Standard G\"odel modal logics are not realized by G\"odel justification logics

Nicholas Pischke

arxiv: 1907.04583 · v1 · pith:6JHFEANQnew · submitted 2019-07-10 · 🧮 math.LO

Standard G\"odel modal logics are not realized by G\"odel justification logics

Nicholas Pischke This is my paper

Pith reviewed 2026-05-24 23:33 UTC · model grok-4.3

classification 🧮 math.LO

keywords Gödel modal logicsGödel justification logicsrealizationforgetful projectionmodal logicjustification logic

0 comments

The pith

Standard Gödel modal logics are not realized by the basic Gödel justification logics despite their connection via forgetful projection.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the standard Gödel modal logics, as defined by Caicedo and Rodriguez, fail to be realized by the basic Gödel justification logics. Realization is a technical relation that requires justification terms to be mapped back to modal formulas while preserving all theorems exactly. This matters because the forgetful projection, which simply drops the justification terms, does relate the two systems, yet the stronger realization link does not hold. A sympathetic reader cares because it reveals a gap in how these two families of logics are connected under standard definitions.

Core claim

The standard Gödel modal logics are not realized by the basic Gödel justification logics, although they are related by the forgetful projection.

What carries the argument

The distinction between realization (a specific theorem-preserving mapping from justification formulas to modal formulas) and the weaker forgetful projection (which erases justification terms).

If this is right

The basic Gödel justification logics cannot serve as a realizing counterpart for the standard Gödel modal logics.
The forgetful projection alone does not guarantee realization in the Gödel case.
Different or extended Gödel justification logics would be needed to achieve realization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar gaps between projection and realization may exist for other combinations of modal and justification logics in the Gödel setting.
One could test whether adding specific axioms or rules to the basic justification logics restores realization.
The result isolates the Gödel case as potentially requiring separate treatment from classical justification logics.

Load-bearing premise

The technical definition of realization used in the justification logic literature applies exactly as stated, and the modal logics match the original Caicedo-Rodriguez formulations without changes.

What would settle it

An explicit function that maps theorems of the basic Gödel justification logics onto precisely the theorems of the standard Gödel modal logics would falsify the result.

read the original abstract

We show that the standard G\"odel modal logics, as initially introduced by Caicedo and Rodriguez in \cite{CR2009,CR2010}, are not realized by the basic G\"odel justification logics although being related by the forgetful projection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows that the standard Gödel modal logics are not realized by the basic Gödel justification logics despite the forgetful projection link.

read the letter

The key point here is that the standard Gödel modal logics introduced by Caicedo and Rodriguez are not realized by the basic Gödel justification logics, although the forgetful projection relates them. This negative result stands out as the main contribution. What is new is the explicit separation on realization. Prior work had the projection link, but this shows realization does not hold under the standard technical definition from justification logic literature. The paper handles the definitions consistently with the cited sources. It does a good job keeping the two notions distinct and avoiding any overclaim. The argument relies on the usual setup without introducing new parameters or fitting. Soft spots are limited. The abstract states the claim clearly, and the stress-test finds no internal contradiction or hidden assumption. However, without the full proof details in this review, one would want to check how the specific modal axioms fail to be realized. The citation pattern is appropriate and points back to the foundational papers. This paper targets researchers in Gödel modal logics and justification logics. A reader working on realization theorems or non-realization results will find it directly relevant. It resolves an open question in the subfield. I think it deserves a serious referee. The evidence for the claim looks solid enough on the available information to warrant review rather than desk rejection. Recommendation: Send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the standard Gödel modal logics introduced by Caicedo and Rodriguez satisfy the forgetful projection onto the basic Gödel justification logics but are not realized by them.

