Default conditionals are statements that express a condition of normality, in the form ‘if φ then normally ψ’ and are of primary importance in Knowledge Representation. There exist modal approaches to the construction of conditional logics of normality. Most of them are built on notions of preferen ce among possible worlds, corresponding to the semantic intuition that φ ⇒ ψ is true in a situation if in the most preferred (most ‘normal’) situations in which φ is true, ψ is also true. It has been noticed that there exist natural epistemic readings of a default conditional, but this direction has not been thoroughly explored. A statement of the form ‘something known to be a bird, that can be consistently believed to fly, does fly’ involves well-known epistemic attitudes and allows the possibility of defining defaults within the rich framework of Epistemic Logic. We pursue this direction here and proceed to define conditionals within KBE, a recently introduced S4.2-based modal logic of knowledge, belief and estimation. In this logic, knowledge is a normal S4 operator, belief is a normal KD45 operator and estimation is a non-normal operator interpreted as a ‘majority’ quantifier over the set of epistemically alternative situations. We define and explore various conditionals using the epistemic operators of KBE, capturing φ ⇒ ψ in various ways, including ‘according to the agent’s knowledge, an estimation that φ is true implies the estimation that (φ∧ψ) is true’ or ‘if φ is known and there is no reason to believe ¬ψ then ψ can be plausibly inferred’. Overall, we define here three nonmonotonic default conditionals, one conditional satisfying monotonicity (strengthening the antecedent) and two nonmonotonic conditionals that do not satisfy the ubiquitous axiom ID (reflexivity). Our project provides concrete evidence that the machinery of epistemic logic can be exploited for the study of default conditionals.

Abstract We develop a logic for reasoning about semi-public environments , i.e. environments in which a process is executing, and where agents in the environment have partial and potentially different views of the process. Previous work on this problem illustrated that it was problematic to obtain both an adequate semantic model and a language for reasoning about semi-public environments. We here use program models for representing the changes that occur during the execution of a program. These models serve both as syntactic objects and as semantic models, and are a modification of action models in Dynamic Epistemic Logic, in the sense that they allow for ontic change (i.e. change in the world or state). We show how program models can elegantly capture a notion of observation of the environment. The use of these models resolves several difficulties identified in earlier work, and admit a much simpler treatment than was possible in previous work on semi-public environments.

In this paper we attempt to shed light on the concept of an agent’s knowledge after a non-deterministic action is executed. We start by making a comparison between notions of non-deterministic choice, and between notions of sequential composition, of settings with dynamic and/or epistemic character; namely Propositional Dynamic Logic (PDL), Dynamic Epistemic Logic (DEL), and the more recent logic of Semi-Public Environments (SPE). These logics represent two different approaches for defining the aforementioned actions, and in order to provide unified frameworks that encompass both, we define the logics DELVO (DEL+Vision+Ontic change) and PDLVE (PDL+Vision+Epistemic operators). DELVO is given a sound and complete axiomatisation.

The actions of an agent operating in a complex environment are based on her knowledge and beliefs. Epistemic states are typically incomplete, thus the agent has often to estimate whether a fact is true or false before proceeding to actions. We introduce a modal framework for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent’s decision-making process. In this modal account, Knowledge and Belief are captured by |$\mathbf{S4.2}$| which has been advocated as a ‘correct’ logic of knowledge by W. Lenzen and R. Stalnaker. Estimation is a non-normal modal operator interpreted as a ‘majority’ quantifier; the approach employs the ‘weak filters’, a general notion of ‘big’ subsets introduced within KR by K. Schlechta and V. Jauregui. Its axiomatization reveals that this ‘majority’ operator has been introduced by J. Burgess (1969) and used also by A. Herzig (2003). We provide a full account of the logic |$\mathbf{KBE}$| in which estimation is complete: exactly one of |$\varphi$| and |$\neg\varphi$| is estimated to be true. We prove soundness and completeness with respect to a class of frames combining relational Kripke frames with Scott-Montague semantics in which neighbourhoods are ‘weak ultrafilters’ and present a tableaux proof procedure. It comes out that believing |$\varphi$| can be equivalently defined in |$\mathbf{KBE}$| as ‘estimating that |$\varphi$| is known’, an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The assumption on complete estimation is rather strong; yet, it is not hard to define |$\mathbf{KBiE}$|⁠, a relaxed variant in which estimation is still consistent but not complete. We provide the technical details for soundness and completeness with respect to the larger class of frames where neighbourhoods are weak filters. Finally, it is proved that |$\mathbf{KBiE}$| invalidates a rule introduced in W. Lenzen’s probabilistic analysis of weak belief and thus the weak filter semantics is essentially different than its probabilistic counterpart.