Translate and Test for Validity

Translate the following arguments into PL and test them for validity using the truth-tree method:

Note: Instead of a check-mark, checked wffs are highlighted in yellow.

  1. Either Sue is tired or if Jude is hungry, then Owen is at home. But Sue isn't tired and Owen is not at home. Therefore Jude is not hungry.


     

    dictionary: S: Sue is tired.
    J: Jude is hungry.
    O: Owen is at home.
     
    translation: (S v (J ⊃ O))
    ~S & ~O
    ~J
     
    truth tree: (S v (J ⊃ O))
    ~S & ~O
    ~~J
    J
    ~S
    ~O

    /         \
           S              (J ⊃ O)
    x                   /       \
                         ~J       O
                           x               x
     
    answer: valid argument

     

  2. Either Sue is tired or Jude is hungry, if Owen is at home. But Sue isn't tired and Owen is not at home. Therefore Jude is not hungry.
     

    dictionary: S: Sue is tired.
    J: Jude is hungry.
    O: Owen is at home.
     
    translation: (O ⊃ (S v J))
    ~S & ~O
    ~J
     
    truth tree: (O ⊃ (S v J))
    ~S & ~O
    ~~J
    J
    ~S
    ~O
    /             \
       ~O         (S v J)
                        /          \
                          S              J
                         x               
                
    answer: invalid argument


     

  3. If Elfride is lonely then Stephen will appear. If Stephen will appear, then Oak will stop the fire. Therefore, if Oak will not stop the fire, Elfride is not lonely.
     

    dictionary: E: Elfride is lonely
    S: Stephen will appear
    O: Oak will stop the fire.
     
    translation:
    (E ⊃  S)
    (S  ⊃ O) 
    (~O ⊃  ~E)
    truth tree:  (E ⊃  S)
    (S  ⊃ O)  
    ~(~O ⊃  ~E)
    ~O
    ~~E
    E
    /          \
    ~E            S
      x            /      \
                     ~S         O
                      x             x 
    answer: valid argument


     

  4. Elfride is lonely only if Stephen will appear. If Stephen will appear, then Oak will stop the fire. Therefore, Oak will stop the fire, if Elfride is lonely.
     

    dictionary: E: Elfride is loney.
    S: Stephen will appear.
    O: Oaks will stop the fire.
     
    translation: (E ⊃ S)
    (S ⊃ O)
    (E ⊃ O)
     
    truth tree: (E ⊃ S)

    (S⊃O)

      ~(E ⊃     O)  
    E
    ~O
       /       \
        ~S       O  
       /     \         x
    ~E         S            
    x          x           
                
                
    answer: valid!


     

  5. Neither Jude nor Sue want to get married. But Sue will eat tomatoes just in case Oak visits. Therefore Oak visits unless Jude does not want to get married.
     

    dictionary: J: Jude wants to get married.
    S: Sue wants to get married.
    T: Sue will eat tomatoes.
    O: Oak visits
     
    translation: ~J & ~S
      T ≡ O
     O v ~J
    truth tree:      ~J & ~S
    T ≡ O
            ~(O v ~J)
    ~O
    ~~J
    J
    ~J
    ~S
    x
          
    answer: valid!

     

  6. A sufficient condition for Cytherea to get mad is Manston asking for her hand in marriage. Manston will ask Cytherea's hand in marriage if either Miss Aldclyffe requests it or Owen remains ill. If Owen does not remain ill, Cytherea will get mad. Therefore Miss Aldclyffe does not request that Manston asks for Cytherea's hand in marriage.

     

    dictionary: M: Manston asks Cytherea for her hand in marriage.
    C: Cytherea gets mad.
    A: Miss Aldclyffe requests that Manston ask Cytherea to marry him.
    O: Owen remains ill.
     
    translation:

    M ⊃ C
    (A v O) ⊃M

     

    M ⊃C
    (A v  O) ⊃ M
    ~O ⊃ C
    ~A
     
    truth tree: click here for pdf        
    answer: invalid!


     

  7. Mr. Swancourt is hungry only if Elfride isn't. Either Elfride is hungry or Knight is lurking. But knight isn't lurking. Therefore Mr. Swancourt isn't hungry.

     

    dictionary: S: Mr. Swancout is hungry.
    E: Elfride is hungry.
    K: Knight is lurking.
     
    translation: S ⊃ ~E
    E v K
    ~K   
    ~S
     
    truth tree: click here for pdf 
    answer: valid!


     

  8. Oak loses his sheep if his dog either chases them or the bank will repossess them. But the bank will repossess them only if Oak fails to pay his bills. Bethsheba will see to it that the fields are well cared for if Oak does not lose his sheep. So the bank will repossess the sheep if Oak's dog doesn't chase them.

     

    dictionary: O: Oak loses his sheep.
    D: Oak's dog chases the sheep.
    B: The bank will repossess Oak's sheep.
    F: Oak fails to pay his bills.
    S: Bethsheba will see to it that the fields are well cared for
     
    translation: (D v B) ⊃ O
    B ⊃ F
    ~O     ⊃ S
    ~D     ⊃ B
     
    truth tree: click here for pdf 
    answer: invalid!


     

  9. If both Oak is at home and Bethsheba is in the field, then Jude is studying Latin. But neither is Jude studying Latin nor is Stephen eating raw eggs. Therefore Bethsheba is in the field.

     

    dictionary: O: Oak is at home.
    B: Bethsheba is in the field.
    J: Jude is studying Latin.
    S: Stephen is eating raw eggs.
    translation: (O & B) ⊃ J
    ~J & ~S
    B
     
    truth tree: click here for pdf
    answer: invalid!


     

  10. Manston works for Miss Aldclyffe and is not related to Owen. Cytherea is related to Owen but not to Manston or Miss Aldclyffe. Miss Aldclyffe will be happy just in case Cytherea is related to Manston. Therefore Miss Aldclyffe will not be happy.

     

    dictionary: M: Manston works for Miss Aldclyffe.
    O: Manston is related to Owen.
    C: Cytherea is related to Owen.
    R: Cytherea is related to Manston.
    A: Cytherea is related to Miss Aldclyffe.
    H: Miss Aldclyffee will be happy.
     
    translation:
    M & ~O
    C & (~R & ~A)
    H     ≡ R
    ~H
     
    truth tree: click here for pdf
    answer: valid!