Subtracting Self-Reference

Brandon recently linked his old post on the Propositional Depth Response to the Liar Paradox:

Every meaningful statement must be assumed to have a determinate propositional depth...

L: [L] is false.

This has no determinate propositional depth. If we assume that L has a propositional depth of n, we find that, since L embeds itself, it must have a propositional depth of n+1.

I like this sort of account. (Alex recently suggested that one might arbitrarily choose whether or not to believe the proposition P: I believe [P]. My immediate response was to doubt that there really is any proposition here. The 'depth' account can explain why.)

A standard objection to this sort of view is that there would seem to be some true self-referential propositions. In an old guest post, Rad Geek suggested:
EM: [EM] is true or [EM] is false.

Now, it's not entirely clear to me that the above constitutes a wholly meaningful claim. (What is this '[EM]' it speaks of? I get stuck in an infinite loop if I try to fill it out.) But perhaps we can apply a lesson from Yablo and say that it is partly true. (I doubt his truthmaker account can actually accommodate this, but never mind that for now.)

Subtract out the self-reference, and what remains ("__ is true or __ is false") is true about logical forms, i.e. insofar as it claims that the law of excluded middle holds. The particular application is meaningless, but we can abstract away from that part of what's said.

Another puzzle case:
M: [M] is true and grass is green.

We certainly don't want to say that this is wholly meaningless. It's partly true: grass is green. Again, it seems that the thing to do is simply to subtract away the meaningless self-referential component.

P.S. Towards the end of his post, Brandon worries that the propositional depth solution commits us to the view that "whether the sentence has the same meaning, or any meaning at all, depends on purely contingent facts about the world that we may not be aware of." We should embrace this result, though, as Michael Sprague once pointed out to me:

It's worth noting that all liar sentences are dependent on context. For example, an instance of the liar sentence next to an arrow pointing to another sentence (like, say, "All ravens are orange") may be true. Context determines the sentence to which "this sentence" refers; the truth or falsity of that sentence then determines the truth of the liar.

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Logical Subtraction and Partial Truth

Stephen Yablo gave a very impressive series of lectures here last week. His core insight was that often we can communicate important truths by way of assertions that are literally false. The rough explanation for this is that the assertion, though false, may be "partly true", or "true insofar as it concerns a certain subject matter." For example, 'the number of dragons is zero' may be literally false if numbers do not exist. But it is partly true, i.e. true in what it says about dragons (namely, that there aren't any). If you take the literal meaning, and subtract the claim that there are numbers, the remaining content is wholly true.

Sometimes logical subtraction seems unproblematic, as the subtracted element is "perfectly extricable" from what's being said. Other cases, however, are more problematic. When you subtract the redness from scarlet, what's left? 'Tom [the tomato] is scarlet - Tom is red' does not leave any remainder that we can make sense of. It seems perfectly inextricable. Then there may be inbetween cases, such as Wittgenstein's famous question: "what is left over if I subtract the fact that my arm goes up from the fact that I raise my arm?" We have some grasp of this, but as Jaeger has pointed out, "it is not the case that there is exactly one statement R such that 'R & my arm goes up' is logically equivalent to 'I raise my arm'."

Yablo's solution is to say that "P-Q always exists, but it doesn't always project very far out of the Q-region [of logical space]. Inextricability simply means that it is hard [or impossible] to evaluate P-Q in worlds where Q fails."

Intuitively, we can say that:
(i) P-Q is false iff P adds falsity to Q.
(ii) P-Q is true iff not-P adds falsity to Q.
(iii) If neither P nor its negation adds falsity to Q, then P-Q is undefined (lacks a truth value).

Yablo systematizes our intuitive judgments here by appeal to truthmakers, or the reasons why a proposition is true/false. P "adds falsity" to Q if it is false for a Q-compatible reason, i.e. there is a Q-compatible falsity-maker for P.

Example: Let P = 'The King of France is bald' and Q = 'France has a King'. Then P-Q is false, because of the following Q-compatible falsity-maker for P: the list of all the bald people, none of whom is a King of France. This falsity-maker could exist, and so make P false, even if Q were true and France did have a King. This shows that P is false for reasons over and above the falsity of its presupposition Q.

Here is a bit more technical detail. Let R be a potential candidate for P-Q. Yablo suggests that R is a successful candidate, i.e. R extrapolates P beyond Q, iff the following three conditions are satisfied:
- "Equivalence: within Q, R is true (false) in a world iff P is true (false)." That is, if R = P-Q, then it had better be the case that R&Q = P.
- "Reasons: within Q, a world is R (~R) for the same reasons it is P&Q rather than ~P&Q (...)" This is equivalence as applied to subject matter, rather than just truth conditions.
- "Orthogonality: outside Q, R is true (false) for the same reasons as within." This is the key principle, which really gets at the intuitive notion that we are genuinely extrapolating P rather than simply gerrymandering a proposition that happens to overlap with P in the Q-region (and then becomes wildly different beyond that point).

Example: The material conditional 'if Q then P' fails the orthogonality condition. Outside the Q region, it is true for the simple reason that Q is false, regardless of P. Compare the visual aid below: 'if Q then P' has truth conditions 'P or ~Q', so would include all the white region in R. The gerrymandering is visible in the fact that the R region would then turn a sharp 'corner' once it left the P & Q region. It should instead extrapolate cleanly as shown.

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Is Logic Overrated?

Good reasoning is invaluable; it's what philosophy is all about. But I'm more skeptical of formal logic's value. Logic is a powerful tool we can use to structure our reasoning and highlight entailment relations. But - like any tool - it can be misused. In particular, I worry that it's just too easy for the manipulation of symbols to substitute for careful thought.

Modal logic is especially susceptible to misapplication, in my experience. The most famous example would have to be the S5 modal argument for God's existence. But it's also not uncommon to come across blog posts where the employed logical apparatus merely serves to build in misunderstandings. The formal steps of the argument may be flawless, but that's all for naught if the entire argument is based on a mistake -- due to failing to understand precisely what all those formalisms really mean.

If one opts to engage in formalism, the hard (philosophical) work lies in interpretation, i.e. ensuring that the formalism adequately captures the intuitive ideas we started with. It's easy to neglect this point, and so produce a formal 'proof' that doesn't really speak to the issue at hand. That's the risk of formalism. The advantages are more well known: they force us to make explicit intermediate steps in our reasoning, and allow these to be easily checked for validity. Do the risks or benefits tend to be greater in practice, do you think?

