## Monday, August 22, 2011

Propositions generally seem to be about things. The proposition that Tibbles is on the mat is about Tibbles and a mat. The proposition that 2+2=4 is about some numbers. The proposition that chips are on the counter is about chips and a counter. And so on.

Some propositions (statements, sentences, beliefs, etc.) are intuitively about themselves. For example, the proposition that all propositions merit investigation is intuitively about itself.

Now for a paradox. Consider the following proposition P: every proposition that is not about itself is mundane.

We can make the paradox more acute by stipulating that 'x is about y' means 'x quantifies over instances of a kind of which y is an instance'. We can then ask whether or not P is about P in that precise sense. (If you think you see a way out of the paradox, ask yourself if there's a way to re-write the paradox that avoids your solution, and I'm guessing you'll see that there is.)

This paradox will remind you of Russell's paradox concerning the set of all sets that aren't members of themselves. But I believe the paradox of propositions is much harder to solve. Concerning sets, we can, if we like, treat "set" talk as plural reference talk (thereby eliminating the existence of sets altogether), or else we may carefully craft axioms of sethood (such as ZFC) that preclude the existence of sets that are members of themselves.

But such solutions are not nearly as promising when it comes to propositions. If you think we can simply eliminate propositions, then run the paradox in terms of sentence tokens: the sentence token represented by P surely exists (or at least there are things arranged P-wise...). We might try to craft axioms of aboutness to get out of this, but such axioms won't take away the deep feeling that P should be about itself if and only if it is not. (With sets, by contrast, there is something right about supposing that no sets contain themselves.) So, we have a paradox on our hands that appears to be more serious than previous ones of its kind.

The paradox could perhaps be viewed as evidence against the reliability of our a priori faculties (though, of course, we'd have to rely on those same faculties to "see" this!) Or, more drastically, someone could view it as evidence that reality is at bottom absurd. I think it should be viewed as an invitation to gain a deeper understanding of the nature of propositions and aboutness.

Suggested ways of resolving the paradox are welcome. (I have a solution, but before I share it, I'd like to see how others might solve the problem.)

1. Suppose propositions are neo-Russellian entities representable by sets or classes (satisfying Foundation). Suppose that a proposition p is about something x just in case x is a constituent of p. (Constitution is membership in p or perhaps in some other class (e.g. the transitive closure of p) that contains p as a subclass.) Then no proposition is about itself and the paradox goes away.

In the same way the propositional liar goes away since no proposition can "refer to itself" (assuming Foundation). "This sentence is not true" expresses a proposition that refers to a sentence, not a proposition. The propositional liar resists formulation.

On a neo-Russellian view of propositions, any purported propositional paradox just is a set-theoretic one (though not necessarily conversely). And if we assume the coherence of e.g. the cumulative hierarchy, we can likewise assume coherence of our theory of propositions.

2. In my view a proposition should consist of four things: a series of terms standing for objects (definitely or indefinitely), a word expressing a relation, a relation of the terms to the relation word (assertion?) and an ordering of the indefinite terms determining subordinatiion among quantifiers.

The "aboutness" of a proposition comes in with the terms, so to say that a proposition is about something would be to say some one of its terms stands either directly or indirectly for that thing.

All propositions not about themselves are dull.

The paradox is derived as follows. If that proposition itself is about itself, then its only term "all propositions not about themselves" refers to it, in which case it is not about itself, as per the term. But if it is not about itself, the term refers to it, and so it is about itself. Paradox.

Now I think this paradox could be resolved simply by rejecting the premise -- what's wrong with that? In this way the proposition is similar to propositions like "I am thinking about all and only those people who are not thinking of me" or "I am thinking about all and only those times when I am not thinking of this moment" or "I am thinking of the set of times when the sets I am thinking of do not contain the times at which I think of them" -- all of which are resoled by denying the premise.

As for modeling propositions for sets, how will you model the proposition, "1 is in the set of natural numbers"?

3. firezdog: "As for modeling propositions for sets, how will you model the proposition, "1 is in the set of natural numbers"?"

<{{}},epsilon,N>, where {{}} is the von Neumann ordinal 1, epsilon is set membership, and N is the set of naturals.

4. First I’ll suggest that the paradox becomes clearer if one substitutes ‘every proposition that is not about itself is mundane’ for ‘every proposition that is not about itself is about something else’ (P*).

