Semantics 1, WiSe 1718

General information

Course description

Semantics is the study of the (literal) meaning of words and sentences. The meaning of a sentence is usually predictable from the words in the sentence and its syntactic structure. Yet, this relationship between form and meaning is not a simple one-to-one mapping. Instead, it is rich in ambiguities, pleonastic marking and elements without any identifiable meaning contribution. We will work on an account that is founded on classical tools of semantic research but still directly addresses these empirical challenges. After the class, the participants will be able to identify - and partly analyze - interesting semantic phenomena in naturally occurring texts. They will have acquired a basic working knowledge in formal logic, which they will be able to apply in the description of meaning

Time and place

• Tuesday 08:15-9.45
• Starting: 17.10.2017
• Room: IG 3.201 (IG-Farben-Haus)

Modules

• Lehramt Englisch (L2/5, L3): FW2
• BA English Studies: 3.4(1)
• BA Empirische Sprachwissenschaft: K 6.1, En 4.1, DH 6.1

Contact

Manfred Sailer
e-mail: sailer@em.uni-frankfurt.de
office: IG 3.214
office hours: contact via e-mail!
www: http://user.uni-frankfurt.de/~sailer/index.htm

Course requirements

L2 and L5

• regular attendance
• pass all assignment sheets
• Modulprüfung (optional): 90 min written exem (2 CP)

L3

• regular attendance
• pass all assignment sheets
• Modulprüfung (optional):
  • 20 min. oral exam
  • not possible: kleine Hausabeit

BA English Studies

• regular attendance
• pass all assignment sheets
• literary scenario:

Part 1: Extract 15 ambiguous sentences from the text such that all types of ambiguity covered in class are represented provide unambiguous paraphrases of the readings determine the type of ambiguity

Part 2:

Define a formal model consisting of 3 characters from your text, which contains 2 properties, 1 2-place relation

Formulate 2 atomic formulae and compute their truth value.

Formulate 4 complex formulae with at least 1 logical connective in each and compute their truth value.

Formulate 1 complex formula with at least 2 logical connectives in

it and compute its truth value.

BA Empirische Sprachwissenschaft

K 6.1

• regular attendance
• Modulprüfung (obligatory): 90min. written exam

En 4.2

• regular attendance
• pass all assignment sheets
• literary scenario

DH 6.1

to be added

Erasmus 6 CP

• regular attendance
• pass the assignment sheets
• 90min. written exam
• small literary scenario:

Part 1: Extract 4 ambiguous sentences from the text such that different types of ambiguity covered in class are represented provide unambiguous paraphrases of the readings determine the type of ambiguity

Part 2:

Define a formal model consisting of 3 characters from your text, which contains 2 properties, 1 2-place relation

Formulate 2 atomic formulae and compute their truth value.

Formulate 2 complex formulae with at least 1 logical connective in each and compute their truth value.

Formulate 1 complex formula with at least 2 logical connectives in

it and compute its truth value.

The grade will be determined by the result of the written exam.

Assignment sheets

Assignment sheet 1: File:WiSe1718-assignment-logic.pdf

Assignment sheet 2: File:WiSe1718-assignment-lrs.pdf

Mock exam: File:WiSe1718-mockexam.pdf

Mock exam

The examples in the text are based on Shakespeare's play Macbeth. The full text of the play is available on Projekt Gutenberg.

We will use the TV show The Fresh Prince of Bel-Air for the final exam this term.

Task 1: Ambiguity

Consider the following ambiguous sentences. For each of them, provide an unambiguous paraphrase for the possible readings.

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

Check your answer

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.

b. Every king trusts a thane.

Check your answer

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.

b. Macbeth and Macduff are married.

Check your answer

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.

b. Macbeth killed a king with a dagger.

Check your answer

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.

Task 2: Model and Interpretation

1. Define a universe that consists of Macbeth and Banquo.

Check your answer

U = { Macbeth, Banquo }

2. Define the interpretation of the names macbeth and banquo in an intuitively plausible way.

Check your answer

I(macbeth) = Macbeth,
I(banquo) = Banquo

3. Define the interpretation of the properties thane₁, king₁, and witch₁ is such a way that Macbeth is a king, both are thanes and neither is a witch.

Check your answer

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

4. Define the interpretation of the 2-place relations mistrust₂ and kill₂ in such a way that Macbeth and Banquo mistrust each other and Macbeth kills Banquo.

