scholarly journals Inference rules for probability logic

2016 ◽  
Vol 100 (114) ◽  
pp. 77-86 ◽  
Author(s):  
Marija Boricic
Keyword(s):  
Twentieth Century ◽  
Inference Rules ◽  
Probability Logic ◽  
Elimination Rule ◽  
Probability Models ◽  
Intended Meaning ◽  
Deductive Systems ◽  
Traditional Proof

Gentzen?s and Prawitz?s approach to deductive systems, and Carnap?s and Popper?s treatment of probability in logic were two fruitful ideas of logic in the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized by means of inference rules, we introduce a system of inference rules based on the traditional proof-theoretic principles enabling to work with each form of probabilized propositional formulae. Namely, for each propositional connective, we define at least one introduction and one elimination rule, over the formulae of the form A[a,b] with the intended meaning that ?the probability c of truthfulness of a sentence A belongs to the interval [a,b] ?[0,1]?. It is shown that our system is sound and complete with respect to the Carnap-Poper-type probability models.

scholarly journals Probability logic: A model-theoretic perspective

2020 ◽  
Author(s):  
M Pourmahdian ◽  
R Zoghifard
Keyword(s):  
Characterization Theorem ◽  
Expressive Power ◽  
Probability Logic ◽  
Theoretic Analysis ◽  
Compactness Property ◽  
Probability Models ◽  
Finitely Additive Probability ◽  
Additive Probability ◽  
Basic Probability ◽  
Abstract Logics

Abstract This paper provides some model-theoretic analysis for probability (modal) logic ($PL$). It is known that this logic does not enjoy the compactness property. However, by passing into the sublogic of $PL$, namely basic probability logic ($BPL$), it is shown that this logic satisfies the compactness property. Furthermore, by drawing some special attention to some essential model-theoretic properties of $PL$, a version of Lindström characterization theorem is investigated. In fact, it is verified that probability logic has the maximal expressive power among those abstract logics extending $PL$ and satisfying both the filtration and disjoint unions properties. Finally, by alternating the semantics to the finitely additive probability models ($\mathcal{F}\mathcal{P}\mathcal{M}$) and introducing positive sublogic of $PL$ including $BPL$, it is proved that this sublogic possesses the compactness property with respect to $\mathcal{F}\mathcal{P}\mathcal{M}$.

Representation of symmetric probability models

1969 ◽  
Vol 34 (2) ◽  
pp. 183-193 ◽  
Author(s):  
Peter H. Krauss
Keyword(s):  
Mathematical Theory ◽  
Preceding Paper ◽  
Probability Logic ◽  
Probability Models ◽  
Additional Material ◽  
First Order ◽  
Order Probability ◽  
Joint Publication

This paper is a sequel to the joint publication of Scott and Krauss [8] in which the first aspects of a mathematical theory are developed which might be called “First Order Probability Logic”. No attempt will be made to present this additional material in a self-contained form. We will use the same notation and terminology as introduced and explained in Scott and Krauss [8], and we will frequently refer to the theorems stated and proved in the preceding paper.

scholarly journals A logic with higher order conditional probabilities

2007 ◽  
pp. 141-154 ◽  
Author(s):  
Zoran Ognjanovic ◽  
Nebojsa Ikodinovic
Keyword(s):  
Conditional Probability ◽  
Probability Space ◽  
Possible Worlds ◽  
Axiomatic System ◽  
Probability Logic ◽  
Conditional Probabilities ◽  
Unconditional Probability ◽  
Possible World ◽  
Intended Meaning ◽  
Probability Operators

We investigate probability logic with the conditional probability operators This logic, denoted LCP, allows making statements such as: P?s?, CP?s(? | ?) CP?0(? | ?) with the intended meaning "the probability of ? is at least s" "the conditional probability of ? given ? is at least s", "the conditional probability of ? given ? at most 0". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: P(? | ?) = P(?^?)/P(?), P(?) > 0, by the (unconditional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.