Suppose then that P* is about itself, then it must be a proposition that is about something else, because P* is about all and only those propositions that are not about themselves.

Suppose to the contrary that P* is not about itself, then it must be about itself, because P* is a proposition about all and only those propositions that are not about themselves.

For all propositions, if a proposition is not about itself, it is about something else.

Now there is no paradox, because we cannot start the procedure of assuming that P is not about itself (or contrarivise), because it would be absurd to suppose that a proposition that is about all propositions, is not about itself.

Note that P can be about itself and something else, by being about all members of the set of which itself is a member. However, it cannot be about itself and something else, if it is about all the members of a set of which it is not a member.

5. Samuel,

What is the meaning of the notation "<...>"?

6. The angle brackets, <, >, are used to denote ordered tuples of objects. So is the ordered pair whose first member is x and second member y. Are you asking purely for clarificatory reasons or because you think there is some real trouble lurking about?

7. I see that the expression \langle x,y\rangle (in LaTeX code) didn't get rendered in my previous post. It should have read "So \langle x,y\rangle is the ordered pair..."

8. Here's my objection to representing the membership relation the way we represent other relations.

As I understand it, a relation is to be modeled as a set of ordered pairs (triples, etc.). So membership would have to be a set of ordered pairs, the first of the pair being the member, the second of the pair being that of which it is a member. And this set will account for all such pairs.

But certainly these pairs must be members of the set in question. And so given any one of them, there will be another pair the first of which is the former pair, the second of which is the very set representing membership, in the set in question.

Now there are various ways of representing ordered pairs. One way is to say that the ordered pair, say, (1,2), is the set {{1},{1,2}} (Kuratowski, apparently) -- and I do not think the others are so very different in style, for a pair will inevitably be reduced to a set of sets. So the first concern is that the set representing membership will contain sets which contain that very set. Maybe this set is supposed to be a class. I'm very fuzzy on classes, so perhaps that could help.

But continuing with the previous observation, that the set representing membership will have to contain an additional ordered pair for any given simple ordered pair there would be a regress. For example, the set as I envision it would contain the ordered pair of 1 and the natural numbers, since 1 is a natural number -- but because it will contain this pair, it must also contain a pair of the pair of 1 and the natural numbers, and the membership relation -- and so must contain a pair of the pair of the pair of 1 and the natural numbers, and the membership relation -- and the membership relation. And so forth ad infinitum. This is a regress. I am not sure if it is vicious, but this membership relation will be a very large set indeed!

I noticed of course you would not model membership as a pair but as a triple. Offhand, I'm not sure if I see the rationale. Does it help if we have a triple? Is your epsilon something other than the set of all these triples? For if it is, I think the same problems would arise.

Finally a note about classes. I do not know much about them, but from what I know of them, I understand they can be very large. That is fine, but the whole rationale of set theory, I thought, was to limit the size of sets in such a way that we do not get paradoxes. What guarantees do we have, when we start talking about classes in addition to sets, that we are not heading into the deep end?

9. One apology is in order -- I recognize now that by your triple <1, the membership relation, the set of natural numbers>, you meant to represent the proposition, "1 is a member of the set of natural numbers" and not, as I for a moment had erroneously supposed, the membership relation itself. But I nonetheless convinced, at least for the moment, that the membership relation itself, when understood to be a set, will land us in these difficulties.

10. Finally to clarify the larger structure of my argument, I should be saying that propositions are bigger than relations (they involve them, whereas relations do not involve propositions) -- and relations cannot all be reduced to set theory -- therefore propositions cannot all be reduced to set theory. I have no direct argument that modeling propositions as tuples leads to a paradox.

I'm sorry for my stupidity. I get frustrated when I don't understand things and blame everyone and have to hash them out again and again. I hope you won't (mis)take me for a troll :(

11. These are nice suggestions. But before I reveal my own solution, I feel it is my duty to explain why the ones proposed so far don't work. :)

Samuel, your proposal is clever and solves the problem on one level... But it seems to me the problem returns if we stipulate that 'p is about x' means 'p quantifies over instances of a kind of which x is an instance' (or 'p says of everything that has a certain property q that it has some property t, and x has q'). Then ask whether P is about P in this technical sense, and you'll see the paradox returns.

firezdog, you suggested escaping the paradox by rejecting a certain premise. I wasn't sure which premise you were referring to exactly. Could you elaborate?

