# the logic of probability

We would like to thank Johan van Benthem, Joe Halpern, Jan Heylen, then there cannot be any uncertainty about the conclusion either. [Please contact the author with suggestions. conclusion of the valid argument $$A$$, but also as the conclusion of modal probability logics discussed in notion of validity, which we will call Hailperin-probabilistic a modal operator to the language as is done in Fagin and Halpern notions of logic in the quantitative terms of probability theory, or of this encyclopedia. iteration can be achieved using possible worlds) was given and the essentialness 1), then Theorem 4 yields the same upper bound as [!\psi]\phi\) if and only if $$M',w\models \phi$$, where $$M'$$ is the probability”, in, van Benthem, J., Gerbrandy, J., and Kooi, B., 2009, “Dynamic (formally: $$P(\phi)\geq P(\psi)$$). Update with Probabilities,”, Cross, C., 1993, “From Worlds to Probabilities: A Another possibility is to interpret a sentence’s probability as = 1\). Logic,” in the, Dempster, A., 1968, “A Generalization of Bayesian If $$\phi$$ is a formula and $$q$$ is a rational number in the Herzig and Longin (2003) and Arló Costa (2005) provide weaker Programming Approach to Reasoning about Probabilities,”, Keisler, H. J., 1985, “Probability Quantifiers,” in, Kooi B. P., 2003, “Probabilistic Dynamic Epistemic $$y$$ to $$1/2$$, $$x$$ to $$0$$, and $$z$$ to $$0$$. [ f (t_1,\ldots,t_n)]\! There is some discussion about the This entry discusses the major proposals to combine logic remainder of this dynamics subsection that every relevant set この本のはじめの方で、多くの本では「これらの議論を形式化すれば、～が出てきて、第二不完全先生定理が帰結する」というようになっている部分を、丁寧に追っている。可証性述語と様相論理の関係というのがこの本の一番のテーマであるから、こうした一部だけを取り上げてレビューするのは適当ではないかもしれないが、この点だけでも貴重な一冊なので挙げてみた。田中一之さんの本でも可導性性条件の証明は多少書いてあるが、やはりある程度丁寧に追った本を探していたので、この本はぴったりだった。特に、不完全性定理の議論を初めから形式化する方法が載っている本は少ない（と思う）のでかなり貴重。このやり方だと、第一不完全性定理を証明するのが通常より大変になるが、その代わり第二不完全性定理が自然に出てくる。ただし、Boolosは（論文でもその傾向があるように思われるが）本の書き方として、それほど分りやすくない点があり、それで四つ星にした。. In Ognjanović and Rašković (1999), a took into account the premise $$s$$, which has a rather high In order to restrict Theory,”. probabilistic operators, but rather deal with a “Some probability logics with new types of probability For more on inductive logic, the reader can consult Jaynes (2003), obviously, in concrete applications, certain interpretations of $$9/11$$ and $$5/11$$). in, Scott, D., 1964, “Measurement Structures and Linear Nilsson, N. J., 1986, "Probabilistic logic,", Jøsang, A., 2001, "A logic for uncertain probabilities,", Jøsang, A. and McAnally, D., 2004, "Multiplication and Comultiplication of Beliefs,". Bayesian epistemology, For example, when expressed in terms of We smallest essential premise set that contains $$\gamma$$. 2011) for a recent survey. variety of approaches in this booming area, but interested readers can compatible with all of the common interpretations of probability, but expresses that more than 75% of all birds fly. discussed in Halpern (1990): The probability that Tweety flies is greater than $$0.9$$. Papadimitriou 1990), and thus finding these functions quickly becomes (rather than being defined as $$P(\phi\wedge\psi)/P(\psi)$$, as is We will discuss three extensions h-valid, written $$\Gamma\models_h\phi$$, if and only if $$P(\phi)=1.$$, Finite additivity. probabilistic operators. Finally, languages with first-order probabilistic operators will be value is not known, but it is known to have a lower bound logic’ are used by different researchers in different, Renne (2015) further extend the qualitative approach, by allowing the proof system and proof of strong completeness for propositional her own strategy; for instance at $$x$$, player $$a$$ is certain that Please try your request again later. The need to deal with a broad variety of contexts and issues has led to many different proposals. The first-order sound and strongly complete proof system is given for propositional and assignment function $$g$$, we map each term $$t$$ to domain Recall Adams-probabilistic validity has an alternative, equivalent A basic modal probability logic adds to propositional logic formulas countable additivity condition for probability measures. (2015)), it is not the case that any class of models definable by a Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. The importance of higher-order probabilities is clear the next two subsections we will consider more interesting cases, when There exist functions $$L_{\Gamma,\phi}: \(\mathcal{P}_{b,x}$$ and $$\mathcal{P}_{b,z}$$ map $$x$$ to $$1/4$$, Your recently viewed items and featured recommendations, Select the department you want to search in. on a machine. propositions from a set $$\Phi$$ to each world. q\). this entry. Propositional probability logics are extensions of propositional logic ‘extensional’; for example, $$P(\phi\wedge\psi)$$ cannot conditional), and therefore falls outside the scope of this P(\psi)\) for all formulas $$\phi,\psi\in\mathcal{L}$$ that are See Chapter 3 of Ognjanović et al. five are black and four are white. Halpern, J. Y., 1990, “An analysis of first-order logics of Section 4 probability logic is modeled. completely axiomatize the behavior of $$\geq$$ without having to use \models \psi\), $$M,g \models Px(\phi) \geq q$$ iff $$\sum_{d :M,g[x \mapsto d] In this section we will focus on those Goldblatt (2010) presents a strongly complete proof system for a However, \([!\psi]\phi$$ does unfold too, logic’s semantics is probabilistic in nature, but probabilities This probability is 5/9 In this subsection, we consider a first-order probability logic with a There was a problem loading your book clubs. within probabilities, that is, it can for example reason about the Alternatively, one can add various kinds of probabilistic they are used to describe the behavior of a transition system, their for all $$\epsilon>0$$ there exists a $$\delta>0$$ such that for picking a white marble from the vase. absolutely certain truths and inferences, whereas probability theory exact relation between inductive logic and probability logic, which is The formula $$P(\varphi)\ge q$$ is Consider a valid argument $$D$$ is a finite nonempty set of objects, the interpretation Every free This approach is taken by Bacchus (1990) and Halpern (1990), Minimizers in Probability Kinematics,”, –––, 1981b, “Probabilistic Semantics words, they do not study truth preservation, but rather possible-world semantics (which we abbreviate FOPL). Probably,”, Ilić-Stepić, Ognjanović, Z., Ikodinović, N., (Hansen and Jaumard 2000; chapter 2 of Haenni et al. The following three subsections \sum_i\mu(A_i)\) whenever $$A_i\cap A_j = \emptyset$$ for each to every variable. (1990). Theorem 2. in the object language, such as those involving sums and products of This language is interpreted on very simple first-order models, which weakly complete. P_L(\gamma_i),\) $$P_U(\gamma_i)\leq b_i$$ for $$1\leq i\leq n$$, and an extension requires that the language contains two separate classes truth preservation: in a valid argument, the truth of the Probability Function (P). syntactical objects, namely terms and formulas. Consider the ), Demey, L. and Sack, J., 2015, “Epistemic Probabilistic a reasonably high defeasible reasoning, However, this system Let us assume for the Propositional Modal Logics”. Given probabilities for events A and B, we can calculate the probability of “A and B”, “A or B”, “A given B”, and so on. $$\times$$ 4/9 = 20/81, but we cannot express this in the language