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Questions tagged [metalogic]

For questions related to metalogic. It is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

Metalogic, the study and analysis of the semantics (relations between expressions and meanings) and syntax (relations among expressions) of formal languages and formal systems. It is related to, but does not include, the formal treatment of natural languages.

Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory.

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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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What is the difference between Gödel's completeness and incompleteness theorems?

What is the difference between Gödel's completeness and incompleteness theorems?
varun
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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…
FNH
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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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Why do people speak about truth value of undecidable propositions?

I have heard people say things like "If the Goldbach conjecture was proven to be independent of PA, then it would follow it's true." The reasoning behind this is that, if it was false, we could explicitly provide the counterexample, thus proving it…
Elvis
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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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Is the distinction between language and metalanguage strictly parallel the distinction between logic and metalogic?

The distinction language/metalanguage is often used to explain the difference between logic and metalogic. My question is to know whether this explanation is sufficient. More precisely: : is it true that being formulated in metalanguage is a…
user655689
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Are there infinitely many metalogics?

Given the definitions of material implication, logical implication, and what a tautology is, we can prove: $$\mathcal B\text{ logically implies }\mathcal C\text{ if and only if }(\mathcal B\rightarrow\mathcal C)\text{ is a tautology.}$$ Clearly,…
GDGDJKJ
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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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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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