I read the paper and I have a simple question: Is DT a science and as a related question Is Psychology a science? They are preliminary questions that determine all the discussion.
Let me start frome the beginning. In 1921 Keynes published A Treatise on Probability (TP), it is the well known book, always quoted by DTs, but he devoted the first part only to epistemology and the remain four parts were about math and statistics. TP received many reviews, mostly from very famous philosophers (Russell), statisticians, mathematicians and economists (Edgeworth and Ramsey). If the TP was such a relevant book, what made it invisible to science? Aldrich (2008) and Scheneider (2021) gave some reasons of TP disappearance: initial reviews ranged from dismissive to extremely positive; extensive logical notation, instability and multi definitions of symbols; no worked examples and not surprisingly the tone adopted by Keynes
Ramsey criticized Keyens probability theory from an epistemic point of view, but even if too oftren forgetted by a math point of view, also: the crcuial criticism is about isomorphism between beliefs and numerical probabilities.
Keynes and Ramsey have the same math. framework, they act in a Boolean algebra, but as you know in such aframework Ramsey's criticism hold.
Epistemologically, Keynes's position can be supplemented by Carnap, Kyburg, Nagel, Levi etc and some weak notion of consitency can be defined. Math the Ramsey's criticism cannot be solved in a boolean algebra.
The '20 were very important for all disciplines: math, phys, phylos. :Einstein-Podolsky-Rosen vs Bohr's controversy, Intuitionism vs Pragmatism, Hilbert vs Brower and in DT Keynes vs Ramsey.
What appened in Phis, Phil- and math? They faced Quantum Revolution and uncertainty was introduced in science.
In the seminal paper Birkhoff and Von Newmann (1933) showed that quantum mechanics cannot be reduced to Standard Mechanics because of: non commutativity and failure of set complementation. So doing they showed that the framework to represent uncertainty was a non-boolean lattice that is a non distributive and orthocomplemented pseudo-boolean lattice.
What appened in DT? We continue to assume a boolean algebra or a boolean lattice and assuming set complementation. In such a farmework Upper and lower probabilities or Multiple priors or core are not different from what Keynes did in TP, when facing the algebra detrermined by x and y given h, he said "xy∣h lies between x∣h and (x∣h + y∣h-1). We thus have limits for xy∣h, close enough sometimes to be useful, which are available whether or not x∣h and y∣h are independent arguments" (TP, 162). A parametric solution!
We have a lot of theories about imprecise probabilities, but they do not face the complementation problem explicitly and so they refer to psychogogy. There are some relevant exceptions that move to non boolean framework: Narens and Quiggin et al. introduced Heyting algebra and relative pseudo-complementation, but I think that the relative pseudo complemnt is a very particular notion.
I think that we could follow B-vN (1933) and introduvce a weak notion of complementation, non distributivity and incompatibility as a version of non commutativity. Of course we move to a non boolean lattice, but in such a framework all is simple and clear and psychology is not necessary.
About the representation theorem, we have to face Economists requests, it is interesting that the question was considered by von Newmann and solved in an afternoon, because of Morgenstern explicit demand for the pleasure of economists, in their seminal book.
Psychology is relevant for DT? As a straring point, perhaps, but I have another question Hilbert is anough for a theory or we are looking for the Truth?
All the best dear friends
Marcello
