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Saturday, July 10, 2010

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Hello Dr. Vallicella,

If I am not mistaken, Cantor's argument was not that there was not a set of a real numbers. There surely was a set R in his mind. He just argued that there was not a one-to-one correspondence between sets N and R. From this, and the fact that N is a subset of R, he concluded that the cardinality of the set R is larger than that of N. Part of the proof was a listing of R such that it was indexed by N. He showed that such a listing is incomplete by creating a real number not in the list. So, one may argue from Cantor's argument that one cannot list every true proposition, but this is not to say that there cannot be a set of all truths. The set may just be denumerable, and therefore not listable.

Brian

Hi Brian,

I'm afraid I don't understand your comment at all. It seems to have nothing to do with what I wrote. You seem to think I am claiming that for Cantor there is not a set of real numbers. But nowhere do I say that or imply it or suggest it.

You also seem not to appreciate that the argument given is not an argument from Cantor, but an argument that employs as a premise Cantor's Theorem. I make clear in the post that the argument is from Patrick Grim.

Dr. Vallicella,

You are right. Please forgive me. My comments were not pertinent at all.

Sincerely,

Brian

Great argument, it's a fascinating result! I get the same weird vibe from it that I get from your premise 5 in the "Nothing" post (as per my last comment on that thread). But this time in a positive sense: I don't know how exactly to put it, perhaps one way to put it is that your argument shows the danger of allowing an object-language to include its metalanguage (when of course the object-language is just a fragment of the metalanguage). Sorry to be vague, I wish I could be clearer.

I don't actually hold the view that possible worlds are sets of propositions. So, I'm not at all familiar with the view that possible worlds are maximally consistent sets of propositions, and below are some questions that could help me better appreciate the view and your argument against it.

Is T supposed to contain all truths including all atomic truths? Or not all but some atomic truths, and all the propositions that follow as matter of logical consequence from those atomic truths?

Assume that T contains all truths, including all atomic truths. Then it will contain atomic truths about some truth t in T and the sets to which it belongs. Then tautologies of the form "either t is in this set or it is not" can be derived as logical consequences of these atomic truths. So T, being maximally consistent, will contain these tautologies. But your argument proves the assumption false: T contains n number of truths, but the power set of T contains 2*n subsets, so that we can generate at least 2*n tautologies. Thus your argument shows that T cannot contain all truths, including atomic truths about its own members.

So, assume that T is supposed to contain not all but some atomic truths, and all the propositions that follow as a matter of logical consequence from these atomic truths. In particular, assume that T does not contain any atomic truths about its own members, hence none about which sets its members belong to. Then t* cannot be derived as a logical consequence of atomic truths in T. But that's OK. After all, T is not supposed to contain all truths, but all and only those that follow as a matter of logical consequence from whatever atomic truths it contains.

Your argument, then, shows that it is the second (not the first option) that should be assumed by people who endorse the view that possible worlds are maximally consistent sets of propositions.

(p.s. Not that it matters much, but what I have assumed in my previous comments in the other thread is that there is some domain or other containing objects that individual constants refer to, and variables range over. I have identified possible worlds with such domains or sets, but I could have just as well identified a possible world with the contents of such domains or sets.)

I'm also tempted to say that all tautologies express just one truth, which becomes apparent if we use the truth table notation. I'm not sure but Wittgenstein might have held this view. I'm also tempted to say that tautologies do not need to be derived from atomics....Am I wrong?

>>Then tautologies of the form "either t is in this set or it is not" can be derived as logical consequences of these atomic truths.<<

I'm afraid I didn't present the argument clearly enough. T is the set of all truths. P(T) is the power set of T, the set of all of T's subsets. Consider some truth t1 in T. For each subset s of T, either t1 will be a member of s or t1 will not be a member of s. Either way, we get a further NONTAUTOLOGICAL truth, e.g., t1 is a member of {t1, t2}. It is all these further truths that cause trouble and lead to a contradiction.

>>Your argument, then, shows that it is the second (not the first option) that should be assumed by people who endorse the view that possible worlds are maximally consistent sets of propositions.<<

But then isn't the maximality condition on worlds violated? Whatever a world is, it has to be maximal. If the actual world is the set of all true propositions, then ALL true propositions must be in it. One cannot exclude some atomic truths from the world and call it a world.

Either way, we get a further NONTAUTOLOGICAL truth, e.g., t1 is a member of {t1, t2}. It is all these further truths that cause trouble and lead to a contradiction.

