Explain the rules of inference used to obtain each conclusion from the premises (Rosen 8th Ed):












1












$begingroup$


I'm having a difficult time with the following - seems like a lot of work and I'm unsure if my conclusions makes sense when translated back to english.





  1. If I work, it is either sunny or partly sunny (Premise):
    $forall x (W(x) to (S(x) lor P(x))$

  2. I worked last Monday or I worked last Friday (Premise):
    $W_{Monday} lor W_{Friday}$

  3. It was not sunny on Tuesday (Premise):
    $lnot S_{Tuesday}$

  4. It was not partly sunny on Friday (Premise):
    $lnot P_{Friday}$




$5. W_{Friday} to (S_{Friday} lor P_{Friday})$ (Universal Instantiation 1.)



$6. P_{Friday} lor (lnot W_{Friday} lor S_{Friday} )$ (Logical Equivalence, Commutative & Associative Laws)



$7.lnot W_{Friday} lor S_{Friday}$ (Disjunctive Syllogism 4. & 6.)



$8.W_{Monday} lor S_{Friday}$ (Resolution 2. & 7.)



$9. W_{Monday} to (S_{Monday} lor P_{Monday})$ (Universial Instantiation 1.)



$10. lnot W_{Monday} lor (S_{Monday} lor P_{Monday})$ (Logical Equivalence)



$11.S_{Friday} lor (S_{Monday} lor P_{Monday}) equiv lnot S_{Friday} to (S_{Monday} lor P_{Monday})$ (Resolution 8. & 10.)



$12. exists x exists y(lnot S(x) to (S(y) lor P(y))$ (Existential Generalization 11. with 3.) Conclusion: There exists a day when it was not sunny, while on another day it was either sunny or partly sunny.



Seems rather inconclusive.






proof-verification logic







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I'm having a difficult time with the following - seems like a lot of work and I'm unsure if my conclusions makes sense when translated back to english.





    1. If I work, it is either sunny or partly sunny (Premise):
      $forall x (W(x) to (S(x) lor P(x))$

    2. I worked last Monday or I worked last Friday (Premise):
      $W_{Monday} lor W_{Friday}$

    3. It was not sunny on Tuesday (Premise):
      $lnot S_{Tuesday}$

    4. It was not partly sunny on Friday (Premise):
      $lnot P_{Friday}$




    $5. W_{Friday} to (S_{Friday} lor P_{Friday})$ (Universal Instantiation 1.)



    $6. P_{Friday} lor (lnot W_{Friday} lor S_{Friday} )$ (Logical Equivalence, Commutative & Associative Laws)



    $7.lnot W_{Friday} lor S_{Friday}$ (Disjunctive Syllogism 4. & 6.)



    $8.W_{Monday} lor S_{Friday}$ (Resolution 2. & 7.)



    $9. W_{Monday} to (S_{Monday} lor P_{Monday})$ (Universial Instantiation 1.)



    $10. lnot W_{Monday} lor (S_{Monday} lor P_{Monday})$ (Logical Equivalence)



    $11.S_{Friday} lor (S_{Monday} lor P_{Monday}) equiv lnot S_{Friday} to (S_{Monday} lor P_{Monday})$ (Resolution 8. & 10.)



    $12. exists x exists y(lnot S(x) to (S(y) lor P(y))$ (Existential Generalization 11. with 3.) Conclusion: There exists a day when it was not sunny, while on another day it was either sunny or partly sunny.



    Seems rather inconclusive.






    proof-verification logic







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I'm having a difficult time with the following - seems like a lot of work and I'm unsure if my conclusions makes sense when translated back to english.





      1. If I work, it is either sunny or partly sunny (Premise):
        $forall x (W(x) to (S(x) lor P(x))$

      2. I worked last Monday or I worked last Friday (Premise):
        $W_{Monday} lor W_{Friday}$

      3. It was not sunny on Tuesday (Premise):
        $lnot S_{Tuesday}$

      4. It was not partly sunny on Friday (Premise):
        $lnot P_{Friday}$




      $5. W_{Friday} to (S_{Friday} lor P_{Friday})$ (Universal Instantiation 1.)



