Abstract
Linguistic models of vagueness usually record contexts of possible precisifications. A link between such models and fuzzy logic is established by extracting fuzzy sets from context based word meanings and by analyzing standard logical connectives in this setting. In this manner the three fundamental t-norms (Łukasiewicz, minimum, and product) highlighted in Petr Hájek’s approach to fuzzy logic reappear as bounds for degrees extracted from contextual models. In a further step, two semantic scenarios for fuzzy logic—similarity based reasoning and Giles’s dialogue game—are adapted to obtain t-norm based truth functions from the point of view of context update semantics.
Dedicated to Petr Hájek
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Notes
- 1.
The term ‘mathematical fuzzy logic’ has been successfully propagated by students and colleagues of Petr Hájek only well after the appearance of Hájek (2001). Hájek, like others at that time, referred to ‘fuzzy logic in the narrow sense’, following Zadeh’s distinction between a wider and a narrow sense of fuzzy logic, where the latter meant the study of deductive systems of logics that are based on the real unit interval as set of truth values.
- 2.
Some fuzzy logicians seem tempted to argue that models like that of Barker, that we will take as starting point here, compare unfavorably with fuzzy logic, even if the aim is to model the semantics of natural language. But, as our brief review of Barker’s model in Sect. 5.2 will indicate, context based models are usually much more fine grained than those offered by fuzzy logic. They indicate that the mentioned methodological principles successfully support contemporary linguistic research in various ways. We thus take the idea that linguists should replace their own approach to formal semantics of vague language by that of fuzzy logic as a quixotic move, that hardly deserves serious debate. On the other hand, the claim that there is no relation between natural language semantics and fuzzy logic at all seems dubious. After all, both fields attempt to model the successful processing of vaguely stated information. In this endeavor they frequently refer to the same natural language examples and moreover rely both on tools from mathematical logic.
- 3.
- 4.
By a filter we (here) just mean a function \(f\) which maps any set to one of its subsets, i.e. \(f: \fancyscript{P}(S) \rightarrow \fancyscript{P}(S)\), with \(f(X)\subseteq X\) for all \(X \in \fancyscript{P}(S)\).
- 5.
Note that \({\delta }(w)\) is polymorphic: for simple predicates such as tall it has only one argument. However, if the first argument is a reference to a modifier like \({[\![{{{ very}}}]\!]}\) or \({[\![{{{ clearly}}}]\!]}\) then a reference to a predicate is expected as second argument.
- 6.
Note that nevertheless both, very and definitely, are understood as vague adjectives, in the sense of being systematically context dependent.
- 7.
For brevity we focus on monadic predicates, but the concepts can easily be extended to relations of higher arity.
- 8.
As already implicitly assumed above (following Barker), we stipulate that the relevant element is in the universe of the context to which the filter is applied. (Otherwise the result simply remains undefined.)
- 9.
Of course, the approach can be generalized to infinite contexts by imposing suitable probability measures on possible worlds. We will implicitly use such a model in Sect. 5.4, below. In any case, we do not claim any originality, but rather follow a well established concept here.
- 10.
In natural language one can also find exclusive disjunction, e.g. \({{ Jana \,is\, either}}\, {{ tall \,or}}\) \({{ clever \,(but \,not \, both)}}\), but note that exclusive disjunction can be modeled as well in the obvious way.
- 11.
As is well known, it is questionable whether material implication has a natural language equivalent. We include this logical connective here mainly for the purpose of comparison.
- 12.
For the sake of readability we write \([{X}]_{C}\) instead of \([{{[\![{{{ X}}}]\!]}}]_{C}\).
- 13.
Note that risk, here, refers to expected payments and not to guaranteed bounds. If I am unlucky then, for a final state \(\left[ p \mid p\right] \), the experiment associated with \(p\) might yield a negative answer for my assertion that \(p\), but might nevertheless yield a positive answer for your assertion that \(p\). Accordingly I have to pay 1€ to you, although my corresponding total risk remains \(0\), independently of \(\left\langle p\right\rangle \).
- 14.
Note that the mentioned literature does not address this problem. There, the similarity relation over worlds is assumed as given and remains independent of the structure of the worlds themselves (at least in principle). However, for our current purpose, we have to take into account that similarities typically depend on the particular valuations of atomic formulas that characterize the individual worlds.
- 15.
In the case where there is no uncertainty about Jane’s height, this definition can be simplified by changing the numerator to \(|{\delta }(u)({{^\uparrow {}{{{ tall}}}}}) - {\delta }(w)({{^\uparrow {}{{{ tall}}}}})|\).
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Acknowledgments
Research partially supported by FWF project LOGFRADIG P25417-G15.
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Fermüller, C.G., Roschger, C. (2015). Bridges Between Contextual Linguistic Models of Vagueness and T-Norm Based Fuzzy Logic. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_5
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