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Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
We define the continuous modeling property for first-order structures and show that a first-order structure has the continuous modelling property if and only if its age has the embedding Ramsey property. We use generalized indiscernible…
Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega_1$ is preserved by any proper forcing. We…
A relatively new topic in computability theory is the study of notions of computation that are robust against mistakes on some kind of small set. However, despite the recent popularity of this topic relatively foundational questions about…
In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was…
By operations on models we show how to relate completeness with respect to permissive-nominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissive-nominal…
We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games.
It is well known that the completeness theorem for $\mathrm{L}_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces Tarski semantics with boolean valued…
We show that the intuitionistic propositional logic with a Galois connection (IntGC), introduced by the authors, has the finite model property.
'Capsule' models try to explicitly represent the poses of objects, enforcing a linear relationship between an object's pose and that of its constituent parts. This modelling assumption should lead to robustness to viewpoint changes since…
Building on ideas of Gurevich and Shelah for the G\"odel Class, we present a new probabilistic proof of the finite model property for the Guarded Fragment of First-Order Logic. Our proof is conceptually simple and yields the optimal…
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
We consider ergodic $\mathrm{Sym}(\mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $\mathbb{N}$ (for $L$ a countable relational language), and call such a measure a properly ergodic structure when no…
The intuitive notion of evidence has both semantic and syntactic features. In this paper, we develop an {\em evidence logic} for epistemic agents faced with possibly contradictory evidence from different sources. The logic is based on a…
Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize…
Argumentation is a promising model for reasoning with uncertain knowledge. The key concept of acceptability enables to differentiate arguments and counterarguments: The certainty of a proposition can then be evaluated through the most…
We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously…
In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.
We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.
We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some…