Here's roughly what I'm thinking at the moment.

The Liar purports to predicate falsehood of itself, but as asserting and denying the Liar come to the same thing this is no real predication at all, but only a sort of pretense.

If we force the Liar into the same sort of form Russell analysed, we get for the logical form a somewhat better result than we might have expected -- something like this:

There is a unique sentence that predicates falsehood of itself, and is false.

Better because, if we make the case that the Liar does not predicate but only purports to, then that description is indeed vacuous, and for the very good reason that just as we cannot consistently affirm or deny the Liar, neither can the Liar itself. If it admits it's not really predicating, we're done; if it lies to itself, and relies on that lie, it fails.

I'm still going round and round on this, but that's where I am at the moment.

ADDENDUM:

I was thinking of definite descriptions but I suppose you could do something like

(Li) is a member of the class of sentences that predicate falsehood of themselves, and (Li) is false.

Then you deny that this class has any members.

Either way, you end up claiming, as with the present king of France, that the sentence is false not because it's true but because no sentence actually does predicate falsehood of itself.

It's a puzzle, to be sure.

So the first test I'd advocate would be, does that work for other formulations - such as "This sentence is true".

There is a unique sentence that predicates truth of itself, and is true.

So would one make the case that this sentence does not predicate but only purports to?

Well, one might, but it's not so clear as to why. Yet if we do not, then it seems we are engaged in special pleading, for we do it to the Liar, but not to this other, very similar, sentence.

It's a curious issue.

I'm thinking that "This sentence is true" is correct relativism, while "This sentence is false" is improper relativism

"This sentence is true" is correct relativism, — Gregory

What is relativism here?

I believe it's a an infinite thought that is not spurious.

And what would you do differently as a result of "This sentence is true"?

It is also, presumably, meaningless? — Banno

Yes. I couldn't hold the belief that "this sentence is true" because there's nothing I'd be inclined to do differently to if it were false. Part of that is meaninglessness (lack of meaningful referent), but part is the self-referentiality. It goes nowhere. "This sentence has five words" is also self-referential, but is affects other beliefs (how easily I could fit it on a page, what number I'd reach if were to count each word out cardinally, how much ink I should need to write it...), so it's not semantically closed.

As a third problem (not that we need a third reason why it's nonsense), such a closed self-referential sentence gives us no context. Is the sentence 'true' like an arrow, 'true' like a true-gentleman, 'true' like straight ruler, 'true like the capital of France being Paris? Is the sentence 'false' like 2+2=5, or 'false' like "make one false move and the bomb will go off".

I believe it's a an infinite thought that is not spurious. — Gregory

And... what is an infinite thought?

I'm not following at all.

Well when I read the two paradoxes you presented, my mind did something infinite each time. One positive, the other negative. Btw, is "This sentence is true" your personal creation or did you read it somewhere?

when I read the two paradoxes you presented, my mind did something infinite each time. — Gregory

Well how come it's finished then?

How can someone enter eternity you might ask in that case.

How can someone enter eternity you might ask in that case. — Gregory

It doesn't seem like the sort of thing I'm likely to ask, no.

is "This sentence is true" your personal creation or did you read it somewhere? — Gregory

It's pretty standard to contrast this with the Liar.

"Who created the world?" asks young Heidegger .

"Who? Huh? The potential to be does that" responds Hegel

"Then the world is inferior to potentiality!" says Heidegger

"Potential is the slave of the actual. " Hegel

"That sounds backwards." Heidegger

" Because you need to see mind as superior to world. " Hegel

"Then potential is subservient to mind!!" Heidegger

" No. Mind is in a different place" responds Hegel

"Then, so what is real?" Gasps Heidegger

"The world!" Hegel

"How?" Heidegger

" Because this sentence is true" chimes in Banno

Sure. Cheers.

That seems to me to work.

The prosentential theory also throws out 'This sentence is true', I believe, on the grounds that this is only purported anaphora; being your own antecedent, you cannot inherit your content from anywhere, so you have no content.

My little thing sees them differently. The argument is more or less:

If a sentence could assert its own falsehood,
and if S were such a sentence,
then there would be no difference between affirming and denying S,
therefore there would be no difference between S asserting its own falsehood and not,
therefore S can only purport to assert its own falsehood,
therefore no sentence can assert its own falsehood.

I find the no-content approach pretty persuasive, but I like that this approach recognizes that the mess we're in with 'I'm false' is different from whatever is odd about 'I'm true.'