Check your answer

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

Task 3: Formulae

Write down logical formulae that express the meaning of the following sentences.

1. Banquo is a thane.

Check your answer

thane₁(banquo)

2. Macbeth is king and Macbeth mistrusts Banquo.

Check your answer

king₁(macbeth) ∧ mistrust₂(macbeth,banquo)

3. If Banquo is king then Macbeth does not kill Banquo.

Check your answer

king₁(banquo) ⊃ ¬ kill₂(macbeth,banquo)

Task 4: Interpreting formulae

Compute the interpretation of the following formulæ step by step.

1. 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.

2. ¬king(banquo)

Check your answer

[[¬ king₁(banquo)]] = 1
iff [[king(banquo)]] = 0
iff < [[banquo]]> is not in [[king₁]]
iff < I(banquo> is not in I(king₁)
iff < Banquo > is not in { <x> | x is king } = { <Macbeth>}

Since this is the case, [[¬ king₁(banquo)]] = 1

3. witch₁(banquo) ⊃ king₁(macbeth)

Check your answer

[[witch₁(banquo) ⊃ king₁(macbeth))]] = 1
iff [[witch₁(banquo)]] = 0 or [[king₁(macbeth) = 1
iff < [[banquo]] > is not in [[witch₁]] or < [[macbeth]] > is in [[king₁]]
iff < I(banquo) > is not in I(witch₁) or < I(macbeth) > is in I(king₁)
iff < Banquo > is not in { <x> | x is a witch} = { } or < Macbeth > is in { <x> | x is king} = { <Macbeth>}.

Since both are the case, [[witch₁(banquo) ⊃ king₁(macbeth))]] = 1.

Task 5: Variables

Provide a g-function that maps the variables x, y, and z to individuals from the universe and compute the interpretation of the following formula with respect to the model and your g.

(i) kill₂(z,x)

Check your answer

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

g(x) = Macbeth,
g(y) = Banquo,
g(z) = Banquo.

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

[[kill₂(z,x)]]^g = 1
iff < [[z]]^g, [[x]]^g > is in [[kill₂]]^g
iff < g(z), g(x) > is in I(kill₂)
iff < Banquo, Macbeth > is in { <x,y> | x killed y} = { <Macbeth, Banquo> }.

Since this is not the case, [[kill₂(z,x)]]^g = 0.

Task 6: Quantifiers

Provide logical formulae that expresse the meaning of the following sentences. Are the formulae true in your model (not in the entire play)? Give a short reason (you don’t need to compute the truth value).

1. Banquo was killed by a king.

Check your answer

∃x (king(x) : kill(x, banquo))

The formula is true in my model, because there is only one king, Macbeth, and Macbeth 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))

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.

Task 7: Analysis

Provide the lexical entries for the words in the sentence Banquo mistrusted Macbeth. Use the features PHON, HEAD, SUBJ, SPR, COMPS, DR, PARTS, and EX-CONT.

Check your answer

	Banquo	mistrusted	Macbeth
	[1]		[2]
PHON	< Banquo >	< mistrusted >	< Macbeth >
HEAD	noun	verb	noun
SUBJ	< >	< NP[DR [a]] >	< >
SPR	< >	< >	< >
COMPS	< >	< NP[DR [b]] >	< >
DR	[a]banquo	[c]mistrust2	[b]macbeth
EX-C	??	??	??
PARTS	< [a]banquo >	< [b]mistrust2, [b]([a],[c]) >	<[c]macbeth >

Task 8: Analysis

Provide the full HPSG and LRS analysis of the sentence Banquo mistrusted Macbeth. Use the features PHON, HEAD, SUBJ, SPR, COMPS, DR, PARTS, and EX-CONT. You only need to mention the EX-CONT value at the highest node in the tree.

