Lets attack the easier part first:

If a sentence can say that it is false then there is a sentence x such that

1) x = "x is false".

But according to Leibniz law we then have that

2) x is true IFF "x is false" is true.

And by the definition of truth we may simplify and we get

3) x is true IFF x is false

We have contradicted the assumption so we conclude that

There is no x such that x = "x is false". (QED)

Now to the more general question:

Is there a sentence that says Z about itself?

If so then there is a sentence x such that

1) x = "Zx"

Suppose now that

2) Zx

Then we have by substitution that

3) Z"Zx"

And we conclude that

4) IF (x = "Zx") THEN (Zx IFF Z"ZX")

So a statement x can say Z about itself only if Zx has the same truth value as Z"Zx".

These ideas are my own and I assume I may post them wherever I find them relevant!

At least until I have been proven wrong.

  • According to the standard Tarskian formalism it can not, languages are not allowed to contain their truth predicates, but there are self-referential languages specifically designed to allow it, see e.g. Smullyan's Languages in Which Self Reference is Possible. More generally, there is vast literature on paradoxes of self-reference and ways to block them. – Conifold May 23 '18 at 21:12
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    Possible duplicate of What formal logical systems "resolve" the Liar Paradox? – Conifold May 23 '18 at 21:15
  • Yes its me writing in both places. Truths can be used everywhere... cant they? – Sigurd Vojnov May 23 '18 at 21:50
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    I read it but all of this is already discussed at length in encyclopedias, and even on this site. You are not "refuting" Tarski, maybe objecting to his motivation for introducing the hierarchy of languages, but there are plenty of alternatives to that, see the links. And please do not use three comment lines for what fits into one. – Conifold May 23 '18 at 22:21
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    Already discussed since a couple of millenia under the heading Liar paradox. The issue is : "Can a sentence say about itself that it is false?" Obviously YES in natural language: you have done it. But this "construction" produce a contradiction. – Mauro ALLEGRANZA May 24 '18 at 6:47

The following putative case of a sentence validly saying something about itself is perhaps worth considering :

Consider the sentence:

M: 'This sentence has no truthmaker.'

A simple argument [from P. Milne] purports to show that M is a truth without a truth maker:

Suppose that M has a truthmaker. Then it is true. So what it says is the case is the case. Hence M has no truthmaker. On the supposition that M has a truthmaker, it has no truthmaker. By reductio ad absurdum, M has no truthmaker. But this is just what M says. Hence M is a truth without a truthmaker.

Dan Lopez Da Sa & Elia Zardini, 'Does this sentence have no truthmaker?', Analysis, Vol. 66, No. 2 (Apr., 2006), pp. 154-157 : 154 quoting P. Milne, 'Not every truth has a truthmaker. Analysis 65: 221-23: 222.

For reasons given in your own text and other answers I cannot see how a sentence can say about itself that it is false.

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There are many problems with your statements :

1°) syntax What is a sentence? Is it just a sequence of words or does it follow rules of construction?

2°) semantic What means 'x is true' ? How do you determine the truth value of a sentence? This looks dumb but the truth of the statement "x is false" might not imply that x is indeed false. For example, the statement "x is false" could mean that the number of characters of x is a multiple of 5.

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  • If you have references to others who take a similar position they would help support your answer and give the reader a place to go for more information. Welcome! – Frank Hubeny May 10 '19 at 14:41

What CAN a sentence say about itself?

Nothing at all--not in anything resembling a formal mathematical proof anyway.

A formal mathematical proof is just an ordered list of statements. A statement is comprised of logical proposition and a citation giving the rule of inference used to generate that proposition. The citation may also point to one or more previous statements. In the standard logic of mathematical proofs, there is no rule of inference that can point to another statement and declare that the proposition there is false (or true). There is simply no need for such a rule in mathematics. Good luck trying to invent one!

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  • IF a certain class of statements (mathematical) cant say anything about themselves then we are done with them . Great work there! But what about all the other statements? – Sigurd Vojnov May 24 '18 at 17:38
  • @SigurdVojnov Mathematical statements in this context will include statements in propositional and predicate logic. – Dan Christensen May 25 '18 at 2:36

I'm not sure whether your arguments work but their conclusions seem correct to me. I can never get my head around the reasons why self-referential statements are seen as important philosophical problems. They seem to be just badly-formed statements.