In his book (p. 81), Scharp mentions that the following triad is inconsistent for a logic L:

(i) L accepts modus ponens and conditional proof;
(ii) L accepts the standard structural rules for derivability (in particular, it accepts cut and contraction);
(iii) The theory consisting of capture (from S infer T("S")) and release (from T("S") infer S) is non-trivial in L.

Scharp argues that the culprit is (iii). But, as Ripley argues in his review of Scharp's book, it may be that the culprit is (ii). In order to understand what is going on, it helps to recast your derivation in terms of the sequent calculus. For those who don't know, the sequent calculus is a calculus that instead of operating with sentences, operates with sequents, or sequences of sets of sentences. The basic idea is this: we interpret a sequent S : R as saying that the disjunction of the sentences in R is derivable from the conjunction of the sentences in S. So it allows us to study the structural properties of the derivability relation. Here are a couple of important rules (I'll use S, R as variables for sets of sentences and A, B, C for sentences):

Structural Rules

Weakening: From S : R, infer S, A : R; from S : R, infer S : A, R (i.e. if a disjunction of a set of sentences is derivable from S, it is derivable from S and A; if a disjunction of a set of sentences is derivable from S, then adding a further disjunct preservers derivability);

Cut: From S : A, R and S', A : R', infer S, S' : R, R' (i.e. if A implies B and B implies C, then A implies C);

Contraction: From S, A, A: R, infer S, A : R; from S : A, A, R, infer S : A, R (i.e. we can reuse premises during a derivation).

Identity: A : A can always be inferred.

Rules for negation:

~L: From S : A, R, infer S, ~A : R (if A v B and ~A, then B);

~R: From S, A : R, infer S : ~A, R (if A & B implies C, then B implies ~A or C)

Rules for truth:

Capture: From S, A : R, infer S, T("A") : R;

Release: From S : A, R, infer S : T("A"), R.

Moreover, a contradiction is symbolized in this system by the empty sequent, : .

Using these rules, we can show that the liar implies a contradiction as follows:

L : L (Identity)
T("L") : L (Capture)
: ~T("L"), L (~R)
: L, L (Definition of L)
: L (Contraction)

L : L (Identity)
L : T("L") (Release)
L, ~T("L") : (~L)
L, L : (Definition of L)
L : (Contraction)

And from : L and L : , we may derive, by cut, : .

This derivation is obviously more complicate, but, on the other hand, it makes clear what are the structural principles involved in Scharp's (ii): Cut, Identity, and Contraction (it also has the advantage of making clear that conjunction is not involved, so that the only logical connective involved is negation). Now, Identity is unimpeachable. What about Cut and Contraction? Now, it is well known that Cut and Contraction are strange rules. In particular, they are the only rules whose premisses are more complicated than the conclusion. In particular, they are the only rules that allow for a formula to "disappear" from the conclusion (that the interaction of Cut, Contraction, and quantification produces anomalies is well-known. Cf., among others, the comments from Jean-Yves Girard on Contraction in his The Blind Spot and this interesting paper by Carbone and Semmes). So we know from logical investigations alone that there are problems with Cut and Contraction.

But there is more. Cut and Contraction are not just responsible for the Liar. As Ripley notes in his review of Scharp (linked above), they are also responsible for a whole host of paradoxes. So, if we get rid of those, we get rid not only of the liar, but also of those other beasts as well (Ripley elaborates a bit in this paper). So why is Scharp so sure that the inconsistent concept is truth? Maybe the inconsistent concept is validity or derivability, if we think of Cut and Contraction as constitutive of those...

The prosentential theory also throws out 'This sentence is true', — Srap Tasmaner

Fair point.

I have not read through the entire thread, so apologies if this point has already been made.

Maybe I'm being naive or missing the point, but I use the word "truth" pretty much as it is used in a court of law. When you swear to tell the truth, the whole truth, and nothing but the truth? Basically you are saying that your words and sentences will - to the best of your ability - describe facts. I'm not super knowledgeable about all the different schools of philosophy, but I'm pretty certain that this is some variation of the Correspondence Theory.

So when you say "This sentence is false"? In order for for this sentence to have any meaning, the pronoun "this" must refer to some statement that makes a factual assertion about reality/existence/the universe/etc. In this case, no such assertion is being made, hence the sentence is meaningless and cannot take a truth value.

This would equally apply to many variations.

