Topics Logic

Question In paradoxes such as the Epimenides 'liar' example, is it not sufficient to say that all such sentences are inherently contradictory and therefore without meaning? Like Chomsky's 'the green river sleeps furiously', it's a sentence, to be sure, but that's all it is. Thanks in advance :)

Accepted:
May 8, 2014

Comments

I think you're right to suspect that the Liar (or Epimenides) sentence, "This sentence is false," is meaningless, i.e., that the sentence fails to express a proposition. But I wouldn't say that the sentence is meaningless because it's self-contradictory, like the sentence "God exists and doesn't exist." The latter sentence is surely false, in which case the sentence expresses a (false) proposition and hence isn't meaningless.

If the Liar sentence is meaningless, then it doesn't assert of itself that it's false (because it doesn't assert anything), and therefore one of the premises used to generate the Liar paradox is false.

Some philosophers have said that the Strengthened Liar sentence, "This sentence is not true," blocks such a solution to the paradox, on the grounds that a meaningless sentence is not true. The proper reply, I think, is to agree that a meaningless sentence is not true but to deny that the Strengthened Liar sentence asserts of itself that it's not true (again, on the grounds that the sentence asserts nothing at all).

The harder part is to explain why the Liar and Strengthened Liar sentences are meaningless (beyond pointing out that a contradiction follows from assuming that they're meaningful). It can't simply be that they're self-referential sentences, since plenty of self-referential sentences are meaningful. The explanation has to be more complicated than that.

One last point: Even if the Liar and Strengthened Liar paradoxes can be solved by claiming that the sentences involved are meaningless, much more argument would be needed to show that all semantic paradoxes, let alone all paradoxes of any kind, can be solved that way.

Chomsky's sentence was actually: "Colorless green ideas sleep furiously". Several people have argued that, embedded in the right kind of context, it can be taken as meaningful. For some examples, see a handout from one of my courses here.

PLEASE NOTE: Professor Maitzen's responses here and below were originally offered in colloquy with Professor Heck, who has since chosen to remove his contributions. [Alexander George on 6/6/2014.]

But then it is a simple step of disjunction introduction to (S) itself.

This simple step works only if (S) is the disjunction of (S') and "(S) is false," each of which disjuncts is meaningful. But if (S) is meaningless, then (S) isn't the disjunction of two meaningful disjuncts, and in particular it's not the disjunction of (S') and "(S) is false." I agree that this response to the Strengthened Liar implies that the meaningfulness of a sentence-token won't always be facially obvious. That implication seems less dire than the implications of some other responses.

Me too, but that was my point: Despite appearances, (S'), which I endorse, isn't the first disjunct in (S). Similarly, despite appearances, the Epimenides sentence doesn't assert of itself that it's false. It follows that the meaningfulness of a sentence-token depends on more than the string of words it contains, but that result isn't surprising in light of other things we know about language.

Thanks, Richard, for your replies. Nice colloquy we're having. I hope anyone else is interested!

Is there or is there not a sentence that is the disjunction of (S') and the sentence "(S) is false"?

There is, and we can token it, but not by way of the sentence-token that you labeled "(S)" in your example. That's been my point all along: two type-identical sentence-tokens can be such that one is meaningful and the other isn't. The context in which a sentence-token is uttered can deprive it of propositional content. I say that's not surprising given other things we know about language. It won't do to respond "But I'm talking only about the syntax!" because you can't generate a liar paradox without assuming things about the truth-conditions of particular strings of words.

Presumably you would also regard the sentence "(S) is false" as itself meaningless, on the ground that (S) is...

Goodness, no! That would be terrible reasoning. To say of a meaningless sentence-token that it's false is to say something false. So "(S) is false" is false. (To head off an objection: Negation is an operation applying to propositions, strictly speaking. The proposition expressed by "(S) is false" isn't the negation of the proposition expressed by (S), because again (S) doesn't express a proposition.) Discomfort avoided.

Can't we be quite sure, even if we are ignorant of the status of the continuum hypothesis, that (T) is not true?

Yes, we can. The sentence-token you labeled "(T)" is meaningless, hence not true. If Jones believes that sentence-token (T) is false, then Jones's belief is false.

Even if we allow the idealized counting required by the example involving (N) and (D), the example trades on an equivocation, and it dissolves once we recognize the equivocation. "Nixon has said 100 things about Watergate other than (N), and exactly half of them have been true" is true only if "said" means "explicitly said and not merely implied," because the number of things implied by what Nixon said about Watergate would be infinite. But of course (N) and (D) generate a paradox only if "said" means "explicitly said or implied": you need the implications of what's said, and not just the explicit saying, to get the paradox going. Problem dissolved.

Regarding sentence-tokens (1) and (2), I have no difficulty in declaring them meaningless. On the contrary, I think anyone bears the burden of proof who would look at those two sentence-tokens and say "Obviously they're meaningful." Obviously?

This is turning out to be an easy way of upping our response-counts!

Surely I can name that sentence (S) if I so choose?

