Is there a distinction between a qualitative /comparative probabilistic structure, ie which admits of a
1. strong numerical probabilistic representation and said representation is unique (given probabilism there is only one set of probability values that the ;and events represented can have)
2. A strong probabilistic representation that is unique up to linear or affine transformations, as are constructible using Luce's ratio scales.
Is this supposed to mean ' that the values of all probabilistic representations are merely linear transformations of each other (in which case, its weaker than 1)', or is it say 'that all metrical real valued representations are (nearly) uniquely probabilistic, they are all linear transformations'
That is the distinction between a probabilistic representation being the only and only possible probabilistic presentation (1), and
(2) the representation is uniquely or nearly uniquely probabilistic
(2) Does not talks about not 'the values of the probabilities being unique given 'probabilism or givenan additive normalized representation which is what (1) is saying, but that ,fact that 'the representation is probabilistic' is unique,
ie there is only one real value representation and it is probabilistic, or only one set of representations, one or more being probabilistic, and all of the remainder being linear transformations of them (and if it satisfies (1), then there is a set, only one of members of which is probabilistic, as the probability representation, given the ordering, is unique, and the remainder are linear tranformations of that very same unique set of values and whose axioms, if you will are also just linear transformations of the probability axioms
(which is stronger than 1, or at least with regard to justifying probabilism, and strictly stronger if it satisfies (1) as well, that there is only one probabilistic representation)
Initially I thought that the latter (2) were weaker uniqueness results, although I believe now that I may be mistaken. I believe the first (1)just says there is only one set of values consistent with probabilism (the axioms of probabilism and your ordering), so that on the assumption of those axioms, you get unique values.
I presume the latter (unique up to affine transformations) is something stronger, halfway between a standard unique representation and Cox's theorem, which is that the one and only probabilistic representation is unique (any two probabilistic representations, by which I mean representations that satisfy the numerical axioms of the probability calculus, are identical, they give the same probability values to all propositions) plus, the representation is almost uniquely probabilistic; that is all of the other numerical representations, also either have a structure that is nearly additive, and or nearly, give the same probability values.
That is all possible real valued numerical representations are either a additive probability measures (of which there is only one), or some linear transformation of it.
This would be a stronger justification if you will, (
(A)for probabilism, as it also not only restricts the number and type of possible non probabilistic representations, but requires them (the non-probabilistic representations) that to be quasi-probabilistic, similar to probabilistic representations (linear transformations of probability functions)
(B) For the uniqueness result. For it would provide a stronger justification derivation for the uniqueness of the values given probabilism;
(A)for not only is there only one set of values consistent with probabilism if (1) holds and correct if i am wrong as to whether (1) s generally a consequence of (2) where (1) is that there is only one probability function uniquely given as for example we see in gleasons result, there is only one values each event in the space can have consistent with probabilism)
but also (B) all of the other representations, have similar values, or, are closely related (via a linear transformation) to a probabilistic function; so ifone requires that the representation be probabilistic for the values to be unique, this would restrict the number of possible ways that the representation may not be probabilistic, and thus may not give those unique values (which are unique given probabilism, if (1) is proven in addition or is a consequence of (2)).
So by there existing a unique strong representation I mean that the qualtiative ordering is such that 1.A>=B iff P(A)>= P(B)
Where I used >= for great or equal to, which is to say that if the qualitative ordering is such that A is seen as qualitatively (equal to, or more likely than B), then the numerical probabilistic representation will likewise assign a greater or equal number to A then it does for B. THis also ensures that the strictly qualitative equalities, 'greater thans' ,'less thans' are also represented in the numerical representation (an event A is more likely then B iff the numerical addivite normalized probabilistic representation assigns a higher number to A than it does to B)
Which for a strong representation requires that for every event X in the field of F events.
1. Tautology (sample space) > contradiction
2.X >=
3. X>=y or Y>=X
4. scotts axiom
(satisfies Scotts axiom, it is a complete, ordering, satisfying qualitative additivity, monotonicity, transitivity, cancellation, non triviality', and or an archimedean condition, or monotone continuity, tightness/scotts infinite axiom or Luce structured sequence axiom, in the case of infinite space) and or fine-ness, scotts infinite axiom) and is unique (for example is fine and tight, ; and a qualitative /comparative probabilistic structure whose representation is unique up to (except for)inear Theories of Probability
: An Examination of Foundations: Terrence L. Fine unique up to linear/affine transformations:
Fine (1973)
See the corrollary and theorem mentioned by Terrence Fine in the third last article from page 26 of his book 1973 theories of probability (page 7 of the attached article) concerning a unique (up to a multiplicative constant finitely additive P). The last article adds some context to qualitative probability ( the second article by Narens talks about the issue of uniqueness and what is required to show that a representation must be probabilistic not just that there exists a unique probabilistic representation (or that there exists a probability representation of your ordering (representation part) and that is the only probabilistic representation (uniqueness part). The last article and this by the same author (Colin Howson) http://www.sciencedirect.com/science/article/pii/S1570868307000857; looks at the distinction between these scale variant ratio sale derivations (as in (2) ) and coxs theorem which purports to be a total derivation of probabilism.
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