We have encountered this question when we try to read Heim and Kratzer's book. This following picture is taken from Heim & Kratzer (1998: 40).

enter image description here

Our answers are simply based on the subscripts:

(a)  <<e,t>, <e,t>>
(b) <<e,<e,t>>, <e,t>>
(c) <e,<<e,t>, <e, t>>>
(d) <<e,t>,t>
(e) invalid?
(f) <<e,t>,<<e,t>,t>>

However, we could hardly translate those λ-calculus into natural language. Were formal semantics in general meant to be used for syntax-to-semantics mapping but not the other way around? See more comments in this blog.

Any clarification or comments are welcome!

  • @Keelan I am not sure if I understand. Do you mean it is simply a unit of a constituent rather than a well formed phrase. So, for a, it actually denotes gray in the constituent of big gray cat?
    – Yili Xia
    Commented Apr 28, 2023 at 12:35
  • 1
    Sorry, my earlier comment was a bit confused. I have written an answer that should be more helpful.
    – Keelan
    Commented Apr 28, 2023 at 13:26
  • Millions of thanks! Keelan! I am starting to read the book and cannot believe that those questions embedded in the first few chapters have already indicated a lot of gems!
    – Yili Xia
    Commented Apr 28, 2023 at 14:03

1 Answer 1


(a) is the denotation for gray proposed in (11) on p. 66. Its type is (et)et so that when applied to cat with type et we get gray cat with type et as well.

In (b) we have a predicate f of which Ann is the subject but the object position is left unspecified, for example Ann verbed ___. The denotation is the set of entities which Ann verbed. It is thus somewhat like the passive predicate was verbed by Ann. Such an account of the passive is given by Dowty and briefly discussed by Heim & Kratzer in note 18 on pp. 59–60.

(c) is like (a), but gray has been replaced with in y, and y has been made an argument. It is the denotation for in on p. 66. If in is of type e(et)et, then in the box is of type (et)et, like gray above.

(d) is the denotation of something on p. 141.

Since Mary is type e, (e) is of type (et)e. We could imagine a determiner actuallymary which maps any NP to Mary: the denotation of actuallymary cat would be Mary (compare: the cat maps the predicate cat to a single entity). There must of course be a reason why such determiners do not exist. H&K suggest that this is a syntactic constraint rather than a semantic one (see pp. 126–127). In the semantic system, the formula is well-formed and has meaning, but you would not use it as the denotation for any expression in natural language.

(f) is the denotation of the quantifier no on p. 146.

  • Thanks very much Keelan! Very detailed and informative answer! Many thanks for providing extra notes on the page of the book. Heim & Kratzer's book is super technical and requires a lot of familiarities on the functions.
    – Yili Xia
    Commented Apr 28, 2023 at 13:50
  • 1
    @Xia.Yili no problem. Of course it’s fine to take the question at this point just as an exercise in understanding the technical machinery, without translating them to natural language. But it’s fun to do so nevertheless :)
    – Keelan
    Commented Apr 28, 2023 at 20:44
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    Why do you assume that Mary is of type et, and not e? Commented Apr 28, 2023 at 21:04
  • @NatalieClarius Thanks, corrected.
    – Keelan
    Commented Apr 29, 2023 at 5:39
  • I was confused too. Thanks for pointing it out.
    – Yili Xia
    Commented Apr 29, 2023 at 13:14

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