205 research outputs found
Ethics and economics in Karl Menger: how did social sciences cope with Hilbertism
This paper deals with the contributions made to the social sciences by the mathematician Karl Menger (1902-1985), the son of the more famous economist, Carl Menger. Mathematician and a logician, he focused on whether it was possible to explain the social order in formal terms.1 He stressed the need to find the appropriate means with which to treat them, avoiding recourse to historical descriptions, which are unable to yield social laws. He applied Hilbertism to economics and ethics in order to build an axiomatic and formalized model of the individual behavior and the dynamics of social groups.
Two-layered logics for probabilities and belief functions over Belnap--Dunn logic
This paper is an extended version of an earlier submission to WoLLIC 2023. We
discuss two-layered logics formalising reasoning with probabilities and belief
functions that combine the Lukasiewicz -valued logic with Baaz
operator and the Belnap--Dunn logic. We consider two probabilistic
logics that present two perspectives on the probabilities in the Belnap--Dunn
logic: -probabilities and -probabilities. In the first case,
every event has independent positive and negative measures that denote
the likelihoods of and , respectively. In the second case, the
measures of the events are treated as partitions of the sample into four
exhaustive and mutually exclusive parts corresponding to pure belief, pure
disbelief, conflict and uncertainty of an agent in . In addition to that,
we discuss two logics for the paraconsistent reasoning with belief and
plausibility functions. They equip events with two measures (positive and
negative) with their main difference being whether the negative measure of
is defined as the \emph{belief in } or treated independently
as \emph{the plausibility of }. We provide a sound and complete
Hilbert-style axiomatisation of the logic of -probabilities and
establish faithful translations between it and the logic of
-probabilities. We also show that the satisfiability problem in all the
logics is -complete.Comment: arXiv admin note: text overlap with arXiv:2303.0456
Fuzzy Description Logics with General Concept Inclusions
Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived
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