In Chatzikyriakidis and Luo (2012), we have only studied how to give generic typing to coordinating connectives such as “and”, but not how to characterize their semantic behaviors. For example, as described in section 2.3.2, we have introduced the universe LTYPE with which “and” can be interpreted as And : ΠA:LTYPE. A → A → A, but we have not described ways to define the semantic operator And.

This appendix is based on the second author’s notes (Luo 2018b). It uses And as an example to describe how to define behaviors of coordinating connectives for each type in LTYPE. Formally, the introduction rules of LTYPE are given in Figure A4.1 – it contains the following three kinds of types as its objects:

• 1) the predicate types of the form Πx₁:A₁...Πx_n:A_n.P rop;
• 2) the types representing CNs (i.e. the types in CN);
• 3) the universe CN itself.

For each A : LTYPE, the semantic behavior of And(A) : A → A → A is characterized by case analysis on A as follows:

1) If A = Πx₁:A₁...Πx_n:A_n.Prop then, for any f, g : A,

And(A, f, g)(x₁, ..., x_n)=_df f(x₁, ..., x_n) ∧ g(x₁, ..., x_n) : Prop.

When n = 0, the above reduces to (2.36) in section 2.3.2.

2) If A : CN then, for any distributive predicate P : A → Prop,¹

3) If A = CN, then for A₁,A₂ : CN and for C : CN such that A_i ≤ C (i = 1, 2),

And(CN, A₁, A₂)=_df Σx:C. IS_C (A₁,x) ∧ IS_C (A₂,x),

where IS is the operator for propositional forms studied in section 3.2.3 and section 7.1.

Then, the examples (2.25-2.32) in section 2.3.2 are given semantics as follows:

(A4.1) (sentences)

• [[John walks and Mary talks]]
• = And(Prop, walk(j), talk(m)) = walk(j) ∧ talk(m).

(A4.2) (verbs)

• [[John walks and talks]]
• = And(Human → Prop, walk, talk)(j) = walk(j) ∧ talk(j).

(A4.3) (adjectives)

• [[Mary is pretty and smart]]
• = And(Human → Prop, pretty, smart)(m) = pretty(m) ∧ smart(m).

(A4.4) (adverbs)

• [[The plant died slowly and agonizingly]]
• = And(T, slowly, agonizingly)(Plant, die, the_plant)
• = slowly(Plant, die, the_plant) ∧ agonizingly(Plant, die, the_plant), where T = ΠA:CN.(A → Prop) → (A → Prop).

(A4.5) (quantified NPs)

• [[Every student and some professors came]]
• = And((Human → Prop) → Prop, ∀(Student), ∃(P rofessor))(come) = ∀(Student, come) ∧∃(P rofessor, come).

(A4.6) (quantifiers)

• [[Some but not all students got an A]]
• = And(T, some, notAll)(Student, getA)
• = ∃(Student, getA) ∧¬∀(Student, getA), where some and notAll are of type T = ΠA:CN.(A → Prop) → Prop and defined as some(A, P) = ∃(A, P) and not All(A, P) = ¬∀(A, P).

(A4.7) (proper names)

• [[John and Mary went to Italy]]
• = goItaly(And(Human, j, m)) = goItaly(j) ∧ goItaly(m).

(A4.8) (CNs)

• [[A friend and colleague came]]
• = ∃(And(CN, Friend, Colleague), come)
• = ∃(Σx:Human.IS(Friend, x) ∧ IS(Colleague, x), come)
• ⇔ ∃x:Human. IS(Friend, x) ∧ IS(Colleague, c) ∧ come(x).