Cut-Free Gentzen Sequent Calculi for Tense Logics

Round 1

Reviewer 1 Report

This paper is difficult and subtle stuff and checking all the details of the proofs would cost me too much time, several days I am afraid. I trust that all the proofs are correct, without having checked them. The presentation of the material is very abstract. The word "possible world" is not used, while the semantics of modal logics is usually in terms of possible worlds.

line 11: is concerned [add: is]

line 11: do you mean: have been taken [instead of are taken]?

line 18: this sentence [Deep inference ...] has no meaning to me

line 20: literature [instead of: literatures]

line 20: in the sense of Gentzen [instead of: in the road of Gentzen]?

line 21: elimination of the cut rule [instead of: eliminating the cut rule]

line 28: calculi [instead of: calculus]

Definition 1: The idea behind the set $W$ is that it is a set of possible worlds, the idea behind the relation $R$ that it is an accessibility relation and the idea behind $V(p)$ is that it is the set of all possible worlds in which $p$ is true. Would not it make sense to add these underlying ideas? Given a frame $F = (W, R)$ and a valuation $V$, the traditional definition of $F, V \models \Box p$ seems to me easier to understand than the abstract algebraic definition given in the paper.

line 85: You say that  a tense logic $L$ is Kripke complete if $L = Th(Fr(\Sigma))$. Which $\Sigma$? However, in Theorem 1 it is stated clearly what it means that $K_t$ is Kripke complete.

line 143: The following [instead of: followings]

In general the English is okay with a few minor exceptions which I have mentioned above.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors introduce cut-free single-succedent Gentzen sequent calculus GKt for the minimal tense logic Kt, which satisfies the displaying property. Their argument proceeds via a Kolmogorov translation and three intermediate sequent systems. Lastly, it is shown that tense logics axiomatized by strictly positive implication have cut-free Gentzen sequent calculi uniformly.

The authors give a well-written introduction which covers all the relevant material and provides sufficient motivation. Overall the paper is well-written and the main results are of interest. The results look to be correct and well justified.

Author Response

Thanks are given to the reviewer for the positive comments. 

Reviewer 3 Report

The paper is quite well written, though complicated for reading and only specialists are to be addressed.

Author Response

Thanks are given to the reviewer for the comments.