Valdi, I'm grateful for your observations. And the "charitable" reading idea actually hints in the direction of the solution I have in mind (I feel you should get a prize or something), but the trick is to explain why no paradoxical reading makes sense (apart from the fact that any paradoxical reading is of course paradoxical).

12. "Samuel, your proposal is clever and solves the problem on one level... But it seems to me the problem returns if we stipulate that 'p is about x' means 'p quantifies over instances of a kind of which x is an instance' (or 'p says of everything that has a certain property q that it has some property t, and x has q'). Then ask whether P is about P in this technical sense, and you'll see the paradox returns."

What I suggested is that propositions are representable by sets but also that so is aboutness. When spun this way, there's just no chance of paradox arising.

I want some substantive meat put on a notion of aboutness, "saying/signifying that", etc. so that it is obvious to me that a paradox is essentially unavoidable (like with naive truth, for instance). Until then it just looks like the neo-Russellian has an easy way out. (And I'm happy with an easy way out!)

13. Josh, I suggest we reject the proposition: propositions that are not about themselves are mundane. What is wrong with that?

14. In fact it seems no proposition which contains the term "propositions not about themselves" would be true, because from all such propositions a paradox can be derived. (Even the proposition: "Propositions not about themselves are not about themselves" would be false, along with the proposition, "Propositions not about themselves are about themselves.". Perhaps this in itself, a seeming violation of the law of noncontradiction, is enough to invalidate my solution.)

15. But negation is it seems very deeply integrated into your term, Josh -- the "not" is not sentential negation, is it? It seems it must be term negation.

16. Samuel, forget about aboutness. Just ask yourself whether P stands in relation R to itself, where R is a relation a proposition bears to an x iff it quantifies over instances of a kind of which x is an instance. Is your complaint that 'R' is not a well-defined relation?

firezdog, the paradox doesn't require the premise that P could be true--only that P exists.

17. In fact, I think you need the negation to be term negation, otherwise this proposition is about propositions about themselves.

Why do you say it only requires the proposition exist? The derivations of the paradox we've seen so far all assume it is true -- or what am I missing?

18. I think I see what you mean about the proposition only needing to exist. Any proposition which contains the term "propositions not about themselves" is about itself (that is, contains a term referring to itself) if and only if it is not about itself, regardless of whether or not it is true.

19. Samuel, I also have a lesser worry concerning your proposal, namely that it seems to me to be false. Here's one of my reasons. Let S be the proposition that Scott Soames no longer exists. It seems to me that S can be true. Here’s an argument for that:
1. Suppose for reductio that S cannot be true.
2. Then: it is not possible that S is true.
3. Thus: it is necessary that it is not true that S is true. (by the inter-definability of modal operators)
4. Necessarily, If the proposition that P is true, then it is true that P. (premise schema)
5. Thus: it is necessary that it is not true that it is true that Scott Soames no longer exists. (3,4)
6. Necessarily, it is not true that it is true that P iff P. (premise schema)
7. Thus: it is necessary that Scott Soames no longer exists. (5,6)
But (7) is not a happy result—moreover, it’s false. So, it seems to me that S really is possibly true. But that’s hard to square with the hypothesis that S has Scott Soames as a constituent—assuming constituents are essential. (There are truth-in truth-at issues here, which I'm prepared to discuss.)

20. Josh, can you please spell out how you are deriving 5 from 3 and 4?

Further, what do you mean by 6? Do you mean --

It is necessary that (it is not true that something is true if and only if that thing is true).

Now if it is not true that something is true, I suppose that thing is not true. So then it seems you are saying:

It is necessary that (something is not true if and only if it is true).

How could that be?

***

Is your main objection that if if a proposition is an ordered series of objects and the relation holding between them it will not be possible to truly deny that something exists?

21. "Is your main objection that if if a proposition is an ordered series of objects and the relation holding between them it will not be possible to truly deny that something exists?"

That's the heart of it. (To get 5 from 3 and 4 substitute "P is true" in 3 with "it is true that Scott Soames no longer exists", keeping in mind that P is the proposition that Scott Soames no longer exists. 6, like 4, is a schema. Treat it as a rule for substituting expressions--as modeled by what I said here about 4.)

