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Assertion, Rejection, and Semantic Universals

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Logic, Rationality, and Interaction (LORI 2021)

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Abstract

Natural language contains simple lexical items for some but not all Boolean operators. English, for example, contains conjunction and, disjunction or, negated disjunction nor, but no word to express negated conjunction *nand nor any other Boolean connective. Natural language grammar can be described by a logic that expresses what the lexicon can express by its primitives, and the rest compositionally. Such logic for propositional connectives is described here as a bilateral extension of update semantics. The basic intuition is that a context can be updated by assertion or by rejection, and by one or multiple propositions at once. These distinctions suffice to characterize the logic of the lexicon.

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 758540) within the project EXPRESS: From the Expression of Disagreement to New Foundations for Expressivist Semantics.

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Notes

  1. 1.

    A competing hypothesis is based on the possibility of expressing missing operators by means of scalar reasoning [6, 12, 15, 22]. For a criticism of this approach, see [14].

  2. 2.

    In fact, the story holds for any many-place connectives with more than one argument. The focus on two-place connectives is for ease of discussion.

  3. 3.

    In fact, these definitions are slightly simpler, since the language under consideration does not contain “genuinely dynamic” operators, such as modals or conditionals, which are sometimes argued to deserve special treatment [9, 25]. The relations of support and validity (and therefore equivalence too) are relative to an update system, but I will not use subscripts for simplicity. The context will make clear which system is under consideration.

  4. 4.

    The definition of ‘Encoding’ requires that the arguments of a connective be atomic. This is necessary, for failures of a formula to encode its classical interpretation (as it is the case with negated conjunction) obviously percolates upwards.

  5. 5.

    It can also be shown that the monotonicity properties for \(\wedge ,\vee ,\) and nor follow from the properties of the update system [14]. Thus the important observation that natural language lexical operators are monotone follows from the logic of speech acts [3].

  6. 6.

    This also implies that U is not “genuinely” dynamic [16]. This should come as no surprise, given that L does not make provision for anaphora, modals, conditionals, or other expressions with dynamic potential. It would be possible to extended U to a theory of presupposition projection, but this is left for another occasion.

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Authors and Affiliations

  1. ILLC, University of Amsterdam, Amsterdam, The Netherlands

    Giorgio Sbardolini

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  1. Giorgio Sbardolini
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Correspondence to Giorgio Sbardolini .

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Editors and Affiliations

  1. Indian Statistical Institute, Chennai, Tamil Nadu, India

    Sujata Ghosh

  2. Department of Philosophy, Stanford University, Stanford, CA, USA

    Thomas Icard

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Sbardolini, G. (2021). Assertion, Rejection, and Semantic Universals. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_15

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