Abstract
We present a first algebraic approximation to the semantic content of linguistic ambiguity. Starting from the class of ordinary Boolean algebras, we add to it an ambiguity operator \(\Vert \) and a small set of axioms which we think are correct for linguistic ambiguity beyond doubt. We then show some important, non-trivial results that follow from this axiomatization, which turn out to be surprising and not fully satisfying from a linguistic point of view. Therefore, we also sketch promising algebraic alternatives.
In the originally published version there is an error in the proof of Lemma 1. The erratum to this chapter is available at https://doi.org/10.1007/978-3-662-53042-9_19
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Notes
- 1.
This roughly distinguishes ambiguity from cases of vagueness [8].
- 2.
[10] delivers another example for the “disjunction fallacy” based on intensional predicates: given a sentence S that is ambiguous between P and Q, the following should hold if S meant \(P \vee Q\): \([A \text { means that } S] = [A \text { means that } [P \vee Q]] = [[A \text { means that } P] \vee [A \text { means that } Q]]\). However, this is obviously not the case. See also [11] and [9] for similar ideas and examples.
- 3.
This also makes clear that ambiguity cannot be interpreted, algorithmically speaking, as non-deterministic choice (which corresponds to disjunction), as we cannot pick an arbitrary meaning.
- 4.
\(\le \) in Boolean algebras is defined in terms of \(\wedge \) (or \(\vee \)): \(a\wedge b=a\) iff \(a\le b\) iff \(a\vee b=b\).
- 5.
Here we use connectives \(\wedge ,\ \vee ,\ \sim ,\ \rightarrow \) in their Boolean meaning, but in principle this would not make a difference.
- 6.
Linguistically, it is of course unclear how to determine such a context. We would (vaguely) say it is a discourse, but of course this is arguable. Instead of being vague we could also be circular and say: such a context is a discourse where ambiguous terms are used consistently in one sense.
- 7.
We do not explain these concepts here, as, to the algebraist, they are clear anyway, and for the non-algebraist they are of no relevance in this paper.
- 8.
This term is very important for the following proofs. One might argue that ambiguity of this kind does not arise in natural language, an argument which leads to partially ambiguous algebras, which we discuss shortly later on. On the other side, there are words such as sacré in French which – though in different contexts – can both mean ‘cursed’ and ‘holy’.
- 9.
From the results in this paper, decidability results easily follow, but for reasons of space we do not include them here.
- 10.
However, one can say such a thing as: I do not need such a bank, I need the other bank! For us, this would count as a disambiguation, hence strictly speaking it takes the ambiguity from the semantics.
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Wurm, C., Lichte, T. (2016). The Proper Treatment of Linguistic Ambiguity in Ordinary Algebra. In: Foret, A., Morrill, G., Muskens, R., Osswald, R., Pogodalla, S. (eds) Formal Grammar. FG FG 2015 2016. Lecture Notes in Computer Science(), vol 9804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53042-9_18
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