I wouldn't say "surely." (After all, in bygone days we thought "Surely there's a set corresponding to any well-defined predicate we choose.") It may turn out that in this case, on pain of contradiction, the name attaches only to the token and not to the type.

...if you allow that "(S) is false" is false if (S) is meaningless...

I trust that you too allow it -- indeed, that you insist on it.

...then it is hard to see why one would ever regard (S) as meaningless: It is a disjunction of meaningful disjuncts.

Here you're simply repeating the claim I've been denying. (Furthermore, I don't see what how the antecedent of your conditional supports the consequent.)

That, not the reasoning you took me to be attributing to you, is why I was assuming you would deny that "(S) is false" is meaningful.

The reasoning I took you to be attributing to me is the reasoning I quoted from you: "Presumably you would also regard the sentence '(S) is false' as itself meaningless, on the ground that (S) is [meaningless]." As I said, it's terrible reasoning.

But what the example is supposed to show is that, even if (T) is paradoxical, Jones can still believe it, in which case it has to be meaningful.

(T) is a sentence (leave aside whether type or token). In your example, Jones forms the belief that (T) expresses a falsehood. It's a false belief about (T), since (T) expresses no proposition at all. Jones's forming that belief about (T) doesn't imply that (T) is meaningful.

But the mere observation that there is context-dependence in natural language is no help here.

Really? Surely it lends credence to the idea that context can make two type-identical sentence-tokens differ in whether they're meaningful.

One who thinks there is context-dependence in (S) needs to say where it is and how it functions.

I agree, and I haven't explained how it works, but this probably isn't the place. Yet surely something is weird about a context in which "(S)" is used to label the sentence "Either (S) is meaningless, or else (S) is false"!

Yes, those were my words. But the argument I was attributing to you...was NOT supposed to be: If (S) is meaningless, then "(S) is false" is meaningless.

Then you can see why I was misled by what you actually wrote. In any case, the argument you say you meant to attribute to me contains a premise I deny, namely, that the first disjunct in (S) is "clearly meaningful." I've been claiming that (S) is meaningless, and I deny that any part of (S) says that (S) is meaningless, just as I deny that the sentence "This sentence is meaningless" (or any part of it) says of itself that it's meaningless.

I understand that you wish to resist the claim that the conjunction of those two things (which happens to be (T) itself)...

You know I deny the claim in parentheses. Every conjunction has truth-conditions, but (T) has no truth-conditions. Neither do "dog" and "cat," and we don't produce a conjunction by writing an ampersand between those two inscriptions.

The answer cannot just be, "Well, we get contradictions if we say that!"

Avoiding contradictions is surely a good reason not to say something that implies them. But I conceded in my original reply to the questioner that the hard part is explaining why the liar sentences fail to express propositions. Mere self-referentiality, for instance, can't be the reason.

(U) Either the continuum hypothesis is true or else (U) is false.

This example is no more persuasive than (T) was. I say that (U) is meaningless, in which case (U) doesn't follow from CH by any rule of inference, including disjunction introduction.

For what it's worth, people who defend context-dependence approaches to the liar, such as mentioned earlier, generally do not think that utterances of the liar are meaningless.

Glad to hear that the approach I'm defending isn't totally unoriginal.

Why does "true" make all the difference?

It doesn't. "This sentence is meaningless" expresses no proposition, on pain of contradiction, without containing "true." What's the deeper explanation? I have a marvelous answer, but it's too long to fit in the margin of this book.

(V) No token of (V) expresses a true proposition.

Earlier you insisted that "(S)" named a given sentence-type, and I replied, "It may turn out that in this case, on pain of contradiction, the name attaches only to the token and not to the type." Now you propose to give the name "(V)" to the sentence-type "No token of (V) expresses a true proposition." If you can try and fail to name a type once, you can try and fail a second time.

Okay, I'll defend my main claims in detail. Following Charles Parsons, you offered the following Strengthened Liar sentence:

(S) Either S is meaningless, or else S is false.

I claimed, and still claim, that S is meaningless. Reasoning:

(1) If S is meaningful, then (a) S expresses the proposition that: Either S is meaningless or else S is false. [What else could S express if it were meaningful?]
(2) If (a), then (b) S is true or S is false. [Bivalence for propositions]
(3) If S is meaningful, then (b). [From (1), (2)]
(4) If S is true, then S is true and not true. [Strengthened Liar reasoning]
(5) If S is false, then S is true and not true. [Strengthened Liar reasoning]
(6) If (b), then S is true and not true. [From (4), (5)]
(7) Not (b). [From (6) by contradiction]
(8) S is meaningless (i.e., not meaningful). [From (3), (7)]

Before you say that (8) commits me to S by disjunction introduction, recall that I distinguish tokens from types. What (8) commits me to is the meaningful and true token

(8*) Either S is meaningless, or else S is false,

which is a different token from S. (I have an argument that S isn't a sentence-type, available on request.)