Significance. If correct, the result supplies a clean separation between forgetful projection and realization in the Gödel setting, using the standard technical definitions from the justification-logic literature. This clarifies the precise scope of realization theorems for Gödel modal logics and strengthens the conceptual distinction between the two notions.

minor comments (2)

[Abstract] The abstract and introduction could briefly recall the precise syntactic form of the realization map used in the literature (distinct from the forgetful projection) to make the negative claim immediately accessible.
[Introduction] A short table or diagram contrasting the axioms of the Caicedo-Rodriguez modal logics with the basic Gödel justification logics would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment. The report correctly identifies the main result: the standard Gödel modal logics satisfy the forgetful projection onto the basic Gödel justification logics but are not realized by them. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; separation result relies on external definitions

full rationale

The paper establishes a negative result: the Caicedo-Rodriguez Gödel modal logics satisfy the forgetful projection but fail to be realized by the basic Gödel justification logics. This distinction is drawn from standard technical definitions in the justification-logic literature (external to the present work) and the cited formulations in CR2009/CR2010. No self-definitional equations, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior author work appear in the derivation. The claim is a standard non-realizability proof that remains independent of its own inputs once the two notions (projection vs. realization) are kept separate.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a pure proof of non-realization relying on standard mathematical logic background and domain definitions from the two cited papers; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Definitions of standard Gödel modal logics as given in Caicedo and Rodriguez 2009 and 2010
The claim depends on these being the correct reference definitions for 'standard'.
domain assumption Definition of realization relation between justification logics and modal logics
Core technical notion from justification logic literature that distinguishes realization from forgetful projection.

pith-pipeline@v0.9.0 · 5551 in / 1123 out tokens · 20060 ms · 2026-05-24T23:33:34.642173+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

S. Artemov. Operational Modal Logic. Technical Report M SI 95-29, Cornell University, 1995. Ithaca, NY

work page 1995
[2]

S. Artemov. Explicit Provability and Constructive Sema ntics. The Bulleting of Symbolic Logic , 7(1):1–36, 2001

work page 2001
[3]

A Godel Modal Logic

X. Caicedo and R. Rodriguez. A Godel Modal Logic. ArXiv e-prints , 2009. arXiv, math.LO, 0903.2767

work page internal anchor Pith review Pith/arXiv arXiv 2009
[4]

Caicedo and R

X. Caicedo and R. Rodriguez. Standard G¨ odel Modal Logic s. Studia Logica, 94(2):189–214, 2010

work page 2010
[5]

Caicedo and R

X. Caicedo and R. Rodriguez. Bi-modal G¨ odel logic over [ 0, 1]-valued Kripke frames. Journal of Logic and Computation , 25(1):37–55, 2015

work page 2015
[6]

M. Dummett. A propositional calculus with denumerable m atrix. Journal of Symbolic Logic , 24(2):97–106, 1959

work page 1959
[7]

M. Fitting. The logic of proofs, semantically. Annals of Pure and Applied Logic , 132(1):1–25, 2005

work page 2005
[8]

M. Ghari. Justiﬁcation Logics in a Fuzzy Setting. ArXiv e-prints , 2014. arXiv, math.LO, 1407.4647

work page arXiv 2014
[9]

M. Ghari. Pavelka-style fuzzy justiﬁcation logics. Logic Journal of the IGPL , 24(5):743–773, 2016

work page 2016
[10]

K. G¨ odel. Zum intuitionistischen Aussagenkalk¨ ul.Anzeiger der Akademie der Wissenschaften in Wien , 69:65–66, 1932

work page 1932
[11]

H´ ajek.Metamathematics of Fuzzy Logic , volume 4 of Trends in Logic

P. H´ ajek.Metamathematics of Fuzzy Logic , volume 4 of Trends in Logic . Kluwer, Dordrecht, 1998. STANDARD G ¨ODEL MODAL LOGICS ARE NOT REALIZED BY G ¨ODEL JUSTIFICATION LOGICS 9

work page 1998
[12]

A. Horn. Logic with Truth Values in a Linearly Ordered He yting Algebra. Journal of Symbolic Logic , 34(3):395–408, 1969

work page 1969
[13]

R. Kuznets. On the complexity of explicit modal logics. In International Workshop on Computer Science Logic, Proceed ings, pages 371–383. Springer Berlin Heidelberg, 2000

work page 2000
[14]

R. Kuznets. Complexity Issues in Justiﬁcation Logic . PhD thesis, City University of New York Graduate Center, 20 08

work page
[15]

Mkrtychev

A. Mkrtychev. Models for the logic of proofs. In Proceedings of Logical Foundations of Computer Science LFC S’97, volume 1234 of Lecture Notes in Computer Science , pages 266–275. Springer, 1997

work page 1997
[16]