22. I remember seeing the "about" thing, either in an article by John Post or in Grim's book.

Anyway, start with a fairly minor issue. Read "All Fs are Gs" in the FOLish way as a quantified material conditional. Then for your argument to work, you need to take <(x)(Fx → Gx)> to be about all and only the xs that satisfy F. But (x)(Fx → Gx) is equivalent to, and perhaps even defined as, (x)(~Fx or Gx). And why think this quantified disjunction is about all and only the xs that satisfy F? Surely it's about all xs.

Now, we may be able to get out of this by stipulation. We can stipulate that <(x)(~Fx or Gx)> is about all and only those xs that satisfy F. This would require, however, that <~Fx or Gx> be a different proposition from <Gx or ~Fx>, and that is far from clear.

Anyway, let's allow that we get to stipulate that <(x)(Fx→Gx)> is about all and only those things that satisfy F.

So, we have an About predicate: About(p,y) iff there are predicates F and G such that p = <(x)(Fx→Gx)> and Fy. This requires substitutional quantification. But we can eliminate the substitutional quantification in favor of metalinguistic objectual quantification:
About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy')).

But now notice that the paradox uses significant metalinguistic machinery, both syntactic and semantic. On the syntactic side, we need concatenation and some quoted strings (both are implicit in "'(x)(Fx→Gx)'", which we should probably logically represent as something like Concat(Concat(Concat(Concat('(x)(', F), 'x→'),G),'x)'). On the semantic side we have both "expresses" and "True".

Once we have concatenation, a sufficient number of quoted strings, "True" and standard first order logical stuff with equality, we can form a strengthened liar sentence. So it's not particularly surprising that with the ingredients at hand one can get paradoxical sentences.

Nor is the paradox really that far from the liar in substance, but that's just an intuition here.

How to get out of it? I think one has to either deny a classical rule of inference, or get rid of an unrestricted truth predicate or deny unrestricted compositional formation rules in logic. (I go for the last of these.)

23. Actually, I would prefer to replace "True('Fy')" with "Satisfies(y,F)". But of course if you can form a liar sentence with "True", you can form a liar-like sentence with "Satisfies".

24. We can stipulate that <(x)(~Fx or Gx)> is about all and only those xs that satisfy F

Or: we can stipulate that it's about all and only those xs that satisfy either F or ~G. Then we won't need to assume that <~Fx or Gx> is a different proposition from . So we have:

About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy') or True('~Gy')).

It should be no surprise that this paradox is connected with others (like Liar). After all, it's conclusion is certainly false. You've got to be right, Alex, that to get out of it, one must make restrictions in the formation rules of logic. But part of the puzzle is in seeing precisely how and why to make those restrictions in non-arbitrary ways.

Also, I'm not sure we need a meta-linguistic formulation. Here's a formulation that has intuitive appeal:

About(p,y) iff there are properties F and G, such that p says that everything that has F has G, and p has F.

What we would like is a solution that get's underneath the intuitions (however vague they might be) that generates the paradox. Finding faults with particular rules of logic formulation doesn't seem to me to do that. That's not fully satisfying.

(I'll give my "fully satisfying" solution eventually; I first want people to hunger for it. :) )

25. Josh:

I'd love to see your fully satisfying solution. Just let me give ahead of time my necessary condition for a fully satisfying solution: it needs to handle, strengthened liar sentences, truth-teller sentences, contingent strengthened liar sentences, contingent truth-teller sentences, and coordination cases (e.g., n propositions arranged in a circle, each saying of the next that it's false). :-)

I've played with the idea that we need to distinguish between the logic of sentences and the logic of propositions. The formation rules for the logic of sentences in natural languages are unsurprisingly messy, and I don't expect a good account. In formalized languages they can be neater, but it is not surprising if a formalized language fails to do the work we need it to do, say handle both semantic and syntactic stuff together.

Now, the best way to get self-reference in a language (you can consider the collection of propositions as a trivial language, where each proposition counts as expressing itself) is to have the language have enough machinery to allow it to parse itself. Once we can self-parse, we can form liar sentences by a diagonal lemma or by a constructive method like here.

