"This sentence is false" - impossible premise

Skalidris

This is probably hard to believe but I do not have the intuitions necessary to see the “mysteries” of some paradoxes. For example, the liar paradox “this sentence is false” simply appears meaningless to me and I do not enter the logic of: If 'This sentence is false.” is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Language conveys information and I can’t extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it.

This is how I visualize the information contained in the following sentences. If the set is correct, it is green, if it is false, it is red.
The first one is the sentence “The sun is yellow”, and the second one is “this sentence is false”.

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To me, the second one is simply meaningless because the sentence conveys information for an empty set and attributes a truth value to it, which isn’t possible since it is empty.

Now, from my understanding, the paradox is misleading because of its grammar. “This sentence is” implies that the set contains something, while it doesn’t. And I’m imagining people see it like this:

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They see true or false as both an element of the set and the validity of the set. So, if the set is valid, it needs to have the true element in it, and if it is not valid, it also needs to have it.

If you find this paradox mind boggling, does this visualization make sense?

To me, this “paradox” is actually a problem that contains an impossible premise, which is that the validity of a set is also an element of the set.

I found this article that also points out a problem in truth attribution:
https://link.springer.com/article/10.1007/s10516-023-09666-2

The crocodile paradox also contains an impossible premise, which is a condition implied by the crocodile that if he eats the child, he will give it back alive.

To me, this shows how much we want to keep our intuitions, as if there were some holy concepts, instead of questioning and dismissing them.

Pantagruel

This is probably hard to believe but I do not have the intuitions necessary to see the “mysteries” of some paradoxes. For example, the liar paradox “this sentence is false” simply appears meaningless to me and I do not enter the logic of: If 'This sentence is false.” is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Language conveys information and I can’t extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it. — Skalidris

I agree. That is what comes of attempting to abstract logical form from content. There is a formalization in set theory involving the set of sets that are not members of themselves (normal, versus abnormal sets). Essentially, this recognizes exactly the real language constraint that a claim be about something.

Michael

↪Skalidris

Your approach seems to be the same as that of Kripke. See here.

In general, if a sentence such as (1) asserts that (all, some, most, etc.) of the sentences of a certain class C are true, its truth value can be ascertained if the truth values of the sentences in the class C are ascertained. If some of these sentences themselves involve the notion of truth, their truth value in turn must be ascertained by looking at other sentences, and so on. If ultimately this process terminates in sentences not mentioning the concept of truth, so that the truth value of the original statement can be ascertained, we call the original sentence grounded; otherwise, ungrounded.

Liar sentences are "ungrounded". Them being true or false isn't meaningful.

I think we can show this by considering the complement of a liar sentence:

1. This sentence is true

If (1) is true then there is no paradox. If (1) is not true then there is no paradox. But is (1) true or not true?

Michael

Let's assume the correspondence theory of truth: that a sentence is true is that it corresponds to a fact. We can use this to rephrase the liar sentence:

1. This sentence does not correspond to a fact.

Does (1) correspond to a fact?

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punos

The first one is the sentence “The sun is yellow”, and the second one is “this sentence is false”. — Skalidris

This is how i look at it:
If it is true that the Sun is yellow then the first sentence is a true statement, else it is false regardless of any other sentences that may exist. If we do not know what is or is not true then in any case...

If the second sentence is referring to the first sentence:
If it is true that the first sentence is false then the second sentence is true in stating that the first sentence is false. (= True)

If it is true that the first sentence is true then the second sentence is false in stating that the first sentence is false. (= False)

If the second sentence is referring to itself:
If the sentence is true that it is false then the sentence is true that it is false. (= True)

If the sentence is false that it is false then the sentence is false that it is false. (= False)

RogueAI

Let's assume the correspondence theory of truth1: that a sentence is true is that it corresponds to a fact. We can use this to rephrase the liar sentence:

1. This sentence does not correspond to a fact.

We can also consider:

2. (3) corresponds to a fact.
3. (2) does not correspond to a fact.

Do (1), (2), and (3) each correspond to a fact?

1 Even if it's incorrect, the question above is worth considering. — Michael

Can't you get around that by changing the paradox to "Everything I say is a lie"? In that case, the sentence does correspond to a fact- that I am a liar.

