The region-based spatial logics, where variables are set to range over certain subsets of geometric space, are the focal point of the qualitative spatial reasoning, a subfield of the KR&R research area. A lot of attention has been devoted to developing the topological spatial logics, leaving other systems relatively underexplored. We are concerned with a specific example of a region-based affine spatial logic. Building on the previous results on spatial logics with convexity, we axiomatise the theory of M = 〈 ROQ (R 2 ), conv M , ≤ M 〉, where ROQ (R 2 ) is the set of regular open rational polygons of the real plane; conv M is the convexity property and ≤ M is the inclusion relation. The axiomatisation uses two infinitary rules of inference and a number of axiom schemas.
On 21 August 1912 Christine Ladd-Franklin, by then an established logician, wrote a letter to Bertrand Russell. He replied on 27 September 1912, followed by another letter on 16 November of that year. After a hiatus on his side in 1913–14, they exchanged letters again in 1915. The main topic of their conversations is solipsism: a theme that was important for Russell throughout his writings. In fact, in some of his works he famously mentions his encounters with Ladd-Franklin, hinting at a difference of opinions and her inability to see the inconsistency in what she claimed. After analysing the correspondence, with some letters resurfacing only recently, one sees a completely different picture: Russell not only does not object to what she claims, he even agrees with her! This article aims to show what really transpired as evidenced by the letters, of which seven of the extant eleven are reproduced in full with annotations.
Abstract. This article describes a refutation method of proving the maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experiment that confirms the theoretical results.
We argue that the ideas of Bertrand Russell, a famous English philosopher and logician, have a bearing on the philosophical foundations of one of the sub-fields of AI, called qualitative spatial reasoning. The research conducted within that field focuses on non-numerical reasoning about regions of space designed to mimic human spatial behaviour and thus avoid the artificiality of the numerical approach. After briefly describing the main characteristics of this field, we analyse Russell's works on geometry. We show that despite major differences in how the subject matter is treated, these publications do have a common core that is related to the non-numerical, qualitative parts of geometry. Therefore, we argue that Russell should be viewed as a forefather of qualitative spatial reasoning on par with Whitehead or Leśniewski. Moreover, we believe that the efforts within qualitative spatial reasoning should be geared more towards the types of geometry he describes.
We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work on different dimensions. We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent.
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