The Modal Logic of Aristotelian Diagrams
Round 1
Reviewer 1 Report
The article introduces a formalism based on hybrid logic and a list of deductive systems built over it which can be used to reason over Aristotelian diagrams. Its main conceptual contribution consists in the following perspective shift: while Aristotelian polygons of opposition (and, more recently, Aristotelian diagrams) have traditionally been used to represent certain deductive theories, here the authors propose to represent Aristotelian diagrams themselves as deductive theories. The authors' focus is on modal logic. As they phrase it, while previous work has been concerned with "Aristotelian diagrams for modal logic", they rather investigate "the modal logic of Aristotelian diagrams". This represents a significant point when it comes to assessing the article's originality.
In the first part of the article, the authors provide fundamental notions taken from logical geometry and present their hybrid modal language to build systems of AD-logic. Such a language does not make use of propositional variables, relying on nominals and state variables as its only atomic symbols (hence, all formulas of the language are pure, according to standard terminology in hybrid logic). It is a highly expressive language, given the chosen set of primitive modal and binding operators (two modal operators associated with the Aristotelian relations of contrariety and contradiction, the global modality, the @ operator and the current-state-binder). Moreover, the authors introduce relational structures to interpret their language (AD-frames and AD-models).
The central part of the article corresponds to Section 4, which is crucial in terms of establishing technical results. The authors show that it is possible to define a bijection between the class of all equivalence classes of finite Aristotelian diagrams obtained via Aristotelian isomorphism and the class of all equivalence classes of finite AD-frames obtained via modal isomorphism. Thus, semantic results for AD-logic obtained with respect to classes of AD-frames/models can be translated into semantic results with respect to classes of Aristotelian diagrams.
In Section 5 the authors provide a correspondence theory for AD-logic, according to which a diagram D belongs to a certain family iff it validates (i.e., iff every AD-frame based on D validates, as per Def. 14) a certain characteristic formula.
Finally, in Section 6 the authors provide hints on how to build Aristotelian diagrams for AD-logic, thus turning back to the traditional perspective of using diagrams to represent deductive systems (although, in the light of the role played by AD-logic, this means opening a new level of analysis).
The article is written in very good English and the structure is entirely clear. The arguments made by the authors are sound and their motivation well-grounded. The proofs of the numerous technical results obtained are, as far as I could check, correct. In light of all this, I strongly recommend that the paper be accepted for publication.
There are only a few minor things I would like to suggest as optional improvements:
- on p. 2 the authors mention the distinction between "general" and "generic" definitions/results, saying that in the first case one still refers to some logical systems whereas in the second case one does not. This distinction is employed while commenting, e.g., on Definition 20 and Theorem 3, on p. 15. The authors state that Def. 20 is general but not generic, whereas Theorem 3 is generic. Yet, my doubt is that there is still a reference to system S in Theorem 3, and this seems to undermine the distinction drawn. So I was wondering whether one could address this possible concern by replacing the following part of Theorem 3
"An Aristotelian diagram D for (F,S) is a PCD iff..."
with
"An Aristotelian diagram D is a PCD iff..."
- in Def. 7 and Def. 8 it might be convenient to use the notation peculiar to state variables, since norminals are not mentioned, while the symbol "s" refers to both nominals and state variables (although the restriction to set V is already appropriate). Thus, the authors might consider using "x" in place of "s" throughout Def. 7 and, e.g., "y" in place of "s" throughout Def. 8.
- it could be useful to add some remarks on the reading of the two modal operators [CD] and [C], given that their standard modal reading seems to be broader than the relation they are associated with. For instance, at a state i one could have [CD]\phi, which could be read as saying that \phi is contradictory w.r.t. position i. Yet, from this one could also infer, given that systems of AD-logic are normal, [CD]\phi\vee\psi, which should not be read as saying that \phi\vee\psi is contradictory w.r.t. position i. A few more comments would help in ruling out this possible concern.
Author Response
We would like to thank the reviewer for their detailed and fair report of our paper. Below, we explain how we have incorporated the reviewer's comments and suggestions.
Reviewer remark: on p. 2 the authors mention the distinction between "general" and "generic" definitions/results, saying that in the first case one still refers to some logical systems whereas in the second case one does not. This distinction is employed while commenting, e.g., on Definition 20 and Theorem 3, on p. 15. The authors state that Def. 20 is general but not generic, whereas Theorem 3 is generic. Yet, my doubt is that there is still a reference to system S in Theorem 3, and this seems to undermine the distinction drawn. So I was wondering whether one could address this possible concern by replacing the following part of Theorem 3
"An Aristotelian diagram D for (F,S) is a PCD iff..."
with
"An Aristotelian diagram D is a PCD iff..."