scholarly journals A first-order conditional probability logic with iterations

2013 ◽  
Vol 93 (107) ◽  
pp. 19-27 ◽  
Author(s):  
Milos Milosevic ◽  
Zoran Ognjanovic
Keyword(s):  
Conditional Probability ◽  
Completeness Theorem ◽  
Axiomatic System ◽  
Probability Logic ◽  
Intended Meaning ◽  
First Order

We investigate a first-order conditional probability logic with equality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted LFPOIC=, allows making statements such as: CP?s(?, ?), and CP?s(?, ?), with the intended meaning that the conditional probability of ? given ? is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven.

scholarly journals Introducing Identity

2021 ◽  
Author(s):  
Owen Griffiths ◽  
Arif Ahmed
Keyword(s):  
Standard Approach ◽  
Logical Constant ◽  
Inference Rules ◽  
Elimination Rule ◽  
Leibniz’S Law ◽  
Logical Constants ◽  
Introduction Rule

AbstractThe best-known syntactic account of the logical constants is inferentialism . Following Wittgenstein’s thought that meaning is use, inferentialists argue that meanings of expressions are given by introduction and elimination rules. This is especially plausible for the logical constants, where standard presentations divide inference rules in just this way. But not just any rules will do, as we’ve learnt from Prior’s famous example of tonk, and the usual extra constraint is harmony. Where does this leave identity? It’s usually taken as a logical constant but it doesn’t seem harmonious: standardly, the introduction rule (reflexivity) only concerns a subset of the formulas canvassed by the elimination rule (Leibniz’s law). In response, Read [5, 8] and Klev [3] amend the standard approach. We argue that both attempts fail, in part because of a misconception regarding inferentialism and identity that we aim to identify and clear up.

Probability logic in the twentieth century

1991 ◽  
Vol 12 (1) ◽  
pp. 71-110 ◽  
Author(s):  
Theodore Hailperin
Keyword(s):  
Twentieth Century ◽  
Probability Logic

scholarly journals A propositional p-adic probability logic

2010 ◽  
Vol 87 (101) ◽  
pp. 75-83 ◽  
Author(s):  
Milos Milosevic
Keyword(s):  
Axiom System ◽  
Propositional Formula ◽  
Kripke Models ◽  
Probability Logic ◽  
Intended Meaning ◽  
Logical Language

We present the p-adic probability logic LpPP based on the paper [5] by A. Khrennikov et al. The logical language contains formulas such as P=s(?) with the intended meaning 'the probability of ? is equal to s', where ? is a propositional formula. We introduce a class of Kripke-like models that combine properties of the usual Kripke models and finitely additive p-adic probabilities. We propose an infinitary axiom system and prove that it is sound and strongly complete with respect to the considered class of models. In the paper the terms finitary and infinitary concern the meta language only, i.e., the logical language is countable, formulas are finite, while only proofs are allowed to be infinite. We analyze decidability of LpPP and provide a procedure which decides satisfiability of a given probability formula.

Some Polish Composers of To-Day

Tempo ◽  
1948 ◽  
pp. 25-28
Author(s):  
Andrzej Panufnik
Keyword(s):  
Twentieth Century ◽  
Fertile Soil ◽  
Early Twentieth Century ◽  
Progressive Group ◽  
Karol Szymanowski

It is ten years since KAROL SZYMANOWSKI died at fifty-four. He was the most prominent representative of the “radical progressive” group of early twentieth century composers, which we call “Young Poland.” In their manysided and pioneering efforts they prepared the fertile soil on which Poland's present day's music thrives.

V1214: Edward Loughborough Keyes: An Early Twentieth Century Leader In Urology

2004 ◽  
Vol 171 (4S) ◽  
pp. 320-320
Author(s):  
Peter J. Stahl ◽  
E. Darracott Vaughan ◽  
Edward S. Belt ◽  
David A. Bloom ◽  
Ann Arbor
Keyword(s):  
Twentieth Century ◽  
Early Twentieth Century
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