Nordic Online Logic Seminar - page 2

One quarter of the talk presents background on how facts about entailments and non-entailments can single out the constants in a language, and in particular on an idea originating with Carnap that the standard relation of logical consequence in a formal language should fix the (model-theoretic) meaning of its logical constants. Carnap’s focus was classical propositional logic (CPL), but his question can be asked for any logical language. The rest of the talk gives a very general positive answer to this question for IPL: the usual IPL consequence relation does indeed determine the standard intuitionistic meaning of the propositional connectives, according to most well-known semantics for IPL, such as Kripke semantics, Beth semantics, Dragalin semantics, topological semantics, and algebraic semantics.

Dynamic Epistemic Logic (DEL) can be used as a formalism for agents to represent the mental states of other agents: their beliefs and knowledge, and potentially even their plans and goals. Hence, the logic can be used as a formalism to give agents, e.g. robots, a Theory of Mind, allowing them to take the perspective of other agents. In my research, I have combined DEL with techniques from automated planning in order to describe a theory of what I call Epistemic Planning: planning where agents explicitly reason about the mental states of others. The talk will introduce epistemic planning based on DEL, address issues of computational complexity, and demonstrate applications in cognitive robotics and human-robot collaboration.

Normal modal logics are closed under conjunctive closure. There are, however, interesting non-normal logics that are not, but which nevertheless satisfy a weak form of conjunctive closure. One example is a notion of group knowledge in epistemic logic: somebody-knows. While something is general knowledge if it is known by everyone, this notion holds if it is known by someone. Somebody-knows is thus weaker than general knowledge but stronger than distributed knowledge. We introduce a modality for somebody-knows in the style of standard group knowledge modalities, and study its properties. Unlike most other group knowledge modalities, somebody-knows is not a normal modality; in particular it lacks the conjunctive closure property. We provide an equivalent neighbourhood semantics for the language with a single somebody-knows modality, together with a completeness result: the somebody-knows modalities are completely characterised by the modal logic EMN extended with a particular weak conjunctive closure axiom. The neighbourhood semantics and the completeness and complexity results also carry over other logics with weak conjunctive closure, including the logic of so-called local reasoning (Fagin et al., 1995) with bounded “frames of mind”, correcting an existing completeness result in the literature (Allen 2005). The talk is based on joint work with Yi N. Wang.

The SHACL Shapes Constraint Language was recommended in 2017 by the W3C for describing constraints on web data (specifically, on RDF graphs) and validating them. At first glance, it may not seem to be a topic for logicians, but as it turns out, SHACL can be approached as a formal logic, and actually quite an interesting one. In this paper, we give a brief introduction to SHACL tailored towards logicians and frame key uses of SHACL as familiar logic reasoning tasks. We discuss how SHACL relates to description logics, which are the basis of the OWL Web Ontology Languages, a related yet orthogonal standard for web data. Finally, we summarize some of our recent work in the SHACL world, hoping that this may shed light on how ideas, results, and techniques from well-established areas of logic can advance the state of the art in this emerging field.

We set out five basic requirements for a logical system to be adequate for the regimentation of deductive reasoning in mathematics and science. We raise the question whether there are any further requirements, not entailed by these five, that ought to be imposed. One possible reply is dismissed: that the logical system should allow one to infer any proposition at all from an inconsistent set—i.e., it should have as primitive, or allow one to derive, the rule Ex Falso Quodlibet. We then propose that the logic should be implosive: it should not allow an inconsistent set to have any consequences other than absurdity. This proposal may appear to be very radical; but we hope to show that it is robust against objections.

The notion of (relative) interpretation for first order theories was introduced in a landmark 1953 monograph by Alfred Tarski, Andrzej Mostowski and Raphael Robinson, where it was developed as a powerful tool for establishing undecidability results. By now the domain of interest and applicability of interpretability theory far exceeds undecidability theory owing to its multifaceted interactions with both proof theory and model theory. Special attention will be paid to recent advances in the subject that indicate the distinctive character of Peano Arithmetic, Zermelo-Fraenkel set theory, and their higher order analogues in the realm of interpretability theory. This talk will present a personal overview of the interpretability analysis of arithmetical and set theoretical theories.

One of the major problems in formal epistemology is the difficulty of combining probabilistic and full (dichotomous, all-or-nothing) beliefs in one and the same formal framework. Major properties of actual human belief systems, including how they are impacted by our cognitive limitations, are used to introduce a series of desiderata for realistic models of such belief systems. This leads to a criticism of previous attempts to combine representations of both probabilistic and full beliefs in one and the same formal model. Furthermore, a formal model is presented in which this is done differently. One of its major features is that contingent propositions can be transferred in both directions between full beliefs and lower degrees of belief, in a way that mirrors real-life acquisitions and losses of full beliefs. The subsystem consisting of full beliefs has a pattern of change that constitutes a credible system of dichotomous belief change.

Compositional semantics that acknowledge vagueness by positing degrees of truth intermediate between truth and falsity can retain classical sentential calculus, provided the degrees form a Boolean algebra. A valid deduction guarantees that the degree of truth of the conclusion be at least as great as every lower bound on the degrees of the premises. If we extend the language to allow infinite disjunctions and conjunctions, the Boolean algebra will need to be complete and atomic. If we extend further by adding quantifiers ranging over a fixed domain, we get the supervaluations proposed by Bas van Fraassen.