My tentative (and admittedly under-informed) opinion is that logical formalisms are rarely indispensible, and often well dispensed with. As a rule of thumb, I'd be wary of using formalisms as the central means of making your case. Their best use may instead be to provide a bare-bones outline of the argument's structure, as a supplement to the argument given in prose. Formalism may prove helpful, but it shouldn't be considered sufficient, since there is more to good reasoning than logic alone.

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Conditional Oughts

I think my favourite ever blog discussion would have to be the exchange I had with Jamie Dreier a couple of years back, over the following paradox:

Suppose you have the chance to bet on the toss of a fair coin, but you have to put up $1000 to win $900. I think each of these is true:

(t) If the coin will land tails, then you should bet on tails.
(h) If the coin will land heads, then you should bet on heads.
Furthermore,
(v) The coin will land heads or the coin will land tails.

Now by v-elimination we can conclude that you should write me a check for $1000 and bet. But that is obviously a false conclusion. Yet all the premises are true, and v-elimination is valid.

I've been meaning to blog this ever since, but somehow never got around to it. But that's easily remedied: let me reproduce our exchange (excavated from the 50+ comments thread)...

* * *

RC: distinguish the ‘ought’ of reasons (externalistically construed, as what would be best in objective fact) vs. the more subjective ‘ought’ of rationality (expected utility, etc.).

In the objective sense, the coin argument goes through just fine: you really should bet, and what you should bet on is the winning option. (Go on, what are you waiting for?)

Taking the subjective sense (which is more appropriate for deliberation), the premises (h) and (t) are false, as per Greg’s [comment] #30. Even if the coin will land heads, you rationally shouldn’t bet on it unless this information is actually accessible to you. Jamie’s conditional proof ("Surely you should [bet tails], on that supposition [that the coin lands tails]!") is only convincing if the supposition is supposed to be actually made by the gambler. Certainly if I (qua gambler) may suppose that the coin will land tails, then that’s what I rationally should bet on. But if we’re supposing, abstractly, that the coin lands heads without my having antecedent knowledge of this, then it wouldn’t be rational for my ignorant self to bet at all.

(This line of thought becomes clearer if we append “without your foreknowledge” to the antecedents of (h) and (t).)

*

JD: Whoa! I thought conditional proof was supposed to be valid for every conditional there is. In conditional proof we do not add the extra assumption that anybody knows the discharged premise, so I find it puzzling that you (and apparently Claudio) think that it’s necessary for validity here.
Anyway, I would be quite happy with an admission that conditional proof fails for these practical conditionals. My view is that it succeeds but modus ponens fails, but I’m okay with the contrary view.

*

RC: Hi Jamie, sorry I was unclear: my complaint is not with the logic of conditional proof, but rather your claim that “Surely you should [bet tails], on that supposition [that the coin will land tails]!” Let me clarify.

Suppose it’s true that the coin will land tails (without the gambler’s foreknowledge). Does it follow that the gambler rationally should bet tails? No, of course not. Quite the opposite: given his ignorance of the fact, he shouldn’t bet at all. So the conditional “if the coin will land tails then the gambler should bet on tails” is false. Replacing ‘the gambler’ with ‘I’ doesn’t change this. All it does is allow us to equivocate in a way that gives your argument its misleading plausibility.

The equivocation arises when you conflate the gambler with the logician doing the supposing. Compare:

(t1) If the coin will land tails without my foreknowledge, then I should bet tails.

(t2) If I may suppose that the coin will land tails, then I should bet tails.

I take it that (t1) is obviously false, and (t2) obviously true, when using the ’should’ of rationality. Now, (t) really corresponds to (t1), and so is likewise false. But your argument tempts us to treat (t) like (t2) instead.

“Suppose the coin will land tails. Now, on that supposition, I ask you: shouldn’t you bet on tails? Surely you should, on that supposition!”

This conflates the gambler and the reasoner.

Sure, if I qua gambler get to suppose that the coin will land tails, then that’s what I should bet on. That is, supposing (for sake of CP) that I get to suppose (for sake of deliberation) that the coin will land tails, it follows that I should deliberately bet tails. But applying conditional proof to this will only yield (t2), not (t).

N.B. The gambler should bet tails only if, at the time of decision, they hold that the coin will (probably) land tails. This condition is satisfied if, internal to deliberation, the gambler may suppose that the coin will land tails. So far so good. But if you have an external logical debate with the gambler, and ask them to make this supposition for sake of argument, then the supposition will be discharged by the time they return to their practical deliberation. It will no longer be accessible when they have to decide whether to bet. So the above condition is not satisfied in such a case.

Perhaps a parody-argument would help illustrate. Consider the conditional:

(t*) If the coin will land tails, then you suppose that the coin will land tails.

A groundless claim. But apply Jamie’s reasoning: “Suppose the coin will land tails. Now, on that supposition, I ask you: don’t you suppose that the coin will land tails? Surely you do, on that supposition!” Hmm.

*

JD: Good, thank you, Richard.
I agree with you entirely about the conditionals you mention, and I think you are absolutely right in bringing out the temptation to conflate the supposition that p with the supposition that the supposer supposes that p.

First, in support of what you’ve said, I think that conflation infects indicative conditionals in many places. Compare this somewhat idiomatic form of conditional:

(if-R) If you believe Richard, then the ought has wide scope.

This conditional is intuitively true. But surely it is grossly implausible that whether the scope of the ought could depend in any way on whether the audience believes Richard. Diagnosis: when we consider (if-R), we imagine that we believe Richard, and then ask ourselves whether the ought has wide scope; but imagining that we believe Richard is very hard to distinguish from imagining that Richard is correct, so our answer is that yes, under that supposition, the ought does have wide scope.

Second, though:
You have replaced ‘should’ with ‘rationally should’, and I want to know how the argument works with just plain ‘should’. And in that case, I don’t buy your analysis. Suppose this ticket is, in fact, the winner; in that case, shouldn’t you spend a dollar on it? Yes (I say)! Only you don’t know it, so it’s understandable that you don’t do what you should.
Or try it with ‘better’. If the coin will land heads, then it is better to make the bet and to make it on heads. Suppose the coin will land heads; now, isn’t it better, on that supposition, that you bet on heads?

Finally, is there an equivocation involved in the argument? Well, I think there must be something like an equivocation at some point (this is what I was saying in [26] about Mike’s remark in [17]). But I don’t believe that ‘ought’, ‘should’, ‘better’, and so on, are all literally ambiguous in just the same way.

*

RC: Jamie, if we adopt a more objective sense of 'should', isn’t the conclusion of the coin argument then unproblematically true? (Cf. my comment #31.) You should “write… a check for $1000 and bet” on the winning outcome. It’s a pity you don’t know which that is, but that doesn’t falsify the conclusion any more than it does the premises (t) and (h).

