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We define that statements are objects that fullfill a certain syntax. But this definition itself is a statement. It is a variant of saying: If the object fulfils a certain syntax then it is a statement because it also fulfils a certain syntax. This is again a statement about the statement and so on...

I know this is very informal and I have started to read a book about logic but the author said he will talk about this subject first at chapter 7 and I hoped that I maybe could get an informal but still comprehensible "peek" of the problem: That we are defining an object in a lower language that still is applicable to higher languages. To me it seems like proving something for fields and then saying that it also holds for rings.

New2Math
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  • Have you seen this? https://en.wikipedia.org/wiki/Russell%27s_paradox – Lincon Ribeiro Jul 27 '19 at 22:32
  • What would be the conclusion of the paradox? – New2Math Jul 27 '19 at 22:39
  • He created a new structure for logical objects called "types" so this way it would be possible to separate what is a set and its elements when you have paradoxes like that. https://en.wikipedia.org/wiki/Type_theory – Lincon Ribeiro Jul 27 '19 at 22:43
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    The paradox "there is a set that contains all sets" is about sets but it is analgous to the paradox that "there is a language that describes all languages"? – New2Math Jul 27 '19 at 23:00
  • Language could be a set, just define it in a formal way. For instance, language A is the set of symbols such that the proposition p is true. You are talking about if a language that describes all languages should describe itself or not, right? Then, it seems that the analogy holds. – Lincon Ribeiro Jul 27 '19 at 23:23
  • You have not actually asked a question. – Derek Elkins left SE Jul 28 '19 at 01:58
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    Regardeless, the problem is likely in your second sentence. A definition is not a statement. Certainly when we formalize logic, "statement" usually means (well-formed, and possibly also closed) formula, and a definition is not a formula in this sense. But even without formalization, the sentence "Define $x$ to be $3$." is not a declarative statement but an imperative one. It doesn't make sense to ask whether "Define $x$ to be $3$" is true any more than it makes sense to ask whether "Sit!" is true. A definition is a command to the reader, not an assertion. – Derek Elkins left SE Jul 28 '19 at 02:00
  • See e.g this post and many others on this site about "meta-". – Mauro ALLEGRANZA Jul 28 '19 at 07:16

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See An Elementary Latin Grammar :

The statement "Cesar scribit" [means] Caesar is writing.

We have here the Latin language, which is the object of the study; call it : object language.

And we have the English language, used to perform the study; call it : meta-language.

The statement "Cesar scribit" is a statement in the object language.

The statement "The statement "Cesar scribit" [means] Caesar is writing." is a statement of the meta-language that expresses a fact about the object language statement "Cesar scribit".

And now compare with : D.van Dalen, Logic and Structure, page 7 :

Definition 1.1.2 The set $\text {PROP}$ of propositions is the smallest set $X$ with the properties

(i) $p_i ∈ X(i ∈ \mathbb N), \bot ∈ X$,

<p>(ii) <span class="math-container">$ϕ, ψ ∈ X ⇒ (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ → ψ), (ϕ ↔ ψ) ∈ X$</span>,</p>

<p>(iii) <span class="math-container">$ϕ ∈ X ⇒ (¬ϕ) ∈ X$</span>.</p>

It is a statement in the meta-language : the usual mathematical argot, made of natural language plus symbols used as abbreviations, regarding the syntax of the object language : the language of propositional calculus.

"this definition itself is a statement" ?

Yes; it is a statement in the meta-language defining the formal syntax of the objcet language.

Mauro ALLEGRANZA
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