We present an account of semantics that is not construed as a mapping of language to the world but rather as a mapping between individual meaning spaces. The meanings of linguistic entities are established via a “meeting of minds.” The concepts in the minds of communicating individuals are modeled as convex regions in conceptual spaces. We outline a mathematical framework, based on fixpoints in continuous mappings between conceptual spaces, that can be used to model such a semantics. If concepts are (...) convex, it will in general be possible for interactors to agree on joint meaning even if they start out from different representational spaces. Language is discrete, while mental representations tend to be continuous—posing a seeming paradox. We show that the convexity assumption allows us to address this problem. Using examples, we further show that our approach helps explain the semantic processes involved in the composition of expressions. (shrink)

While “meaning negotiation” has become an ubiquitous term, its use is often confusing. A negotiation problem implies not only a convenience to agree, but also diverging interest on what to agree upon. It implies agreement but also the possibility of disagreement. In this chapter, we look at meaning negotiation as the process through which agents starting from different preferred conceptual representations of an object, an event or a more complex entity, converge to an agreement through some communication medium. We shortly (...) sketch the outline of a geometric view of meaning negotiation, based on conceptual spaces. We show that such view can inherit important structural elements from game theoretic models of bargaining – in particular, in the case when the protagonists have overlapping negotiation regions, we emphasize a parallel to the Nash solution in cooperative game theory. When acceptable solution regions of the protagonists are disjoint, we present several types of processes: changes in the salience of dimensions, dimensional projections and metaphorical space transformations. None of the latter processes are motivated by normative or rationality considerations, but presented as argumentation tools that we believe are used in actual situations of conceptual disagreement. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood as (...) involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)