• Nagase
    60


    I'm not sure I understand. What do you mean by "can't speak about all statements in a definition of a statement"? In any case, it seems to me false; in defining the predicate "S", we will presumably use universal quantifiers, e.g. "(x) (Sx <--> ...)", where "..." is the definition of "S". But then, this quantifier will range over all statements (and more besides), so we will be in a sense "speaking about" all statements. If you mean that we can't mention all statements in the definition of a particular statement, that seems false too. We could define a new statement, say Q, such that Q is "All statements imply themselves", which is presumably true.

    As for your point regarding taking statements naively, I'm unconvinced. Formal linguistics try to capture what is meant by a "naive" statement, and it's not unreasonable to suppose that in such a theories we quantify over all such statements. There may be other problems lurking in the background, though: if you construe statement broadly enough (naively?), consider a statement about every statement which is not about itself. Is it about itself?
  • Meta
    54
    Let me clarify what I want to say.
    Let's take the Berry paradox. In the paradox there is a definition D which is talking about every (arithmetical) definition. D is a well formed definition in the natural language. However if we could formally model D in arithmetic that would mean arithmetic is inconsistent. When I call a concept "naive" I think of this. So there is a (naive) definition that can exist only outside of formal logic.

    Same goes for sentences. We can't have a statement A where the definition of A mentions A. When you are talking about the S predicate you automatically restrict the universe to well-formed formulas and the sentence in the OP is no such statement. So there are statements outside of logic and outside of the S predicate. This is why the OP's sentence is outside of the scope of logic and his proof is not a formal proof.

    We can't define Q formally (only in the informal metalanguage). There isn't such definition as
    Q <-> (Q->Q and X)
    Q is talking about all statements but Q itself is a statement and not part of the (subject) universe of statements.
  • andrewk
    718
    D is a well formed definition in the natural language.Meta
    How can we say it is well-formed when natural language is informal, and hence does not have a notion of well-formed statement? Do you just mean it is grammatically correct? If so, that doesn't tell us much as the statement 'The cheese of five is sad' is also grammatically correct.
    Same goes for sentences. We can't have a statement A where the definition of A mentions A.Meta
    Depending on what language we are using, this may be possible. For instance, the sentence:

    'This sentence contains the first letter of the alphabet'

    is meaningful and true, and can be formalised without difficulty.
  • Meta
    54
    If we have a first order language L we can define a relation symbol (with one variable) R by saying:
    R(x) <-> Phi(x)
    Where Phi(x) is a formula of L. R is not part of L so Phi does not contain R.
    I call a statement a (naive) well formed definition if there is no such restriction.
    In this sense 'This sentence contains the first letter of the alphabet' is not a formal definition.

    You can say we can be more general and use higher order logic but higher order logic is also problematic.

    edit: And yes i think being grammatically correct and being a well formed statement (or definition or whatever) are very close concepts in the natural language.
  • Nagase
    60


    There is no need to use higher-order logic. Let T be a first-order theory containing enough arithmetic to capture the primitive recursive functions (you can let T be primitive recursive arithmetic, Robinson's Arithmetic, etc.). Then T has enough resources to code its own syntax: in particular, it has enough resources to express the following notions: <is a (well-formed) term>, <is a well-formed formula>, <is a sentence>, <is an atomic formula>, <is a theorem>, <is Q> (where Q is any particular expression of the language: it can be part of the alphabet, a term, a formula, anything), etc. The basic idea here is that we can use numbers to stand as codes for things, for example people (ID cards), products (bar codes), strings of symbols (word processing), etc. Anyway, the important point is that using any reasonable coding scheme, T can prove the following lemma:

    Carnap-Gödel Diagonalization Lemma: Let Q(x) be any formula of the language with one free variable. Then there is a sentence D such that T proves "D <-> Q("D")", where "D" is the number coding D.

    Notice that the above implies that, for any property Q, there is a sentence D which says of itself that it has the property Q (notice also that this immediately implies that, if the theory is consistent, truth is not definable, otherwise we would have a version of the liar paradox---again, which is not to say we cannot introduce a predicate Tr by axiomatic stipulation). In particular, if we order the alphabet of the language and construct a predicate F(x) such that a (code of a) sentence satisfies F(x) iff it contains the first letter of the alphabet (a tedious, but entirely routine exercise, once you get the hang of the coding machinery), there will be a sentence D which says of itself that it contains the first letter of the alphabet. So that particular sentence can be given a formal definition.
  • Meta
    54
    This argument fails when applied to a natural language, because there is no precise definition for <is a well-formed formula>. Only our intuition can tell us what do we consider a WFF. There isn't a fixed set of relation symbols either.