Regarding your example

(U) Either the continuum hypothesis is true or else U is false,

I can show that if U is meaningful, then the continuum hypothesis is true:

(9) If U is meaningful, then (d) U expresses the proposition that: Either the continuum hypothesis is true or else U is false. [What else could U express if it were meaningful?]
(10) If (d), then U is true or U is false. [Bivalence for propositions]
(11) If U is meaningful, then U is true or U is false. [From (9), (10)]
(12) If U is false, then U is false and not false. [Strengthened Liar reasoning]
(13) U is not false. [From (12) by contradiction]
(14) If U is meaningful, then the continuum hypothesis is true. [From (9), (13)]

Thus, if you claim that U is meaningful, you're committed to the soundness of a silly "proof" of the continuum hypothesis (or anything else I put as the first disjunct!). I, who claim that U is meaningless, am not.

Lastly, I'll show that your example

(V) No token of V expresses a true proposition

is meaningless even if (as you insist) it's a sentence-type:

(15) If V is a sentence-type, then V is meaningful only if (e) V expresses the proposition that no token of V expresses a true proposition. [What else could V express if it were meaningful?]
(16) If (e), then V is true or V is false. [Bivalence for propositions]
(17) If V is a sentence-type, then V is meaningful only if V is true or V is false. [From (15), (16)]
(18) If V is a sentence-type and V is true, then V is not true. [No sentence-type having no true tokens is true.]
(19) If V is a sentence-type, then V is not true. [From (18)]
(20) If V is a sentence-type, then V is meaningful only if V is false. [From (17), (19)]
(21) If V is false, then some token of V is both true and not true. [Strengthened Liar reasoning]
(22) V is not false. [From (21) by contradiction]
(23) If V is a sentence-type, then V is meaningless. [From (20), (22)]

Paracomplete theories, which are perhaps the most popular these days (though I do not myself incline to them) would reject (2), (10), and (16). There are many other choice points as well.

Yeah, yeah. There are also theories that (claim to) reject (7), (13), and (22). Are we to think that those moves are even remotely plausible? Maybe they're plausible and not plausible, or neither?

Perhaps you mean something like...

No, I mean exactly this:

(S) Either S is meaningless, or else S is false.

(24) S is meaningless. [Repetition of (8), already established]
(25) Either S is meaningless, or else S is false. [From (24) by disjunction introduction]
(26) If S is a sentence-type, then S is a meaningful sentence-type or S is a meaningless sentence-type. [If P, then (P & Q) or (P & not Q).]
(27) If S is a meaningful sentence-type, then (f) the token of it labeled "S" above is meaningful.
(28) Not (f). [From (24)]
(29) If S is a sentence-type, then S is a meaningless sentence-type. [From (26), (27), (28)]
(30) If S is a meaningless sentence-type, then S has no meaningful sentence-tokens. [What else could a meaningless sentence-type be?]
(31) If S is a sentence-type, then (25) is a meaningful (indeed, true) token of that type. [From (25)]
(32) S isn't a sentence-type. [From (29), (30), (31)]

So you are committed to denying that (V) is a sentence-type, as well.

No. I'm committed to denying that V is a meaningful sentence-type, which is the correct stance anyway: see (15)-(23).

Never mind which steps others may reject. Which steps do you reject?

Thank you for the argument for that claim, but your reasons for it do not particularly interest me.

Wow. How very philosophical. We philosophers aren't interested in each other's reasons, after all. Now, am I supposed to be interested in the reasons you're giving for your claims?

I've given a numbered-step argument for a claim about S, in particular, that you've been denying, viz. (32). You've responded by referring me to work that you say bears on a sentence that you say is "like" S. I'm not asking you to take my say-so. If (32) is false, then there's a mistake in my (1)-(8) or (24)-(32). Surely a professional logician can tell us what it is.

You, Richard, claim to have established something by your (24)-(28), but your (24) and (25) both lack justification:

(24) If (V) is a sentence-type, then no token of (V) expresses a proposition. (No token of a meaningless sentence expresses a proposition.)

The justification you provide simply doesn't justify (24). You haven't established that V is meaningless, and I doubt it can be done. All that can be established in that regard is my (23).

(25) If (V) is a sentence-type, then no token of (V) expresses a true proposition. [~p --> q |- ~(p --> q & r)]

The justification you provide for (25) isn't even a theorem. Your reasoning wasn't "more compact" than mine; it was just sloppier.

...you are also committed to denying that (V) is a sentence-type, by the reasoning I gave.

Sloppy reasoning doesn't commit me to anything.

Next you say:

If (V) is not a meaningful type, then what else could it mean for the type (V) to be meaningless except that: No token of (V) expresses a proposition.

I suspect more sloppiness. From the claim that V isn't a meaningful type, it doesn't follow that V is a type that's meaningless. It follows merely that V is either not a type, or not meaningful, or both: i.e., my (23). Certainly "No token of V expresses a proposition" doesn't imply that V is a meaningless sentence-type. Proof: Let V be the word-type "while."