Stephan van der Waart van Gulik - Fuzzy concepts, non-scalar hedging and context-adaptive interpretations

In [3], George Lakoff analyzed the semantics of a specific kind of hedges, namely those terms or phrases which modify the concept of the lexical item on which they operate. I call these modifiers ‘non-scalar hedges’. Examples are “Technically, Nixon is a Quaker”, or “Strictly speaking, Nixon is a Quaker”. According to his analysis, each concept C on which non-scalar hedges can operate must be fuzzy and complex, i.e. C consists of a set of associated (non-)complex concepts each of which has a specific level of importance for the meaning of C, cf. also [4]. An example of an associated concept of ‘Quaker’ is ‘Pacifist’. A non-scalar hedge selects sets of associated concepts in function of their level of importance and combines these sets in a specific logical way. For example, ‘technically’ demands that all ‘definitionally important’ concepts hold and at least one concept of ‘relatively high’ importance does not.    The setting is that of an agent who needs to interpret a predicate in light of both the logical constraints of the set of conveyed sentences as well as its corresponding concept stored in his personal memory. I will present several related logics which allow for context-adaptive (re)interpretations of complex fuzzy concepts by taking into account the possibility of implicit non-scalar hedging. The logics are in the standard format of the ‘Adaptive Logic Programme’, cf. [1]. An adaptive logic (AL) oscillates between a ‘Lower Limit Logic’ (LLL) and an ‘Upper Limit Logic’ (ULL). It interprets a specific type of formulas, called ‘abnormalities’, as false until proven otherwise, thereby allowing applications of ULL-exclusive rules until their conditions for valid application are violated.   In this context, the logic G~h is used as LLL. G~h is a modified version of the first-order fuzzy Gödel logic G~, cf. [2], which incorporates the Lakoff-style semantics of the hedges ‘technically’, ‘strictly’, and ‘loosely’. The modification involves a new type of object called a ‘concept matrix’ (CM) which simulates the concept structures that Lakoff uncovered with respect to non-scalar hedges. Semantically, CM’s are used to calculate the membership values of instances for both hedged and non-hedged predicates denoting complex concepts. Proof-theoretically, CM’s are consulted during interpretative actions, i.e. when substituting a (non)-hedged complex (sub)formula by its less complex, CM-equivalent variant (and vice versa).    Depending on the type of hedge that is incorporated in the abnormalities, different AL’s emerge. For example, when incorporating ‘technically’, AG~ht is generated. AG~ht can be used to check whether some instance of a predicate C denoting a complex concept can or cannot be interpreted in its technical sense. If the CM’s in question allow for a technical interpretation of C that is consistent with the set of sentences, then AG~ht will allow for this interpretation. Note that these AL’s can exhibit context-adaptive interpretational dynamics. When extra sentences are conveyed, the AL’s may reject earlier interpretations.

References

Batens, D. (2007): A Universal Logic Approach to Adaptive Logics Logica Universalis 1: pp. 221-242

Hájek, P. (1998): Metamathematics of Fuzzy Logic, Kluwer Academic Publishers.

Lakoff, G., Hedges (1973): A Study in Meaning Criteria and the Logic of Fuzzy Concepts, Journal of Philosophical Logic 2: pp. 458-508.

Rosch, E. and C. Mervis, Family Resemblances (1975): Studies in the Internal Structure of Categories, CognitivePsychology 7: pp. 573-605.