubien needs to do is generate an atomic proposition such that it or its negation is not included in the pw.

Also the pw theorist may only want/need to include truths about determinates and not determinables: that the shirt is scarlet is enough and entails that it is red, so this further truth need not be included in the set of atomic propositions the pw theorist needs to appeal to.

on January 27, 2007 at 5:56 pm |Andrew BaconJared and Craig mention evading the problem by appealing to proper classes, but I’m not sure that always does evade the paradox. Especially in Lewis’s case, because of the mereological principles he espouses. If Kaplan’s paradox really does give us an injection then we can form a proper class of ordered pairs, the second element giving us the *fusion* of a class of worlds that represents a proposition and the first giving us the world in which that proposition is uniquely thought by person X. This proper class acts as a class function, f, and if you consider the class C of worlds, w, such that w is not a part of f(w) you can show that: w is a part of f(w) iff w is a part of the fusion of worlds which constitutes a proposition uniquely thought by x, iff w *belongs* to that class (since worlds are mereologically disjoint according to Lewis), iff w is not a part of f(w) (by the definition of C). This gives an analogous contradiction.

Of course, if you reject some of the mereological principles you can see that this doesn’t form a contradiction. The other crucial premise is that worlds are disjoint. Cantor’s theorem doesn’t hold for class *or* classes + mereological principles – it only holds in this particular case because worlds are disjoint. So people who believe in transworld identity do not necessarily have a similar problem. (Also note that it is possible to construct ordered pairs – the appendix to Lewis’s ‘Parts of Classes’ shows how this is possible.)

on January 27, 2007 at 6:00 pm |Andrew BaconSorry – I missed out something crucial – w in the argument should be the worlds such that f(w) = the fusion of C.

on January 29, 2007 at 4:58 am |Show-Me the Argument » Philosophers’ Carnival #42[…] A Trouble for Possible Worlds as Maximally Consistent Sets of Propositions … 11 Florida Student Philosophy […]

on January 29, 2007 at 8:33 am |Jason ZarriPatrick Grim has used arguments very similar to this to show that there that there can be no totality of propositions, among other things. There’s an interesting series of exchanges on the subject between Grim and Plantinga here:

http://www.sunysb.edu/philosophy/faculty/pgrim/exchange.html

If you’re interested in these issues you should check it out.

on January 29, 2007 at 2:26 pm |Rico VitzThanks to the “Philosophers’ Carnival” for the link to this discussion. I encourage our readers to check out the work of the Mizzou grads at Show Me the Argument — see our links page.

on February 5, 2007 at 5:29 pm |TomYou guys might be interested in this exchange:

PLANTINGA, ALVIN and GRIMM, PATRICK,

Truth, Omniscience, and Cantorian Arguments: An Exchange , Philosophical Studies, 71:3 (1993:Sept.) p.267

Not-so-great text version available here:

http://www.sunysb.edu/philosophy/faculty/pgrim/exchange.txt

on February 20, 2007 at 8:18 am |Lee WaltersI’m just reading Scott Soames’ paper “Actually” and cam across this “Details aside, a Carnapian state description [which Soames uses to define a pw] is a complete, consistent set of atomic sentences of L, or their negations (resulting in a complete assignment of truth values to atomic sentences). Truth values of complex sentences relative to a state description are determined using familiar recursive clauses for quantifiers, truth functions, and modal operators.

This is more or less what I said above. This problem for pw is only a problem because the notion of a pw was incorrectly specified.

on February 20, 2007 at 1:46 pm |Jesse ButlerI think Michael Jubien — it was his lecture that I tried to summarize in the first part of the original post — claimed only that the result caused a difficulty for possible worlds understood as “maximal consisent sets of propositions” (if I remember his exact turn of phrase correctly). In the seminar, he was warming up to present his view of modal semantics and wanted to flag difficulties with various other views in preparation. He first attacked (his version of) a Lewisian approach: possible worlds as concrete universes unrelated spatio-temporally, then turned to consider other ways in which one might reasonably construe a possible world. The suggestion that possible worlds could be understood as maximal, consistent sets of propositions may have been roughly the position of Platinga and Stalnaker. I haven’t read Soames “Actually”, but if he is indeed basing his notion of a possible world on that of Carnap’s state descriptions, then, for Soames, a possible world will just be a set of atomic sentences which includes for every predicate term ‘P’ and singular term ‘a’ either ‘Pa’ or ‘~Pa’.

For this understanding of possible worlds, it doesn’t seem that a criticism like the one presented in the first part of this post will find any purchase. Of course, other, different criticisms of this view will be available to someone in Jubien’s position…

In sum, there are different ways to understand exactly what “possible worlds” are. One may have a personal preference for which construal to hold. To offer a competing proposal for modal semantics that’s convincing, one must consider (and hopefully find reason for rejecting) various positions. In the particular case of modal semantics, doing so requires considering each of the ways of understanding possible worlds — or at least a representative sample of the mainstream views on the subject.

on February 21, 2007 at 5:07 pm |Lee WaltersUnderstood. My point is that Jubien is not giving a charatible reading of “maximal”.

on April 23, 2007 at 11:50 pm |Andrew BaconActually, I think I remember that there were also cardinality objections against the Carnapian version of possible worlds. I think Lewis, but I can’t remember where, said that, since there are countably many atomic propositions, and hence continuum many sets of atomic propositions, there will be at most continuum many maximally consistent sets of atomic propositions. However the possibility space is much larger than this, for example there should be a possible world in which a given region of space is occupied with matter but everything else is unoccupied. Since there are 2^continuum many regions of space there should be more possible worlds than Carnapian possible worlds.

on April 24, 2007 at 7:04 am |Lee WaltersThat’s right, but this is a different objection to Jubien’s. In any case Lewis later realised that his objection rests on assuptions the Carnapian need not accept and so fails. See OTPOWs pp. 143-144

on June 3, 2009 at 3:53 pm |Laureano LunaI don’t think the set theoretic problems encountered by the interpretation of possible worlds as maximal consistent sets of propositions can be avoided by the Carnapian turn.

Grim in ‘The Incomplete Universe’ MIT Press, 1991, 95, gives an argument against the existence of a set of all atomic contingent propositions.

Assume such set exists and call it SAC = {p, q, r, s, … }. Let C be the set of connectives {~, v, Exists}. Consider SAC union C. Call it M. Consider P(M). There you will find as members sets like {p, v, ~q} or {Exists, p}. It is easy to see that with minor changes you can get the set of all (atomic and non atomic) contingent propositions from P(M).

But we know that set can’t exist. Assume it exists and call it CP. Take P(CP) and any contingent proposition c; assign to each member s of P(CP) the proposition ‘c and c is in s’.

These propositions are contingent and there as many of them as members of P(CP). Hence there are more contingnet propositions than propositions in CP, which is absurd.

Therefore, the existence of the set SAC of all atomic contingent propositions seems incompatible with the axioms of the standard set theory.

Regards