Semantics 1, WiSe 2016/17, Week 4: Difference between revisions

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<embedvideo service="youtube" dimensions="400">http://youtu.be/8HGCB9urmbg</embedvideo>
<embedvideo service="youtube" dimensions="400">http://youtu.be/8HGCB9urmbg</embedvideo>
== 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 [[ExerciseFOModels-d|a previous exercise]].
* '''father-of-someone'''('''paul''','''lisa''')
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">
<nowiki>[[</nowiki>'''father-of-someone'''('''paul''','''lisa''')]] = ''true'' iff<br />
< <nowiki>[[</nowiki>'''paul''']], <nowiki>[[</nowiki>'''lisa''']] > &isin; <nowiki>[[</nowiki>'''father-of-someone''']] iff<br />
< I('''paul'''), I('''lisa''') > &isin; I('''father-of-someone''') iff<br />
< ''Paul'', ''Lisa''> &isin; {<''Paul, Tom''>,<''Paul, Lisa''>}.
Since this is the case, the formula is true.
</div>
</div>
* '''blonde'''('''walter''')
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">
<nowiki>[[</nowiki>'''blonde(walter)''']] = ''true'' iff<br />
< I('''walter''') > &isin; I('''blonde''') iff <br />
< ''Walter'' > &isin; {< ''Alice'' >,< ''Lisa'' >}.
Since this is not the case, the overall formula is false.
</div>
</div>
* '''enjoy-watching-football-together'''('''alice''','''tom''')
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">
<nowiki>[[</nowiki>'''enjoy-watching-football-togehter(alice,tom)''']] = ''true'' iff<br />
< I('''alice'''), I('''tom''') > &isin; I('''enjoy-watching-football-together''') iff<br />
< ''Alice'', ''Tom'' > &isin; {<''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.
</div>
</div>
<br/>
<br/>
== Interpretation of formulae with logical connectives ==
Consider these two natural language sentences. While keeping in mind the scenario given in [[ExerciseFOModels-d|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.
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">Sentence: Alice is a dog and Lisa and Tom enjoy watching football together.
Here the interpretation in predicate logic notation:
<nowiki>[[</nowiki>'''dog (Alice) Ʌ enjoy-watching-football-together (Lisa,Tom)''']] = ''false''<br/>
because <nowiki>[[</nowiki>'''dog (Alice)''']]= ''false'' <br/>
::because I('''Alice''')= <''Alice''> and <''Alice''> is NOT an element of  I('''dog''') <br/>
and <nowiki>[[</nowiki>'''enjoy-watching-soccer-together (Lisa,Tom)''']] = ''true'' <br/>
::because I('''Lisa''')= <''Lisa''>, I('''Tom''')= <''Tom''> and <''Lisa,Tom''> '''is''' in the set of I('''enjoy-watching-football-together'''). <br/>
'''Conjunction (Ʌ)''': Both atomic formulae have to be true in order for the complex formula to be true.
</div>
</div>
'''b.)''' Tom is not Paul's daughter or Tom is tall.
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answers
<div class="mw-collapsible-content">
Sentence: Tom is not Paul's daughter or Tom is tall.
Here the interpretation in predicate logic notation:
<nowiki>[[</nowiki>'''¬daughter-of-someone (Tom,Paul) v tall(Tom)''']] = ''true'' <br/>
because <nowiki>[[</nowiki>'''¬daughter-of-someone (Tom,Paul)''']]= ''true'' <br/>
::because I('''Tom''')= <''Tom''>, I('''Paul''')= <''Paul''> and <''Tom,Paul''> is NOT in the set of I('''daughter-of-someone''') <br/>
and <nowiki>[[</nowiki>'''tall(Tom)''']] = ''false'' <br/>
::because I('''Tom''')= <''Tom''> and <''Tom''> is NOT an element of I('''tall'''). <br/>
'''Disjunction (v)''': At least one of the atomic formulae has to be true in order for the complex formula to be true.
</div>
</div>
<hr />
Back to
* the [[Exercise-ch2|exercises for chapter 2]]
* the material for [[Textbook-chapters#Chapter_2:_Predicate_logic|chapter 2]]
* the overview over [[Textbook-chapters|all chapters]]


<hr>
<hr>
Back to the [[Semantics_1,_WiSe_2016/17_(Sailer) | course page]].
Back to the [[Semantics_1,_WiSe_2016/17_(Sailer) | course page]].

Revision as of 23:53, 31 October 2016

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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