Logic and self referential statements

[Bomb#20]

"preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.
This doesn't make much sense. It's confounded even further by the fact it's a grammatically incorrect sentence containing a grammatically incorrect sentence, not to mention that the sentence minus the quote is identical to the quote. It's a mess!

I don't understand why you think it doesn't make much sense, why you think it's grammatically incorrect, what you think it contains that's a grammatically incorrect sentence, or why you think the fact that the sentence minus the quote is identical to the quote confounds it. As far as I can see by direct inspection, none of these things are true. Can you explain?

(Of course there's one sense in which it doesn't make sense: it's paradoxical. It's true if we assume it's not true, and it's not true if we assume it's true. That's the whole point of the construction in the first place. But I don't think that's what you're referring to.)

(Also, for those of you who think what I'm saying is obviously wrong and I'm just being an idiot for not seeing it, this isn't my work. What I'm presenting in these sentences is called Quine's paradox. It was invented by one of the most famous logicians of the 20th century.)

 

[Bomb#20]

True; but the full sentence doesn't claim the part in quotation marks is a sentence, or call it true or untrue. It only claims doing certain operations to the part in quotation marks gives you an untrue sentence.

No it does not. The point was that your example is not a sentence at all. Therefore it is also not an untrue sentence.

What evidence do you have that my example is not a sentence at all? What you argued earlier was "What is between the quotation marks in your version is a phrase but not a sentence.". That's true. But "your example" and "What is between the quotation marks in your version" are two different noun phrases referring to two different objects.

There is also no "operations" at all (except to read the phrase and try to interpret it). So, it's just a phrase and then no amount of reading and interpretation will make it a sentence.

The "operations" I was referring to were first putting it in quotation marks and then putting that in front of the original phrase. Perhaps "operations" isn't the right word for that; I haven't yet thought of a better.

You tried to evade the problem by pretending your example was a proper sentence when it wasn't.

It is a proper sentence as far as I can see. Do you have an explanation for why you think it isn't a sentence?

I showed you what would be a proper sentence. I accept the example doesn't work but at least this is grammatically correct.

You could show me a lot of proper sentences. That your sentence is proper does not make mine improper. That yours is grammatically correct does not make mine grammatically incorrect. What rule of grammar are you claiming my sentence violates?

Now you can try to make it work if you think that there should be a solution. Me I think there is no formal solution, i.e. something cannot refer to itself just by itself, i.e. only on the basis of formal rules (grammatical, logic, etc.).

The following example, I think already suggested, works:

"The sentence immediately following is false. The sentence immediately preceding is true."​

But it's not self-referential.
EB

If that's acceptable to you, how about this?

"Pigs fly. The sentence immediately following the sentence immediately preceding is false."​

 

[fast]

Edited out

 

[fast]

"preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.
This doesn't make much sense. It's confounded even further by the fact it's a grammatically incorrect sentence containing a grammatically incorrect sentence, not to mention that the sentence minus the quote is identical to the quote. It's a mess!

I don't understand why you think it doesn't make much sense, why you think it's grammatically incorrect, what you think it contains that's a grammatically incorrect sentence, or why you think the fact that the sentence minus the quote is identical to the quote confounds it. As far as I can see by direct inspection, none of these things are true. Can you explain?

(Of course there's one sense in which it doesn't make sense: it's paradoxical. It's true if we assume it's not true, and it's not true if we assume it's true. That's the whole point of the construction in the first place. But I don't think that's what you're referring to.)

(Also, for those of you who think what I'm saying is obviously wrong and I'm just being an idiot for not seeing it, this isn't my work. What I'm presenting in these sentences is called Quine's paradox. It was invented by one of the most famous logicians of the 20th century.)

After closer inspection, I retract my statement.

 

[Speakpigeon]

No it does not. The point was that your example is not a sentence at all. Therefore it is also not an untrue sentence.

What you argued earlier was "What is between the quotation marks in your version is a phrase but not a sentence.". That's true.