Perhaps your point is as follows: granted, it is better to bet on the winning outcome. But it is not better to bet, simpliciter, for that leaves open a 50% chance of losing.

That seems right, but may not connect with the original coin argument as you presented it. The most you can strictly infer from (t), (h), and (v) alone is that “you should bet on tails or you should bet on heads”. To obtain your more general conclusion (that you should bet, simpliciter), you need the further premise(s): if you should bet on tails (heads), then you should bet, simpliciter. But this is simply the denial of the above point. So if we take that point seriously, we should reject this new premise.

*

JD: Hm, you’re right, Richard, but I feel like that’s a loophole that I just have to be more careful about rather than a serious problem.

Well, here’s a try at repairing with minimal amendment. We can use the principle, “If you ought to do A, and the only way you can A is by doing B, then you ought to do B.”

Now, in each branch of the OR-elimination, I’ll use that principle to conclude, “You ought to bet”, so the overall conclusion resting on the disjunction (but not separately on the disjuncts) will be “You ought to bet”.

How’s that? (I hope that instrumental principle is true! I think it is. I’m less sure that it supports modus ponens itself, of course, but if it doesn’t that works for me.)

*

RC: the puzzle then seems to come down to this: suppose that it’s best to do B and A, but worst to do B and not-A, compared to the neutral option of not-B. As such, is it better to do B?

I’m not sure if there’s really a determinate answer to this. It seems like the right thing to say is, “It depends whether you’ll go on to do A!”

Similarly for your instrumental principle, and its application to the betting case. You should bet on the winning option. So should you bet? Well, it depends whether you’ll be doing so on the winning option or not…

*

JD: I agree with all that except for the bit about there being no determinate answer. Aren’t there lots and lots of cases in which whether P or Q is the case depends on whether R is true, but this plainly doesn’t mean that whether P or Q is the case is indeterminate?

Still, when you put it in that nice Boolean way instead of the overtly instrumental way that I put it, I agree that what seemed obviously true to me suddenly seems very questionable. It seems to me that there must be plenty of cases in which you ought to do A&B, but, since you are not in fact going to do B, you ought not to do A, since A&~B is a lot worse than ~A&~B.

* * *

Good stuff -- there's something deeply satisfying about making sense of such logic puzzles. [See also my short essay on the idle argument, which discusses a similarly paradoxical argument for foolhardy battle tactics, which I also owe to Jamie.]

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The Logic of Indeterminacy

Alex's comment to my previous post reminded me that I need to say a bit more about my treatment of indeterminacy. He wonders how the following claims of mine could be consistent:
(1) It is indeterminate whether 'Bob' denotes Bob1 or Bob2.
(2) It is likewise indeterminate whether 'Mirror-Bob' denotes Bob1 or Bob2.
(3) But it is determinate that 'Bob' and 'Mirror-Bob' do not co-refer.

After all, if you took 1 and 2 to be the fundamental semantic facts in this scenario, then there is nothing to rule out their being coreferential -- it would simply appear to be indeterminate whether that's actually the case. To supervaluate: there are four ways* to resolve the indeterminacy, and on two of them the names co-refer, and on the other two they don't. So we cannot settle the question whether the terms co-refer.

* = Those four ways are:
(a) 'Bob' denotes Bob1, 'Mirror-Bob' denotes Bob1 [co-refer]
(b) 'Bob' denotes Bob1, 'Mirror-Bob' denotes Bob2
(c) 'Bob' denotes Bob2, 'Mirror-Bob' denotes Bob1
(d) 'Bob' denotes Bob2, 'Mirror-Bob' denotes Bob2 [co-refer]

But that's the wrong way to go about things. While 1 and 2 offer a partial description of the semantic situation, such an atomistic approach cannot capture everything that's going on. It is whole scenarios that are instead fundamental. Compare 1 and 2 with:
# (4) It is indeterminate which of the four semantic scenarios, a-d, obtains.

This is the situation as Alex took it to be. But it is not what I had in mind. And this way of presenting things brings out some alternative possibilities, such as:
# (5) It is indeterminate whether semantic scenario a or d obtains.
(6) It is indeterminate whether semantic scenario b or c obtains.

This is my claim. Crucially, it is 6, not 1 and 2, which provides the fundamental account of the situation I had in mind. (1 and 2 are consistent with any of 4, 5, or 6. That is why they are merely partial descriptions.) To derive more particular claims -- e.g. my 1, 2, and 3 -- from the fundamental account (6), we simply supervaluate. That is, a claim is:
(I) determinately true if it is true in all allowed scenarios;
(II) determinately false if it is false in all allowed scenarios; and
(III) indeterminate if it is true in some allowed scenarios and false in others.

Since my allowed scenarios are b and c only, we obtain the following results:
My claim 1 is true because 'Bob' denotes different people in scenarios b and c.
Likewise for claim 2 and 'Mirror-Bob'.
3 is true because in both b and c, 'Bob' and 'Mirror-Bob' denote different people.

Does that all make sense?

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Are Contradictions So Bad?

Granting the law of non-contradiction, do we necessarily have reason to avoid holding inconsistent beliefs (as such)? If so, then our pre-existing beliefs, no matter how absurd, would 'bootstrap' reasons into existence. But that seems dubious: just because one believes that the world is 6000 years old, it does not follow that one has any reason whatsoever to refrain from believing that the world is over a billion years old. (One ought to revise the former belief instead!)

In 'The Myth of Instrumental Rationality', pp.20-1, Raz addresses the grounds of our lingering hostility to contradictions:

When we learn that there is a contradiction among our beliefs we learn (1) that some of our beliefs are false, and (2) that we hold some beliefs that if used together as premises in an argument may lead us astray in a special way [i.e. logical 'explosion']. Big deal! We hope that we all know that some of our beliefs are false anyway. And the risk that we will actually be led astray not by the logical implications of our false beliefs, but by their contradictory features, is, for all practical purposes, negligible...

To conclude: There is nothing wrong with holding contradictory beliefs as such, and the fact that one does is no reason to change one's beliefs. At most we could say that we should abandon our false beliefs. But that is so not because of the contradiction. Knowing that a set of propositions is contradictory has epistemic relevance: It tells us that the contradictory set contains a falsehood. It may be part of a case for believing that one particular proposition is false. But it is no such case by itself. Without such a case we have no reason to abandon any of them. For all we know, we may then abandon a true belief and remain with false ones. Nor do we have reason to suspend belief in all the propositions in the contradictory set. The cost, epistemic and otherwise, of doing so may be too great. That is why the logical paradoxes are rightly not generally taken as a reason to suspend our acceptance of the principles that generate them.

What do you think?

