# Questions tagged [metalogic]

98 questions

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### How are metalogic proofs valid?

I'm reading some materials on mathematical logic. I wonder how we can "prove" metalogical properties (soundness, completeness, etc.)? As at this point, the proof system has not been verified yet. Isn't this a chicken-and-egg question (we may then…

A.Stone

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### Has the Gödel sentence been explicitly produced?

I do not pretend to know much about mathematical logic. But my curiosity was piqued when I read Hofstadter's Gödel, Escher, Bach, which tries to explain the proof of Gödel's first incompleteness theorem by using an invented formal system called…

StrangerLoop

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### Formalizing the meta-language of First order Logic and studying it as a formal system

We've a formal system say First order Logic, we reason about it in our meta-language using our meta-logic. We study its properties as a mathematical object. We prove theorems like group theory. This makes us able to know the limits and the strength…

Fawzy Hegab

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### In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things?

In logic, do the $\Longrightarrow$ and $\rightarrow$ signify different things? Are there contexts where one is more appropriate than the other?
I had believed that the $\Longrightarrow$ was for metalogic, and the $\rightarrow$ was for logic.…

Hal

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### What mathematics cannot be done in proof assitants (Isabelle/HOL, HOL) and how proof assistants should be improved?

What mathematics cannot be done in proof assitants (Isabelle/HOL, HOL) and how proof assistants should be improved?
I am working on the project of mechanizing algorithms in Isabelle/HOL and I would like to know the limits and be prepared for the…

TomR

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### Expressing "does not imply''

In ordinary discourse, when we say that "$A$ implies $B$", we shall formalize it by writing the following:
$$A\rightarrow B$$
But when we say that "$A$ does not imply $B$", we cannot formalize it as the following:
$$\neg(A\rightarrow B)$$
Because by…

Fred

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### Judgment-level negation $\nvdash$

I have a basic question about the use of judgment-level negation $\nvdash$. Though I meet it in some of my courses on proof theory, I usually treat it as a metalevel expression and I don't know how to handle it if it appears in the object language.…

ferdinand

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### Do definitions have to fit axioms in logic?

One thing I find confusing in propositional logic is that we have things like axioms and inference rules but then we seem to be able to define whatever we want in syntax that doesn't necessarily adhere to the axiom formats.
For…

user525966

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### $\to$ vs. $\vdash$ in logic

I am really lost trying to understand the difference between the logical connective "implies", $\to$, and the metalogical symbol (or maybe it's also a connective?) $\vdash$. (This is all focusing on prepositional logic here).
In metalogical terms,…

user525966

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### Are there situations where "isomorphism implies elementary equivalence" fails?

I'm puzzled by a sentence in an old paper on the history of the notions of categoricity and completeness in mathematical logic.
In a paragraph near the end, the author, John Corcoran, takes some writers to task for failing to appreciate the relation…

symplectomorphic

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### Are soundness and completeness a part of proof theory, model theory or something else?

I have a question that I hope can clarify the scopes of model theory and proof theory. I have the following naïve understanding of the two areas (please correct me if I'm wrong):
Model theory is about the relationship between languages and…

dumbo

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### (Solution verification and criticism) If wff $\phi$ has no repetition of sentence letters, then $\nvDash \phi$

Edit: Here is attempt number 2. I've been much more terse, which I think flows much better. I've also changed the binary connective to disjunction to improve readability as well. By truth table $\lnot\phi \to \psi$ is equivalent to $\phi \lor \psi$,…

Ten O'Four

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### Meta-Definition of Convergence

So, just recently I realized that the idea of convergence is not "all encompassing"... Let me explain.
I thought that the topological definition of convergence was the most basic one in the following sense. If I had a metric space and defined…

Davi Barreira

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### In what formal system is Godel's Incompleteness Theorem (or similar statements of undecidability) proven?

For example, consider the proof using Rosser's trick as shown on wikipedia.
https://en.wikipedia.org/wiki/Rosser%27s_trick#The_Rosser_sentence
That proof isn't inside the arithmetical theory T, but in some higher level meta-theory. So if I wanted to…

Hank Igoe

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### How can you derive a formula without premises?

What is the difference between $\vdash A $ and $\models A$?
I'm not asking about the general difference between syntactic entailment and semantic entailment.
I specifically don't understand the difference in the case where the antecedents are…

Colin Hicks

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