Ok.

There is also no "operations" at all (except to read the phrase and try to interpret it). So, it's just a phrase and then no amount of reading and interpretation will make it a sentence.

The "operations" I was referring to were first putting it in quotation marks and then putting that in front of the original phrase. Perhaps "operations" isn't the right word for that; I haven't yet thought of a better.

Yes, I understood what you meant, but these operations are irrelevant to the logic of your example since all we need to assess the sentence is to read it.

You tried to evade the problem by pretending your example was a proper sentence when it wasn't.

It is a proper sentence as far as I can see. Do you have an explanation for why you think it isn't a sentence?

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.

What is the sentence is the whole, i.e. <"X" X>. But this is not what the sentence refers to. It doesn't refer to itself. The sentence <"X" X> doesn't refer to <"X" X>. It refers to X. It says: X preceded by itself is an untrue sentence. So, it refers to X and since X is not a sentence then it's not an untrue sentence.

I showed you what would be a proper sentence. I accept the example doesn't work but at least this is grammatically correct.

You could show me a lot of proper sentences. That your sentence is proper does not make mine improper. That yours is grammatically correct does not make mine grammatically incorrect. What rule of grammar are you claiming my sentence violates?

See above.

Now you can try to make it work if you think that there should be a solution. Me I think there is no formal solution, i.e. something cannot refer to itself just by itself, i.e. only on the basis of formal rules (grammatical, logic, etc.).

The following example, I think already suggested, works:

"The sentence immediately following is false. The sentence immediately preceding is true."​

But it's not self-referential.
EB

If that's acceptable to you, how about this?

"Pigs fly. The sentence immediately following the sentence immediately preceding is false."​

Sure it works but the statement is still not a self-referential statement. The second sentence is self-referential but you need the preceding sentence for it to work. But, Ok, I should have said there is no "stand-alone" self-referential statement.
EB

 

[Speakpigeon]

The following example, I think already suggested, works:

"The sentence immediately following is false. The sentence immediately preceding is true."​

But it's not self-referential.
EB

A duplicitous statement is not true or false, it's both.

T= True= not duplicitous or false.
F1= False = not duplicitous and not true.
F2= False = duplicitous or not true.
D= Duplicitous= true and false and neither
N= No truth value= not T, F, or D

The paradox relies on incomplete definitions of true and false which don't contain the condition of duplicity.

The sentence immediately following is F2. The sentence immediately preceding is T.

Does order of evaluation matter?

A "duplicitous" statement is a knowingly false one wilfully presented as true. Not quite what you say. Further, I don't see any interest in using "duplicitous" in the sense you suggest. We already have a term "contradiction" and this says exactly what it is.

Paradox is a different idea. To call a contradictory statement a paradox is to suggest that it is superfcially understandable while only revealing itself as contradictory or undecidable on close examination. So, the statement" "This statement is false" is both a contradiction and a paradox.

Order of evaluation does not matter in classical logic but maybe it does in some other logic.

The definitions of truth-values in classical logic only apply in the context of the assumption that classical logic applies, and classical logic does not even pretend to model entirely the way we think. It certainly does not apply to all statements. It's really a pragmatic system to be used essentially by engineers and particularly some computer sciences. I don't think even most mathematicians need any training in formal logic. They need to know the terminological conventions, if that. Some philosophers may use classical logic quite a lot but most don't pay it much attention anymore than mathematicians. They just assume their arguments are logical.
EB

 

[Kharakov]

A duplicitous statement is not true or false, it's both.

T= True= not duplicitous or false.
F1= False = not duplicitous and not true.
F2= False = duplicitous or not true.
D= Duplicitous= true and false and neither
N= No truth value= not T, F, or D

The paradox relies on incomplete definitions of true and false which don't contain the condition of duplicity.

The sentence immediately following is F2. The sentence immediately preceding is T.

Does order of evaluation matter?

A "duplicitous" statement is a knowingly false one wilfully presented as true. Not quite what you say.

