Semantics 1, SoSe 2016 (Sailer): Assignment Sheet 1

On this page you find an example solution to the first assignment sheet.

Download the assignment sheet: File:SoSe16-assignment-logic.pdf

Task 0: Choose a text

The answers to this assignment sheet will be based on Shakespeare's Macbeth.

You can find a summary of the play here: http://www.sparknotes.com/shakespeare/macbeth/summary.html

Task 1: Ambiguity

1. Write down two ambiguous sentences with respect to the book’s content.
2. For each of these, provide an unambiguous paraphrase for the possible readings.
3. Classify the type of ambiguity.
4. For each of the readings: Is one of them more plausible in the context of the book than the other?

Here are four example sentences.

(1) a. Duncan trusted Macbeth because he was a thane.

Check your answer

1. Example: Duncan trusted Macbeth because he was a thane.
2. Reading 1: he refers to Macbeth. Paraphrase: Duncan trusted Macbeth because Macbeth was a thane.
   Reading 2: he refers to Duncan. Paraphrase: Duncan trusted Macbeth because Duncan was a thane.
3. Type of ambiguity: referential ambiguity
4. The first reading is maybe slightly more plausible because Duncan is not only a thane, but also the king. However, in the context of the play, both readings would make sense.

b. Every king trusts a thane.

Check your answer

1. Example: Every king trusts a thane.
2. Reading 1: every takes scope over a. Paraphrase: For every king there is at least one thane such that the king trusts that thane.
   Reading 2: a takes scope over every. Paraphrase: There is one particular thane such that each king trusts this thane.
3. Type of ambiguity: scope ambiguity
4. The sentence is false under both readings, because Macbeth is king at some point, but he does not trust anyone. Normally, the first reading is prefered, because different kings may consider different thanes trustworthy.

b. Macbeth and Macduff are married.

Check your answer

1. Example: Macbeth and Macduff are married.
2. Reading 1: collective reading. Paraphrase: Macbeth and Macduff are married to each other
   Reading 2: distributive reading. Paraphrase: Macbeth and Macduff are both married, but not to each other.
3. Type of ambiguity: collective-distributive ambiguity
4. In the play, the second reading is preferred because the spouses of both Macbeth and Macduff appear in the play.

b. Macbeth killed a king with a dagger.

Check your answer

1. Example: Macbeth killed a king with a dagger.
2. Reading 1: the PP with a dagger is a modifier of the verb kill Paraphrase: Macbeth used a dagger to kill a king.
   Reading 2: the PP with a dagger is a modifier of the noun king. Paraphrase: Macbeth killed a king who had a dagger.
3. Type of ambiguity: structural ambiguity
4. In the play, the first reading is more plausible because Macbeth uses a dagger to kill Duncan. During the assassination, Duncan is asleep and we do not know nor care whether he has a dagger as well.

Task 2: Model and Interpretation

(Note: For this task you do not need to use the eventuality variable)

1. Define a universe consisting of four main characters from your book.

Check your answer

U = { Macbeth, Banquo, LadyMacbeth, Duncan }

2. Introduce names for your characters and provide their interpretations.

Check your answer

I(macbeth) = Macbeth,
I(banquo) = Banquo
I(lady-macbeth) = LadyMacbeth,
I(duncan) = Duncan

3. Introduce three property symbols relevant to the plot and provide their interpretations.

Check your answer

I(thane₁) = {<Macbeth>, <Banquo>, <Duncan>},
I(king₁) = {<Macbeth>, <Duncan>},
I(witch₁) = {}

4. Introduce two 2-place relation symbols relevant to the plot and define their interpretations.

Check your answer

I(mistrust₂) = {<Macbeth, Banquo>, <Banquo, Mactbeth>},
I(kill₂) = {<Macbeth,Banquo>, <Macbeth, Duncan>, <LadyMacbeth,LadyMacbeth>}

Task 3: Atomic formulæ

Write down two atomic formulæ and compute their truth value with respect to your model as defined in Task 2.

