Mathematics Stack Exchange
Mathematics Stack Exchange

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.

194 questions
1
vote
1 answer

Seeking intuition for LK sequent calculus ${\rightarrow}L$ inference rule

This Wikipedia article states the LK (${\rightarrow}L$) rule as: $$\Gamma \vdash A, \Delta \qquad \Sigma, B \vdash \Pi \over \Gamma, \Sigma, A \rightarrow B \vdash \Delta, \Pi$$ I'm trying to find some confirmation or intuition of why this is…
joseville
  • 1,477
1
vote
1 answer

Is there a way to prove the cut rule in LK sequent calculus without converting to implications?

The cut rule is given as follows by these two wikipedia articles 1, 2: $$\Gamma \vdash \Delta, A \qquad \Sigma, A \vdash \Pi \over \Gamma, \Sigma \vdash \Delta, \Pi$$ I have the following proof which converts to and from implications. Starting with…
joseville
  • 1,477
1
vote
0 answers

Confused about the ground rules of sequent calculus (LK): cut or no cut?

There are many books and internet tutorials about sequent calculus for classical first-order logic (LK – which, if I understood correctly, is the one that allows for any number of formulae on the right side). I am actually only concerned with…
pglpm
  • 831
0
votes
0 answers

understanding when to use left right rules sequent calculus

I'm trying to understand this classical Sequent Calculus proof. These are the rules Of Sequent Calculus(version 1) in respect to this question How do we know to use [v R2] and [v R1] instead of [v L] which I originally used and also on why we…
user1068052
0
votes
1 answer

understanding the implication rule in sequent calculus

I'm trying to understand a proof on sequent Calculus Note: This proof is not yet finished. I don't understand how the 3rd statement(counting from bottom up) is converted into the fourth statement using [ -> L] By this rule: I would assume r = P,Q…
user1068052
0
votes
0 answers

What does it mean for a sequent to be equivalent to a set of sequents?

Adam Přenosil's article "Cut elimination, identity elimination, and interpolation in super-Belnap logics" contains this proposition: Proposition 3.2. Each sequent is equivalent in the Gentzen relation $\mathrm{G}\mathcal{B}$ to a finite set of…
jdonland
  • 197