Semantics 1, WiSe 2016/17, Week 4

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,macbet)

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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• the exercises for chapter 2
• the material for chapter 2
• the overview over all chapters

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