Corvus

If 'This sentence is false.” is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Language conveys information and I can’t extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it. — Skalidris

For example, the liar paradox “this sentence is false” simply appears meaningless to me and I do not enter the logic of: If 'This sentence is false.” is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on. — Skalidris

The statement is unclear to be true or false. "This sentence" doesn't indicate which sentence it is describing or declaring about. From the statement, it is implied that there must another sentence before it, for the statement to be qualified to conclude "False", but it is not clear, whether it is the case, or "This sentence" means the sentence itself.

If it is the sentence before it, then it is missing, and if it is the sentence itself, then it doesn't indicate why it is false.

Therefore, if someone uttered the statement, it would beg the question, "Which sentence do you mean?"

Alkis Piskas

For example, the liar paradox “this sentence is false” simply appears meaningless to me and I do not enter the logic ... — Skalidris

The term "paradox" is overrated and abused. Most "paradoxes" are simply self-contradictory, self-refuting or circular statements or statements based on a false hypotheses. In short, invalid statements.
The statement in question --“This sentence is false”-- is a classic example of a self-contradictory statement. It's also circular. It indicates two opposite things coexisting, an impossibility: if this sentence is true, then it is also false. There's nothing more to it. It's a dog chasing its tail, a snake swallowing itself. It does not leave room for any interpretation. It just can't stand. It's not a paradox.

The word "paradox" comes from ancient Greek "para" (= besides, contrary to) + "doxa" (= opinion). Indeed, it indicates something that exists or happens which is contrary to what one expects or believes to be true or happen. For example, a paradox would be raining without any cloud in the sky. Yet, it is possible, if there are very strong winds that bring rain from some other place than where we are.

Philosophim

Its just a bad contraction. If we break out the sentence into its full meaning, its fine.

A. This is a sentence. True
B. The sentence in point A is a false sentence. False.

There ya go.

Michael

Its just a bad contraction. If we break out the sentence into its full meaning, its fine.

A. This is a sentence. True
B. The sentence in point A is a false sentence. False.

There ya go. — Philosophim

This sentence contains 36 characters

Should we break the above sentence into the below?

A. This is a sentence
B. The sentence in point A contains 36 characters

Philosophim

This sentence contains 36 characters

Should we break the above sentence into the below?

A. This is a sentence
B. The sentence in point A contains 36 characters — Michael

That's another way to break it down if you would like. Same idea.

Michael

↪Philosophim

Except you can’t break it down that way because “This sentence contains 36 characters” is true but “The sentence in point A contains 36 characters” is false.

Philosophim

Except you can’t break it down that way because “This sentence contains 36 characters” is true but “The sentence in point A contains 36 characters” is false. — Michael

You didn't tag what was true and false in your breakdown, so I assumed that A was true and B was false in isolation. If your intention is that the break down accurately fits the intention of the primary sentence, it does not. My example was the breakdown of a contraction, yours is not.

Banno

Most "paradoxes" are simply self-contradictory, self-refuting or circular statements or statements based on a false hypotheses. In short, invalid statements. — Alkis Piskas

Trouble is, the paradox is right there in the initial version of Principia Mathematica; that is, an "invalid" statement was implied by the formalisation of mathematics in a first order logic. It looked as if the whole edifice would collapse.

jorndoe

Can't recall where I saw this treatment...

p) this sentence is false
is implicitly the same as
q) "this sentence is false" is true
and
r) this sentence
refers to the same by self-reference, so we have both
p) this sentence is false
and, via the above
s) this sentence is true
which is an ordinary contradiction, implying anything

In a way, implicity and self-reference allow unpacking a regular contradiction, which, if not much else, isn't as mystifying.

Alkis Piskas

the paradox is right there in the initial version of Principia Mathematica — Banno

Unfortunately, I'm not knowledgeable on the subject.
But, as I said, there are real paradoxes, which are quite perplexing or structured in a way that cannot be easily refuted or explained, or even not at all. There are such factors as perspective and relativity, which alone leave certain paradoxes "open" or "unsolvable". E.g. The Ship of Theseus paradox (thought experiment).

Skalidris

What if you may already intuitively understand that the statement is lacking substance? — Vaskane

Yes that is probably the case.