N. Pischke. A note on strong axiomatization of G¨ odel Ju stiﬁcation Logic. ArXiv e-prints , 2018. arXiv, math.LO, 1809.09608

work page arXiv 2018
[17]

T. Studer. Decidability for some justiﬁcation logics w ith negative introspection. Journal of Symbolic Logic , 78(2):388–402, 2013. Hoch-Weiseler Str. 46, Butzbach, 35510, Hesse, Germany E-mail address : pischkenicholas@gmail.com

work page 2013

[1] [1]

S. Artemov. Operational Modal Logic. Technical Report M SI 95-29, Cornell University, 1995. Ithaca, NY

work page 1995

[2] [2]

S. Artemov. Explicit Provability and Constructive Sema ntics. The Bulleting of Symbolic Logic , 7(1):1–36, 2001

work page 2001

[3] [3]

A Godel Modal Logic

X. Caicedo and R. Rodriguez. A Godel Modal Logic. ArXiv e-prints , 2009. arXiv, math.LO, 0903.2767

work page internal anchor Pith review Pith/arXiv arXiv 2009

[4] [4]

Caicedo and R

X. Caicedo and R. Rodriguez. Standard G¨ odel Modal Logic s. Studia Logica, 94(2):189–214, 2010

work page 2010

[5] [5]

Caicedo and R

X. Caicedo and R. Rodriguez. Bi-modal G¨ odel logic over [ 0, 1]-valued Kripke frames. Journal of Logic and Computation , 25(1):37–55, 2015

work page 2015

[6] [6]

M. Dummett. A propositional calculus with denumerable m atrix. Journal of Symbolic Logic , 24(2):97–106, 1959

work page 1959

[7] [7]

M. Fitting. The logic of proofs, semantically. Annals of Pure and Applied Logic , 132(1):1–25, 2005

work page 2005

[8] [8]

M. Ghari. Justiﬁcation Logics in a Fuzzy Setting. ArXiv e-prints , 2014. arXiv, math.LO, 1407.4647

work page arXiv 2014

[9] [9]

M. Ghari. Pavelka-style fuzzy justiﬁcation logics. Logic Journal of the IGPL , 24(5):743–773, 2016

work page 2016

[10] [10]

K. G¨ odel. Zum intuitionistischen Aussagenkalk¨ ul.Anzeiger der Akademie der Wissenschaften in Wien , 69:65–66, 1932

work page 1932

[11] [11]

H´ ajek.Metamathematics of Fuzzy Logic , volume 4 of Trends in Logic

P. H´ ajek.Metamathematics of Fuzzy Logic , volume 4 of Trends in Logic . Kluwer, Dordrecht, 1998. STANDARD G ¨ODEL MODAL LOGICS ARE NOT REALIZED BY G ¨ODEL JUSTIFICATION LOGICS 9

work page 1998

[12] [12]

A. Horn. Logic with Truth Values in a Linearly Ordered He yting Algebra. Journal of Symbolic Logic , 34(3):395–408, 1969

work page 1969

[13] [13]

R. Kuznets. On the complexity of explicit modal logics. In International Workshop on Computer Science Logic, Proceed ings, pages 371–383. Springer Berlin Heidelberg, 2000

work page 2000

[14] [14]

R. Kuznets. Complexity Issues in Justiﬁcation Logic . PhD thesis, City University of New York Graduate Center, 20 08

work page

[15] [15]

Mkrtychev

A. Mkrtychev. Models for the logic of proofs. In Proceedings of Logical Foundations of Computer Science LFC S’97, volume 1234 of Lecture Notes in Computer Science , pages 266–275. Springer, 1997

work page 1997

[16] [16]

N. Pischke. A note on strong axiomatization of G¨ odel Ju stiﬁcation Logic. ArXiv e-prints , 2018. arXiv, math.LO, 1809.09608

work page arXiv 2018

[17] [17]

T. Studer. Decidability for some justiﬁcation logics w ith negative introspection. Journal of Symbolic Logic , 78(2):388–402, 2013. Hoch-Weiseler Str. 46, Butzbach, 35510, Hesse, Germany E-mail address : pischkenicholas@gmail.com

work page 2013