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Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section. This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

This paper introduces a logical characterization of multi-adjoint algebras with a twofold contribution. On the one hand, the study of multi-adjoint algebras, from a logical perspective, will allow us to discover both the core and new features of these algebras. On the other hand, the axiomatization of multi-adjoint algebras will be useful to take advantage of the properties of the logical connectives considered in the corresponding deductive system. The mechanism considered to carry out the mentioned axiomatization follows the one given by Petr Hajek for residuated lattices. Specifically, the paper presents the bounded poset logic (BPL) as an axiomatization of a bounded poset, since this algebraic structure is the most simple structure from which a multi-adjoint algebra is defined. In the following, the language of BPL is enriched with a family of pairs, composed of a conjunctor and an implication, and its axiomatic system is endowed with new axioms, giving rise to the multi-adjoint logic (ML). The soundness and completeness of BPL and ML are proven. Finally, a comparison between the axiomatization of the multi-adjoint logic and the one given for the BL logic is introduced.

In this paper we introduce a new set of axioms that characterize uniform random dictatorship (URD) as a randomized impartial nomination rule. Unlike earlier work we use a variable population model which allows us to employ axioms that reflect consistency and proportionality–axioms that ensure that the rule behaves well with respect to the combinatorial structure of the nomination profile. Earlier work characterizing URD employed strong symmetry axioms and so it is surprising that our axioms characterize the same rule.

Fuzzy rough approximation operators are the underlying concepts in fuzzy rough set theory . There are at least two approaches to develop these primary concepts, i.e., the constructive approach and the axiomatic approach . Single axiomatic characterizations of fuzzy rough approximation operators have got tons of attention. In this paper, considering L being a GL-quantale, we will develop the theory of L -valued rough sets with an L -set as the basic universe of defining L -valued rough approximation operators. Adopting the idea of single axiomatic characterizations of fuzzy rough sets , we will present the axiomatic characterizations of L -valued upper and lower rough approximation operators on an L -set with respect to reflexive, symmetric, transitive L -valued relations on an L -set as well as their compositions. Choosing an L -set as the universe will break the rules that adopting Zadeh’s fuzzy sets as the universe. By these results, we will further provide a new framework of axiomatic research of fuzzy rough set theory.

I present the proof of Gödel's First Incompleteness theorem in an intuitive manner assuming college-level computational background, while covering all technically challenging steps. I also discuss Gödel's Second Incompleteness theorem, their connection to Gödel's Completeness theorem, and conclude with brief remarks on implications for mathematics, computation, theory of mind and AI.

Behavior-related neuronal signals often vary between neurons. Despite the unreliability of individual neurons, brains are able to accurately represent and drive behavior. The notion may also apply to economic (‘value-based’) choices and the underlying reward signals. Reward value is subjective and can be defined by nonlinear weighting of magnitude (utility) and probability. Using a wide variety of reward magnitude and probability, we assessed subjective reward value at choice indifference between safe and risky rewards as prescribed by the continuity axiom that provides stringent criteria for meaningful choice. We found that individual neurons in the orbitofrontal cortex (OFC) of monkeys carry unreliable and heterogeneous neuronal signals for subjective value that largely fails to match the animal’s choice. However, the averaged neuronal signals matched well the animals’ choices, suggesting reliable subjective economic value encoding by the observed population of unreliable neurons.HighlightsDifferent from widely held views, reliable neuronal information processing may not require reliable processors.Neurons in monkey orbitofrontal cortex (OFC) process reward magnitude and probability heterogeneously and unreliably.Despite unreliable neuronal processing, OFC population activity codes choices reliably.Reliability systems performance from unreliable elements seems to be a broad feature of neuronal reward coding.In briefUsing stringent concepts of behavioral choice, Ferrari-Toniolo and Schultz describe unreliable individual reward neurons in monkey orbitofrontal cortex whose activity combines to a reliable population code for economic choice.