Therefore, if someone uttered the statement, it would beg the question, "Which sentence do you mean?" — Corvus

Yes, my reaction exactly. The most intriguing thing about this paradox is that a lot of people don't mind reasoning with something that is empty of meaning... Probably because they did not check that it actually has meaning prior entering this logic loop.

The term "paradox" is overrated and abused. Most "paradoxes" are simply self-contradictory, self-refuting or circular statements or statements based on a false hypotheses. — Alkis Piskas

Yes, I agree. And I find it quite unbelievable that no discipline has managed to reach a consensus about all of these "fake paradoxes".

There are such factors as perspective and relativity, which alone leave certain paradoxes "open" or "unsolvable". E.g. The Ship of Theseus paradox (thought experiment). — Alkis Piskas

The Ship of Theseus paradox looks more like a philosophical or linguistic issue than a paradox.

Corvus

Yes, my reaction exactly. The most intriguing thing about this paradox is that a lot of people don't mind reasoning with something that is empty of meaning... Probably because they did not check that it actually has meaning prior entering this logic loop. — Skalidris

An ambiguous statement disguised as a paradox.

Alkis Piskas

The Ship of Theseus paradox looks more like a philosophical or linguistic issue than a paradox. — Skalidris

Right. That's why I added "thought experiment" in parentheses.

Michael

I think we can show this by considering the complement of a liar sentence:

1. This sentence is true

If (1) is true then there is no paradox. If (1) is not true then there is no paradox. But is (1) true or not true? — Michael

Curry's paradox is an interesting extension of this.

1. Let (a) be the sentence "if this sentence is true then Germany borders China"
2. If (a) is true then Germany borders China
3. Given that (2) is true, and given that (a) and (2) are materially equivalent, then (a) is true
4. Therefore, Germany borders China

In formal logic:

1. X := (X → Y)
2. X → X
3. X → (X → Y)
4. X → Y (from 3 by contraction)
5. X (substitute 4 by 1)
6. Y (from 4 and 5)

Brendan Golledge

↪Michael

When you used text, I disagreed that a and 2 are equivalent. Just substitute a into 2 and you'll see that it's not. It's the difference between saying "..." and '"..." is true'. When you used formal logic, you didnt prove that x is true, or that x->y is true. If you assert that X is false, then it doesn't imply Y. I don't think you could prove this unless the logic was a tautology, which it clearly isn't. At any rate, the original poster argued that an the validity of an argument cannot be an element of that argument, which would mean that your example sentence is also meaningless.

I think the OP made a good argument. I don't think I can add anything to it.

Michael

disagreed that a and 2 are equivalent — Brendan Golledge

Two statements are materially equivalent if either both are true or both are false:

1. A if and only if B

If (1) is true then "A" and "B" are materially equivalent.

So, in the above case:

A) if this sentence is true then Germany borders China
B) if (A) is true then Germany borders China

If (B) is true then (A) is true. If (B) is false then (A) is false. Therefore, (A) and (B) are materially equivalent.

When you used formal logic, you didnt prove that x is true — Brendan Golledge

Are you referring to step 5? As it explains, it simply takes step 4 and replaces X → Y with X, which is allowed given the definition in step 1.

or that x->y is true — Brendan Golledge

Are you referring to step 4? As it explains, it follows from step 3 given the rule of contraction.
X → (X → Y) entails X → Y.

Brendan Golledge

If A is false, then B is not false. Given the definition of the sentence you are using, A is false (or meaningless) and B is true.

"A" is not the same as B: "if A is true, then the statement given by A is true". B as written here is true regardless of the truth value of A. I could just as well write, A: "The sky is pink" and B: "if A is true, then the sky is pink". This A is false and this B is true.

As for your formal logic, I think I am confused about whether you are asserting logic or truth. For instance, I cant tell whether you mean, "if X is true, then Y is true" (I agree with this logic) or "X IS true, and therefore Y is true" (I disagree with this because I think X is either false or meaningless).

Michael

If A is false, then B is not false. Given the definition of the sentence you are using, A is false (or meaningless) and B is true. — Brendan Golledge

Consider these sentences:

1. if this sentence is true then Germany borders China
2. if (2) is true then Germany borders China

Do you accept that (1) and (2) are materially equivalent?

If so then consider these sentences:

2. if (2) is true then Germany borders China
3. if (2) is true then Germany borders China

Do you accept that (2) and (3) are materially equivalent?