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Parity of Value: a formal model

Over at Ethics Etc, S. Matthew Liao presents Ruth Chang's argument for a fourth relation of comparative value - 'on a par' - to supplement the standard 'better than', 'worse than', and 'equal to' relations:

1. Mozart is neither better nor worse than Michelangelo, with respect to Creative Genius (CG).
2. A Mozart who has some small improvement that bears on CG (Mozart+) is better than Mozart, with respect to CG.
3. Mozart+ is not better than Michelangelo, with respect to CG.
4. Therefore, Mozart and Michelangelo are not related by any of the standard trichotomy of relations, with respect to CG. (This is the Small Improvement Argument)

5. Mozart is better than Talentlessi, a very bad music composer, with respect to CG.
6. Michelangelo is also better than Talentlessi, with respect to CG.
7. Therefore, Mozart and Michelangelo are comparable, with respect to CG. (This is the Chaining Argument)

8. Given 4 and 7, there must be a fourth comparative relation; Chang calls it the parity relation. (This is the Parity Conclusion)

The premises all seem intuitively plausible, yet it may be initially puzzling how they could all be true -- at least if we conceive of value as a point on a scale.

The idea seems to be that CG is some kind of holistic value, constructed from a composite of various partly-commensurate dimensions (e.g. music and art). That could explain why Mozart+ beats Mozart (the slight increase is on the same dimension, so Mozart+ strictly dominates Mozart, being better in some ways and worse in none) yet Mozart+ does not beat Michelangelo, being better in some ways but worse in others, with the tradeoff being, in some sense, "too close to call". But note also that the two dimensions are at least comparable at the extremes: Michelangelo's artistic genius outweighs Talentlessi's sorry musical skills. In sum: Mozart and Michelangelo both excel along different dimensions of Creative Genius, which places them 'on a par' in such a way as that a minor improvement to either would not affect their relative standing.

It's an intuitive enough picture, but is it theoretically consistent? Liao expressed doubts. But I think we can construct a formal model which exhibits all the theoretical properties Chang needs here, i.e. showing the premises (1, 2, 3, 5, 6) to be mutually consistent. Here is my model:

(A) Let 'x' and 'y' denote two dimensions of Creative Genius, and let Proto-CG be composite value combining x and y but with some vagueness as to their relative weightings.

Assign base values:
* Mozart = 100x + 0y
* Mozart+ = 101x + 0y
* Michelangelo = 0x + 100y
* Talentlessi = 1x + 0y

Hence, the following facts hold concerning ordering relations with respect to the proto-CG scale:
1-p. It is not determinate whether Mozart is either better or worse than Michelangelo.
2-p. Mozart+ is determinately greater than Mozart.
3-p. It is not determinate that Mozart+ is better than Michelangelo.
5-p. Mozart is determinately greater than Talentlessi.
6-p. Michelangelo is also determinately greater than Talentlessi.

Liao raised an important objection to my model at this point:

If it is vague as to whether Mozart is better or worse than Michaelangelo or equally good, then, it is not true that Mozart is neither better nor worse than Michaelangelo or equally good.

Granting this point, it is important for me to emphasize that the #-p facts hold merely with respect to proto-CG, and do not yet speak to the ultimate CG relations which we are interested in. What we need is some schema to translate these vague Proto-CG relations into the determinate CG relations stated in the original premises. That is the role of the second part of my model.

(B) We may now construct CG orderings from Proto-CG orderings as follows:

For the standard trichotomy of positive ordering relations ('better than', 'worse than', and 'equal to'), let us say that the relation holds with respect to CG iff it is determinate that the relation holds with respect to proto-CG. (I'll call this the "axiom of determination" unless anyone can think of a spiffier name.)

This axiom establishes entailment relations from each #-p to the corresponding original premise #. For example, from the fact (1-p) that it is not determinate whether Mozart is either better or worse than Michelangelo with respect to proto-CG, we can infer from the axiom of determination that (1) Mozart is neither better nor worse than Michelangelo, with respect to CG.

Closing Remarks: My formalization raises some intriguing questions of philosophical methodology. E.g. what philosophical interest can such a formal model have? What does this style of argument really show? It's not as though the process I've described is meant to literally reflect the fundamental metaphysics of values. It's merely a model. (In particular, it seems implausible that my internal 'proto-CG' variable corresponds to any significant value in reality! I employ it as a purely technical 'fix', to get my model to yield the right outputs.)

But I think it has philosophical worth in the following respect: it establishes that Chang's premises about value are mutually consistent. This model shows one possible way that they could all be true. Perhaps reality provides another. But at least we can dispel our initial skepticism about whether they were consistent at all.

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The Traveller's Dilemma

Two people independently pick a dollar value between 2-100. If they pick the same number, then both are paid this amount. Otherwise, the lowest value is paid out to each, with the greedier person paying a $2 penalty to the other player in addition. If you want to maximize your earnings, what number should you pick? Cue the game theorists:

To see why 2 is the logical choice, consider a plausible line of thought that Lucy might pursue: her first idea is that she should write the largest possible number, 100, which will earn her $100 if Pete is similarly greedy. (If the antique actually cost her much less than $100, she would now be happily thinking about the foolishness of the airline manager's scheme.)

Soon, however, it strikes her that if she wrote 99 instead, she would make a little more money, because in that case she would get $101. But surely this insight will also occur to Pete, and if both wrote 99, Lucy would get $99. If Pete wrote 99, then she could do better by writing 98, in which case she would get $100. Yet the same logic would lead Pete to choose 98 as well. In that case, she could deviate to 97 and earn $99. And so on. Continuing with this line of reasoning would take the travelers spiraling down to the smallest permissible number, namely, 2. It may seem highly implausible that Lucy would really go all the way down to 2 in this fashion. That does not matter (and is, in fact, the whole point)--this is where the logic leads us.

Jason Kuznicki has an interesting response:

The smallest figure only becomes a reasonable strategy when neither Pete nor Lucy have any guidance whatsoever about how the other traveler might respond. In the world of actual prices, this never happens. In other words, the $2 solution is only plausible when it is entirely divorced from economics, and when neither player has any cues at all for giving an answer.

I don't think market cues are relevant here. There's a perfectly salient "default option" even in the abstract case, namely, the maximum value of $100. Suppose the market price is $50. The recursive "race-to-the-bottom" reasoning will apply just as disastrously from 50-49 as it originally did from 100-99. Changing our starting point doesn't really change anything. It's the reasoning that's the problem.

