Exercise Semantics of Predicate Logic: Difference between revisions

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{{CreatedByStudents1213}} ''Involved participants: [[User:Lisa| Lisa]], [[User:Marthe| Marthe]], [[User:Elisabeth.krall| Elisabeth]], and [[User:IsaB|Isabelle]].''
== 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]].


'''2.'''
* '''father-of-someone'''('''paul''','''lisa''')
'''a)''' 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. To check your answers click on single expressions.
<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>


'''*[[AnswerEX2-a1|family dog]]'''


'''*[[AnswerEX2-a2|blonde(Alice,Paul)]]'''


'''*[[AnswerEX2-a3|father-of-someone(Alice,Lisa)]]'''
* '''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'' >}.


'''*[[AnswerEX2-a4|tall(Alice)]]'''
Since this is not the case, the overall formula is false.
</div>
</div>


'''*[[AnswerEX2-a5|enjoy-watching-football-together]]'''




For a general explanation of formulae [[General_Explanation_Formulae|Click here]]
* '''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.


'''b)''' Interpret the following formulae as true or false. If you have not defined these relations or properties in your World use the ones given in the World of exercise 1d). [[AnswerEX1-d|See here]]
</div>
To check your answers click on the single formulae.
</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.


'''* [[AnswerEX2-b1|father-of-someone(Paul,Lisa)]]'''


'''* [[AnswerEX-b2|blonde(Walter)]]'''
'''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.
 


'''* [[AnswerEX-b3|enjoy-watching-football-together(Alice,Tom)]]'''
Here the interpretation in predicate logic notation:




'''3.''' Consider these two natural language sentences. While keeping in mind the scenario given in exercise 1d) [[AnswerEX1-d|See here]], create complex formulae with logical connectives and compute the interpretation, respectively.
<nowiki>[[</nowiki>'''dog (Alice) Ʌ enjoy-watching-football-together (Lisa,Tom)''']] = ''false''<br/>




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


'''[[AnswerEX3-a|Check your solution]]'''
'''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.
'''b.)''' Tom is not Paul's daughter or Tom is tall.


'''[[AnswerEX3-b|Check your solution]]'''
<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>




Latest revision as of 08:01, 14 April 2015

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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