A Meta-Logical Framework for the Equivalence of Syntactic and Semantic Theories

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper starts with an exhaustive and very informative introduction into the discussion of the question "What is a Scientific Theory?" within philosophy. 

I'm fully convinced that the concept of institution is quite appropriate to attack this question and to give a very general formal proof of the equivalence of the liberal syntactic view and the liberal semantic view of theories.  
It is, however, very disappointing to see how sloppy and even inaccurate this idea has been worked out.

I would gladly accept the paper for publication in  Philosophies but only after a major revision taking into account the corrections, criticisms and proposals below. (See the attached PDF.)

One of my concerns is that we must spend much more effort to present, explain and propagate the idea of the paper to philosophers. The minimal requirement is, that the presentation should be consistent and correct! Especially, throughout the paper you should always use the same notation for the same thing!!

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

We feel very grateful for your precise and helpful comments on our manuscript; as you remarked, the draft paper was, at traits, not technically sound; your report helped us improving our manuscript. As you can see reading the revised draft, all your remarks have been addressed, except the last one (as explained below); in the following we comment only those points that need an actual reply.

- p. 3, l. 94 This whole part has been removed, following the suggestion of reviewer#2, but yes, we meant what you are suggesting.

- p. 12, Def. 3 We came to the conclusion that it is probably better to define what an opposite category is. This is now done in footnote 17, showing why, in Figure 1, the two functors have opposite directions.

- p. 15, l. 574 You are perfectly right and we agree with your suggestion; the paper has been modified accordingly.

- p. 15, l. Def. 5 “Just to be sure: For isomorphisms in Th the underlying signature morphisms are isomorphisms in Sign right?” Yes, that is right. We made both the suggested modifications.

- p. 16, l. 630 Thank you again for this comment. Yes, by ψ we wrongly meant Mod(ψ)^{−1}, as it is now specified in the manuscript.

- p. 19, l. Def. 9 The reason why, in the manuscript, definition 9 is not symmetric is that it has only been proven that direction (from syntax to semnatics), while there is no proof of the opposite direction. Hence, we prefer to keep the definition as it is.

Reviewer 2 Report

Comments and Suggestions for Authors

Summary
The article introduces a new formal framework for the rational reconstruction of scientific theories. The authors propose using the concept of institutions from category theory combining a syntactic with a semantic specification of a theory. The syntactic representation consists of a signature and a set of sentences E in terms of this signature, while the semantic specification comprises the same signature and a set of models M. The interrelation is established by a satisfaction relation, which requires that every model from M satisfies the conjunction of sentences of E.
Using this framework, the authors provide a rigorous proof of the equivalence of the syntactic and the semantic representation which they associate with the syntactic and the semantic view on theories in their liberal forms.

Scope
The manuscript deals with a topic from the philosophy of science and thus fits perfectly into the scope of the journal. Despite relying on a formal framework, it is accessible also to non-specialists in formal philosophy of science, since all formal concepts used in the paper are introduced and well explained.

Novelty
The proposed framework differs from (the few) other approaches that combine the syntactic and the semantic view on theories (e.g., E. Scheibe, H. Andreas) in two regards:
Firstly, while the others attempted to combine syntactic and semantic elements into an integral view, the authors' concept keeps the syntactic and semantic reconstruction separate. In this respect it resembles Muller's exuberant view on theories (F. A. Muller, Structures for Everyone, section IV.3), which has only been sketched and not further developped.
Secondly, the previous hybrid views make use of set theory and set-theoretical structures, while this paper draws on category theory. The authors cite an article which applies the concept of institutions to semantic and syntactic theories in computer science, but to my knowledge this manuscript is the first application of this concept in philosophy of science.

The second novelty is the proof of the equivalence of the formal frameworks of the syntactic and semantic view. Even though it seems to be taken for granted nowadays that (the generic versions of) both views are equivalent, the rigorous proof is a new contribution.

Significance
Both the proof of the equivalence of the syntactic and the semantic view, as well as the alternative framework for rational reconstruction of theories have a high significance for philosphers of science. The debate on the alleged superiority of the semantic view over the syntactic view received renewed interest during the last decade and this article is surely a valuable contribution.

Quality and scientific soundness
The manuscript fully meets the quality standards for scientific publications. Its argumentation is clear and sound. It is well structured, all technical terms are precisely defined and used consistently throughout the paper. It includes and discusses the relevant literature in an appropriate manner, and the writing is absolutely adequate for a philosophical article.

Verdict
Acceptance after Minor Revisions

critical remarks:

1) It is a common and good practice to begin a philosophical paper with an introductory section which states the main claim(s), motivates its significance, and gives an overview on the structure of the article. The abstract to this paper contains most of this information. However, in my view there is tension regarding the primary aim of the paper. While title and conclusion indicate that it is the introduction of the new framework, section 1 (which is called "Introduction" but is in fact directly starting with the content) centers on the proof of equivalence. Clarifying the main objective of the article would probably require a brief introductory section.