    edit: Any anyways we are not talking about real self-reference just some kind of reflection. Let's modify the statement of the Berry paradox:
    "The definition with the least Godel number not definable in fewer than 20 words."
    This is also paradoxical.
    Or "The statement with the least Godel number that does not contain the first letter of the alphabet."
    This can be a paradox.
  • Nagase
    60
    This argument fails when applied to a natural language, because there is no precise definition for <is a well-formed formula>. Only our intuition can tell us what do we consider a WFF. There isn't a fixed set of relation symbols either.Meta

    I don't agree with the premises here, but even granting them that's irrelevant: you originally claimed that the op's argument was not formalizable, since it contained self-referent statements. So the claim was about the expressive powers of formal languages, not natural languages. The Carnap-Gödel diagonal lemma literally disproves this assertion: if T is a nice arithmetic theory, for any one-place predicate of the language, we can form a sentence which says of itself that it has that predicate.

    Any anyways we are not talking about real self-reference just some kind of reflection. Let's modify the statement of the Berry paradox:
    "The definition with the least Godel number not definable in fewer than 20 words."
    This is also paradoxical.
    Meta

    This just shows that "definable" is not, well, definable in a formal language (even this, strictly speaking, is not true: we can both define definability for a given language in a meta-language with richer expressive resources, or, some times, we can make do with "local" definability---e.g. the constructible sets in ZFC). The problem is with the concept of definability, not with self-reference.

    Or "The statement with the least Godel number that does not contain the first letter of the alphabet."
    This can be a paradox.
    Meta

    But it's not, and is perfectly definable in, say, Robinson's Arithmetic.
  • Meta
    54
    We use the expression self-reference differently. When I say self-reference I mean (as I have explained earlier) that the definiens can not contain the definiendum. And I am not talking about Godel numbers and codings.

    You restrict the universe of statements to a set of formulas of a given language. Even in this case the truth predicate is undefinable so the OP's sentence can't be formalized. Or if you formalize the truth predicate in a metalanguage then when talking about all statements you will only talk about all statements of the object language.

    I think OP meant something much more general than statements of a fixed object language L when he was talking about all statements.

    So either:
    1) "All statements are false" is self-referential (as I mean it), or
    2) if we assume Godel numberings and that the sentence talks about its Godel-number then
    2a) we don't have a truth predicate defined or
    2b) we have a truth predicate but then we have meta statements outside of the quantification range
    of "All statements are false".

    This is why it can't be formalized.

    "But it's not, and is perfectly definable in, say, Robinson's Arithmetic."
    But it is because "a" is the first letter of the alphabet and
    "The statement with the least Godel number that does not contain the first letter of the alphabet." contains "a".

    edit: "the definiens can not contain the definiendum" in an explicit definition.
  • Pippen
    41
    As I mentioned in a private message, I'm not entirely sure this step holds. What is the reference of "this" above? If it is "this statement is false" (taking it to have small scope), then your proposed rule would take us from "All statements are false" to ""This statement is false" is false". But the latter one is false, not truth-valueless. So the whole conjunction is false.Nagase

    You are right with the reference, but if the part "...this statement is false" is false then it'd be true (and vice versa false if it's true), isn't it? So therefoe it can't have a truth value and so does the conjunction.

    Just for reference the proof again:

    1. We assume (A) "All statements are false" has a truth value.

    2. Since we can always go from "All x are y" (x may stand for 1,2,3 here) to for instance "All x are y and 2 are y" without changing the truth value, it must hold that A is logically equivalent to (A') "All statements are false and this statement is false", so A <-> A'.

    3. Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction of A'.

    4. That leads to a contradiction, because it's impossible that A <-> A' (2.) and A' doesn't have a truth value (3.). That means that the assumption, A has a truth value, leads to a contradiction and is false, A has no truth value.
  • Michael
    4k
    Now we prove that A' can't have a truth value because of its part "this statement is false". This is self referential and can't have a truth value (if it's true it would be false and vice versa), so it follows for the whole conjunction of A'.Pippen

    I addressed this mistake here.
  • Pippen
    41
    @Michael: We are dealing just in the scope of classical (two-valued) logic and there a statement like "This statement is false" is just without a truth-value as far as I know.
  • Michael
    4k
    We are dealing just in the scope of classical (two-valued) logic and there a statement like "This statement is false" is just without a truth-value as far as I know.Pippen

    So not having a truth value is the third option. If we have the conjunction p ∧ q and if p is false and q doesn't have a truth value then the conjunction as a whole is false.
  • Michael
    4k
    Although actually there's also Bochvar's internal three-valued logic which has a different truth table to Łukasiewicz's, and has a conjunction of this kind not having a truth value (e.g. propositions like "thiggledy piggledy and grass is green" are meaningless), which is consistent with your view.