Check your answer

Tree structure:

	Banquo	mistrusted	Macbeth
	[1]		[2]
PHON	< [4] Banquo >	< [5] mistrusted >	< [6] Macbeth >
HEAD	noun	[3] verb	noun
SUBJ	< >	< [1] NP[DR [a]] >	< >
SPR	< >	< >	< >
COMPS	< >	< [2] NP[DR [b]] >	< >
DR	[a]banquo	[c]mistrust2	[b]macbeth
EX-C	??	[d]	??
PARTS	<banquo >	<mistrust2, mistrust2([a],[c]) >	<macbeth >

	VP: mistrusted M.	S: B. mistrusted M.
PHON	< [5], [6] >	< [4], [5], [6] >
HEAD	[3]	[3]
SUBJ	< [1] NP>	< >
SPR	< >	< >
COMPS	< >	< >
DR	[c]	[c]
EX-C	[d]	[d]
PARTS	<mistrust2, mistrust2([a],[c]) ,	<mistrust2, mistrust2([a],[c]),
	macbeth >	macbeth, banquo >

Task 8': Principles of syntax

1. How is the COMPS value of the VP determined by the lexical entries of the words and the principles of grammar?

Check your answer

The VP is licenced as a head-complement structure. The constraint on head-complement structures requires that the head daughter have a non-empty COMPS list and the mother have an empty COMPS list.
(It is also required that the non-head daughters are identical to the elements on the head daughter's COMPS list, but this is not relevant for the question at hand.)

2. How is it guaranteed that the PHON values of the words all appear in the PHON value of the sentence?

Check your answer

The Phonology Principle specifies that the PHON value of a mother is the concatenation of the PHON values of its daughter(s). Therefore, a element of the PHON value of a word in a sentence will always be part of the PHON values of every phrase that dominates this word. Since the overall sentence dominates all its component words, its PHON value comprises the PHON values of all words of this sentence.

3. How is it achieved that the HEAD value of the sentence is verb?

Check your answer

The HEAD value of the lexical verb, mistrust, is verb. In the VP, the lexical verb is the syntactic head of the phrase. According to the Head Feature Priniciple, the HEAD value of the mother node is the same as that of its head daughter, i.e., verb. Since this VP is the headdaughter of the sentence, the sentence's HEAD value should also be the same, i.e., verb again.

Task 9: General mechanisms of LRS

Explain how the following formulae are excluded from occurring as EX-CONT values of the sentence from Task 7.

(a) mistrust₂(macbeth,banquo,banquo)

(b) mistrust₂(banquo,banquo)

(c) macbeth(mistrust₂,banquo)

(d) mistrust₂(macbeth,banquo)

Check your answer

(a) The expression cannot be a possible logical form, because it is not a well-formed formula: the predicate mistrust can only combine with two arguments, not with three. (This is indicated with the element mistrust₂(...,...).

(b) The formula does not use all expressions from the PARTS list: the expression macbeth is missing.

(c) macbeth denotes an individual , mistrust₂ is a predicate. Therefore, macbeth cannot function as predicate, not can mistrust₂ function as its argument.

(d) The subject, Banquo, should be linked to the first semantic argument slot of the predicate mistrust₂ - analogously for the complement and the second argument slot of the predicate. This is specified in the lexical entry of the verb, where the DR values of the subject ([a]) and the complement ([b]) are identified with the first and the second argument slots of mistrust₂ respectively.

Material for week 14 (30.1.2018)

Possible EX-CONT values

Given the following PARTS lists, what are possible EX-CONT values (if we do not assume other restrictions)

1. PARTS < pat, alex,like, like(__,__) >

Check your answer

like(pat,alex)
like(alex,pat)

2. PARTS < alex,snore, snore(__), ¬(__) >

Check your answer

¬(snore(alex))

3. PARTS < alex,alex,snore >

Check your answer

There is no possible EX-CONT value because the three elements on the PARTS list cannot be combined.

3. PARTS < alex,alex,snore, snore(__) >

Check your answer

snore(alex)

4. PARTS < alex,alex,snore, snore(__), __ ∧ __ >

Check your answer

snore(alex) ∧ snore(alex)

Analysis of simple sentences

Indicate the missing values of the VAL and the HEAD features using tags ([1], ...) or "-" for empty lists.

Alex snored.

syntactic structure:

Words:                                                                                                   Phrase:

Alex                                                             snored                                    S: Alex snored.

HEAD [4]noun                                  HEAD [5]verb                                    HEAD

SUBJ < >                                  SUBJ < >                                    SUBJ < >

SPR   < >                                  SPR < >                                     SPR < >

COMPS < >                              COMPS < >                               COMPS < >

Indicate the missing values of the VAL and the HEAD features using tags ([1], ...) or "-" for empty lists.

