Gentzen Calculi for the Existence Predicate

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.     

Using temporal logics of knowledge for specification and verification—a case study

Temporal logics of knowledge are useful for reasoning about situations where the knowledge of an agent or component is important, and where change in this knowledge may occur over time. Here we use temporal logics of knowledge to reason about the game Cluedo. We show how to specify Cluedo using temporal logics of knowledge and prove statements about the knowledge of the players using a clausal resolution calculus for this logic. We discuss the advantages and disadvantages of using this logic to specify and verify the game Cluedo and describe related implementations.     

A general method for proving decidability of intuitionistic modal logics

We generalise the result of [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34] on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in monadic second order logic. In [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34], such constraints apply to one relation at a time. We modify their proof to obtain decidability for constraints involving several relations. Now we can use this result to prove decidability of multi-modal modal logics where conditions on accessibility relations involve more than one relation. Our main application is intuitionistic modal logic, where the intuitionistic and modal accessibility relations usually interact in a non-trivial way.     

Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator.     

Abstract rationality in education: from Vygotsky to Brandom

Abstract rationality has increasingly been a target of attack in contemporary educational research and practice and in its place practical reason and situated thinking have become a focus of interest. The argument here is that something is lost in this. In illustrating how we might think about the issue, this paper makes a response to the charge that as a result of his commitment to the ‘Enlightenment project’ Vygotsky holds abstract rationality as the pinnacle of thought. Against this it is argued that Vygotsky had a far more sophisticated appreciation of reason and of its remit. The paper proceeds first by examining the picture of Vygotsky that is presented in the work of James Wertsch, and especially his claim that Vygotsky was an ambivalent rationalist, goes on to provide an account of Vygotsky that corrects this picture, and develops this in the light of the work of Robert Brandom, who shares Vygotsky’s inheritance of Hegel. The conclusion towards which this piece points is that the philosophical underpinnings of Vygotsky’s work provide a radically different idea of rationality and epistemology from that characterised as abstract rationality and that this has significance for education studies.

Interpolation in non-classical logics

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation. Supported by PRIN project 2006/2007 ‘Large-scale development of certified mathematical proofs’.     

Determining who owns what: do children infer ownership from first possession?

A basic problem of daily life is determining who owns what. One way that people may solve this problem is by relying on a ‘first possession’ heuristic, according to which the first person who possesses an object is its owner, even if others subsequently possess the object. We investigated preschoolers’ use of this heuristic in five experiments. In Experiments 1 and 2, 3- and 4-year-olds inferred that an object was owned by the character who possessed it first, even though another character subsequently possessed it. Two-year-olds also showed this bias, but only when the object was placed between the characters when children were asked about ownership. Experiment 3 ruled out the possibility that children’s bias to select the first possessor results from a tendency to select the character first associated with the object. Experiment 4 showed that 3- and 4-year-olds have difficulty disregarding the first possession heuristic, even when provided with evidence that the character who first possessed an object is not its owner. But Experiment 5 found that children can disregard the heuristic in at least some situations. These five experiments suggest that the first possession heuristic guides children’s ownership inferences. The findings provide the first evidence that preschoolers can infer who owns what, when not explicitly told, and when not reasoning about objects with which they are personally acquainted.     

Permission from an Input/Output Perspective

Input/output logics are abstract structures designed to represent conditional obligations and goals. In this paper we use them to study conditional permission. This perspective provides a clear separation of the familiar notion of negative permission from the more elusive one of positive permission. Moreover, it reveals that there are at least two kinds of positive permission. Although indistinguishable in the unconditional case, they are quite different in conditional contexts. One of them, which we call static positive permission, guides the citizen and law enforcement authorities in the assessment of specific actions under current norms, and it behaves like a weakened obligation. Another, which we call dynamic positive permission, guides the legislator. It describes the limits on the prohibitions that may be introduced into a code, and under suitable conditions behaves like a strengthened negative permission.     

Routes to Triviality

It is known that a number of inference principles can be used to trivialise the axioms of naïve comprehension – the axioms underlying the naïve theory of sets. In this paper we systematise and extend these known results, to provide a number of general classes of axioms responsible for trivialising naïve comprehension.