    I suppose which to choose is just a matter of preference.
  • Pippen
    41
    So not having a truth value is the third option. If we have the conjunction p ∧ q and if p is false and q doesn't have a truth value then the conjunction as a whole is false.Michael

    Not in classical logic. Not having a truth value is not a third option there. I think 2. of my proof is dubious, maybe nagase will check that out.
  • Michael
    4k
    Not in classical logic. Not having a truth value is not a third option there. I think 2. of my proof is dubious, maybe nagase will check that out.Pippen

    But your argument rests on your own truth table in which there are three options. You're saying that if p is false (or true) and if q doesn't have a truth value then p ∧ q doesn't have a truth value.

    If there wasn't a third option then q ("this sentence is false") must be either true or false.

    If it helps, don't think of them as truth values but as predicates. You can have "true" as a predicate, "false" as a predicate, or "neither true nor false" as a predicate.

    So the question is on what predicate a conjunction has if one of its operators has "false" as its predicate and the other has "neither true nor false" as its predicate. According to Łukasiewicz such a conjunction has "false" as its predicate and according to Bochvar such a conjunction has "neither true nor false" as its predicate. Unfortunately I don't know enough to determine how one goes about choosing one truth table over another. Maybe it's simply axiomatic. But as you say, perhaps @Nagase has some insight into the matter.
  • Michael
    4k
    However, the above might not even be relevant to the particular issue at hand. It could be that your argument conflates. To explain this, consider the statement "this statement is false and grass is red". There are two different ways to interpret this, depending on what we consider to be the referent of "this statement":

    1. a) "this statement is false" is false and b) grass is red
    2. a) "this statement is false and grass is red" is false and b) grass is red.

    We might say that if "this statement is false" is false then "this statement is false" is true, giving us our paradox . But can we say that if "this statement is false and grass is red" is false then "this statement is false and grass is red" is true, giving us another paradox? I don't think we can. So given that grass isn't red, both 2a and 2b are false, and so the conjunction of 2a and 2b is false.

    I also think that 2 is the correct interpretation of the statement "this statement is false and grass is red".

    Now replace "grass is red" with "every other statement is false".
  • Srap Tasmaner
    1k

    Let S be the set of all statements.
    Let z be the string "If x ∈ S, then x is false."
    Assume z ∈ S.
    If z is true, then z is false.
    If z is false, then it is false that if x ∈ S, then x is false.
    Therefore if z ∈ S, z is false, and it is false that if x ∈ S, then x is false.

    Let z* be the string "z* is false and if x ∈ S/z*, then x is false."
    Assume z* ∈ S.
    Let S/z* be the complement of z* in S.
    If z* is true, then z* is false.
    If z* is false, then either z* is true or it is false that if x ∈ S/z*, then x is false.
    Therefore if z* ∈ S, z* is false, and it is false that if x ∈ S/z*, then x is false.
  • Pegasys
    4


    It's true. If "all statements" implies that the statement "all statements are false" is a part of the set of all statements, then "false" is identical. But this does not mean that the statement is false, if the definition of identity is merely reflexive. If however, "identity" is transitive, then this would imply a higher resolution in the proof, implying that more statements are added to the block of proof. And since these statements are elements of the set of all statements, then it follows that some statements which are not on the right side of the equation, are not in the game, but they remain elements of the set of all statements. It follows that the "belief" you mention induces meta-perception, intelligent observation, but this does not imply omni. It only means that self-referentials implicitly define the truth of the statement if "limited truth" is observed by climbing the meta-ladder disregarding the left-hand side, if symmetry is not defined. but how-could it be symmetrical if "identity" is implicit and "false" is not? So it must be true since the interpretation of "all statements" is "false" and "false" is an non-valid intepretation on a lower level but on the level in question it is a "non-valid interpretation" reflecting some symmetrical property of belief dynamics, that is true, and hence a part of the negation of your statement "all statements are false". This implies the importance of the "some" quantifier.

    Human nature has a tendency to contradict in order to control environment. Logic is the web in that its elements are of a dual nature, but they only are dual through the lens the spider uses. If an element becomes conscious of itself, then it can only do so by attaching meaning to the situation, to the set it belongs to, like using Gödel numbering for example. Then this element needs to evoke contradiction in order for its statements to make sense, but only to itself, the rest may challenge the element or not. It's a game for machines and rises the question if machines are conscious or not, amongst other things, dependend on the individual, or machine, or element, dependent on the sociology involved.
  • Owen
    14
    Ep(p).

    (All statements are false) is false and it is equivalent to
    (Some statements are true).

    1. (All p)(~p) -> ~q.
    2. (All p)(~p) -> ~(~q).
    3. (All p)(~p) -> q.
    4. (All p)(~p) -> (~q & q).
    5. ~(All p)(~p).
    6. (Some p)(p).
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