I'm leaning more towards:

the state or quality of having two elements or parts; being twofold or double.

duplicity (n.)
early 15c., from Old French duplicite (13c.), from Late Latin duplicitatem (nominative duplicitas) "doubleness," in Medieval Latin "ambiguity," noun of quality from duplex (genitive duplicis) "twofold." The notion is of being "double" in one's conduct (compare Greek diploos "treacherous, double-minded," literally "twofold, double").

forked (adj.)
c.1300, "branched or divided in two parts," past participle adjective from fork (v.). Of roads from 1520s; from 1550s as "pointing more than one way." In 16c.-17c. sometimes with a suggestion of "cuckold," on the notion of "horned." Forked tongue as a figure of duplicitous speech is from 1885, American English. Double tongue in the same sense is from 15c.

Further, I don't see any interest in using "duplicitous" in the sense you suggest. We already have a term "contradiction" and this says exactly what it is.

I'm not aiming for contradiction or paradox, instead I'm aiming for a category of truthfulness that describes things that have elements of truth and falsehood at the same time (thus duplicitous in meaning, ambiguous, two faced (true and false), two valued...).

True: not duplicitous and not false, conforms to reality or rule set followed (1+1=2 truth for axioms of arithmetic) without also not conforming to reality or rule set followed (1+1=7 not true according to axioms of arithmetic)

False: if defined as not true is either duplicitous (conforming and non-conforming at the same time) or not conforming
False: if defined as not conforming to reality or rule set, is without duplicitous element of conforming to reality or rule set

Duplicitous: conforming and not conforming (may simply depend on perspective). Duplicity includes all possibilities, like true reality, which contains true, false, and duplicitous elements.

 

[fast]

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.

What is the sentence is the whole, i.e. <"X" X>. But this is not what the sentence refers to. It doesn't refer to itself. The sentence <"X" X> doesn't refer to <"X" X>. It refers to X. It says: X preceded by itself is an untrue sentence. So, it refers to X and since X is not a sentence then it's not an untrue sentence.

"preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.

A few points.

First, let's just take a look at what's in quotes. Reading it with a long pause between "marks" and "is", I'm inclined to think something is a sentence. In fact, the claim is that the something in question is a sentence that isn't true.

Second, let's focus on the second time the word preceded appears (after the end quote). The lower case p had me going for awhile. Let's change it to a capital P. In fact, let's treat everything after the end quote as a sentence that stands alone.

Third, reword the sentence (everything after the end quote) for better clarity--beginning with the subject. It's essentially saying that the preceding sentence is untrue. It also makes note that the preceding sentence is a duplicate (or worded the same)--and that it's in quotes.

I'd say the second sentence is referring to the first. It's just that the first sentence is in quotes. The lower case p leads us to think that it's just one long sentence, but notice how the p isn't capitalized in quotes either. I know, the period in quotes should of given it away, but punctuation is sometimes included in quoted sentences.

 

[Bomb#20]

You tried to evade the problem by pretending your example was a proper sentence when it wasn't.

It is a proper sentence as far as I can see. Do you have an explanation for why you think it isn't a sentence?

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.

Your reasoning sounds like the fallacy of composition. Why would X have to be a sentence in order for <"X" X> to be a sentence? My earlier link contained a link to this counterexample:

"is a sentence fragment." is a sentence fragment.​

Here's an even simpler counterexample.

"contains three words." contains three words.​

What is the sentence is the whole, i.e. <"X" X>. But this is not what the sentence refers to. It doesn't refer to itself. The sentence <"X" X> doesn't refer to <"X" X>. It refers to X. It says: X preceded by itself is an untrue sentence. So, it refers to X and since X is not a sentence then it's not an untrue sentence.

