PLIN0009 Semantic Theory

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PLIN0009 Semantic Theory

Coursework 2

Instructions

-  You have until 17 May 2024, 12 noon to complete the questions below.

-  Answer all questions in entirety. Label your answers on your document by section number + letter (e.g., ‘1a)’ for question a) in section 1).

-  Pay careful attention to all instructions. If anything is unclear, please post your question on the Moodle forum.

-  You  may handwrite  (part of) your answers, as long as you scan them and submit the file on Moodle, but please make sure that they are legible and display correctly on Turnitin.

-  Do not use AI to answer the questions.

-  If you consult other sources, please indicate them in a bibliography and in the text.

1 - Properties of Quantifiers

Conservativity

The conservativity of a natural language quantifier Q can be defined as follows: given two sets A and B, ∈ [Q]M  iff   ∈ [Q]M.

As an example, consider the following sentence:

(1)       Every child plays.

This sentence is true in M iff <[child]M, [plays]M> ∈ [every]M. If every is conservative, we also expect that <[child]M, [child]M ∩ [plays]M> ∈ [every]M. This could be paraphrased as Every child is a child that plays.

a) Show that every is conservative by constructing a model M in which <[child]M, [plays]M> ∈ [every]M (i.e., in which (1) is true). To do so, provide the extensions of [child]M and [plays]M in list notation, and briefly explain why <[child]M, [plays]M> ∈ [every]M  holds. (You will need to go back to the extension of the quantifier). Then, explain why it also holds that <[child]M, [child]M ∩ [plays]M> ∈ [every]M.

b) Show that some is conservative. Assume that the following sentence is true in M:

(2)       Some books are interesting.

c) Show that no is conservative. Assume that the following sentence is true in M:

(3)       No book is boring.

d) Now consider the following hypothetical quantifier:

[wug]M  = { : |X| = |Y|}

Discuss whether wug is conservative, taking as a starting point the following sentence:

(4)       Wug book is interesting.

Hint: consider both conditions that must be met for wug to be conservative.

I.e., for any two sets X and Y, it must hold that  ∈ [wug ⟧M and that  ∈ [wug ⟧M. If you can come up with a model in which one of the two conditions holds but the other does not, then you have shown that wug is non-conservative.

e) Consider  another  hypothetical   quantifier  (notice  a  crucial  difference  with   respect  to  the denotation of every):

[wis⟧M  = { : Y ⊆ X}

Discuss whether wis is conservative, following the same hint provided for wug and taking as a starting point the following sentence:

(5)       Wis student left.

Monotonicity

Determine the monotonicity patterns of the following quantifiers. To do so, for each quantifier Q construct a model M such that given an arbitrary NP and an arbitrary VP, <[NP ⟧M, [VP ⟧M> is a member of [Q⟧M. This implies that a certain relation must hold between [NP ⟧M and [VP ⟧M. To work on monotonicity, you may want to refer to a subset and a superset of [NP ⟧M and a subset and a superset of [VP ⟧M.

Then, determine whether the quantifier is ↑MON, ↓MON, MON↑, or MON↓ . Remember that it is generally easier to prove that a certain property does not hold. For the purposes of this assignment, it will be sufficient to show that a given relation does hold in at least one situation to show that the relevant quantifier is monotonic in the relevant direction (e.g. ↑MON).

As an example:

(i) Every student left.

↑MON: every is not ↑MON. Every student left does not entail every person left. There may be a situation in which some people who are not students did not leave.

↓MON: every is ↓MON. Every student left entails every undergraduate student left.

MON↑: every is MON↑ . Every student left entails every student moved.

Now consider the following ambiguous sentence:

(14)     Every politician is not corrupt.

Here the scope relations are between the quantifier and negation.

d) Represent the two readings in predicate logic; assume that P = politician and C = corrupt.

e) Does the surface scope reading entail the inverse scope reading? If it does not, briefly present a situation in which the surface reading is true but the inverse reading is false.

f) Does the inverse scope reading entail the surface scope reading? If it does not, briefly present a situation in which the inverse reading is true but the surface reading is false.

4 - Tense and Aspect

a) Based on the following data:

(15)     a.         ?Mary builds a house.

b.         Mary is building a house.

(16)     a.         Mary knows French.

b.         *Mary is knowing French.

What  generalisation  can  you  make  about  a  verb’s  ability  to  appear  in  the  present  tense  ((a) sentences)  or in the progressive  ((b)  sentences)  depending  on  that verb’s Aktionsart (aspectual class)?

b) In the Reichenbach-Klein theory of tense-aspect, there are three main coordinates: Utterance Time (UT), Reference Time (RT), and Event Time (ET).

In the present tense, we can assume that UT = RT and RT = ET. In the simple past, on the other hand, RT < UT and RT = ET:

(17)     [Assume now = 6pm]

Mary ate at 4pm.         UT: now; RT: 4pm; ET: 4pm

Now consider the following tenses:

(18)     At 4pm, Mary had eaten.

(19)     At 4pm, Mary will have eaten.

What precedence (<) relations between ET and RT, and between RT and UT, can be established in

(18) and (19), respectively?