Fido chased a mouse.

syntactic structure:

Words:

Fido                                                             chased                                    a                                                              mouse

HEAD [8]noun                                  HEAD [9]verb                                    HEAD [10] det                                   HEAD [11] noun

SUBJ < >                                  SUBJ < >                                    SUBJ < >                                   SUBJ < >

SPR   < >                                  SPR < >                                     SPR < >                                     SPR < >

COMPS < >                              COMPS < >                               COMPS < >                                COMPS < >

Phrases:

NP: a mouse                               VP: chased a mouse                                S: Fido chased a mouse.

HEAD                                      HEAD                                        HEAD

SUBJ < >                                  SUBJ < >                                    SUBJ < >

SPR   < >                                  SPR < >                                     SPR < >

COMPS < >                              COMPS < >                               COMPS < >

Indicate the missing values of the VAL and the HEAD features using tags ([1], ...) or "-" for empty lists. Don't use spaces.

Pat gave Alex a ride.

syntactic structure:

Words:

Pat                                                             gave                                         Alex                                                              a                                      ride

HEAD [9]noun                                  HEAD [10]verb                                   HEAD [11] noun                                HEAD [12] det                                HEAD [13] noun

SUBJ < >                                  SUBJ < >                                    SUBJ < >                                   SUBJ < >                                 SUBJ < >

SPR   < >                                  SPR < >                                     SPR < >                                     SPR < >                                   SPR < >

COMPS < >                              COMPS < >                        COMPS < >                                COMPS < >                            COMPS < >

Phrases:

NP: a ride                               VP: gave Alex a ride                                S: Pat gave Alex a ride.

HEAD                                      HEAD                                        HEAD

SUBJ < >                                  SUBJ < >                                    SUBJ < >

SPR   < >                                  SPR < >                                     SPR < >

COMPS < >                              COMPS < >                               COMPS < >

Feel free to send feedback on this exercise to Manfred Sailer.

Basic combinatorics: Canonical examples

(the following exercises are adapted from the textbook material to [Chapter 5].

1 Sentence: Pat snored.
Logical form: snore(pat)
Which parts of the logical form are contributed by which word?

pat ¦	snore ¦	snore(pat)
			Pat
			snored

2 Sentence: Pat likes Chris.
Logical form: like(pat,chris)
Which parts of the logical form are contributed by which word?

pat ¦	chris ¦	like ¦	like(pat,chris)
				Pat
				likes
				Chris

Possible EX-CONT values

Given the following PARTS lists, what are possible EX-CONT values (if we do not assume other restrictions)

1. PARTS < pat, alex,like, like(__,__) >

Check your answer

like(pat,alex)
like(alex,pat)

2. PARTS < alex,snore, snore(__), ¬(__) >

Check your answer

¬(snore(alex))

3. PARTS < alex,alex,snore >

Check your answer

There is no possible EX-CONT value because the three elements on the PARTS list cannot be combined.

3. PARTS < alex,alex,snore, snore(__) >

Check your answer

snore(alex)

4. PARTS < alex,alex,snore, snore(__), __ ∧ __ >

Check your answer

snore(alex) ∧ snore(alex)

Material for week 12 (16.1.2018)

Basic syntactic notions

Parts of speech

Determine the part of speech of the words in the sentences.
Use the following part of speech labels: A, Adv, Conj, Comp, Det, N, P, V

a. Alex/ talked/ to/ my/ best/ friend/ .

b. You/ might/ suspect/ that/ Pat/ is/ a/ genius/ .

c. The/ title/ of/ a/ book/ largely/ determines/ whether/ it/ will/ be/ successful/ or/ a/ flop/ .

Feel free to send feedback on this exercise to Manfred Sailer.

Syntactic categories

Determine the syntactic categories of the following groups of words in the sentences.
Use the following labels: AP, AdvP, NP, PP, VP. Write "-" if the group of words does not form a constitutent.
Example: [S: Pat [VP: will [VP: wait [PP: for Alex]]]]

a. [ Alex [ talked [ to [ my best friend]]]]

b. [ [ The president] [ announced [CP: that [ there [ will [ be [ no further taxes]]]]]]].

Feel free to send feedback on this exercise to Manfred Sailer.

Lexical entries as Attribute-Value Matrix

Provide the required information on the lexical properties of the underlined words in the following sentences.
Note:

• Put a minus ("-") if a slot should not receive any filling
• Use det, noun, prep or verb for the HEAD values.