You appear to be misparsing my sentence. Grammatically it has the same structure as "Seven multiplied by two is an even number.", in which the predicate, "is an even number", is asserted to be true of 14; it isn't asserted to be true of 7. The main verb of the sentence is "is", not "multiplied". Likewise, the main verb of my sentence from post #70 is "is", not "preceded". The subject of the sentence is the part before the main verb:

"preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks​

or, if you want to work from your simplified version,

X preceded by itself​

A sentence refers to whatever its subject refers to. So the referent of <X preceded by itself> would be <X X>. Likewise, the referent of <"X" preceded by a copy of itself in quotation marks> is whatever you get if you take X and precede it with a copy of itself in quotation marks, i.e. <"X" X>. So yes, my sentence does refer to itself (or to a sentence identical to itself, if you metaphysically prefer that formulation).

If that's acceptable to you, how about this?

"Pigs fly. The sentence immediately following the sentence immediately preceding is false."​

Sure it works but the statement is still not a self-referential statement. The second sentence is self-referential but you need the preceding sentence for it to work. But, Ok, I should have said there is no "stand-alone" self-referential statement.
EB

Hmm. How about this then?

If there is a sentence immediately preceding then the sentence immediately following the sentence immediately preceding is false; contrariwise, if there is no sentence immediately preceding then the first sentence is false.​

Now it doesn't need any preceding sentence.

 

[fast]

"is a sentence fragment." is a sentence fragment.

Awe, this I can wrap my head around.

What I see is one sentence.

I'm going to break this down into three parts, compare it to your simpler counter example, compare it to an altered counter example and finally relate this to the original example.

First, everything in quotes will be referred to as A.
Everything after the quotes will be referred to as B.
Finally, everything altogether will be referred to as C.

Thus:
A = "is a sentence fragment."
B = is a sentence fragment.
C = "is a sentence fragment." is a sentence fragment.

A is not a sentence.
B is not a sentence.
C is a sentence.

Between A, B, and C, there is only one sentence, and that's why I said that what I see is one sentence.

Now, I'm turning to your simpler counter example:

Instead of using A, B, and C, I will use D, E, and F:

Thus:

D = "contains three words."
E = contains three words.
F = "contains three words." contains three words.

D is not a sentence
E is not a sentence
F is a sentence.

Between D, E, and F, there is (again in this example) one sentence.

Now I will create another example:

"water is wet." is a true saying.

G = "water is wet."
H = is a true saying.
I = "water is wet." is a true saying.

There are a couple differences here. One difference, unlike the other examples, is that what's in the quotes is not the same as what's after the quotes (but more on that in a bit), and the second difference is that unlike the other examples, there are two sentences and not just one:

G is a sentence.
H is not a sentence.
I is a sentence.

So, when I look at I, although it is a sentence, it's not just a sentence but a sentence within a sentence. In other words, the sentence (I) contains a quote (G) which itself is a sentence.

Now, let's turn to the original example broken down in the same fashion:

J = "preceded by a copy of itself in quotation marks is a sentence that isn't true."
K = preceded by a copy of itself in quotation marks is a sentence that isn't true.
L = "preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.

Now, let's make some sense of this (some comparative notes, if you will). I originally thought that L was a sentence containing a sentence (like I). Later, I thought L was two consecutive sentences. Now, I've come full circle and believe what I once thought (based on the assumption that your counter examples were meant to mimic the structure), so I believe L is a sentence. In fact, the third notation in every example, mine included, is a sentence; hence, C, F, I, and L are all sentences.

What I find particularly interesting is if J is a sentence, as purported by L, how come the first notation in every counter example (except mine) is not a sentence? Well, I know why mine isn't (not withstanding the quotes, G and H don't match), but that's a different matter.

I suppose what I'm getting at is if J is a sentence, then so is K, so the example has not one, not two, but three sentences (one main sentence containing two sentences, one of which is quoted).

Your thoughts?

 

[Bomb#20]

"is a sentence fragment." is a sentence fragment.

<examples A-I snipped>
Now, let's turn to the original example broken down in the same fashion:

J = "preceded by a copy of itself in quotation marks is a sentence that isn't true."
K = preceded by a copy of itself in quotation marks is a sentence that isn't true.
L = "preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.

