CARDINAL, ORDINAL, AND ‘#’-CONSTRUCTIONS

Cardinal, ordinal, and ‘#’-constructions express concepts of cardinality, numerical rank, and numerical label. Cardinal measure constructions (as opposed to cardinal counting constructions) include concepts of measure functions as part of their meaning. The different numerical concepts are integrated in the semantic representation of complex number word constructions. In the following representations, a is an empirical object (in the case of cardinal constructions the empirical object a is a set), p is a progression, s is a set, and /θriː/ is an element of the English counting sequence C (that is, /θriː/ is an element of a set of numerical tools).

Semantic representations for the different classes of number word constructions

• Cardinal counting construction: three pens: ɛa[PEN⊕(a) ∧ NQ(a, /θriː/)]

• Cardinal measure construction: a pumpkin of 3 kg: ɛa[PUMPKIN1(a) ∧ NQ(KG(a), /θriː/)]

• Ordinal construction: the third player: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (ordinal): player #3: ιa ∃p[PLAYER1(a) ∧ NR(a, p, /θriː/)]

• ‘#’-construction (nominal): player #3: ιa ∃s[PLAYER1(a) ∧ NL(a, s, /θriː/)]

Explanations of the elements used in the formulae

1. ɛ: The ‘ɛ’-operator is used to model nominal terms: F(ɛx(G(x))) =df. ∃x(G(x) ∧ F(x)).

2. ι: The ‘ι’-operator is used to model definite terms: ‘ιx(F(a))’ stands for the most salient F, that is, the most salient realisation of a concept F.

3. F1: ‘ɛa(F1(a))’ stands for ‘one F’, that is, F1(a) is true iff a is a singleton of realisations of F; a singleton of realisations of F is defined as a set of Fs with the numerical quantity ‘one’;

4. hence, employing our definition of NQ and of English words as an instance of numerical tools, we can state that F1(a) is true iff NQ(a, /w∧n/) is true.

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