1 Alex read a book yesterday.

PHON < >

HEAD

SUBJ < >

SPR < >

COMPS < >

2 Alex talked to a friend.

PHON < >

HEAD

SUBJ < >

SPR < >

COMPS < >

3 Pat liked this new documentary on African wild life.

PHON < >

HEAD

SUBJ < >

SPR < >

COMPS < >

4 Alex talked to a friend.

PHON < >

HEAD

SUBJ < >

SPR < >

COMPS < >

Feel free to send feedback on this exercise to Manfred Sailer.

Material for week 10 (19.12.2017)

Determiners/quantifiers

Watch the following video on logical determiners:

Exercises

Task 1 Variable assignment function
Start with the following variable assigment function g: g(u) = Romeo, g(v) = Juliet, g(w) = Romeo, g(x) = Laurence, g(y) = Mercutio, g(z) = Juliet

Provide the changed variable assignment function g[v/Paris].

Check your solutions here:

g[v/Paris](u) = g(u) = Romeo
g[v/Paris](v) = Paris
g[v/Paris](w) = g(w) = Romeo
g[v/Paris](x) = g(x) = Laurence
g[v/Paris](y) = g(y) = Mercutio
g[v/Paris](z) = g(z) = Juliet

After having watched the video, work on the following tasks.

Task 2 Identify the determiners in the following sentence.

(a) Juliet talked to some stranger at the party.

(b) Every Capulet is an enemy to some Montague.

(c) Many people in Verona are not happy about the Capulet-Montague feud.

Check your solutions here:

(a) some

(b) every, some

(c) many

Task 3 Identify the formula that corresponds to the translation of the sentence.

Some Montague who was at the party fell in love with Juliet.

	∃x (montague1(x) : (at-party1(x) ∧ fall-in-love-with2(x,juliet)))
	∃x ((montague1(x) ∧ at-party1(x)) : fall-in-love-with2(x,juliet))
	∃x (montague1(x) : (at-party1(x) ∧ fall-in-love-with2(x,juliet))
	∃x ((montague1(x) ∧ fall-in-love-with2(x,juliet)) : at-party1(x))

Task 4 The sentence: Some Tybalt loved some Montague. is translated into the formula
∃ y (montague₁(y) : love₂(tybalt,y)).

Mark all the cells in the table that stand for a true statement.

montague1(y) zwisch	love2(tybalt,y)zwisch
		Romeo
		Mercutio
		Juliet
		Tybalt
		Laurence
		Paris

Given this table, is the overall formula true or false? (Give a reason for your answer.)

Check your solutions here:

The formula is false, because there is no individual in our model for which both the restrictor and the scope are true.

Material for week 9 (12.12.2017)

Video

Truth tables

Truth tables are also useful to compute the truth value of complex formulae. This is shown in the following podcast, created by Lisa Günthner.

Material for week 6 (21.11.2017)

Video

(Note: the videos contain connectives that we have not talked about in class yet!)

The following video presents the step-by-step computation of the truth value of two formulae with connectives. The example uses a model based on Shakespeare's play Macbeth. The two formulae are:

• ¬ king(lady-macbeth)
• king(duncan) ∨ king(lady-macbeth)

The next video shows how the truth value of a more complex formula can be computed. The example contains two connectives:

kill(malcom,lady-macbeth) ∨ ¬thane(macbeth)

The video shows two different methods: top down and bottom up.

Material for week 5 (14.11.2017)

Video

The following video presents the step-by-step computation of the truth value of two atomic formulae. The example uses a model based on Shakespeare's play Macbeth. The two formulae are:

• kill(macbeth,duncan)
• kill(lady-macbeth,macbet)

Material for week 4 (7.11.2017)

Our literary scenario

Literary scenario: The Fresh Prince of Bel-Air
Wikipedia entry: https://en.wikipedia.org/wiki/The_Fresh_Prince_of_Bel-Air

Why it is too difficult to go directly from language to the world

The following architecture is extremely useful when talking about semantics:

1. A natural language expressions: Daenerys loves Drogo.
2. ... is mapped to some expression from a formal language (here: predicate logic): love2(daenerys,drogo)
3. This logical expression is then interpreted with respect to our scenario/world: The formula love2(daenerys,drogo) is true, because, in our scenario, Daenerys loves Drogo.

The following properties of natural language make it useful to use the intermediate step of a logical language:

1. The same expression can have different meanings (ambiguity).
2. Different expressions can have the same meaning (synonyms, paraphrases)

Find examples for the above-mentioned properties (ambiguity, synonymy, paraphrases).