Now, let's make some sense of this (some comparative notes, if you will). I originally thought that L was a sentence containing a sentence (like I). Later, I thought L was two consecutive sentences. Now, I've come full circle and believe what I once thought (based on the assumption that your counter examples were meant to mimic the structure), so I believe L is a sentence. In fact, the third notation in every example, mine included, is a sentence; hence, C, F, I, and L are all sentences.

My counterexamples from post #89 were not meant to mimic the structure. Sentence L has a different and more complicated structure than those counterexamples. They were intended only to demonstrate that Speakpigeon's statements:

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.​

are an unsound argument. So far I've only offered one example sentence with the same structure as L:

Seven multiplied by two is an even number.​

Let's name the parts of that too: M="Seven", N="multiplied by two", O="is an even number", P = M N O..

The grammatical correspondence between the two sentences is as follows:

Seven               "preceded by a copy of itself in quotation marks is a sentence that isn't true."
multiplied          preceded
by                  by
two                 a copy of itself in quotation marks
is                  is
an even number      a sentence that isn't true

The reason this is important is that there's a big difference between the grammatical structure of L and that of C, F, and I: in C, F, and I, the quotations at the beginning of those sentences, i.e. A, D and G, are the subjects of the respective sentences. J, in contrast, is not the subject of L. J is just one component of the subject of L, the same way "Seven" is just one component of the subject of P, "Seven multiplied by two". This is a complication that appears to be throwing people off.

What I find particularly interesting is if J is a sentence, as purported by L, how come the first notation in every counter example (except mine) is not a sentence?

J is not a sentence, any more than M is an even number. But L does not purport J to be a sentence, any more than P purports M to be an even number. Rather, P purports MN to be an even number. Likewise, L purports the part of sentence L corresponding to MN to be a sentence; i.e., it purports

J preceded by a copy of itself in quotation marks​

(which works out to "J" J), to be a sentence. And "J" J is a sentence -- the same sentence as L.

 

[Speakpigeon]

A "duplicitous" statement is a knowingly false one wilfully presented as true. Not quite what you say.

I'm leaning more towards:

the state or quality of having two elements or parts; being twofold or double.

duplicity (n.)
early 15c., from Old French duplicite (13c.), from Late Latin duplicitatem (nominative duplicitas) "doubleness," in Medieval Latin "ambiguity," noun of quality from duplex (genitive duplicis) "twofold." The notion is of being "double" in one's conduct (compare Greek diploos "treacherous, double-minded," literally "twofold, double").

forked (adj.)
c.1300, "branched or divided in two parts," past participle adjective from fork (v.). Of roads from 1520s; from 1550s as "pointing more than one way." In 16c.-17c. sometimes with a suggestion of "cuckold," on the notion of "horned." Forked tongue as a figure of duplicitous speech is from 1885, American English. Double tongue in the same sense is from 15c.

Further, I don't see any interest in using "duplicitous" in the sense you suggest. We already have a term "contradiction" and this says exactly what it is.

I'm not aiming for contradiction or paradox, instead I'm aiming for a category of truthfulness that describes things that have elements of truth and falsehood at the same time (thus duplicitous in meaning, ambiguous, two faced (true and false), two valued...).

True: not duplicitous and not false, conforms to reality or rule set followed (1+1=2 truth for axioms of arithmetic) without also not conforming to reality or rule set followed (1+1=7 not true according to axioms of arithmetic)

False: if defined as not true is either duplicitous (conforming and non-conforming at the same time) or not conforming
False: if defined as not conforming to reality or rule set, is without duplicitous element of conforming to reality or rule set

Duplicitous: conforming and not conforming (may simply depend on perspective). Duplicity includes all possibilities, like true reality, which contains true, false, and duplicitous elements.