Actually, that part of your argument though perhaps implicit was clear and so I had understood it. There have to be those further truths, and as far as I can see they have to be atomic truths (unless someone finds a way to reduce set-membership talk to say part-whole talk). We need them to generate the 2^n times n number of tautologies (e.g., via disjunction intro), which is clearly a greater number than n, and so I figured that's why you couched your argument in terms of those tautologies. Just taking the atomic truths of the form "ti is in si", you may be able to show that they have to be of some number m > n... I just don't have the math-smarts or the patience to work that out for myself.

But then isn't the maximality condition on worlds violated? Whatever a world is, it has to be maximal. If the actual world is the set of all true propositions, then ALL true propositions must be in it. One cannot exclude some atomic truths from the world and call it a world.

This is the feature of the view that I'm not clear about. I thought "maximality" meant:

(1) "maximal consistency", taking this to mean "containing all (not just some but not all) truths that follow as a matter of first-order logical consequence from atomic truths, and none that do not, and logically independent atomic truths that neither entail nor contradict one another."

But if "being maximal" means:
(2) "maximal completeness", understanding this to mean "taking all atomic propositions (or wffs), asking of them whether they are true or not (satisfied or not) in a given world, and then including all atomic truths and all logical consequences thereof,"

then I think your proof indeed shows that there cannot be any maximal set in sense (2). For any set T proposed to be the maximal set, there is the larger set T1 containing additional atomic truths about the set-membership of atomic truths in T (i.e., membership in the subsets of the power set of T), and likewise for T1 and the power set of (T1), and so on ad infinitum.

So if one seeks to construct any possible world as a set of propositions true in that world, it will be misguided to try to construct a set that is maximally complete. Just say that a possible world is to be defined or uniquely characterized in terms of the set of basic atomic truths, taking "basic atomic truths" to mean atomic truths that are not made true by the existence of other atomic truths. The set of basic atomic truths, given the tools of first order logic and set theory, will determine the existence of all the other truths, and will suffice to answer all the questions about that possible world that we can formulate in first order logic and set theory.

Correction (2nd para. of lst comment): Yes, I now see that you've provided a way of constructing n^2 times n number of non-tautological truths about the n members of T. Anyway, thank you for the discussion, I should go back to writing about ethics.

Correction (2nd para. of last comment): Yes, I now see that you've provided a way of constructing n^2 times n number of non-tautological truths about the n members of T. Anyway, thank you for the discussion, I should go back to writing about ethics.

Boram,

You should write about everything. Your comments are good, and in any case blogging is open-ended and exploratory. I go out on limbs all the time, and on occasion they get sawed off by an astute commenter.

Suppose we think of a world as a proposition. Obviously, not every proposition is a world. To be a world, a proposition need not be true, but it must be possibly true, and it must be maximal. A proposition is maximal just in case it entails every proposition with which it is BL-consistent. Accordingly, the actual world is the true proposition that entails every true proposition.

Maximality as I understand it combines both consistency and completeness.

You seem to be defining maximality as a property of sets. Furthermore, I am not grasping your distinction between (1) and (2).

Yes I misunderstood the bandied-about phrase "maximally consistent set of propositions", but I'm not sure what "BL" means. For what it's worth, what I meant by a "maximally consistent set" is one that contains all and only the logical consequences of some subset of logically independent propositions (taking members of that subset to be logical consequences of themselves). But that subset need not contain all propositions that are logically independent of one another. Our beliefs are not "maximally consistent" in this sense, because we have some inconsistent beliefs, and also we do not believe all the logical consequences of our beliefs. And what I meant by a "complete set" is one that tells us, of any proposition, whether it is true or false, so that it gives a yes or no answer to any question we might have. Actual works of fiction are not "complete" in this sense. But I don't rule out the possibility that a complete set might be a paraconsistent set, that it can give a yes-and-no answer to some question we might have. (E.g., it might tell us that at midnight between today and tomorrow, it is both today and not today.) So, a complete set need not be a consistent one.

Anyway, since you are urging me to be adventurous, I will venture one defense of the view that possible worlds are complete and consistent sets of true propositions. Take the revised view I put forward near the end of Comment # . According to that view, we cannot identify a possible world with a complete and consistent set of true propositions, but we can nevertheless uniquely characterize a possible world in terms of all the basic truths. By "basic" I mean propositions whose truth value does not depend on the existence of other truths, thus ruling out compound wffs constructed out of atomic wffs, and also ruling out atomic wffs of the form "This truth belongs to that set". Nevertheless, once we have all the basic truths, that seems to determine the existence or the non-existence of all the other truths, and we can work out the answer to questions of the form "Does this truth belong to that set?" with the aid of first order logic and the set theory of your choice.

