Exercise First Order 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, and Isabelle.

Take a look at the following story:

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.

Using predicate logic terms and notation we now want to define the world described in the story. First we need our individuals, their relations and possible properties. You will need a pen and paper to write down your answers!

Note: As in the textbook, the expressions of the World will be written in italics and the ones of the predicate logic in bold.

(a) Define the universe described in the scenario. Introduce appropriate name symbols and define their interpretation. Make sure you use the correct notation.

Check your answers

These are the individuals of the story:
Alice
Paul
Lisa
Tom
Walter

This is how is this is stated in predicate logic:

U= {Alice, Paul, Lisa, Tom, Walter}

Name symbols: alice, paul, lisa, tom, walter

Interpretation of the name symbols:

I(alice) = Alice
I(paul) = Paul
I(lisa) = Lisa
I(tom) = Tom
I(walter) = Walter

(b) Choose three individuals from those mentioned in the story and map them to three properties mentioned in the story. Don't forget: empty sets are possible!

Check your answers

These are some possible answers:

rather-small: { <x> | x is rather small} = {<Tom>,<Alice>}

tall: { <x> | x is tall} = {<Paul>, <Lisa>}

blonde:{ <x> | x is blonde} = {<Alice>, <Lisa>}

female: {<x> |x is female} = {<Lisa>,<Alice>}

male: {<x> | x is male} = {<Tom>,<Paul>}

bored-watching-soccer: {<x> | x gets bored watching soccer} = {<Tom>,<Paul>}

enjoys-watching-soccer: {<x> | x enjoys watching soccer} = {<Alice>,<Lisa>}

enjoys-watching-football: {<x> | x enjoys watching football} = {<Tom>,<Alice>, <Lisa>, <Paul>}

doesn't-care-about-sports: {<x> | x doesn’t care about sports} = {<Walter>}

likes-eating-shoes: {<x> | x likes eating shoes} = {<Walter>}

lives-in-Berlin: {<x> | x lives in Berlin} = {<Tom>, <Alice>, <Lisa>, <Paul>}

want-to-live-in-Munich: {<x> | x wants to live in Munich} = {<Tom>, <Alice>, <Paul>, <Lisa>}

lives-in-Munich: {<x>| x lives in Munich} = {}

(c) Write down the possible relations mentioned in the story and map them to the individuals you wrote down in (a).

Check your answers

Here are examples of some relations:

son-of-someone: {<x,y> | x is the son of y} = {<Tom, Paul>, <Tom, Alice>}

father-of-someone: {<x, y> | x is the father of y} ={<Paul, Tom>, <Paul,Lisa>}

daughter-of-someone: {<x, y> | x is the daughter of y} = {<Lisa, Paul>, <Lisa, Alice>}

mother-of-someone: {<x, y> | x is the mother of y} ={<Alice, Lisa>, <Alice, Paul>}

brother-of-someone: {<x, y> | x is the brother of y} = {<Tom, Lisa>}

sister-of-someone: is the sister of y} = {<Lisa, Tom>}

dog-of-someone: {<x, y> | x is the dog of y} = {<Walter, Alice>, <Walter, Paul>, <Walter, Tom>, <Walter, Lisa>}

married-to-eachother: {<x, y> | x is married to y} = {<Alice, Paul>, <Paul, Alice}

enjoy-watching-football-together: {<x, y> | x enjoys watching football with y} = {<Alice, Paul>, <Paul, Alice>, <Alice, Lisa>, <Lisa, Alice>, <Alice, Tom>, <Tom, Alice>, <Paul, Lisa>, <Lisa, Paul>, <Paul, Tom>, <Tom, Paul>, <Tom, Lisa>, <Lisa, Tom>}

(d) Write down the I-functions (interpretation functions) for (a), (b) and (c).
Check your answers

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

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