You may be able to work out an actual formalisation of this sort of logic, like other people have done, and you would need to produce truth-tables, unless it's not a truth-functional logic, but in any case the crucial point would be to show how it could apply to engineeering applications, computers, decision processes, etc. You might even be able to make some actual money. Or is this something that's already been done by any chance?
EB

 

[Speakpigeon]

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.

What is the sentence is the whole, i.e. <"X" X>. But this is not what the sentence refers to. It doesn't refer to itself. The sentence <"X" X> doesn't refer to <"X" X>. It refers to X. It says: X preceded by itself is an untrue sentence. So, it refers to X and since X is not a sentence then it's not an untrue sentence.

"preceded by a copy of itself in quotation marks is a sentence that isn't true." preceded by a copy of itself in quotation marks is a sentence that isn't true.

A few points.

First, let's just take a look at what's in quotes. Reading it with a long pause between "marks" and "is", I'm inclined to think something is a sentence. In fact, the claim is that the something in question is a sentence that isn't true.

What is in quotes is definitely not a sentence (even if we capitalise the first "p". The part <preceded by a copy of itself in quotation marks> is not identifiable as a subject (of "is").

Also, the whole sentence does not show what is the status of the part between quotes (say, X1). Do we have to understand that the two quotation marks are in fact part of X1 as a mere nonsensical sequence of alphanumerical characters, or that X1 is quoted, and then as what? A sentence? A phrase? That's why I suggested earlier to put "The phrase" before it.

Second, let's focus on the second time the word preceded appears (after the end quote). The lower case p had me going for awhile. Let's change it to a capital P. In fact, let's treat everything after the end quote as a sentence that stands alone.

Third, reword the sentence (everything after the end quote) for better clarity--beginning with the subject. It's essentially saying that the preceding sentence is untrue. It also makes note that the preceding sentence is a duplicate (or worded the same)--and that it's in quotes.

I'd say the second sentence is referring to the first. It's just that the first sentence is in quotes. The lower case p leads us to think that it's just one long sentence, but notice how the p isn't capitalized in quotes either. I know, the period in quotes should of given it away, but punctuation is sometimes included in quoted sentences.

Here you are trying to move towards a different kind of example. So I tried to write it down
but it doesn't seem to work. You can't have a capital letter in the middle of a sentence.

But you're right, the first "p" in lower case is a problem and if you capitalise it you have to capitalise the second one as you suggested but then I don't know what is the status of such a mongrel phrase.
EB

 

[Speakpigeon]

Your example can be reduced to: <"X" X>. What I meant was that X is not a sentence. So, even preceded by itself it's still not a sentence.

Your reasoning sounds like the fallacy of composition. Why would X have to be a sentence in order for <"X" X> to be a sentence?

What is the subject of the second X in <"X" X>? Is it <X>, i.e. the X between the quotation marks, or is it <"X">, i.e. a string of characters that includes the two quotation marks? We just don't know (that's why I suggested earlier to put the "The phrase" before the quoted part to remove the ambiguity.

Now, I suspect that somehow your example needs both at the same time but then you can't.

My earlier link contained a link to this counterexample:

"is a sentence fragment." is a sentence fragment.​

It's different from your example in that it is self-explanatory. The second part explains properly how to take the quoted part, i.e. as a sentence fragment, which it is. So we have a way of understanding what the whole means, although it's also not a sentence (no capital letter). However, the example also does not pretend to be a sentence (unlike in your own example). So, although it's not a sentence, it is still both understandable and true. Yet, still not a true sentence.

Also, unlike your example, there is not suggestion of this thing being self-referential. There's no logical issue here at all.

Here's an even simpler counterexample.

"contains three words." contains three words.​

This thing is true, although not a sentence. It's Ok since it doesn't pretend to be one and if nobody says it is one.
Also nothing self-referential in this example.
EB

 

[Speakpigeon]

What is the sentence is the whole, i.e. <"X" X>. But this is not what the sentence refers to. It doesn't refer to itself. The sentence <"X" X> doesn't refer to <"X" X>. It refers to X. It says: X preceded by itself is an untrue sentence. So, it refers to X and since X is not a sentence then it's not an untrue sentence.