The real solution, then, is to affirm norms of global rationality: look at the big picture, and reason according to a decision-procedure that will predictably yield better results. That means ditching the economist's "local rationality" of backwards induction and its race to the $2 bottom. The SciAm writer has it right:

If I were to play this game, I would say to myself: "Forget game-theoretic logic. I will play a large number (perhaps 95), and I know my opponent will play something similar and both of us will ignore the rational argument that the next smaller number would be better than whatever number we choose." What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler's Dilemma.

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Infinity ain't everything

Steve Esser drops an interesting comment at FQI:

If the actual universe is infinite, wouldn’t it contain all possibilities?

Not necessarily. A collection may be infinite whilst making systematic exclusions. Consider the set of even numbers. It's infinite in number, but it doesn't contain all the numbers. Granted, the odd numbers never really had a chance, in this artificial case. So perhaps the thought is instead that an infinite expanse will contain every finite pattern that it possibly could. This seems more intuitive. But in fact this too turns out to be false.

Consider an infinite sequence of coin-flips. Suppose it is a fair coin, so for each individual flipping event, there is a 50/50 chance of it landing heads or tails. So one obvious possibility is for a "tails" event to occur. But is it guaranteed to occur, at some point in the infinite sequence? Well, no. There is, after all, some (infinitesimal) chance that the coin will land heads on every single flip whatsoever! In this case, all the other possibilities that might have obtained, won't. So, infinity is no guarantee of actualizing every possibility.

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The Shooting Room Paradox

Here's a fun puzzle from the same guys that introduced me to the infinite spheres of utility problem. This one's more along the lines of the Doomsday argument, or perhaps the Sleeping Beauty paradox. Anyway, here's the set up: a group of people are sent into a room. Two dice are rolled, and if they land double sixes then everyone in the room gets shot. Otherwise, they're released and the whole procedure is repeated again with a new group of ten times as many people. (And so on, until a group gets shot.) You're sent into the room. What is your chance of being shot?

Well, obviously 1/36, right? That's the chance of double sixes being rolled. But note that the vast majority (~90%) of people who enter the room get shot. (This is stipulated -- assume there is an unlimited stock of people, ammo, etc. There's no risk of "running out", no matter how many rounds the game goes on for.) You have no reason to consider yourself one of the lucky 10%. You have the same chance of being shot that anyone else who enters the room does. So you must all have a 9/10 chance of being shot.

Very puzzling. I'm inclined to insist that in fact each person only has a 1/36 chance of being shot, even though 90% of the people will be shot. Chance is about causal bases (or something), not frequencies.

Curiously, this means that an immortal with an indefinite overdraft could make a lot of money by betting with every single individual about to enter the room. But then, an immortal with an indefinite overdraft can strategically gamble and profit from almost any repeated game, no matter how poor the odds. (Just keep doubling your bet until you win, then pocket the winnings and start over.) So maybe the same kind of thing is going on here.

Categories:

• logic
• probability
• infinity

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P iff Actually P

Let '@' be the rigidifying 'actually' operator. "P iff @P" is a classic example of an a priori contingency. We can know a priori that P has the same truth value as it does in this world. But there are other possible worlds where its truth value differs from ours, and so "P iff @P" is false at such a world (recall, the extension of "@P" is always its truth value in our world), and hence merely contingent.

Now, Dave Chalmers [PDF] has used this to cleverly argue that no non-actual metaphysically possible world-state is also epistemically possible. The reductio (p.19):

Let W be one such world-state, and let P be a contingent statement that is true at @ but false at W. By the reasoning above, <P iff @P> is a priori, so true at all epistemically possible world-states, so true at W. It follows that @P is false at W.

But @P is a necessary truth, given that P is in fact true at @. So this contradicts our assumption that W is a metaphysically possible world. So, by reductio, there can be no such W that is non-actual, metaphysically possible, and also epistemically possible.

This highlights one way in which epistemic space behaves differently from metaphysical modal space. In particular, [where "true(P,W)" means "P is true at W"] the logical axiom:
(iv) ∀P ∀W (true(@P,W) ≡ true(P,@))
fails to hold for epistemic space. There are epistemically possible scenarios which represent the actual truth value of P to be other than what it truly is. This should be unsurprising when we recall that in epistemic modality we are considering each world as actual (rather than counterfactual). As such, rigid designation - as a feature of a term's secondary intension - gets ignored.

Still, one might think this a mere semantic trick. In epistemic modality we can't treat the term 'actual' rigidly; but surely we can stipulatively introduce another term to play the crucial role. This will have to involve super-rigidity, i.e. terms whose primary and secondary intensions are both invariant. We might take 'S' as our new super-rigid operator, and say that 'Sp' is the claim that p is true of some super-rigidly designated scenario S. Even if other epistemically possible scenarios deny that S is actual, surely none would wish to misdescribe the intrinsic content of S. So we should accept the modified principle, even for epistemic space:
(iv*) ∀p ∀W (true(Sp,W) ≡ true(p,S))

But can't we now super-rigidly designate the scenario which corresponds to actuality? Can't we demonstratively identify S as this (de re, rigidly designated), actual, scenario? And if so, can't we know a priori that "p iff Sp"? This would then provide a counterexample to the core thesis of epistemic two-dimensionalism, i.e. that Q is a priori iff Q is true in all epistemically possible scenarios (≡ Q has a necessary primary intension). For "p iff Sp" will be false in those non-actual scenarios W where the truth-value of p differs from its truth-value in scenario S.

Update: on second thought, the above clearly won't work, since what gets demonstrated will vary depending on which scenario we consider as actual. Demonstratives can't base super-rigidity, in other words. The best you can get from them is standard (secondary) rigidity, which doesn't apply to epistemic space, as already explained. I'll leave in my alternative response below, albeit bracketed, but I think I must have mistraced the background to the Soames-Chalmers dialectic.

[[ Scott Soames puts the proposal as follows [PDF, p.14]:

[A]gents in @, but not those in other world-states, can move apriori from any truth p in @ to the proposition that p is true in @, and vice versa, simply by identifying @ demonstratively, as “this very state that actually obtains.”

However, as Chalmers notes in his response to Soames, once we recognize that the actual world is not a possible world, it's no longer so clear that scenario S is ostendible after all. We can point to the actual world itself, of course. But we can't directly point to the scenario or possible world which accurately represents actuality. So we cannot know a priori that "p iff Sp"; not for any scenario S. (Without ostensive identification, S might could be a non-actual scenario, for all we know.) ]]

Let's conclude by contrasting three different interpretations of "P iff actually P".

1) No rigidity: here we use the term 'actually' as a fluid indexical, so that "P iff actually P" is logically equivalent to "P iff P" and hence both necessary and a priori. No problems here.

