A proof-theoretic universal property of determiners

Determiners are the natural language (NL) analogue of quantifiers in logic. In model-theoretic semantics (MTS), their denotations are taken as binary relation over subsets of the domain of the model (see [3] for an extensive discussion). When combined with a noun-meaning, a subset of the domain, they give rise to determiner phrase (dp), which, according to the generalised quantifiers theory [1], a cornerstone of MTS, denotes a generalised quantifier (GQ), a subset of the power set of the domain D. It is generally assumed (and empirically attested), that the only GQs that can be denotation of dps are the conservative ones, satisfying, for every X,Y⊆D: [[D]](X)(Y ) iff [[D]](X)(X∩Y ). Example: every girl smiles iff every girl is a girl who smiles. Since there exist many non-conservative binary relation on subsets of a domain, conservativity serves as a selection criterion for possible determiner denotations. On the other hand, in proof-theoretic semantics (PTS), an approach to semantics according to which meaning is determined by means of the rules of a meaning-conferring natural-deduction (ND) proof-system (see [4] as a general reference and [references suppressed] for PTS for NL), independently of models and truth-conditions. Thus meaning is captured by inferential role. In [reference suppressed] it is suggested that the proof-theoretic meaning of a determiner D is the following (with some details suppressed): [[D]] = λz1λz2λΓ. ⋃ j1,...,jm∈P ID(z1)(z2)(j1) · · · (jm)(Γ), where the notation means (with more detail in the paper): – z1 ranges over (proof-theoretic) noun meanings and z2 over (proof-theoretic) vp meanings. – Γ is a collection of NL sentences, from which a conclusion sentence S including a dp headed by D (here, in subject position only, for simplicity) can be inferred. –The jks are individual parameters. – ID is the introduction-rule (I-rule) for D in the meaning-conferring ND-system. The dp headed by D is introduced into the subject position of S similarly to introducing a connective or quantifier into a logical formula, and similarly for elimination. Based on this characterisation of determiners’ meanings and on a suitable adaptation of conservativity to a proof-theoretic setting, it was proved [reference suppressed] that every determiner is conservative. Thus, conservativity cannot serve as a PTS criterion for classifying determiners meanings. In this paper, I want to argue for another classifying property of determiners meanings, based on their prooftheoretic characterization. This criterion is the well-known harmony property of the I/E-rules, the absence of which disqualifies an ND-system from being considered as meaning-conferring. The paper is structured as follows. – A presentation of a simplification of the extensional fragment of English and its accompanying ND meaning-conferring proof-system, in terms of which the issue is discussed. – A definition of harmony [2], and proof of harmony of the meaning-conferring rules for the fragment’s determiners every, some and no. – A proof-theoretic definition of a “pathological” determiner, donk, by means of I/E-rules, that in spite of being conservative cannot be an admissible NL determiner. – A proof of the disharmony of the I/E-rules for donk, that in terms of truth-conditions, the effect of donk in a sentence like donk girl smiles, is that either every girl smiles or no girl smiles, a trivialising effect. – Conclusion: harmony of a determiner’s I/E-rules is a necessary condition for the determiner being NL-admissible.