"This sentence is true"
"The sentence 'Quadruplicity drinks procrastination' is false".
etc

"This sentence is false" has made a difference to reality by eliciting this thread.

Let us suppose you are right and the Liar is meaningless. This raises the question: why is it meaningless? Let us suppose, for definiteness, that the liar is "This sentence is not true". It is composed of meaningful parts meaningfully put together. That is, "This sentence", "is not" and "true" are each meaningful expressions and the sentence is grammatical. So why does it fail to be meaningful?

One possible answer to this is 's: the problem is with self-reference. Self-reference is a meaningless construction. This is tempting, but, as I have pointed out in my first post in this thread, self-reference is built in our best syntactic and arithmetic theory. So unless we are also willing to throw out arithmetic, self-reference must be considered unimpeachable. But if it is not self-reference, then what is the culprit?

One possible answer to this is ↪Janus
's: the problem is with self-reference. Self-reference is a meaningless construction. This is tempting, but, as I have pointed out in my first post in this thread, self-reference is built in our best syntactic and arithmetic theory. So unless we are also willing to throw out arithmetic, self-reference must be considered unimpeachable. But if it is not self-reference, then what is the culprit? — Nagase

I have little knowledge of syntactic and arithmetic theory, so I would appreciate it if you could explain how self-reference is built into them.

I haven't said that self-reference always leads to meaninglessness, or that it alone is the problem in the case of the Liar. I suggested earlier that maybe self-reference coupled with a truth claim could be the problem.

@Banno then said this was in a nutshell Kripke's response, and linked an SEP article, which I lack the predicate logic background to understand without considerable work, so I'm still in the dark as to the tie-in with Kripke.

I don't have much to add besides what I have already mentioned in my first reply to you. Take arithmetic, for instance. It is not difficult (though it is laborious) to show that it can code any syntactic notion. In particular, given any reasonable alphabet and vocabulary, it is possible to code it using numbers (this should be obvious in this digital age). Less trivially, it is possible to code any syntactic operation using predicates about numbers (this is called arithmetization of syntax). Importantly, the operation of replacing a variable x in a formula P(x) by another term, say n, is also expressible in the language of arithmetic. Using these ingredients, we can build, for any formula P(x), a sentence S such that S is equivalent to P("S"), where "S" is the code for S. So S effectively says of itself that it has P. This is called the fixed-point or diagonal lemma in most treatments (for more details, cf. Peter Smith's Gödel book, esp. chaps. 19-20).

Since this construction uses only typical arithmetical resources (addition, multiplication, numerals), it follows that any arithmetic theory will contain self-reference. Hence, self-reference, by itself, is not to blame for the problems resulting from the liar. You now say that the problem is with self-reference coupled with truth. That's not exactly right, since we can construct paradoxes without the use of self-referentiality (see Yablo's Paradox), but let us suppose you are right for the sake of the argument. The question then arises about what it is about truth that, when coupled with self-reference, generates the paradox. That this is a problem specific to truth is clear from the fact that other notions, when coupled with self-reference, do not generate any paradox (e.g. provability). And here we are back to square one.

I wonder if you could clarify something for me: when you talk about getting rid of Cut and Contraction, is the idea that we can build useful formal systems (good enough for mathematics) that are not prey to the Liar -- though natural languages may have to struggle on without relief -- or is it that the sequent calculus, say, or some other system, represents the system that underlies the informal* reasoning we do in natural languages? (I associate -- loosely and perhaps incorrectly! -- the first view with Frege and the second with Montague.)

Are we to say, 'If only Epimenides (and everyone since) hadn't used Cut and Contraction!' or 'At least we don't have to worry about that in our new and cleaner system'?


* Perhaps "unformalized" is better -- the whole point is that such reasoning may have an underlying system we would consider strongly analogous to the formal systems we develop.

A bit of both, I suppose. I'm following Scharp (and Ripley's account of Scharp), here (this is not meant as an endorsement of his position, I'm just trying to explore it). If I got the gist of his position right, he argues that the constitutive principles of truth, as present in our ordinary practice, are inconsistent. Ripley then presents the following suggestion: perhaps it is not our practices regarding truth that are inconsistent, but rather our ordinary practices regarding validity or perhaps derivability which are inconsistent. Now, once we identify a certain conception as inconsistent, either we accept the inconsistency and attempt to live with it (a dialetheist would perhaps adopt this view), or else we may try to reform it (this is Scharp's position). One way of reforming it is by jettisoning the principles that got us in a pickle in the first place. If you accept Ripley's diagnosis, then that means creating a new formal system that gets rid of Cut (I think you can keep Contraction if you get rid of Cut).

