What CAN a sentence say about itself? Can a sentence say about itself that it is false?

Question

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)

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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.

Answer 1

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.

Answer 2

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.

Answer 3

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!

Answer 4

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.

Answer 5

It is simpler to define the set of things that a sentence cannot say about itself because this is a small finite set.

A sentence cannot refer to only its own truth value because this defines an infinitely recursive evaluation such as: "This sentence is true."
Truth_Teller := True(Truth_Teller)
Liar_Paradox := ~True(Liar_Paradox)

The Prolog programming language detects and rejects these expressions:
?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

Because the Prolog Liar Paradox has an “uninstantiated subterm of itself” we can know that unification will fail because it specifies “some kind of infinite structure.” that causes the LP expression to be rejected by unify_with_occurs_check.

To put this in simpler terms

This sentence is not true.
Not true about what?
Not true about being not true.
Not true about being not true about what?
Not true about being not true about being not true.
Ah I see you never get to the point, you are stuck in an infinite cycle.

As you also pointed out the Liar Paradox is also untrue because it is self-contradictory.

A sentence can refer to its own truth value in addition to other claims that it makes about itself:

"It is true that this sentence has nine words."
"It is false that this sentence has thirty-seven words."

Answer 6

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.