=> This is an interesting remark; our paper was indeed not entirely clear on this. We have clarified this in the new version of the paper; in particular, see the newly added/rewritten passages on ll. 650-661 and ll. 690-698.
Reviewer remark: in Def. 7 and Def. 8 it might be convenient to use the notation peculiar to state variables, since norminals are not mentioned, while the symbol "s" refers to both nominals and state variables (although the restriction to set V is already appropriate). Thus, the authors might consider using "x" in place of "s" throughout Def. 7 and, e.g., "y" in place of "s" throughout Def. 8.
=> We have decided to leave Definitions 7 and 8 unchanged, because (i) as the reviewer already indicates, the restriction to the set V already suffices to guarantee that these Definitions are technically correct, and (ii) this is not merely a matter of notation ("s" versus "x" versus "y"), since s is a metavariable (as we mention explicitly on line 147) that ranges over object-level (state) variables like x and y.
Reviewer remark: it could be useful to add some remarks on the reading of the two modal operators [CD] and [C], given that their standard modal reading seems to be broader than the relation they are associated with. For instance, at a state i one could have [CD]\phi, which could be read as saying that \phi is contradictory w.r.t. position i. Yet, from this one could also infer, given that systems of AD-logic are normal, [CD]\phi\vee\psi, which should not be read as saying that \phi\vee\psi is contradictory w.r.t. position i. A few more comments would help in ruling out this possible concern.
=> If we understand it correctly, this remark seems to involve a confusion between (the roles of) the languages L_S and L_{AD}. This is precisely the confusion that we want to avoid in our paper, so we have added some additional explanation at the place we deemed most appropriate, viz., ll. 383 -- 387.
Reviewer 2 Report
It is a high quality, rigorously written paper about specifically designed modal logic AD for reasoning about Aristotelian diagrams of logical opposition. It is shown that there is bijection between the AD frames and Aristotelian diagrams. Families and variations of Aristotelian diagrams (for example, the square, hexagon and octagon of opposition) are defined in terms of AD frames.
Minor remarks:
Page 4, 3rd paragraph from below: in the comment on @, it seems as if the equivalence of \Box (s -> \phi) and \Diamond (s & \phi) is implied?
Page 14, 4th paragraph. A double characterization of Lemma 9 both as a corollary and a lemma might be confusing.
Page 14, 1st paragraph in section 5. It is stated that the correspondences considered do not refer to any underlying logical system. Some explanation would be welcome because, for example, standard modal logic with validity of K (\Box (\phi -> \psi) -> (\Box \phi -> \Box \psi)) seems to be preserved in AD.
Page 15, 1st paragraph in 5.2: typo "that that".
Throughout the text: accent ' is missing on "e" in "Beziau"
Author Response
We would like to thank the reviewer for their detailed and fair report of our paper. Below, we explain how we have incorporated the reviewer's comments and suggestions.
Reviewer remark: Page 4, 3rd paragraph from below: in the comment on @, it seems as if the equivalence of \Box (s -> \phi) and \Diamond (s & \phi) is implied?
=> Indeed, these two formulas are equivalent to each other in hybrid logic. We have slightly reformulated this passage to make this more explicit.
Reviewer remark: Page 14, 4th paragraph. A double characterization of Lemma 9 both as a corollary and a lemma might be confusing.
=> We have slightly reformulated this paragraph, so that we now refer to Lemma 9 only as a "lemma" and no longer as a "corollary".
Reviewer remark: Page 14, 1st paragraph in section 5. It is stated that the correspondences considered do not refer to any underlying logical system. Some explanation would be welcome because, for example, standard modal logic with validity of K (\Box (\phi -> \psi) -> (\Box \phi -> \Box \psi)) seems to be preserved in AD.
=> This is an interesting remark; our paper was indeed not entirely clear on this. We have clarified this in the new version of the paper; in particular, see the newly added/rewritten passages on ll. 650-661 and ll. 690-698.
Reviewer remark: Page 15, 1st paragraph in 5.2: typo "that that".
=> Thanks, this has been taken care of.
Reviewer remark: Throughout the text: accent ' is missing on "e" in "Beziau"
=> We have chosen to leave this unchanged, for the following reason. Somewhere around the year 2016, the name that Jean-Yves Beziau uses in his authored/edited publications, changed from "Béziau" (with accent) to "Beziau" (without accent) (we do not know the reason for this sudden change). In the bibliography, we systematically write the name in exactly the same way as it appeared in the referenced publication; cf. for example "Béziau" (with accent) in references 63 and 64 (from the years 2012 and 2013) versus "Beziau" (without accent) in reference 65 (from the year 2016). However, throughout the main text of the paper, we write "Beziau" without accent, because that is the most recent/up-to-date version of his name that the author himself uses in his publications.
