The Logic ILP for Intuitionistic Reasoning About Probability

• Published in: Studia logica Vol. 112; no. 5; pp. 987 - 1017
• Main Authors: Ilić-Stepić, Angelina, Ognjanović, Zoran, Perović, Aleksandar
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2024
• Subjects: Computational Linguistics; Education; Logic; Mathematical Logic and Foundations; Philosophy; Logic; Intuitionistic
• ISSN: 0039-3215; 1572-8730
• DOI: 10.1007/s11225-023-10084-z

We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form ⟨ H , μ ⟩ that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) μ : H ⟶ [ 0 , 1 ] Q satisfies the following condition: if α , β , α ∧ β , α ∨ β ∈ H , then μ ( α ∨ β ) = μ ( α ) + μ ( β ) - μ ( α ∧ β ) . Since the range of μ is the set [ 0 , 1 ] Q of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability.