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‘Probability dynamics’ (PD) is a second-order probabilistic theory in which probability distribution d X = (P(X 1), . . . , P(X m )) on partition U Xm of sample space Ω is weighted by ‘credence’ (c) ranging from −∞ to +∞. c is the relative degree of certainty of d X in ‘α-evidence’ α X=[c; d X] on U Xm . It is shown that higher-order probabilities cannot provide a theory of PD. PD applies to both subjectivist and frequentist theories. ‘Straight PD’ (SPD) produces associative and commutative mergers of evidence, in which evidences of positive credence are mutually reinforcing. ‘Offsetting PD’ (OPD) sets off conflicting evidences against each other. Subjectivist PD is a quantified second-order logic of action. Frequentist PD relates to descriptions of physical states of affairs. Acceptance of evidence α X1 = [c 1; d X1] at t 1 updates α X0 = [c 0; d X0] at t 0 into SPD-merger (alpha_0^{X}oplus alpha_1^X) or OPD-merger (alpha_0^Xdiamond alpha_1^X). Given ‘co-evidence’ (E_0^{XY} = [c_0;P_0(XY),P_0(Xoverline{Y}),P_0(overline{X}Y),P_0(overline{XY})]) at t 0 < t 1, ‘indirect’ PD accepts evidence (widetilde{alpha}_1^{Y} = [widetilde{c}_1; widetilde{d}_1^{Y}]) at t 1 and produces support (hat{alpha}_1^{X}) for update α X10 = [c10;d X10 α X0 such that (alpha_{10}^X = alpha_{0}^Xoplus hat{alpha}_{10}^X) in SPD and (alpha_{10}^X= alpha_{0}^Xdiamond hat{alpha}_{10}^X) in OPD. For binary X and Y, with α X0 = [c 0; P 0(X)] at t 0 (short-hand for ([c_0;P_0(X),P_0(overline{X});X,overline{X}])) and (widetilde{alpha}_1^{Y}= [widetilde {c}_1;widetilde{P}_1(Y)]) the accepted evidence, the support is (hat{alpha}_1^{X}=[hat{c}_1; widetilde{P}_1(Y)P_0(X|Y)+widetilde{P}_1(overline{Y})P_0(X|overline{Y})]); (hat{c}_1=| ho_0(X, Y)|c_0widetilde{c}_1/[c_0+(1-| ho_0(X, Y)|)widetilde{c}_1]); where ρ0(X, Y) is the correlation coefficient of X and Y, and update α X10 of α X0 is (P_{10}(X)=[c_0P_0(X)+hat{c}_1hat{P}_1(X)]/ (c_0+hat{c}_1); quad c_{10} = lambda(c_0+hat{c}_1)) with ‘accord’ λ = 1 in SPD and ( lambda=1-2|P_0(X)-hat{P}_1(X)|sqrt{c_0hat{c}_1}/(c_0 + hat{c}_1)) in OPD.

DOI: 10.1007/s11229-005-0197-9

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