With this revised view in one mind, one could propose a bolder view, which identifies a possible world with maximally consistent set of truths, and this set with the set of all basic truths. (Not that I think it is correct, I just find the line of thought interesting to pursue.) One could maintain this view if one denies that first-order logical consequences and truths of the form "This truth belongs to that set" are additional truths, in other words, claims that they are just redundant truths and that adding them to basic truths would be a form of double-counting.

The main motivation for this idea comes from considering logically equivalent statements. Are the material conditional "if p then q" and the disjunction "either not-p or q" two different truths, or the same truth stated in two different ways? Truth-functionally speaking, they are notational variants of one and the same truth, and the truth table notation perspicuously shows that they express the same truth. Since the logical connectives are interdefinable, if we take 'or' and 'not' as our primitive connectives, then we can even do without "if-then" and "and".

Can this idea be generalized? Perhaps. As Quine would say, definition is elimination. And it will not do to count the referents of the definiendum in addition to the referents of the definiens. Bachelors are adult males who've never married. If there are 10 bachelors in the room, it will not do to say that there are at least 20 people in the room, 10 bachelors and 10 adult males who've never married. Also, it will not do to say that "There are bachelors in the room" and "There are adult males who've never married in the room" are two different truths. They are the same truth stated in two different ways. So we should count them as one. But we should add among our basic truths the definition "bachelors =df adult males who've never married".

To generalize further, we may note that the truth values of compound wffs are defined in terms of the truth values of atomic wffs. When a proposition is true, I will say that its value v = 1. E.g.,
v(~P) = 1 iff v(P) = 0
v(P or Q) = 1 iff the sum of v(P) and v(Q) is at least 1
v(there is some x such that Fx) = 1 iff there is at least one object c such that v(Fc) = 1

These definitions show us how to eliminate the truth-assignments on the left hand side in favor of the truth-assignments on the right hand side, so one might argue that the compound truths on the left hand side do not add any new truths. We could also add these arguments. Negating any of the basic truths would result in falsehoods, so negating them cannot result in any new truth. Disjoining any of the basic truths results in loss of information, and what information the disjunction does contain is that at least one of its disjuncts is basically true. Existential quantification only conveys the information that there is at least one basic truth. And so on. These are all only vehicles for conveying atomic truths in different and convoluted ways. So the compound statements do not seem to commit us to additional truths. All we need to do is to add the above definitions among the set of basic truths.

Likewise for the axioms and definitions of set theory, which specify the rules for constructing sets. Just include your favored axioms and definitions you like among the basic truths, and various theorems and applied truths will follow as logical consequences of set theoretical axioms and definitions and of other basic truths. But these should not count as additional truths, just as "There are bachelors in the room" should not count as an additional truth above and beyond "There are adult males who've never married in the room" and the definition.

Boram writes,

>>what I meant by a "maximally consistent set" is one that contains all and only the logical consequences of some subset of logically independent propositions (taking members of that subset to be logical consequences of themselves). But that subset need not contain all propositions that are logically independent of one another. Our beliefs are not "maximally consistent" in this sense, because we have some inconsistent beliefs, and also we do not believe all the logical consequences of our beliefs.<<

That's clear, and I have no problem with it. But this definition has no relevance to a discussion of possible worlds for two reasons.
First, whatever worlds are they must be be all-inclusive, or maximal in that sense. For example, suppose you think of the actual world as a set of true propositions. One would have to add: set of ALL true propositions. That is obvious, is it not? Second, worlds cannot be sets because of the Cantorian argument above.

>>And what I meant by a "complete set" is one that tells us, of any proposition, whether it is true or false, so that it gives a yes or no answer to any question we might have. Actual works of fiction are not "complete" in this sense.<<

This is not so clear. Do you mean the following? A complete set S of propositions is a set of propositions which is such that, for any proposition p, either S has p as a member or S has ~p as a member. I agree that actual works of fiction are not complete in this sense. For example, *Captain Ahab suffered from erectile dysfunction* and its negation are neither of them or any logical consequences of them members of the set of sentences corresponding to Moby Dick.

Fictional characters (of the sort we finite beings are capable of producing) are always indeterminate with respect to many properties. You could say that LEM in its propery form does not apply to them. They are, then, incomplete objects in roughly Meinong's sense.

'BL' is short for 'broadly logical.'

More later.

Do you mean the following? A complete set S of propositions is a set of propositions which is such that, for any proposition p, either S has p as a member or S has ~p as a member.

Yes, that's clearer than how I put it.