2) Subjunctive rigidity (the standard view): here we use the term 'actual' as a rigid designator, whose secondary intension is invariant, but the primary intension varies as in (1) above. This is the sense in which "P iff actually P" is contingent a priori. Note that it is true in all epistemically possible scenarios still. No problem for 2D-ism here.

3) Super-rigidity. There's not really any sense in which "actually" is super-rigid, but suppose we were to instead substitute the super-rigid specification of the scenario S (where S happens to be actual). Then "P iff in S, P" is not even a priori. Like any other old contingent a posteriori truth, it poses no problems either.

Categories:

• 2d
• semantics
• modality

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Iterative Modal Theory

David Armstrong argues that there cannot be a set of all possible worlds:

Given a world with n wholly distinct elements, we can combine the elements to form 2^n possible worlds. Then we can construct a single world having non-overlapping parts parts, with a part to duplicate each of the 2^n worlds. And since this procedure is available for any set of possible worlds, there is no all-embracing set of such worlds.

The obvious parallel is with set theory. We know that there is no set of all sets. Nevertheless, there is an iterative procedure which shows us that, given a set, we can go on to form a higher-order set with a higher cardinality than the original set. Iterative set theory, which is contradiction-free as far as we know, provides a respectable way of talking about sets.

What we must accept, therefore, is an iterative conception of possible worlds. Given any world, in particular the actual world, Combinatorial principles deliver further worlds. But any attempt to form the set of all such worlds is defeated by a procedure which uses the given set to form worlds outside the set.

-- D. Armstrong, A Combinatorial Theory of Possibility, p.29.

He goes on to say that we can still speak of "all" possible worlds, so long as the universal quantifier is allowed to range over broader entities than sets (e.g. perhaps there is a "class" of all possible worlds).

Categories:

• modality
• mathematics

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The Limits of Divine Defeation

I previously noted an interesting phenomenon: if you hear from a source that you know with certainty to be infallible, "P is true but you cannot know it," then you'll be thrown into such a state of epistemic confusion that you truly won't know that P, despite having heard it from an infallible source! This suggests that God could undermine or 'defeat' what would otherwise amount to knowledge - and even certain knowledge! - simply by asserting to you that you lack sufficient justification for knowledge. God's assertion here is "self-fulfilling", since it itself is the cause of your epistemic confusion, and is the only reason why your belief fails to constitute knowledge in this case. Speaking generally, we might recognize the phenomenon of "self-fulfilling defeation claims": claims of epistemic defeation in which the claim itself constitutes the defeater. But I'll refer to this phenomenon by the snazzier title of "divine defeation".

Now, I'm wondering just how far God's powers of defeation extend here. (Not that God exists, of course, but it can be a useful philosophical heuristic to pretend otherwise.) His arbitrary proclamations can defeat knowledge, but can they do any more than that? In particular, can they defeat claims of justification, or of high rational subjective probability?

If we stipulate that God cannot utter falsehoods, then he won't be able to say things like, "P is true but you cannot believe it" (for arbitrary P). For that would simply be false; even after hearing God say this, I might go on to believe P (perhaps alongside the second-order belief that I do not believe that P -- hi Moore!). This yields the contradiction that you cannot believe P and you can believe P, thus providing a reductio of the claim that God could utter such a statement.

But what of the divine proclamation, "P is true but you cannot justifiably believe it?" Is this impossible, like the belief case? Or is it instead "self-fulfilling", like the knowledge case? Would hearing such a statement throw a rational agent into such a state of epistemic confusion that they would lose all justification for their belief? That seems implausible to me. You might be confused, but still, testimony from an infallible source has got to count for something, right? If I heard God say that, I would believe that P, and justifiably so! (I might also believe the latter conjunct, i.e. that my former belief is unjustified; but I would simply be mistaken on that point.) So the proclamation would be false, so God could not say it.

The key difference between this and the knowledge case seems to concern the possibility of "meta-defeation". By this I mean the idea that ("meta") doubts about your epistemic status can themselves influence this status. Plausibly, if you doubt whether you know that P, then you actually lack knowledge that P. (This principle seems debatable, however. And if rejected, one might deny that any sort of 'divine defeation' is possible, even for strict knowledge.) But it is far less plausible to hold that such 'meta' considerations defeat justification. One can be justified in believing that P, even if they doubt whether they're so justified.

We could close this gap by building the "meta" considerations into the case. God could say, "P is true, but you cannot justifiably* believe that P (where justification* supplements standard justification with the additional internal requirement that you rationally recognize your belief as being justified)."

Or perhaps that won't work either, since you might believe that P, and recognize this justification ("I heard it from an infallible source!"), all the while mistakenly taking yourself to lack this recognition. That is, much like before, your belief is in fact justified*, even though you don't believe it is justified*. The incongruity is simply pushed back a step. So one-step recursion isn't enough. We're going to need infinite recursion, e.g. by saying: "P is true, but you cannot justifiably** believe that P (where justification** supplements standard justification with the additional internal requirement that you justifiably** recognize your belief as being justified)." If that even makes sense.

(The idea is to close all the "meta" gaps, by rendering it impossible for one to be in this super-justified state without realizing it. In addition to your belief being justified, you must recognize this fact, and also recognize that fact, and then recognize... ad infinitum.)

I'm going to give myself a headache if I go on any further. But it looks like divine defeation is fairly limited in power. Just as well, since I just had an experience as of God telling me that I couldn't justifiably accept the conclusions of this post. That actually isn't true, but you'll never know for sure.

Categories:

• epistemology
• logic

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Infinite Spheres of Utility

Imagine a universe containing infinitely many immortal people, partitioned into two "spheres". In one sphere, all the inhabitants live a blissful existence, whereas the members of the other sphere suffer unbearable agony. Now compare the following two variations:

1) Everyone starts off in the blissful sphere. But each day, one more person gets permanently transferred across to the agony sphere, where they reside for the rest of eternity.

2) Everyone starts off in the agony sphere. But each day, one more person gets permanently transferred across to the blissful sphere, where they reside for the rest of eternity.

Which scenario is better? The answer, paradoxically, appears to be "both". At any moment in time, there will be infinitely many people in the original sphere, and only a finite number who have been transferred across. So option 1 is better.

However, each particular person will spend only a finite amount of time in the first sphere, whereas they will spend an eternity in their post-transfer home. So option 2 is better.

[A clarification is in order. As stated, it remains possible for some people to remain forever in their original sphere. (Suppose we assign each person a natural number. Each day we can transfer across the next even-numbered person. Then the infinitely many odd-numbered people never get transferred at all!) So let us stipulate that no-one is "skipped" in this way, and that every individual will indeed get transferred at some point.]

How are we to evaluate the options without falling into paradox?

