WiSe 2025/26: Semantics 1

Meeting 3

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:

• kill2(macbeth,duncan)
• kill2(lady-macbeth,macbeth)

Syntax of atomic formulae

Exercise 1

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, and Isabelle.

This exercise is based on the following scenario:

At the time Alice, Paul, Tom and Lisa live in Berlin, but they rather want to live in Munich. Alice is married to Paul. They are Tom and Lisa's parents. Both Lisa and her father are tall, while Alice and Tom are rather small. Lisa and her mom share the same hair color, which is blonde. The family enjoys watching American football games together. But while the girls also like watching soccer, the boys get bored of it. Walter, the family's dog, doesn't care about sports at all, he likes to eat the familiy members´ shoes.

Which of the following expressions of predicate logic are formulae? Give an explanation for your decision. If the expression is not a formula try to change it into one.
(Click on the box if the expressionis a formula. When you press the submit button, you will see a suggestion for the second part of the question.)

1

family-dog

2

blonde(alice,paul)

3

father-of(alice,lisa)

4

tall(alice)

5

enjoy-watching-football-together

For a general explanation of formulae Click here

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Exercise 2

For the following exercises we use names and properties from the The Lord of the Rings novels.

Names: frodo, sam, gandalf, aragorn
1-place predicates: hobbit, wizard
2-place predicates: know, help

1 Click on the items that are well-formed expressions of the semantic representation language.

	gandalf
	hobbit
	sauron
	know(gandalf)
	help(aragorn,frodo)

2 Click on the expressions that are well-formed formulae.

	hobbit
	frodo
	hobbit(aragorn)

Interpretation of atomic formulae

Interpret the following formulae as true or false. If you have not defined these relations or properties in your model use the ones given in a previous exercise.

• father-of-someone(paul,lisa)

Check your answers

[[father-of-someone(paul,lisa)]] = true iff
< [[paul]], [[lisa]] > ∈ [[father-of-someone]] iff
< I(paul), I(lisa) > ∈ I(father-of-someone) iff
< Paul, Lisa> ∈ {<Paul, Tom>,<Paul, Lisa>}.

Since this is the case, the formula is true.

• blonde(walter)

Check your answers

[[blonde(walter)]] = true iff
< I(walter) > ∈ I(blonde) iff
< Walter > ∈ {< Alice >,< Lisa >}.

Since this is not the case, the overall formula is false.

• enjoy-watching-football-together(alice,tom)

Check your answers

[[enjoy-watching-football-togehter(alice,tom)]] = true iff
< I(alice), I(tom) > ∈ I(enjoy-watching-football-together) iff
< Alice, Tom > ∈ {<Alice, Paul>,<Paul, Alice>,<Alice, Lisa>,<Lisa, Alice>,<Alice, Tom>,<Tom, Alice>,<Paul, Lisa>,<Lisa, Paul>,<Paul, Tom>,<Tom, Paul>,<Tom, Lisa>,<Lisa, Tom>}

Since this is the case, the formula is true.

Meeting 2

Models

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.

Watch a short podcast what first-order models look like.

Based on this podcast, we can define a scenario as follows:

• Universe: U = {LittleRedRidingHood, Grandmother, Wolf}
  
• Properties:

RedHood = { < x> | x wears a read hood } = { <LittleRedRidingHood> }

Female = { <x> | x is female } = { <LittleRedRidingHood>, <Grandmother> }

BigMouth = { <x> | x has a big mouth } = { <Wolf> }

LiveInForest = { < x> | x lives in the forest } = { <Grandmother>, <Wolf>}

• Relations:

GrandChildOf = { <x,y> | x is y 's grandchild } = { <LittleRedRidingHood,Grandmother > }

AfternoonSnackOf = { <x,y> | x is y 's afternoon snack } = { <LittleRedRidingHood,Wolf > }

From this scenario, we can build a model M = < U, I >

• Universe: U = {LittleRedRidingHood, Grandmother, Wolf}
• Name symbols: NAME = {little-red-riding-hood}
  Note: In our model, only one individual has a name.
• Predicate symbols: PREDICATE = {red-hood1, female1, big-mouth, live-in-forest1, grand-child-of2, afternoon-snack-of2}
• Interpretation function I:

• for name symbols: I(little-red-riding-hood) = LittleRedRidingHood
• for predicate symbols:
  

I(red-hood1) = RedHood = { < x> | x wears a read hood } = { <LittleRedRidingHood> }

I(female) = Female = { <x> | x is female } = { <LittleRedRidingHood>, <Grandmother> }

I(big-mouth1) = BigMouth = { <x> | x has a big mouth } = { <Wolf> }

I(live-in-forest1) = LiveInForest = { < x> | x lives in the forest } = { <Grandmother>, <Wolf>}

I(grand-child-of2) = GrandChildOf = { <x,y> | x is y 's grandchild } = { <LittleRedRidingHood,Grandmother > }

I(afternoon-snack-of2) = AfternoonSnackOf = { <x,y> | x is y 's afternoon snack } = { <LittleRedRidingHood,Wolf > }

Meeting 1

Video

Challenging phenomena at the syntax-semantics interface

Scenario

The Hunger Games (film, 2012): https://en.wikipedia.org/wiki/The_Hunger_Games_(film)