Regarding my last comment, I hadn't worked out the argument fully, and stopped as soon as I came to sets. Now I don't think the argument for identifying possible worlds with a maximally consistent set of true propositions, and that set with the set of basic truths, can succeed. I thought it had some chance of success because it seemed to me that truths about the membership of truths in sets are in some sense trivial and derivative (in the sense that they can be derived from the existence of basic truths including set theory). But just because they are trivially true and derivative, does not mean they are redundant truths. For one thing, set-theoretic axioms seem to commit us to the existence of abstract entities, unlike merely stipulative definitions that do not commit us to the existence of additional entities. And even someone as nominalistically inclined as Quine believes that ontological commitment to sets is indispensable. So it seems that truths about the membership of truths in sets must count as additional truths. For another, set-theoretic axioms express general truths of the form "for all x, there is y...", "there is y such that...", etc., so we cannot include the axioms in the set of atomic truths without eliminating the general truths in favor of satisfied atomic wffs. This means we do have to add at least 2^n times n number of atomic wffs of the form "t is a member of s" among the basic truths.

Anyway, here's a new question regarding what you wrote:
A proposition is maximal just in case it entails every proposition with which it is BL-consistent.
And you wrote earlier:
As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from from being maximally consistent conjunctive propositions.


You characterize maximality in terms of universal quantification over propositions ("every proposition"), which usually would mean quantifying over the domain of propositions, i.e. a set of propositions. So I wonder if Cantorian worries arise for your proposal as well. Patrick Grim discusses this issue with Platinga in the link you provided to his webpage ("Truth, Omniscience and Cantorian Arguments"...). I just quickly scanned through that paper, but I see that Grim presents Cantorian arguments that apply not just to sets, but to properties and such, in short to pretty much any conceivable way of comprehending the totality of truths.

Dr. Vallicella,

I am a little disappointed you are bringing this up again. As I pointed out in http://lifetheuniverseandonebrow.blogspot.com/2009/07/my-oversight-apologies-and-thoughts.html, any collection of maximally consistent propositions of the type you mean must be a proper class. Proper classes don't have cardinality and don't have power sets, so Cantor's Theorem does not apply to them. Thus, there is no inconsistency in discussing a maximally consistent class of propositions.

Boram writes,

>>You characterize maximality in terms of universal quantification over propositions ("every proposition"), which usually would mean quantifying over the domain of propositions, i.e. a set of propositions. So I wonder if Cantorian worries arise for your proposal as well. Patrick Grim discusses this issue with Plantinga in the link you provided to his webpage ("Truth, Omniscience and Cantorian Arguments"...). I just quickly scanned through that paper, but I see that Grim presents Cantorian arguments that apply not just to sets, but to properties and such, in short to pretty much any conceivable way of comprehending the totality of truths.<<

Your objection is a good one. You're right that my definition of maximality involves universal quantification over a domain of propositions. But you identify a domain with a set. Is that obvious? Why couldn't a domain be a nonset? 'All the shoes in this closet are running shoes.' This is a universal quantification over a domain of shoes. Couldn't one deny that there are mathematical sets while admitting domains of quantification? One possibility is that a domain is a mereological sum. Only if you can show that a domain must be a set will your objection go through.

Maybe we understand 'set' differently here is my understanding:

A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set. A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

To sum up:

1. The Cantorian argument shows that there is no set of all truths.
2. Only if there is a set of all truths could possible worlds be identified with sets of propositions. For if possible worlds are sets of propositions, then the actual world is the set of all true propositions. (It is obvious, I hope, that (i) there must be an actual world, and (ii) there can only be one.)
3. Ergo, possible worlds cannot be sets.
4. So I propose that they are maximal propositions, it being understood that a proposition cannot be a set. (Because a prop. is either true or false, but no set is either true or false.)
5. Boram Lee points out, correctly, that my definition of maximality involves universal quantification over a domain of propositions.
6. Lee assumes, plausibly, that a domain is a set and so thinks the Cantorian argument can be deployed against my conception of worlds.
7. But it is not obvious that a domain is a (mathematical) set, and so it is not obvious that the Cantorian argument can be deployed against the view that worlds are maximal propositions.

Hello One Borw,

I am not sure I understand what is disappointing you. Your assertion, "any collection of maximally consistent propositions of the type you mean must be a proper class," is an assertion motivated by the result of Russell's paradox, which is essentially what is being presented in this post. The argument that there cannot be a set of all sets (or a set of all truths) does not say that there cannot be something else - like your proper class of all sets (or truths). The title of this post speaks of the non-existence of maximally consistent *sets* of true propositions, and its proof seems to me to be consistent with your theory of proper classes. So, as I said, it is unclear to me where your disappointment lies.

Sincerely,

Brian

Well, Brian, it looks like One Brow just doesn't have what it takes to replay to your questions in this thread. I did see a web site that discussed this argument, though.

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