(I owe this problem to recent discussion with ANU grad students. I think they in turn got it from Alan Hajek.)

Categories:

• Logic
• utilitarianism

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Moore's Paradox for Desires

Long-time readers may recall my interest in converting belief/truth-based paradoxes into desire/value analogues. So I found it very interesting to talk with David Wall, a grad student here at ANU, who is exploring the idea of a "Moore's paradox" for desire.

Recall that the standard Moore's paradox involves assertions like "p, and I do not believe that p", which sound very odd - even incoherent - despite being consistent and indeed quite often true. The problem is that assertions express beliefs, and the underlying belief expressed above is the Moorean belief:

(1) The belief that [p, and I do not believe that p]

where belief in the first conjunct undermines the second. Now, Dave is interested in the analogous Moorean desire:

(2) A desire that [p, and i do not desire that p]

Note that this is a single conjunctive desire, and should not be confused with a conjunction of atomic desires, e.g.:

(2*) A desire that [p], and another desire that [i do not desire that p]

2* is perfectly comprehensible, just think of Frankfurtian agents whose first-order motivations fail to match up with their second-order desires. Say someone who has a strong craving for drugs, but who wishes that he didn't. Note that they are only comprehensible as two distinct desires. If combined into one, i.e. if the agent wanted it to be the case that he took drugs while no longer wanting to, that just seems bizarre. It's not literally inconsistent, but there does seem something deeply irrational about desires which take this logical form. So there is a good prima facie case for an analogue with Moore's Paradox in this vicinity.

(Incidentally, if you'd like further support for the idea that a conjunctive desire can be irrational even when the conjunction of distinct desires is not, just see this example from the Ethical Werewolf, where a prisoner wants to eat his captor's food, and wants the captor's food to be poisoned, but doesn't want both!)

Dave fleshes this out by trying to show that Sorenson's analysis of Moorean beliefs also applies to Moorean desires. Briefly: holding the Moorean belief guarantees that one will fall short of the ideally true and complete belief set. Dave argues that a similar problem befalls the Moorean desire in (2).

I think this is a misguided strategy. For note that as far as the global coherence or completeness of a mental-state set is concerned, it makes no difference whether the states are considered distinct or conjoined into one. Compare (1) above with its segregated version:

(1*) A belief that [p], and another belief that [I do not believe that p]

In the global sense, (1*) is just as imperfect as (1) is. Either suffices to guarantee that one will fall short of the ideal belief set. So if this sort of global imperfection was the underlying explanation of Moorean paradoxes, we should find (2*) just as wrongheaded as (2). But we don't. The torn agent in (2*) is certainly less than ideal. But they're not suffering from the kind of severe incoherence we find in (2).

We might even say the same about the Moorean beliefs, though I think it less obvious there. But perhaps an agent could be in situation (1*) through having compartmentalized beliefs. Then, although there is an underlying incoherence in his belief set, and thus he is a less than fully ideal doxastic agent, still our agent is not nearly so irrational as someone with the Moorean belief in (1). So the problem here is a local one, concerned with these mental states in particular, and not the total completeness and perfectibility of our state sets considered as a whole.

(Besides, perfection is never a realistic goal to begin with, so it isn't clear why we should find it so shocking when a state guarantees that we will fall short of it. That outcome is pretty well guaranteed in any case!)

So, how else might we try to understand the problem with Moorean desires? I was initially struck by their interesting relation to conditional desires. Note that often we desire some future event (e.g. eating ice-cream) on the condition that we still desire it at the time of occurrence. I don't now desire that in future I eat icecream, if it happens that at the future time I will no longer want it!

But that regrettable mismatch -- between the desire's satisfaction and its (lack of) persistence at the time of satisfaction -- is precisely what we find in case (2). The agent desires that he goes unsatisfied in some sense; that he receives an object which he will no longer appreciate at the time of receipt. This seems very odd indeed.

Alternatively, we might steer closer to the belief-truth // desire-value parallel, as follows: The agent either considers p to be of value to him, or he does not. If not, then he has no reason to desire it in the first place. But if he does, then he shouldn't want this desire to go away, for that would detract from his appreciation of p's value. Thus, the conjunctive desire in (2) displays a kind of internal incoherence; resting on the judgments both that p has value, and that it does not.

This explanation more closely mirrors our understanding of the Moorean belief (wherein the agent appears committed to judgments both that p is true, and that it is not), and hence seems an opportune strategy in light of Dave's goals.

Anyway, Dave said he'd like some more feedback about all this (even from those who think the entire project is misguided, and that there isn't really any Moorean paradox for desire), so do leave a comment!

Categories:

• logic
• mind
• epistemology
• semantics
• philosophy

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Disjunctive Requirements

You shouldn't believe contradictions. For any given proposition, you should disbelieve either it or its negation. This claim is ambiguous. It might mean the wide-scope disjunctive requirement:
(WD) You ought to disbelieve either one of P or not-P (and it doesn't matter which, at least for the purposes of this requirement).

Or it might instead be proposing a disjunction of narrow scope requirements:
(ND) It is either the case that you ought to disbelieve P, or that you ought to disbelieve not-P.

For ND, it does matter which option you choose. One of them is the correct option, we just don't know which! WD, by contrast, can be satisfied equally well by either option. (Footnote 27 of Kolodny's 2005 illustrates this difference nicely. I can quote it if anyone wants further clarification.)

I find wide-scope requirements like WD interesting because, if normative, they open up a new aspect of (non-evidential) epistemic assessment. They might lead us to assess beliefs against internal standards of coherence and consistency, rather than the external standard of truth. I used this idea in an earlier essay to argue that there could be non-evidential reasons for belief: after all, even if there are no reasons at all to think that P is false, the mere fact that you believe not-P would seem to give you reason to disbelieve P, since doing so would bring you to satisfy requirement WD.

But I don't know how plausible that is. If all the evidence supports P, we might think it more plausible to simply insist that you ought to believe P, and - further - that there's nothing at all to be said in favour of disbelief, no matter that it contradicts your prior (ill-founded) belief that not-P. If you're more sympathetic to these claims, then you'll likely prefer ND to WD as the proper form of our non-contradiction rule.

Incidentally, this should not be confused with the simple narrow-scope conditional requirement:
(NC) If you believe that P, then you are rationally required to disbelieve not-P.

NC is plainly false as a universal principle. Sometimes rationality requires us to reject our prior beliefs in favour of their negations. This lends support to the wide-scope reading of the conditional:
(WC) Rationality requires that: if you believe that P, then you disbelieve not-P.