1. king₁(macbeth)

Check your answer

[[king₁(macbeth)]] = 1
iff < [[macbeth]] > is in [[king₁]]
iff < I(macbeth) > in I(king₁)
iff < Macbeth > in { <x> | x is king } = { <Macbeth>, <Duncan> }

Since this isthe case, [[king₁(macbeth)]] = 1.

2. mistrust₂(macbeth,macbeth)

Check your answer

[[mistrust₂(macbeth,macbeth)]] = 1
iff < [[macbeth]], [[macbeth]] > is in [[mistrust₂]]
iff < I(macbeth), I(macbeth) > in I(mistrust₂)
iff < Macbeth, Macbeth > in { <x,y> | x mistrusts y } = { <Macbeth, Banquo>, <Banquo, Macbeth> }

Since this is not the case, [[mistrust₂(macbeth,macbeth)]] = 0.

Task 4: Complex formulæ

Combine your two formulæ from Task 3 into two complex formulæ. Use three different connectives for this.

1. ¬king(macbeth)

Check your answer

[[¬ king₁(macbeth)]] = 1
iff [[king(macbeth)]] = 0

As we saw in Task 3, this is not the case. Therefore: [[¬ king₁(macbeth)]] = 0.

2. king₁(macbeth) ⊃ mistrust₂(macbeth,macbeth)

Check your answer

[[king₁(macbeth) ⊃ mistrust₂(macbeth,macbeth)]] = 1
iff [[king₁(macbeth)]] = 0 or [[mistrust₂(macbeth,macbeth) = 1

Since we saw in Task 3 that the first subformula is true but the second formula is false, this is not the case. Therefore, [[king₁(macbeth) ⊃ mistrust₂(macbeth,macbeth))]] = 0.

Task 6: Truth tables

p	q	r	not r	q or (not r)	if p then (q or (not r))
1	1	1	0	1	1
1	1	0	1	1	1
1	0	1	0	0	0
1	0	0	1	1	1
0	1	1	0	1	1
0	1	0	1	1	1
0	0	1	0	0	1
0	0	0	1	1	1

Task 7: Variables

1. Provide a variable assigment function g which maps the variables x1,..., x5 to members of your universe.
2. Provide one formula that contains two occurrences of variables.
3. Compute the truth value of this formula with respect to your assingment function g.

Check your answer

Example solution (other values for g are equally possible).

g(x1) = Macbeth,
g(x2) = Banquo,
g(x3) = Banquo,
g(x4) = LadyMacbeth,
g(x5) = Macbeth.

With this variable assignment we can compute the truth value of the formula:

[[kill(x5,x2)]]^g = 1
iff < [[x5]]^g, [[x2]]^g > is in [[kill]]^g
iff < g(x5), g(x2) > is in I(kill)
iff < Macbeth, Banquo > is in { <x,y> | x killed y} = { <Macbeth, Banquo>, <Macbeth,Duncan>, <LadyMacbeth,LadyMacbeth> }.

Since this is the case, [[kill(x5,x2)]]^g = 1.

Task 9: Quantifiers

1. Write down one formula with a quantifier.
2. For each individual in your universe, indicate whether the restrictor and the scope are true for that individual.
3. Given your results from (b), is the formula true in your model?
4. In which way would your model have to be different to make the formula false (or, in case the formula is false: to make it true in your model)?

An example with an existential and one with a universal quantifier are given here.

1. Banquo was killed by a king.

Check your answer

∃x (king(x) : kill(x, banquo))

g(x)	king(x)	kill(x,banquo)
Macbeth	1	1
Banquo	0	0
LadyMacbeth	0	0
Duncan	1	0

The formula is true in my model, because there is an individual, Macbeth, such that Macbeth is king and killed Banquo.
(Note: The English sentence is in passive, but this has no effect on the logical form.)

2. Macbeth mistrusts every witch.

Check your answer

∀x (witch(x) : mistrust(macbeth, x))

g(x)	witch(x)	mistrust(macbeth,x)
Macbeth	0	0
Banquo	0	1
LadyMacbeth	0	0
Duncan	0	0

The formula is true in my model, because there are no witches in my model. Therefore, the formula with the universal quantifier is trivially true.

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