This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) Boolean algebra of conditionals from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.
This paper is a contribution to the study of two distinct kinds of logics for modelling uncertainty. Both approaches use logics with a two-layered modal syntax, but while one employs classical logic on both levels and infinitely-many multimodal operators, the other involves a suitable system of fuzzy logic in the upper layer and only one monadic modality. We take two prominent examples of the former approach, the probability logics Plin and Prpol (whose modal operators correspond to all possible linear/polynomial inequalities with integer coefficients), and three logics of the latter approach: PrŁ, PrŁΔ and PrPŁΔ (given by the Łukasiewicz logic and its expansions by the Baaz–Monteiro projection connective Δ and also by the product conjunction). We describe the relation between the two approaches by giving faithful translations of Prlin and Prpol into, respectively, PrŁΔ and PrPŁΔ, and vice versa. We also contribute to the proof theory of two-layered modal logics of uncertainty by introducing a hypersequent calculus HPrŁ for the logic PrŁ. Using this formalism, we obtain a translation of Prlin into the logic PrŁ, seen as a logic on hypersequents of relations, and give an alternative proof of the axiomatization of Prlin.
P. Baldi, P.Cintula, C.Noguera. Classical and Fuzzy Two-Layered Modal Logics for Uncertainty: Translations and Proof-Theory, International Journal of Computational Intelligence Systems, https://doi.org/10.2991/ijcis.d.200703.001
We present a logic to model the behaviour of an agent trusting or not trusting messages sent by another agent. The logic formalizes trust as a consistency checking function with respect to currently available information. Negative trust is modeled in two forms: distrust as the rejection of incoming inconsistent information; mistrust, as revision of previously held information becoming undesirable in view of new incoming inconsistent information, which the agent wishes to accept. We provide a natural deduction calculus, a relational semantics and prove soundness and completeness results. We overview a number of applications which have been investigated for the proof-theoretical formulation of the logic.
ASPIC+ is an established general framework for argumentation and non-monotonic reasoning. However ASPIC+ does not satisfy the non-contamination rationality postulates, and moreover, tacitly assumes unbounded resources when demonstrating satisfaction of the consistency postulates. In this paper we present a new version of ASPIC+ – Dialectical ASPIC+ – that is fully rational under resource bounds.
We introduce measures of uncertainty that are based on Depth-Bounded Logics and resemble belief functions. We show that our measures can be seen as approximation of classical probability measures over classical logic, and that a variant of the PSAT problem for them is solvable in polynomial time.
Baldi P., D’Agostino M., Hosni H. (2020) “Depth-Bounded Approximations of Probability”. In: Lesot MJ. et al. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2020. Communications in Computer and Information Science, vol 1239. Springer, Cham DOI: 10.1007/978-3-030-50153-2_45
In this talk, I will discuss logics for social networks, their epistemic extensions, and dynamics in such structure, including diffusion as modeled by threshold models. I will present a selection of recent models for social networks and their epistemics, with a focus on how these may be represented using dynamic term-modal logic (DTML)—a dynamic, quantified modal/epistemic logic, where the subscripts of operators are first-order terms, allowing formulas such as $\exist x K_x N(x,b)$: there exists and agent that knows that it is networked with agent b. DTML is based on an enriched version of action models of dynamic epistemic logic fame, and comes with a complete set of reduction axioms. Modelling social network dynamics in DTML thus directly provide sound and complete logics. Additionally, such logics are decidable when only a finite set of agents is considered.
This paper introduces and investigates Depth-bounded Belief functions, a logic-based representation of quantified uncertainty. Depth-bounded Belief functions are based on the framework of Depth-bounded Boolean logics, which provide a hierarchy of approximations to classical logic. Similarly, Depth-bounded Belief functions give rise to a hierarchy of increasingly tighter lower and upper bounds over classical measures of uncertainty. This has the rather welcome consequence that “higher logical abilities” lead to sharper uncertainty quantification. In particular, our main results identify the conditions under which Dempster-Shafer Belief functions and probability functions can be represented as a limit of a suitable sequence of Depth-bounded Belief functions.
The Logic Group successfully started its Seminar Series online on Teams last week. On 19/03 Pere Pardo gave a talk on “Towards a Tractable Epistemic Logic”. We will reschedule guest lectures and add group’s members lectures. The talks will be open to attend with a shared link. Monitor this website or contact us for updates.