So it is both "If only Epimenides (and everyone since) hadn't used Cut and Contraction!" (the diagnosis part) and "At least we don't have to worry about that in our new and cleaner system!" (the prognosis).

The position is very reminiscent of Carnap's ideal of explication. According to Carnap, it is common for an ordinary concept to be vague and imprecise. This is not troubling---indeed, it may be an advantage---in our day-to-day dealings, but it may hamper scientific progress. Thus, one of the main tasks for the philosopher is to create precise (typically formalized) analogues of those ordinary concepts which are important for science. These analogues need not capture every feature of their ordinary counterparts---if they did, they would not be as precise as we need! But they should capture enough for the scientific purpose at hand. This activity of engineering precise concepts for the needs of science is called by Carnap explication. My point is that, as far as I can see, both Scharp and Ripley are engaged in explication.

Let us suppose you are right and the Liar is meaningless. This raises the question: why is it meaningless? Let us suppose, for definiteness, that the liar is "This sentence is not true". It is composed of meaningful parts meaningfully put together. That is, "This sentence", "is not" and "true" are each meaningful expressions and the sentence is grammatical. So why does it fail to be meaningful? — Nagase

You're equivocating definitions of meaningful and truth. There's two potential conversations here and you can't bridge them in the way you're attempting to.

There the rules of one or more formal systems in which truth means something like [coheres with the rules] and meaningful means something like [uses the accepted syntax of the system]. In mathematics 2+2=4 is true because is coheres with any expression of the same form in the rest of the system and it is meaningful because it uses accepted syntax (in the way 2%4=& wouldn't be).

There are then the uses of the terms in ordinary language and the psychological states and behaviours associated with the beliefs they (sometimes) represent. Here 'truth' might be something more like {works out as I expected when I act as if it were the case}, and meaningful is more like {I know what to do about what you've said - it has some consequence on my behaviour or mental state that is predictable from the expression}.

The two are barely related - we simply do not consult formal systems of the type you describe to infer either truthfulness or meaningfulness, it's nothing more than a game. It's like discussing why the Bishop cannot move from c1 to g1 in chess by invoking it's restriction to diagonal movement in the rules and then expecting the result to have an effect when laying the pieces out before the game commences.

The solution to the liar in formal systems is perhaps a fascinating subject to those invested in those systems, but it has no bearing on the solution to the liar in ordinary language. You cannot invoke a system most people do not understand to explain why most people do not use sentences like the liar in their day-today speech - unless you're suggesting that we are all superlative logicians in our subconscious.

"This sentence is false" has made a difference to reality by eliciting this thread. — Banno

But would "This sentence is true" have elicited any different consequence?

I don't see what you think is true here. — Janus

The sentence itself, is in fact true, because it's a sentence.

Here you have two different sentences which are referring to each other; which seems to be just a more elaborate form of self-referentiality, so I don't see why the same would not apply as with the "Liar". — Janus

Yes, correct.

an entirely different matter. No empirical statement that happens to be true can be proven (in the deductive sense) to be true. How would you prove that water boils at 100 degrees C, for example? — Janus

Sounds kind of like Kantian things-in-themselves... .

I thought I was clear that the problem has nothing to do with self reference. Rather is because the sentence is incoherent - it makes no sense.

it is possible to construct sentences that - while grammatically correct - have no semantic meaning.

Quadruplicity drinks procrastination.
Colorless green dreams sleep furiously.
The unambiguous zebra promoted antipathy.
Etc

We all immediately recognize that these sentences are composed of words which have clear common use definitions, yet we all immediately recognize that under the clear common use definitions of the words these are nonsense sentences.

So now the question is - are such sentences true or false? I am not highly knowledgeable about all the different philosophical movements, but I believe that there are two schools of thought on this topic.

One school of thought basically says (and stealing a Star Trek reference here) "Dammit Jim, quadruplicity does not drink procrastination. That sentence is false"

The other school of thought says that you cannot assign a truth value to such utterances.

I go with that second line of thinking.

I thought I was clear that the problem has nothing to do with self reference. Rather is because the sentence is incoherent - it makes no sense. — EricH

But it is not like the examples you give of nonsense.

"... is not true" is just the sort of thing we say about sentences, and it is here said, with the usual meaning, about a sentence. If it doesn't make sense in this case, why not?