WC, unlike NC, may be satisfied by rejecting the antecedent (belief that P) as an alternative to fulfilling the consequent (disbelieving not-P). In fact, WC is logically equivalent to WD above (well, if we treat the earlier 'ought' as merely meaning "rationally required", leaving open the question of whether these requirements are genuinely normative). But this reminds us that there is a second way to revise NC. Rather than converting it into a wide-scope requirement, we might instead replace it with a disjunction of narrow-scope requirements, as in ND. So the problems with simple "NC"-style narrow-scope requirements need not lead us to accept wide-scope requirements.

Categories:

• rationality
• philosophy

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A Paradox for Subjective Rationality

I've written before about why rationality can't be too subjective. But here's another reason: it leads to contradiction. Consider Kolodny's two "core requirements" of rationality:

C+: If one believes that one has conclusive reason to have A, then one is rationally required to have A; and
C-: If one believes that one lacks sufficient reason to have A, then one is rationally required not to have A.

Now consider someone who believes that they have conclusive reason to do what they believe they lack sufficient reason to do. (Granted, this is a very odd belief to have. But I think it is possible. Perhaps they've been told that their beliefs have been manipulated by an evil demon into being unreliable [update: this is explained further in my comments below]. Or perhaps they're just incredibly irrational. Whatever.)

It would then follow from C+ that they are rationally required to do what they believe they lack sufficient reason to do. Call this action 'X'. That is, we have so far established that they are rationally required to X. But recall that our agent believes that they lack sufficient reason to X. It thus follows from C- that they are rationally required not to X. Putting these two results together, we find that our poor confused agent is both rationally required to X and rationally required not to X!

This violates what I will call the "consistency of rational requirements" principle:
(CRR) It is not possible for one to be both rationally required to A, and rationally required not to A.

In other words, rationality cannot make contradictory demands of us. It cannot demand both that we do something, and that we don't do it. That's just not a fair ask.

If (CRR) is true, as I think it is, then the case I provide above shows that Kolodny's "core requirements" cannot be true.

Categories:

• rationality
• logic
• philosophy

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Traversing the Infinite

Common sense would have us believe that you cannot travel an infinite distance in a finite amount of time. But do we actually have any good reasons to think that this is logically (rather than merely physically) impossible? On the contrary, the sort of exponential acceleration one imagines in accelerating Turing machines would seem to allow for the possibility of infinite traversals.

The idea is seen most simply in Zeno's paradox. Zeno taught us that any movement necessarily traverses an infinite number of spatial intervals (first half the distance, then half the remainder, then half of the new remainder, ad infinitum). But that's no problem, because the time it takes to traverse each interval is "accelerating" in the sense mentioned above - each interval will be traversed in half the time of the previous one. If we're travelling at a constant speed, and cross half the distance in one minute, then we will reach our destination after two minutes. The time it takes to traverse each of the intervals follows the pattern: {1, 1/2, 1/4, ... 1/(2^n) ...} which sums to 2. The paradox only gets off the ground if we make the false assumption that an infinite series cannot yield a finite sum. Contrary to Zeno, movement is possible after all. (What a relief!)

In the above example, we achieved the required 'acceleration' by travelling at a constant speed across decreasing distances. (Recall that each spatial interval was half the length of the one that went before.) But we could achieve the same effect by increasing our speed across constant distances. So let's take the infinite distance we want to traverse, and break it up into (infinitely many) 1 metre intervals. Suppose I am accelerating in such a way that it takes me 1 minute to cross the first metre, 1/2 to cross the 2nd, 1/4 for the third, and so on, in general taking 1/(2^n) minutes to travel the (n+1)th metre. After two minutes, I would be finished, having travelled an infinite distance in that time.

So contrary to common sense (and the old post at Mathetes that inspired this one), it seems logically possible to traverse the infinite. Any objections?

Categories:

• philosophy
• logic
• mathematics

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If the title of this guest post is true, then you should read it.

[By Charles Johnson]

Here's one of the few canonical philosophical puzzles that I had learned about by the age of five. What's the truth-value of the following statement?

(L) This statement (L) is false.

The problem, of course, is that if (L) is true then it's false, and if (L) is false then it's true. Thus, any theory of truth that assigns a truth-value to (L) is internally contradictory, since the theory will (inter alia) include the contradictory truth-ascription:

(TL) L is true if and only if L is false.

Since there are no true contradictions, a theory of truth must not assign any truth-value to (L) at all. But how do you doing it? If a statement hasn't got a truth-value, then the usual take is that they are, in some respect, nonsense; that is, they fail to make an assertion -- just as "Cat mat on the sat the" fails to make an assertion. The canonical approach to (L) in the 20th century has been to try to come up with some principled means of ruling (L) out of the language by means of setting up the right structure of rules beforehand (just as you can point to the preexisting rules of syntax to show that "Cat mat on the sat the" doesn't amount to a complete sentence). The most famous attempt, and the inspiration of many of the subsequent attempts, has been Tarski's attempt to sidestep the Liar Paradox by means of segmenting language into object-language and meta-language layers. The idea being that, if you do this assiduously, you can avoid self-referential paradoxes because self-reference won't be possible in languages whose sentences can be ascribed truth-values; because they can only be ascribed truth-values within a meta-language that contains the names of the object language's sentences and truth-predicates for those sentences. I have a lot of problems with this approach; a full explanation of them is something that I ought to spell out (indeed, have spelled out) elsewhere. But here's a quick gloss of one of the reasons: Tarski and the people inspired by him started setting up ex ante rules to try to rule out self-referential sentences because it's self-reference that makes the Liar Paradox paradoxical (and that makes for similar paradoxes in similar sentences; exercise for the reader: show how "If this sentence is true, then God exists" is both necessarily true and strictly entails the existence of God). But there's an obvious and general problem for the method: there are self-referential sentences which are unparadoxical, and indeed self-referential sentences which are true. Here's an example which may or may not cause trouble for Tarskian theories, depending on the details:

(E) This sentence (E) is in English.

(E) is truth-valuable; and in fact it is true. (If, on the other hand, it had said "This sentence is in French," it would have been false.) Now, this may cause trouble for the Tarskian method and it may not, depending on the details of a particular account. (Sometimes people want to ban all self-referential sentences; sometimes they are more careful and claim that object languages might be able to name their own sentences but only so long as they don't contain the truth-predicates for their own language.) But even if (E) is allowed, you haven't solved the problem. There are plenty of self-referential truth-ascribing sentences that aren't paradoxical, too. Here's one:

(EM) Either this sentence (EM) is true, or this sentence (EM) is false.

Unlike (L), this causes no logical paradoxes. If you suppose that it's false, that means that it turns out to be true -- since the second disjunct, "this sentence (EM) is false" turns out to be true; meaning that it cannot