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Questions tagged [sequent-calculus]

For questions concerning sequent calculus, a formal proof system originally introduced by Gerhard Gentzen in 1933/1935 and studied in the framework of proof theory.

This tag is for questions concerning sequent calculus rules and proofs. Sequent calculus is a tool in proof theory explicitly designed for investigations of logical consequence and derivability.

Sequent calculus is strictly linked to the other Gentzen's big discovery: natural deduction. In sequent calculus systems, there are no temporary assumptions that would be discharged, but an explicit listing of the assumptions on which the derived assertion depends.

The derivability relation, to which reference was made in natural deduction, is an explicit part of the formal language, and sequent calculus can be seen as a formal theory of the derivability relation.

209 questions
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Main differences and relations between Sequent Calculus and Natural Deduction

What are the main differences between the Sequent Calculus and the Natural Deduction (independently of if we're working with classical, intuitionistic or another logic) ? As far as I know : Differences : The sequent calculus is more suitable for…
Boris
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Relationship between sequent calculus and Hilbert systems, natural deduction, etc

I am trying to learn the basics of logic and I'm confused on how these proof systems work together. The big ones I see are Hilbert style, and then Gentzen style which includes natural deduction, and sequent calculus. I also see "intuitionistic…
Brandon L
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Good textbook for learning Sequent Calculus

There are many modern text books teaching logic using Natural Deduction. There are no books teaching logic using the axiomatic method (see Good book for learning and practising axiomatic logic ) Now in another post (Prove by introduction rules (P ⇒…
Willemien
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Does double negation distribute over implication intuitionistically?

Does the equivalence $$\neg\neg (P \rightarrow Q) \leftrightarrow (\neg\neg P \rightarrow \neg\neg Q)$$ hold in propositional intuitionistic logic? In propositional classical logic the equivalence holds obviously since $P \leftrightarrow \neg\neg…
Z. A. K.
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What does conditional tautology mean?

Does conditional tautology mean a tautology which hast the form of $(A)\rightarrow (b)$ and therefore $A \lor \lnot A$ is unconditional tautology? (in regards to the following paragraph from wikipedia about sequent calculus:…
K. Smith
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Proving $\exists x[P(x)] \to \exists y[\exists x[P(x)]\to P(y)]$ in for intuitionistic $\varepsilon$-calculus.

I am researching Mint's paper: Intuitionistic Existential Instantiation and Epsilon Symbol (this is as far as I know unfinished work) In intuitionistic logic, it is not difficult to prove that $$\exists x[P(x)] \to \exists y[\exists x[P(x)]\to…
Tungsten
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Linear logic and linearly distributive categories

1. Context On page two of the introduction to their paper Weakly distributive categories on linearly distributive categories Cockett and Seely write: It turns out that these weak distributivity maps, when present coherently, are precisely the…
Max Demirdilek
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In linear logic sequent calculus, can $\Gamma \vdash \Delta$ and $\Sigma \vdash \Pi$ be combined to get $\Gamma, \Sigma \vdash \Delta, \Pi$?

Linear logic is a certain variant of sequent calculus that does not generally allow contraction and weakening. Sequent calculus does admit the cut rule: given contexts $\Gamma$, $\Sigma$, $\Delta$, and $\Pi$, and a proposition $A$, we can make the…
Sophie Swett
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Distributive property of tensor ($\otimes$) over par (⅋) in linear logic

In the setting of linear logic, does the tensor $\otimes$ distribute over the par $⅋$? That is, is it possible to show that $$ A \otimes (B ⅋ C) \stackrel?\equiv (A \otimes B) ⅋ (A \otimes C) $$ holds? If not, what is a counterexample? The…
Andrea Censi
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Understanding the meaning of $\forall,\exists$ rules in sequent calculus.

I'm stucking in understanding the usage and soundness of the rules for the quantifiers $\forall,\exists$ in sequent calculus. $\forall-L$: $~~~~~\dfrac{\Gamma,\phi[t]\vdash \Delta}{\Gamma,\forall x\phi[x/t]\vdash\Delta}$…
Eric
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Basic: Sequent definition, and-introduction, and iff

I am reading through "Mathematical Logic by Ian Chiswell & Wilfred Hodges"(amazon, and publisher) So far have it has covered $\land$-Introduction and $\land$-Elimination Sadly this text only has answers to selected solutions, which annoys me to no…
cjh
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Classical logic without negation and falsehood

It seems to me that Gerhard Gentzen's sequent calculus could just omit negation and falsehood, and still prove any classical tautology in a suitable form. (For a specific formula, falsehood gets replaced by the conjunction of all relevant…
Thomas Klimpel
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Understanding Shamkanov's Paper on Circular Proofs for Gödel-Löb Logic

I’ve been working through Shamkanov’s paper on cyclic proofs for Gödel-Löb Logic (https://arxiv.org/pdf/1401.4002). While it’s a landmark paper in the field of (cyclic) proof theory, I’m finding that many details are left implicit, especially from…
Rustam Gumnam
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How to establish this generalization rule in sequent calculus for First Order Logic?

I stumbled across this rule: $$\frac{\Gamma\vdash q\rightarrow p(a) }{\Gamma \vdash (q \rightarrow \forall x. p(x))}$$ where $a$ also needs to be a fresh constant, so with that in mind you could re-write the rule as the…
gg gg
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Understanding sequent calculus

Are the following rules correct? $(i)$ $\dfrac{\Phi \Rightarrow \Delta}{\Phi, \psi \Rightarrow \Delta}$ $(ii)$ $\dfrac{\Phi, \psi \Rightarrow \Delta}{\Phi \Rightarrow \Delta}$ Intuitively, I would've said that rule $i$ is incorrect and rule $ii$…
Monika
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