Check your answers

1. one form, two meaingns: Ambiguity: (see earlier in this meeting and the slides of last week's meeting)

1.a Ambiguous words: date (fruit or point in time); bank (financial institute or bank of a river)

1.b. Ambiguous sentences: Sycorax and Prospero were stranded on the island with their children.

2. two forms, one meaning:

2.a Synonymous words: couch - sofa; instant - moment

2.b Paraphrases:

• active-passive pairs: Prospero set Ariel free. - Ariel was set free by Prospero.
• cleft sentences: Prospero set Ariel free. - It was Prospero who set Ariel free.
• different ways to express a possessor: Sycorax was the first inhabitant of the island. and Sycorax was the island's first inhabitant.

Towards a formal model

First steps

The following material is an adapted form of material created by student participants of the project e-Learning Resources for Semantics (e-LRS). Involved participants: Lisa, Marthe, Elisabeth, Isabelle.

You can think of building a formal model like being the producer of a film who has to collect everything that should be included in the film.

Here is a very simple story from which we can derive an example model.

Mark those elements that we need in a model.

	relations
	individuals
	nouns
	properties
	relatives

What is the status of the following entities in the video on Little Red Riding Hood?

individual	property	relation
			Red Riding Hood
			lives in the forest
			Grandmother
			is afternoon snack for
			has a red hood
			has a big mouth
			is grandmother of

The universe and name symbols

Task: Assume three individuals from our Game of Thrones-scenario.

Formally we collect the individuals of our model in a so-called universe (U). For the fairy-tale story, we can define the universe as follows:

U = {Redridinghood, Grandmother, Wolf}

Do a similar definition for your own scenario.

We can introduce name symbols for some of our individuals. For example: redridinghood, grandmother, wolf.

We link the name symbols to the individuals in our modal. To do this, we introduce the interpretation function. We will written the interpretation function as as I.
This function can be defined in the following way:

I(grandmother) = Grandmother
I(redridinghood) = Red Riding Hood
I(wolf) = Wolf

Relations and predicate symbols

In the fairy-tale scenario we express a relation between Little Red Riding Hood and the Wolf, namely that Little Red Riding Hood is the Wolf's afternoon snack. To formalize this, we collect all pairs of individuals which are such that the first element in the pair is the afternoon snack of the second. Note: A pair is written in between pointy brackets.

Formally we can write this down as follows:
{< x, y > | x is y 's afternoon snack} = { < Redridinghood, Wolf >, < Grandmother, Redriding hood >.}

We can also assume empty relations:

{< x, y > | x is y 's father } = { }

Note, if a relation works both ways, two pairs must be added:

{< x, y > | x talks with y} = { <Redridinghood, Wolf >, < Wolf, Redridinghood >}

Just like with names, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols afternoon-snack-of_2 and father-of_2, and talks-with_2. (The number 2 indicates that the interpretation consists of pairs, not just of single individual) There interpretation is defined as follows:

I(afternoon-snack-of_2) = { < x, y > | x is y 's afternoon snack } = { <Redridinghood, Wolf >, <Grandmother, Wolf > }.

Task: For each of your properties, invent an appropriate predicate symbol. Define its interpretation.

Properties and predicate symbols

A property is a specification that either holds of an individual or not. In the little story, having a big mouth is a property of the Wolf, but of noone else in the story. Being female holds of both Little Red Riding Hood and the Grandmother.

We can think of a property as the set of individuals that have this property. Under this view, the property of being female would be the set {Redridinghood, Grandmother}.

Alternatively it is convenient to think of properties as 1-place relations. Under this view, the property of being female would be a set of lists of length 1. This is what the property of being female then looks like: { <Redridinghood>, <Grandmother> }

Task: Using your Game of Thrones universe, define two properties in the format of 1-place relations.

Just like before, we want to have symbols that we can use in the logical language. For our example, let's take the predicate symbols female_1 and has-big-mouth_1. There interpretation is defined as follows:

I(female_1) = { < x > | x is female } = { <Redridinghood>, <Grandmother> }.

Task: For each of your properties, invent an appropriate predicate symbol. Define its interpretation.

Computing the truth value of atomic formulae

The following video presents the step-by-step computation of the truth value of two atomic formulae. The example uses a model based on Shakespeare's play Macbeth. The two formulae are:

• kill(macbeth,duncan)
• kill(lady-macbeth,macbeth)

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