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Faithful and invariant conditional probability in Łukasiewicz logic. (English) Zbl 1163.03016
Makinson, David (ed.) et al., Towards mathematical philosophy. Papers from the Studia Logica conference Trends in Logic IV, Toruń, Poland, September 1–4, 2006. Berlin: Springer (ISBN 978-1-4020-9083-7/hbk; 978-1-4020-9084-4/e-book). Trends in Logic–Studia Logica Library 28, 213-232 (2009).
A version of conditional probability in Łukasiewicz infinite-valued propositional logic $${\mathcal L}$$ is introduced and discussed, thus continuing and expanding the previous work of author in [D. Mundici, Int. J. Approx. Reasoning 43, No. 3, 223–240 (2006; Zbl 1123.03011)]. To each consistent finite set $$\Theta$$ of conditions (one or more formulas in $${\mathcal L})$$, a mapping $$P_\Theta:{\mathcal L}\to Q\cap[0,1]$$ assigns rational numbers to formulas. Basic properties of the mapping $$\Theta\to P_\Theta$$ are the faithfulness, additivity and invariance. Moreover, if $$\Theta=\{\theta\}$$ for a tautology $$\theta(x_1,\dots,x_n)$$, then for any formula $$\psi=\psi(x_1,\dots,x_n)$$, the (unconditional) probability $$P_\Theta(\psi)$$ is the Lebesgue integral over the $$n$$-cube of the McNaughton function represented by $$\psi$$. In final remarks, some open problems for the future work are indicated.
For the entire collection see [Zbl 1153.03004].

##### MSC:
 03B50 Many-valued logic 06D35 MV-algebras 60A05 Axioms; other general questions in probability
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