Probabilistic Extensions of Various Logical Systems

Preface
Contents
List of Contributors
General notations and conventions
Chapter 1 Logics with Probability Operators
	1.1 Introduction
	1.2 The first order probability logic LFOP1 and some related logics
		1.2.1 Syntax
		1.2.2 Semantics
		1.2.3 Axiom system AxLFOP1
		1.2.4 Soundness and completeness
		1.2.5 Versions of the logic LFOP1
	1.3 Related Work
	References
Chapter 2 Formalization of Probabilities with Non-linearly Ordered Ranges
	2.1 Introduction
	2.2 Logics for reasoning about p-adic valued probabilities
		2.2.1 p-adic numbers
		2.2.2 The logic LDQ
		2.2.3 Modeling the double-slit experiment
		2.2.4 Conditional p-adic probability logics
	2.3 Complex valued probability logics
		2.3.1 The logic LCOMPB
	2.4 Measure logic
		2.4.1 The logic LPG
		2.4.2 Extensions of the logic LPG
	References
Chapter 3 Probabilistic branching time logic
	3.1 Introduction
	3.2 Temporal logics
		3.2.1 Linear temporal logic
		3.2.2 Branching time logic
	3.3 Probabilistic branching time logic pBTL
		3.3.1 Syntax and semantics
		3.3.2 Axiomatization
		3.3.3 Completeness
	3.4 Probabilistic reasoning about temporal information
		3.4.1 Syntax and semantics
		3.4.2 The axiomatization of PLLTL
		3.4.3 Completeness of Ax(PLLTL)
		3.4.4 Decidability
	3.5 Temporal reasoning about evidence
		3.5.1 Weight of evidence
		3.5.2 Reasoning about prior and posterior probabilities
		3.5.3 Adding temporal operators
	3.6 Related work
	References
Chapter 4 Probabilistic Modeling of Default Reasoning
	4.1 Introduction
	4.2 Some formal systems for default reasoning
		4.2.1 Default logic
		4.2.2 Autoepistemic logic
		4.2.3 Circumscription
	4.3 Nonmonotonic consequence relations
		4.3.1 System P
		4.3.2 Conditional logic
		4.3.3 Logics with generalized quantifiers
	4.4 Probabilistic first-order logic LP,I
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		4.4.1 Syntax and semantics
		4.4.2 Axiomatization
		4.4.3 Soundness and completeness
		4.4.4 Decidable fragments of LP,I
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	4.5 Some extensions of the logic LP,I
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	4.6 Applications
	4.7 Conclusions
	References
Chapter 5 Some New Probability Operators
	5.1 Introduction
		5.1.1 Qualitative probability
		5.1.2 Conditional probabilities
		5.1.3 Independence and confirmation
		5.1.4 Probabilistic operators QF
		5.1.5 Organization of the chapter
	5.2 Formal languages
	5.3 Completeness and decidability
	5.4 Logics with iterations and nesting of probability operators
	5.5 Hierarchies of probability logics
		5.5.1 Upper hierarchy
		5.5.2 Lower hierarchy
	5.6 Discussion and concluding remarks
	References
Chapter 6 Applications of Logics About Simple Probabilities
	6.1 Introduction
		6.1.1 Related work
		6.1.2 Organization of the chapter
	6.2 Charges generated by t-norms
		6.2.1 Preliminaries on charges
		6.2.2 Classification via e(s)
		6.2.3 Grabisch’s example
		6.2.4 Probabilistic formalization
	6.3 On weighted logics
		6.3.1 Axiomatization
		6.3.2 Applications
	6.4 Big-stepped probabilities
		6.4.1 Syntax and semantics
		6.4.2 Axiomatization
	References
Chapter 7 Probabilized Sequent Calculus and Natural Deduction System for Classical Logic
	7.1 Sequent calculus, natural deductions and probability
	7.2 Natural deductions probabilized – NKprob
	7.3 Sequent calculus probabilized – LKprob
	7.4 Suppes’ style sequent calculus – LKprob(
	7.5 End view on NKprob, LKprob and LKprob (e)
	References
Chapter 8 Justification Logics with Probability Operators
	8.1 Introduction
	8.2 The basic justification logic J
	8.3 Non-iterated probabilistic justification logic
	8.4 Iterated probabilistic justification logic
	8.5 Soundness and strong completeness
	8.6 Complexity bounds
	8.7 Extensions
	References
Index