Scott's theorem/axiom and other qualitative to numerical probability representation theorems give conditions under which a qualitative comparative probability ordering, (under $\geq|\leq $ denotes at least as probable than) can be strongly represented by some probability function . strongly represented $(SR)$ $$(SR)\text{where}\,P: F \to [0,1]; A \geq B \iff P(A) \geq P(B)$$
Generally it requires totality $(T)$in the sense of weak order or total pre-order. $$(T) [A \leq B \vee B\leq A]$$ ).
I presume this implies strict weak order? Suppose, one has something stronger then $(T)$, akin to trichotomy(strongly connex
where $\neg A\equiv A\neq B$:and $A>B\equiv B<A$ and exactly one of $$(T1)[A>B \vee A=B \vee A< B]\text{the ' or'is exclusive or}$$ $(1)\vee $$.
Holds ,where $A\neq B$\iff \neg (A>B \vee AB\vee B>A)\iff A\geq B \land B\geq A\iff A\leq B\land B\leq A$$
**$$\text{where} T1 \text{is the corresponding numerical probability representation}.$$ $$(T1)A > B \iff P(A)> P(B)$$.
$$A < B \iff P(A)< P(B)$$. $$ A=B \iff PR(A)=PR(B)$$**'
$(2)'='$ denotes equal comparative probability and is an equivalence relation that is anti-symmetric (in the sense that sense) like a strict total order. As below. strong anti symmetry
$$A=B \iff[ A \leq B \land B\leq A\land iff \neg [A<B] \land \neg [B<A]\iff A\leq B \land B\leq A $$.
$$A>B \iff A\neq B \land \neg (A<B)\iff A\neq B\land \neg B>A\iff A\neq B\land A\geq B$$
$(3)$ $'='$ and $>$ are basic and both are transitive, negatively transitive (and in relation to each other .
$(4)$ $\neg (A=B)\equiv A\neq B$ and $\geq$ is redundant where it is defined as:.
$$B\geq A\equiv \neg (A>B)\equiv A\leq B\equiv \neg B<A)\equiv [B=A\vee B>A]$$:
$$A>B \land B=C \to A>C$$.
complementary for $=$ and $>$and everything if and only if.$>$ is
assymetric. $$A>B\rightarrow \neg (B>A)$$.
$$A>B\rightarrow \neg (A<B)$$
inversion asymmetric.
$$\text {inversion assymetric} A>B\iff B<A$$.
complementary (everythig if and only if ): $$A=B\iff \neg A= \neg B$$.
$$A>B\iff \neg A< \neg B$$.
which is considered stronger then de-finetti axioms as well where it satisfies the stronger (than de finetti) addivity. $$A>B \land C=D\to A\cup C> B\cup D\text{where}\land A \cup C=\emptyset $$ $$A>B \land C>D\to A\cup C> B\cup D$$ $A\cup C> B\cup D$\to \text{at least 2 of} A>B\land A>D ,C>B C>D$$
as $$A=B \land C=D\to A\cup C= A\cup D\text{where both are mutually exclusive) this is strict > if one of the two are not mutually exclusive,} $$ See chapter 5 in (under the QM section)
Krantz, David H.; Luce, R.Duncan; Suppes, Patrick; Tversky, Amos, Foundations of measurement. Vol. I: Additive and polynomial representations, Mineola, NY: Dover Publications (ISBN 0-486-45314-6). xxvi, 584 p. (2007). ZBL1118.91359.
THe authors consider stronger then de-finettisaxiom
$$\text{de finetti} A\leq B \iff A\cup C\leq B\cup C where A\cup C = B\cup C=\emptyset$$ although it the stronger axiom above apparently implied according to Terrence Fine monogragh
Fine, Terrence L., Theories of probability. An examination of foundations, New York-London: Academic Press. XII, 263 p. $ 14.50 (1973). ZBL0275.60006. by
$$ A\cup C= B\cup D\to A=B \land C= D \vee A>B \vee C<D \vee A<B \vee C>D$$
**Is there a name of this property $(A)$ (strongly connex)?
*And is a probability order where $=$, and $>$ are basic that satisfies $(A)$ (trichotomy so defined for a CP order) stronger than a strict weak order or a comparativee probability relation that satisfies the more standard $(T)$totality?** ?**
And is $(T_1)$ a stronger numerical representation than $SR$ above?**
Fishburn, Peter C., The axioms of subjective probability, Stat. Sci. 1, 335-358 (1986). ZBL0604.60004. as $\neg B>A$ at the end of $(2)$
s)
Fishburn, P.C., SSB utility theory and decision-making under uncertainty, Math. Soc. Sci. 8, 253-285 (1984). ZBL0553.90017.
Fishburn, Peter C., Intransitive indifference in preference theory: A survey, Oper. Res. 18, 207-228 (1970). ZBL0195.21404.
Domotor, Zoltan, Qualitative information and entropy structures, Inform. Inference, 148-194 (1970). ZBL0237.94006.
Krantz, David H.; Luce, R.Duncan; Suppes, Patrick; Tversky, Amos, Foundations of measurement. Vol. I: Additive and polynomial representations, New York-London: Academic Press. XXIV,577 p. $ 16.50 (1971). ZBL0232.02040.
Fishburn, Peter C., Finite linear qualitative probability, J. Math. Psychol. 40, No.1, 64-77 (1996). ZBL0851.60002.
Fishburn, P.C., Utility theory for decision making, Publications in Operations Research. No.18; New York etc.: John Wiley and Sons, Inc. XIV, 234 p. (1970). ZBL0213.46202. and it singles out more easily a unique representation in the presence of other conditions which often imply Scott's condition in any case.
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