But I am not sure propositions have the right kind of parseability. You invoked the idea of a proposition p saying that everything that has F has G. That's a parseability assumption. To keep it at the level of propositions (and properties--after all, propositions are just nullary properties), we need a ternary property like: lambda_x lambda_y lambda_z (x says that everything that has y has z).

I don't know whether this ternary property exists. An initial worry is that it might turn out that <all Fs are Gs> is the same proposition as <all ~Gs are ~Fs>. This particular worry doesn't come up in your case: for your purposes, we can use the ternary property lambda_x lambda_y lambda_z (x says that everything that has y has z or that everything that has ~z has ~y). But there may be further worries along these lines.

In particular, I have a general worry about quantifying into contexts like "x has F". The worry comes from Kripke-type puzzles like this one.

26. my necessary condition for a fully satisfying solution: it needs to handle, strengthened liar sentences, truth-teller sentences, contingent strengthened liar sentences, contingent truth-teller sentences, and coordination cases (e.g., n propositions arranged in a circle, each saying of the next that it's false). :-)

Let's see what we can do. I will begin by giving a solution to the problem that results on an intuitive, undefined understanding of 'about'. I'll then tackle the problem for various stipulated definitions of 'about'. I'll conclude by meeting your necessary conditions. :)

This solution doesn't go deep enough, of course. For we can stipulate other nearby senses of "about" such that P would seem to only be about those propositions that aren't about themselves.

I'll begin with what I take to be the most intuitive definition:

27. About(p,y) iff there are properties F and G, such that p says that everything that has F has G, and y has F.

But I will now argue that there is no such property as ~About(x,x) if it is true that propositions can say anything.

I begin with a definition:
‘x has Q’ =df ‘there is a property f, such that x says that f exists, and x has f’

Now let L = the proposition that the property of ~Q exists

If it makes sense to suppose that a proposition can say something, then L surely says that the property of ~Q exists. But ask yourself: does L have Q, or not? Either answer leads to contradiction. Suppose L has Q. Then by the definition of ‘x has Q’, it follows that L has ~Q, which is inconsistent with its having Q. Suppose, on the other hand, L lacks Q. Then it follows that L doesn’t have ~Q (else it would satisfying the definition for having Q), which is inconsistent with its lacking Q. Either way we have a contradiction.

So, to get out of the contradiction, either L doesn't say anything, or ~Q doesn't exist. I see no other serious options. (Notice that self-reference isn't even in play here!) But if ~Q doesn't exist, then by parity of reasoning, it seems that neither does ~About(x,x): for there seems to be no relevant difference between the respective paradoxes that would explain why ~Q can't exist while ~About(x,x) can. So, we have an independent reason to think that if propositions say anything, then there's no such property as ~About(x,x). And if there's no such property as ~About(x,x), then the paradox dissolves: P is simply not about itself (because it doesn't say anything about ~About(x,x)).

28. This one is trickier. Since it is metalinguistic, it won't suffice to deny the existence of the property of ~About(x,x). The paradox just needs the existence of the predicate '~About(x,x)'. And that exists: just look back a few words!

Thus, the solution is to recognize that if we define 'About(x,x)' meta-linguistically, then asking whether P is about itself (in that same sense) is meaningless. And no contradiction can come from a meaningless question.

So, to review, we've seen that the commonsense notion of aboutness is compatible with supposing that P is about itself. And upon scrutiny, we saw why certain stipulated definitions also fail to generate a contradiction.

29. Now to meet your necessary condition:
1. Strengthened Liar: What I said about semantic circularity applies. "this statement is false" is semantically circular because the meaning of "this" is dependent upon the meaning of "this statement is false" which ultimately implies that the meaning of "this" is dependent on the meaning of "this". So, strengthened liar fails to express anything. (Or it expresses a necessary falsehood--because the circularity problem occurs at the level of referents, not "meanings"; that may make better sense of coordination cases...)

2. Truth-teller: Same as above.

3. Coordination cases. Same as above. :)

I had planned to bring in my theory of propositions (as arrangements of properties) and aboutness (in terms of the exemplifiable parts of a proposition), but it seems that wasn't necessary for a fully satisfying solution. :)

30. (Of course, there may be other Liar sentences, but the question is whether any of them call into question my solution to the aboutness paradox.)

