In this chapter we have worked through a compositional analysis for the meanings of simple sentences like those in (4), (26), and (29). We have developed a rule of semantic interpretation for simple clauses of the form [NP VP] (see rule 22), a similar rule for transitive VPs (rule 28b), and a rule for adjective modifiers (30c). We have shown how these rules can be applied in a stepwise fashion to derive the truth-conditions of a simple sentence from the denotations of the words that it contains and the manner in which those words are combined syntactically.
In discussing the meanings of quantifiers, conditionals, tense markers etc. in later chapters we will focus more on understanding the phenomena than on formalizing the rule system, but we will still draw heavily on the concepts introduced in this chapter. Moreover, an important assumption in everything that follows is that our description of the meanings of these elements must be compatible with the kind of compositional analysis illustrated in this chapter.
Further reading
Good brief introductions to set theory are provided in Allwood et al. (1977: ch. 2), J. N. Martin (1987: ch. 2), Coppock (2016: ch. 2); and McCawley (1981a: ch. 5). Readable introductory textbooks include Halmos (1960) and Enderton (1977). Formal introductions to truth-conditional semantics are provided in Dowty et al. (1981) and Heim & Kratzer (1998). An informal discussion of this approach is presented in E. Bach (1989). A brief introduction to Model Theory is provided by Hodges (2013). Standard textbooks for this topic include Chang & Keisler (1990) and Hodges (1997).
Discussion exercises
A. Set theory Fill in the following tables:
B. Model theory
(1) Sketch a picture of the situation defined by the following model:
a. the set of individuals U = {Able, Baker, Charlie, Doug, Echo, Fred, Geronimo}
b. denotation assignments:
((FISH)) = {Able, Baker, Charlie, Doug}
((SUBMARINE)) = {Echo}
((SEAHORSE)) = {Fred, Geronimo}
((RED)) = {Able, Baker, Fred}
((GREEN)) = {Charlie, Geronimo}
((BLUE)) = {Doug, Echo}
((SWIM)) = {Able, Baker, Charlie, Doug, Fred, Geronimo}
((OCTOPUS)) = ∅
((FOLLOW)) = {⟨Able, Echo⟩, ⟨Doug, Able⟩, ⟨Doug, Echo⟩, ⟨Charlie, Fred⟩}
((a)) = Able
((b)) = Baker
((c)) = Charlie
((d)) = Doug
((e)) = Echo
((f)) = Fred
((g)) = Geronimo
(2) Complete the following table by providing logical formulae and set-theoretic interpretations for sentences (e–i), and evaluate the truth value of each sentence relative to the model provided above.
(3) Draw annotated tree diagrams for the following sentences showing how their truth conditions would be derived compositionally from our rules of interpretation:
a. Henry snores.
b. Henry loves Jane.
c. Henry is a happy man.
a Platypus plus four species of echidna
b Note: there were no female French monarchs.
Homework exercises
A: Assume that the following individuals are included in our universe of discourse:
((b)) = Mrs. Bennet
((c)) = Mr. Collins
((d)) = Mr. Darcy
((e)) = Elizabeth (Bennet)
((l)) = Lydia (Bennet)
((w)) = Mr. Wickham
For each of the following logical formulae, provide an English translation and an interpretation stated in terms of set notation. Then create a model under which sentences (2–3) will be false, and the rest (including 1) will be true.a
1. LOVE(d,e)
2. REJECT(e,c)
3. ∀x [(MAN(x) ∧ WEALTHY(x)) → ADMIRE(b,x)]
4. ∃x [MAN(x) ∧ WEALTHY(x) ∧ ADMIRE(b,x)]
5. ¬∃x [WOMAN(x) ∧ LOVE(x,c)]
6. DECEIVE(w,l) ∧ RESCUE(d,l)
7. ∀x [WOMAN(x) → CHARM(w,x)] ∧∀y [MAN(y) → ANGER(w,y)]
a Patterned loosely after Saeed (2009: 350).
