Mathematics

We introduce so-called consistent posets which are bounded posets with an antitone involution ' where the lower cones of x,x' and of y,y' coincide provided x,y are different form 0,1 and, moreover, if x,y are different form 0 then their lower cone is different form 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication. Finally we show that the Dedekind-MacNeille completion of a consistent poset is a consistent lattice, i.e. a bounded lattice with an antitone involution satisfying the above mentioned properties.

Temporal logics provide a formalism for expressing complex system specifications. A large body of literature has addressed the verification and the control synthesis problem for deterministic systems under such specifications. For stochastic systems or systems operating in unknown environments, however, only the probability of satisfying a specification has been considered so far, neglecting the risk of not satisfying the specification. Towards addressing this shortcoming, we consider, for the first time, risk metrics, such as (but not limited to) the Conditional Value-at-Risk, and propose risk signal temporal logic. Specifically, we compose risk metrics with stochastic predicates to consider the risk of violating certain spatial specifications. As a particular instance of such stochasticity, we consider control systems in unknown environments and present a determinization of the risk signal temporal logic specification to transform the stochastic control problem into a deterministic one. For unicycle-like dynamics, we then extend our previous work on deterministic time-varying control barrier functions.

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new <κ -sequences (for some regular κ ). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cichoń's diagram, plus ℵ 1 <m<p<h<add(N) . (So we get thirteen different values, including ℵ 1 and continuum). We also give constructions to alternatively separate other MA-numbers (instead of m ), namely: MA for k -Knaster from MA for k+1 -Knaster; and MA for the union of all k -Knaster forcings from MA for precaliber.

We prove that if an ω -categorical structure has an ω -categorical homogeneous Ramsey expansion, then so does its model-complete core.

We analyze the causal-observational languages that were introduced in Barbero and Sandu (2018), which allow discussing interventionist counterfactuals and functional dependencies in a unified framework. In particular, we systematically investigate the expressive power of these languages in causal team semantics, and we provide complete natural deduction calculi for each language. As an intermediate step towards the completeness, we axiomatize the languages over a generalized version of causal team semantics, which turns out to be interesting also in its own right.

Given a completely metrizable space X , let par(X) denote the smallest possible size of a partition of X into Polish spaces, and cov(X) the smallest possible size of a covering of X with Polish spaces. Observe that cov(X)?�par(X) for every X , because every partition of X is also a covering. We prove it is consistent relative to a huge cardinal that the strict inequality cov(X)<par(X) can hold for some completely metrizable space X . We also prove that using large cardinals is necessary for obtaining this strict inequality, because if cov(X)<par(X) for any completely metrizable X , then 0 ??exists.

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

We give a survey of current research on Gödel's incompleteness theorems from the following three aspects: classifications of different proofs of Gödel's incompleteness theorems, the limit of the applicability of Gödel's first incompleteness theorem, and the limit of the applicability of Gödel's second incompleteness theorem.

This paper introduces a natural deduction calculus for intuitionistic logic of belief IEL − which is easily turned into a modal λ -calculus giving a computational semantics for deductions in IEL − . By using that interpretation, it is also proved that IEL − has good proof-theoretic properties. The correspondence between deductions and typed terms is then extended to a categorical semantics for identity of proofs in IEL − showing the general structure of such a modality for belief in an intuitionistic framework.

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules.