Exercise Semantics of Predicate Logic

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.

Interpretation of formulae with logical connectives

Consider these two natural language sentences. While keeping in mind the scenario given in a previous exercise, create complex formulae with logical connectives and compute the interpretation, respectively.

a.) Alice is a dog and Lisa and Tom enjoy watching football together.

Check your answers

Sentence: Alice is a dog and Lisa and Tom enjoy watching football together.

Here the interpretation in predicate logic notation:

[[dog (Alice) Ʌ enjoy-watching-football-together (Lisa,Tom)]] = false

because [[dog (Alice)]]= false

because I(Alice)= <Alice> and <Alice> is NOT an element of I(dog)

and [[enjoy-watching-soccer-together (Lisa,Tom)]] = true

because I(Lisa)= <Lisa>, I(Tom)= <Tom> and <Lisa,Tom> is in the set of I(enjoy-watching-football-together).

Conjunction (Ʌ): Both atomic formulae have to be true in order for the complex formula to be true.

b.) Tom is not Paul's daughter or Tom is tall.

Check your answers

Sentence: Tom is not Paul's daughter or Tom is tall.

Here the interpretation in predicate logic notation:

[[¬daughter-of-someone (Tom,Paul) v tall(Tom)]] = true

because [[¬daughter-of-someone (Tom,Paul)]]= true

because I(Tom)= <Tom>, I(Paul)= <Paul> and <Tom,Paul> is NOT in the set of I(daughter-of-someone)

and [[tall(Tom)]] = false

because I(Tom)= <Tom> and <Tom> is NOT an element of I(tall).

Disjunction (v): At least one of the atomic formulae has to be true in order for the complex formula to be true.

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