Kolmogorovian versus Non-Kolmogorovian Probabilities in Contextual Theories

Round 1

Reviewer 1 Report

There are a couple of theories that it would be interesting to consider, because they fall within the framework defined by the author: Mohrhoff and the "manifestation of the world": https://philpapers.org/rec/MOHQMA-3; and lka Event Based QM of Licata-Chiatti: https://www.mdpi.com/2073-8994/11/2/181There are a couple of theories that it would be interesting to consider, because they fall within the framework defined by the author: Mohrhoff and the "manifestation of the world": https://philpapers.org/rec/MOHQMA-3; and lka Event Based QM of Licata-Chiatti: https://www.mdpi.com/2073-8994/11/2/181

Author Response

I thank Reviewer 1 for pointing out the papers by Mohrhoff and by Licata and Chiatti to me. However, in my opinion the views of these authors are only feebly related to mine. Both papers propose a new interpretation of the formalism of quantum mechanics (QM) that is alternative to the standard interpretation, while my results concerning quantum probability hold in every interpretation in which the weak assumptions of the minimal interpretation of QM are accepted (hence, in particular, if the standard interpretation of QM is adopted). Indeed, such results only require that the theoretical notion of microscopic context (μ-context) be added to the theoretical apparatus of standard QM and that some simple membership conditions be satisfied by QM (Section 8). Moreover, I could not find any notion similar to the notion of μ-context in the papers by Mohrhoff and by Licata and Chiatti. Therefore, after introducing some improvements in Section 1, I limited myself to add two footnotes mentioning these papers and briefly commenting on them. 

Reviewer 2 Report

In this paper the author continues his investigation on his epistemic interpretation of quantum probability via contextuality. In previous works, the author already showed in a very convincing way that quantum probabilities can be interpreted as mean conditional statements, and that this allows for their epistemic interpretation. In the present work, the author proposes an even more general approach, this time making no reference to individual entities, showing that everything works well also for minimal interpretations of QM. The author makes extensive connection and comparison with the work of Aerts et al., particularly the notions of universal average, universal measurement and hidden-measurement. This nicely completes the author’s analysis in an interesting way, relating his work to similar but not identical approaches, also distinguishing different typologies of lack of knowledge. Generally speaking, the article is well written and well organized; the different mathematical steps are presented with the clarity and rigor for which the author his know, being one of the recognized experts in the foundations of quantum mechanics and quantum axiomatics. So, I recommend the publication of the article.

I just have a minor point:

When describing Aerts et al. approach as ontic (lines 910-91), I’m not sure the authors would agree with such characterization. Of course, there are different meanings for the terms ‘ontic’ and ‘espistemic’, in relation to probability. If one adopts the very general definition that a probability is epistemic if it refers to a situation of lack of knowledge, then no doubts that in Aerts et al. approach quantum probabilities are epistemic and not ontic. If, on the other hand, ‘ontic’ is used more in the sense of ‘irreducible’, then yes, in that sense Aerts et al. approach refers to quantum probabilities as definitely ontic (the measurement process is like a symmetry breaking process and there is no way, without altering the very measurement, to eliminate them). I think that a clarification of this kind should be included, for clarity (for example in a footnote).

Please consider also the following:

Line 8: “notions” should be “notion”

Line 882: since the notion of “pure state” is used in standard quantum mechanics in relation to vector-states, I would replace “…considering the points within the Bloch sphere as representative of new states, considered as pure rather than mixed, as it would…” with  “…considering the points within the Bloch sphere as representative of genuine new states, rather than statistical mixtures, as it would…”

All the “open quotes” in the paper are done using the bas command in LaTeX, so they appear as “closed quotes”. This needs to be corrected.

Author Response

I thank Reviewer 2 for useful suggestions, which I fully accepted. In particular:

(i) I refined my definition of "ontic" in Section 1 (line 59) and specified that quantum probability is ontic in the sense established by that definition in the Aerts et al. approach (Section 1, line 148, and Appendix, line 911);

(ii) I introduced the replacement suggested by Reviewer 2 in the Appendix (line 882);

(iii) I corrected the typographical errors pointed out by the Reviewer (together with other errors found out by myself).