I think it is important to decide for one primary goal because depending on this choice, the discussion should address different aspects:
If the introduction of the novel framework is the main objective, I would expect a short comparison to other formal framework for rational reconstructions of theories. What are the advantages of this approach compared to the alternatives? E.g., is it useful for analyzing intertheoretical relations? Some forms are mentioned in section 4, but the most interesting kinds like limiting case reduction are not addressed and it is not obvious how those could be treated in this framework. Whereas, if the proof is the main aspect, the discussion should briefly mention how the results relate to other formal comparisons between syntactic and semantic view such as (Lutz, 2013) and (Muller, 2011, Reflections on the revolution at Stanford).

2) In the abstract, the authors claim that the article contains "an extensive presentation of the history of the syntactic-semantic debate" (lines 10/11). On the one hand, a truly extensive presentation would require more than just nine pages and include a discussion of a wider range of literature than that cited in this article. On the other hand, I think that an extensive discussion is not necessary for the purposes of this article, and that section 1 already goes into too many details of the history of the syntactic and the semantic view and the debate on the superiority of the latter. I think this is not necessary for the simple reason that the reminder of the paper only deals with formal aspects of both views. If the revision requires to shorten parts of the text, I think section 1 can be tailored more closely to the objective of the paper.

3) On footnote 21: I would expect at least a short comment on how the authors plan to deal with the discrepancy between their formalization and the fact that the entirety of models is a class and not a set in many semantic conceptions?  Would they (a) change the semantic view such that only set-theoretical predicates are allowed which give rise to a set of models or (b) is there an easy modification to institutions such that M becomes a class? But the proof via Galois connections depends on M being a set ...

remarks on minor issues:
1) I do not understand the list in lines 39-42. Usually the received view is characterized by a (i) vocabulary (divided into logical, theoretical and observation terms), (ii) set of sentences (including correspondence rules), (iii) set of axioms, and (iv) rules of inference. I do not understand why the authors consider the vocabulary of primitive and defined terms and vocabulary of extra-logical terms as two distinct basic constituents, while set of sentences asserted by T, axioms and rules of inference are just one. But as mentioned before, I think pages 2-4 could be condensed to cover only the aspects that are relevant for the later discussion of the formal reconstruction.
2) Some footnotes are no proper sentences (fn. 3, 19, 20) and should be transformed into full sentences.
3) Regarding Suppes' claim that the mathematical representations of matrix and wave mechanics are isomorphic (lines 115-119): Interestingly, the very same example is discussed by Halvorson in his 2012 paper where he concludes that "matrix algebra is obviously not isomorphic to a space of wave functions; hence, a simpleminded isomorphism criterion would entail that these theories are inequivalent." Since Halvorson's objection is discussed later in this paper, I think there should be at least a footnote to warn readers that Suppes' view is not the final word on this matter.
4) Footnote 12 is missing (line 318).
6) There are some type setting errors on the last pages. I think "E**" should be E^{**} (line 548), and there are some missing spaces in (lines 578, 665, 672, 673). Also several instances of the symbols "<"/">" are used instead of the proper latex commands "\langle"/"\rangle".
7) The discussion that followed Halvorson's 2012 paper shows that it does matter how the isomorphism relation is defined. Therefore, the authors should motivate and comment their choice.

Author Response

Dear Reviewer,

Thank you very much for you kind and clear review. We believe that your comments helped us improving the quality of our manuscript and we feel grateful for that. In what follows, you will find a point by point reply to your remarks.

1) We clarified the main objective of the paper, namely the proof of equivalence, by editing the introduction, which now more explicitly introduces the paper, and the conclusion; the paper title has also been modified. Consequently, section 4 now contains a new paragraph relating our work with that of Lutz.

2) We agree with your remark. The subsections on the syntactic and semantic approaches have now been reduced accordingly.

3) There is no need to modify the institution theory: in its original formalization, both E and E^* were intended as classes rather than sets; we opted, as many do, to consider them sets for the sake of simplicity, but now they are considered as classes throughout the manuscript.

Minor issues

1) You are perfectly right, thank you for commenting this. The manuscript has been modified following your suggestion.

2) Corrected

3) Footnote 5 now specifies that the issue is indeed quite debated.

4) Fixed

5) Corrected

6) The last paragraph of subsection 1.3 now more explicitly relates our equivalence notion with H- and M- isomorphisms.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors took into account all my corrections, criticisms and proposals concerning the first version and I gladly accept the paper for publication in  Philosophies after a minor revision taking into account the few new corrections below.

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

thank you for having spotted the pointed typos. All of them have now been fixed.