But others see it differently and perhaps I'm missing something. Perhaps someone could provide an example of such a sentence that cannot simply be dismissed as a party-trick with words. I may ask a question inviting ideas.

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  • Quine gives an example of analyticity; "Bachelor" has less than ten letters. And the following sentence contains exactly six words: This sentence contains exactly six words. – Sigurd Vojnov Jun 9 '18 at 20:05
  • @SigurdVojnov - Hmm. I cannot see any problem with these sentences. There seems to be no self-reference involved. . – user20253 Jun 10 '18 at 12:12
  • @SigurdVojnov - Ah. I see I was wrong. The third is self-referential. At the time of writing 'This sentence' it contained two words so I'm not sure it is legitimate as stated. Properly expressed It would say 'The sentence "This sentence contains six words" contains six words'. Just thinking out loud. It's a slippery topic. . . – user20253 Jun 25 '18 at 10:46

There are no sentences in the field of logic. Grammar deals with sentences. Logic deals with propositions which can be expressed as declarative sentences. You are confusing propositions with grammar.

Propositions are concepts in the mind. In this way you cannot verify with the senses a proposition. You extract propositions.

Propositions being concepts and not literal things cannot refer to themselves. The declarative sentence if it refers to itself is equivocating on the term truth. That is, by definition all propositions are already true or false. The definition is not about the world but an ideal found if you draw out a truth table. Notice I said truth table and not real world correspondence.

The paradox plays with corresponding truth in the real world and not the truth value as found in truth tables. The confusing part for many people is when the truth value of a truth table differs from the corresponding truth value. For example, "if Donald Trump is President, then this sentence is false". The corresponding fact is that Trump is the President. Truth table wise we must assign a value to the consequent of this example. If we assign the consequent TRUE then the conditional is TRUE if we are talking about the truth table. The truth table displays when both antecedent and consequent have truth values set to true. Then the conditional is true.

The corresponding value of the sentence "if Donald Trump is president, then this sentence is false" is false in the real world. At this time Trump is indeed the president and we would have to rewrite the sentence to correct it as: "If Donald Trump is president then this sentence is TRUE."

Let us suppose the consequent of "If Donald Trump is president, then this sentence is false" is valued as false. Conditionals with a true antecedent and a false consequent have a truth value of false according to the truth table. The corresponding value would be different. The corresponding value would be true because the sentence is false in the real world. So the sentence matches the real world and is true in the context of corresponding facts.

The writer of such a sentence is more likely intentionally being deceitful. Once the writer clarifies which context he is expressing then the response wouldn't be so controversial. Also recall in some forms of logic only the form matters and the CONTENT of the propositions do not matter. So which is it? Does the content of the propositions make a difference or does only the form of the proposition and immediate inferences it expresses make a difference? Once you pick one, such as the form, the writer of such deception will pick the other.

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    "There are no sentences in the field of logic." This is straightforwardly false. One can construct a predicate logic over a domain containing sentences, and say useful things about sentence composition in that logic. Indeed, this is a key research programme in the current study of logic. See: Godel, Tarski. – Paul Ross Jul 1 '18 at 8:34
  • @Paul Ross, if you think my proposition is False about there being sentences in logic can you at least give me a single counter example? Please also make a clear distinction between sentences and propositions. Both if these requestt would be would necessary for you to absolutely refute my initial claim. I take sentences and propositions are NOT the same thing. Even though propositions can LOOK like sentences they are not literal sentences of any kind. They are NOT literally declarative sentences as many people think. I can explain further if needed. Many people get the two confused. – Logikal Jul 1 '18 at 15:39
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    en.wikipedia.org/wiki/Sentence_(mathematical_logic) (Was hoping not to have to go there, but as I said, straightforward) – Paul Ross Jul 1 '18 at 15:43
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    "∀x∃y.x = y" is an example of a sentence. It is constructed compositionally using quantifiers, variables and a relation symbol, and is interpreted to represent the proposition ∀x∃y.x = y, which is a feature of the world that may or may not obtain. – Paul Ross Jul 1 '18 at 15:50
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    A proposition does not contain a symbol. – Paul Ross Jul 1 '18 at 15:53

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