If so then (1) and (3) are materially equivalent.

As for your formal logic, I think I am confused about whether you are asserting logic or truth. For instance, I cant tell whether you mean, "if X is true, then Y is true" (I agree with this logic) or "X IS true, and therefore Y is true" (I disagree with this because I think X is either false or meaningless). — Brendan Golledge

Hopefully this is clearer:

1. X means if X is true then Y is true (definition)
2. If X is true then X is true (law of identity)
3. If X is true then if X is true then Y is true is true (switch in the definition of X given in (1))
4. If X is true then Y is true (from 3 by contraction)
5. X is true (switch out the definition of X given in (1))
6. Y (from 4 and 5)

Although one thing to consider is that A → B is equivalent to ¬B → ¬A, and so these are equivalent:

1. if this sentence is true then Germany borders China
2. if Germany does not border China then this sentence is not true

(2) appears to be a more complex version of the standard liar sentence.

Brendan Golledge

I was curious about how it is possible that we can not be understanding each other, so I went to look up Curry's paradox. I was surprised to see that it is supposed to be a legit paradox.

1. X := (X → Y)
2. X → X
3. X → (X → Y)
4. X → Y (from 3 by contraction)
5. X (substitute 4 by 1)
6. Y (from 4 and 5) — Michael

1. X means that if X is true then Y is true (definition)
2. If X is true then X is true (law of identity)
3. If X is true then if X is true then Y is true is true (switch in the definition of X given in (1))
4. If X is true then Y is true (from 3 by contraction)
5. X is true (switch out the definition of X given in (1))
6. Y (from 4 and 5) — Michael

I did not understand number 5, because it seemed obvious to me that X was false (or I was at least very skeptical), so I did not see how substituting it into itself could turn it true. The source I read explained that step 5 is modus ponens, and given the definition (1), it works. But the paper went on further to prove that if 6 is false, then 1 must also be false. So, it is a bad definition. I hadn't worked through all the logic yet to see the paradox, but I did see that it was false.

:

A) if this sentence is true then Germany borders China
B) if (A) is true then Germany borders China — Michael

If A is false, then B is not false. Given the definition of the sentence you are using, A is false (or meaningless) and B is true.

"A" is not the same as B: "if A is true, then the statement given by A is true". B as written here is true regardless of the truth value of A. I could just as well write, A: "The sky is pink" and B: "if A is true, then the sky is pink". This A is false and this B is true. — Brendan Golledge

So, I guess I just never accepted that the sentence was true, and that's why I did not see the paradox.

"A" is not the same as B: "if A is true, then the statement given by A is true". B as written here is true regardless of the truth value of A. I could just as well write, A: "The sky is pink" and B: "if A is true, then the sky is pink". This A is false and this B is true. — Brendan Golledge

Going over this part again, I understood the whole argument to basically be:
A
B: A -> A
Therefore, A

B is true, but we don't know anything about A without more context. I guess this is not what you wrote down formally, and I just didn't get it, because I interpreted your words to mean the A & B I wrote immediately above.

Brendan Golledge

It seems to me that steps 1-4 are circular reasoning. You can't use a definition to prove part of its own definition. You can also do a truth table of X, Y, and X -> Y and see that X <-> (X -> Y) is false.

Michael

But the paper went on further to prove that if 6 is false, then 1 must also be false. So, it is a bad definition. — Brendan Golledge

It’s not that the definition is bad, it’s that when we apply the normal rules of logic to some self-referential sentences then we lead to a contradiction. It’s the paradox of all liar like sentences and there’s no agreed upon resolution.

Brendan Golledge

↪Michael

You can also do a truth table of X, Y, and X -> Y and see that X <-> (X -> Y) is false. — Brendan Golledge

Is this statement false? If I've done the truth table right, then it means that the first line of the proof is wrong.

Michael

Is this statement false? If I've done the truth table right, then it means that the first line of the proof is wrong. — Brendan Golledge

The first line is a definition, not a premise, and so not truth apt. It is simply saying this:

Let "A" mean "if A is true then B is true".

Brendan Golledge

↪Michael

If definitions aren't subject to truth apt, then can I say, "Let 'X' mean a married bachelor," and that this sentence is not truth apt?

"This sentence is false" - impossible premise

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