You appear to be misparsing my sentence. Grammatically it has the same structure as "Seven multiplied by two is an even number.", in which the predicate, "is an even number", is asserted to be true of 14; it isn't asserted to be true of 7. The main verb of the sentence is "is", not "multiplied".

This example doesn’t readily compare the way you assume.

The sentence “X multiplied by Y is Z” is standardly interpreted as “<The result of X multiplied by Y> is Z”, making clear that the subject is the result. In your example, there is no such standard interpretation.

For example, the word “pig” preceded by the phrase “you are such a” is not a sentence. It is just a word, the word “pig”. It happens to be preceded by the phrase “you are such a” but it is still just a word. What is a sentence here is the result of fronting the word “pig” with the phrase “you are such a”.

Same for the sentence <The word “pig” preceded by the phrase “you are such a” is normally taken to refer to a person.>. Here, what is said to refer to a person is the word “pig”. Therefore the subject of “is” is the word “pig”, not the noun phrase <The word “pig” preceded by the phrase “you are such a”>.
EB

 

[Speakpigeon]

A sentence refers to whatever its subject refers to.

Not really.
A sentence will usually refers to a fact (or state of affair, etc.). The sentence “The town prospered in the 19th century” just refers to the fact that the town prospered in the 19th century.

A sentence refers to whatever its subject refers to. So the referent of <X preceded by itself> <snip>.

But <X preceded by itself> is not a sentence. It is a noun phrase.
And it normally refers to whatever the head noun is, i.e. <X> here, although I suspect there are various sorts of counterexamples.

A sentence refers to whatever its subject refers to. So the referent of <X preceded by itself> would be <X X>.

Nothing can precede itself.
But even if we accepted that, <X preceded by itself> would still not refer to <X X>. Instead, it would refer to the second X in <X X>.
EB

 

[Speakpigeon]

If that's acceptable to you, how about this?

"Pigs fly. The sentence immediately following the sentence immediately preceding is false."​

Sure it works but the statement is still not a self-referential statement. The second sentence is self-referential but you need the preceding sentence for it to work. But, Ok, I should have said there is no "stand-alone" self-referential statement.
EB

Hmm. How about this then?

If there is a sentence immediately preceding then the sentence immediately following the sentence immediately preceding is false; contrariwise, if there is no sentence immediately preceding then the first sentence is false.​

Now it doesn't need any preceding sentence.

Doesn't work. You example relies on assumptions you are implicitly asking readers to make. Without them, there is no formal interpretation making the sentence necessarily self-referential.

I put these assumptions in between <>:

If there is a sentence immediately preceding <this one> then the sentence immediately following the sentence immediately preceding is false; contrariwise, if there is no sentence immediately preceding <this one> then the first sentence is false.​

In other words, this sentence may be about any sentence, not just this one.

Your example is identical in its principle to the example <This sentence is false.> To decide that the sentence <This sentence is false.> refers to itself, you need some pivot. Here, it's the context. You present the sentence on its own in the middle of a white page and there's no other sentence it could refer to. The reader is forced to understand what you mean. Yet, it remains that you need the context to interpret it this way. For example:

"The cat is out of the bag."

This sentence is false.​

Here, <This sentence is false.> will be interpreted as referring to "The cat is out of the bag."

EB

 

[Speakpigeon]

E = contains three words.

Well, how does one read that!?

(1) E = <contains three words.>

(2) <E=> contains three words.​

EB

 

[fromderinside]

E = contains three words.

Well, how does one read that!?

(1) E = <contains three words.>

(2) <E=> contains three words.​

EB

(E) = (container)*(three words)

E = a container of three words

 

[fast]

E = contains three words.

Well, how does one read that!?

(1) E = <contains three words.>

(2) <E=> contains three words.​

EB

I was just denoting what's to the right of the equal sign with what's to the left of the equals sign.