31. To be clear: the post above that starts with "this one is trickier" is dealing with the definition "About(p,y) iff EF EG('(x)(Fx→Gx)' expresses p and True('Fy') or True('~Gy'))".

32. Josh:

"So, to get out of the contradiction, either L doesn't say anything, or ~Q doesn't exist."

Just to make sure we're on the same wavelength, let me query where in your dichotomy you include the following solution: The stipulation "let L = the proposition that the property of ~Q exists" fails.

A stipulation of the form "let L = the F" fails when (and only when?) there is more than one F or there are no Fs. In this case, if there is more than one F, that's perhaps less of a problem. We should be able to let L be any one of the Fs, and in effect proceed inside an existential-elimination subproof. But where in your taxonomy is the suggestion that there is no F--no proposition that says that ~Q exists?

33. Josh:

You say "let L = the proposition that the property of ~Q exists". Where in your disjunctive taxonomy of ways out ("either L doesn't say anything, or ~Q doesn't exist") is the suggestion that the stipulation fails?

There are three ways a stipulation of the form "Let L = the F" can fail:
i. There is more than one F.
ii. There are no Fs.
iii. The expression "the F" is nonsense.

In this case, (i) is not a big deal since you can just do existential-elimination, and let L be any one of the Fs.

But (ii) and (iii) are real issues. Are they distinct from your option "L doesn't say anything"? I think so, since if the stipulation of L fails, then it's dubious that it's correct to affirm anything "of L", including that it doesn't say anything.

Here is a technicality. What does "~Q" mean? There are two readings of "~Q". One reading is that it's a name for the property that negates Q. The other reading it's a definite description like "the property that negates Q". If it's a name, then if there is no property that negates Q, "Let L = the F" may fail in way (iii) by containing a non-referring name (this depends on what you think about non-referring names). In this case, your solution that ~Q doesn't exist and (iii) end up both holding.

Suppose "~Q" is a definite description. Then we need to Russelize the definition of L as:
"The proposition that there is a unique property that negates Q and for every property that negates Q, that property exists."
And then your argument that if L lacks Q, then L has Q fails, since L does not say of any property that it exists. (Unless you think that the second conjunct does. But we can also say that the second conjunct says of every property that it exists or doesn't negate Q.)

So I think you need to take "~Q" as a name. So, suggestion for presentation in the future. Instead of using "~Q", introduce a new letter:
Let N be the property that negates Q.
(Or if you're worried--one should be--about uniqueness, do an existential-elimination subproof, and let N be "any one" of the properties that negate Q.)

34. Josh:

"The problem is that we cannot even meaningfully ask if About(P,P) unless the domain of quantification in the definition of 'About(x,x)' includes the pieces of language necessary to express P".

This needs some expansion. Consider this parallel: "The problem is that we cannot even meaningfully ask if Loves(Romeo,Romeo) unless the domain of quantification in the definition of 'Loves(x,x)' includes the pieces of language necessary to express Romeo". But surely the domain of quantification in the definition of 'Loves(x,x)' doesn't need to have any language in it.

35. where in your dichotomy you include the following solution: The stipulation "let L = the proposition that the property of ~Q exists" fails

Perhaps under "there is no such property as ~Q" (because if there is, then surely there is the proposition that ~Q exists). But your points about the meaning of "~Q" are helpful.

But surely the domain of quantification in the definition of 'Loves(x,x)' doesn't need to have any language in it.

Right, but I was dealing with the metalinguistic definition of 'About(x,x)' which explicitly quantifies over predicates... I agree, though, that expansion is deserved.

36. Joshua, you do recognise that when you say " if there is [the property -Q] then surely there is the proposition that -Q exists", commits you to Platonism about propositions, i.e. to the idea that there is a realm of reality outside space and time where 'ideas' roam. I know a lot of philosophers do this routinely, mistakenly believing that it is either intuitively right, or simply convenient and innocent. In 'Truthmakers without Truth' I argue that it is neither intuitively right or innocent (it might be convenient, but if false it is devastating). The paper is available here: http://durham.academia.edu/RÃ¶gnvaldurIngthorsson/Papers/308786/Truthmakers_Without_Truth

37. Thanks Valdi for the link. I read it with interest, and yes I'm well aware of the issues and agree with you. I think independent arguments would need to be given (such as here, pp 62-78).