      $6. P_{Friday} lor (lnot W_{Friday} lor S_{Friday} )$ (Logical Equivalence, Commutative & Associative Laws)



      $7.lnot W_{Friday} lor S_{Friday}$ (Disjunctive Syllogism 4. & 6.)



      $8.W_{Monday} lor S_{Friday}$ (Resolution 2. & 7.)



      $9. W_{Monday} to (S_{Monday} lor P_{Monday})$ (Universial Instantiation 1.)



      $10. lnot W_{Monday} lor (S_{Monday} lor P_{Monday})$ (Logical Equivalence)



      $11.S_{Friday} lor (S_{Monday} lor P_{Monday}) equiv lnot S_{Friday} to (S_{Monday} lor P_{Monday})$ (Resolution 8. & 10.)



      $12. exists x exists y(lnot S(x) to (S(y) lor P(y))$ (Existential Generalization 11. with 3.) Conclusion: There exists a day when it was not sunny, while on another day it was either sunny or partly sunny.



      Seems rather inconclusive.






      proof-verification logic







      share|cite|improve this question











      $endgroup$




      I'm having a difficult time with the following - seems like a lot of work and I'm unsure if my conclusions makes sense when translated back to english.





      1. If I work, it is either sunny or partly sunny (Premise):
        $forall x (W(x) to (S(x) lor P(x))$

      2. I worked last Monday or I worked last Friday (Premise):
        $W_{Monday} lor W_{Friday}$

      3. It was not sunny on Tuesday (Premise):
        $lnot S_{Tuesday}$

      4. It was not partly sunny on Friday (Premise):
        $lnot P_{Friday}$




      $5. W_{Friday} to (S_{Friday} lor P_{Friday})$ (Universal Instantiation 1.)



      $6. P_{Friday} lor (lnot W_{Friday} lor S_{Friday} )$ (Logical Equivalence, Commutative & Associative Laws)



      $7.lnot W_{Friday} lor S_{Friday}$ (Disjunctive Syllogism 4. & 6.)



      $8.W_{Monday} lor S_{Friday}$ (Resolution 2. & 7.)



      $9. W_{Monday} to (S_{Monday} lor P_{Monday})$ (Universial Instantiation 1.)



      $10. lnot W_{Monday} lor (S_{Monday} lor P_{Monday})$ (Logical Equivalence)



      $11.S_{Friday} lor (S_{Monday} lor P_{Monday}) equiv lnot S_{Friday} to (S_{Monday} lor P_{Monday})$ (Resolution 8. & 10.)



      $12. exists x exists y(lnot S(x) to (S(y) lor P(y))$ (Existential Generalization 11. with 3.) Conclusion: There exists a day when it was not sunny, while on another day it was either sunny or partly sunny.



      Seems rather inconclusive.






      proof-verification logic




      proof-verification logic






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 20 at 2:55







      Elliott

















      asked Jan 20 at 1:29









      ElliottElliott

      424




      424






















          1 Answer
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          1












          $begingroup$

          Hint



          The premises refer to three days : Mon, Tue, Fri.



          Thus, it can be useful to use all of them in Universal instantiation of 1) to get :




          5) $W(F) → (S(F) ∨ P(F))$ i.e. $lnot W(F) ∨ (S(F) ∨ P(F))$



          6) $lnot W(T) ∨ (S(T) ∨ P(T))$




          and




          7) $lnot W(M) ∨ (S(M) ∨ P(M))$.




          Then, using Resolution, we get :



          8) $lnot W(T) ∨ P(T)$ --- from 3) and 6)



          9) $lnot W(F) ∨ S(F)$ --- from 4) and 5)



          10) $W(F) lor S(M) ∨ P(M)$ --- from 2) and 7)



          11) $W(M) lor S(F)$ --- from 2) and 9).



          All this is not very useful...



          8) is $W(T) to P(T)$ and from it : $exists x (W(x) to P(x))$.



          In the same way, from 9) : $exists x (W(x) to S(x))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
            $endgroup$
            – Elliott
            Jan 22 at 4:04











          Your Answer





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          1 Answer
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          active

          oldest

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          active

          oldest

          votes









          1












          $begingroup$

          Hint



          The premises refer to three days : Mon, Tue, Fri.



          Thus, it can be useful to use all of them in Universal instantiation of 1) to get :




          5) $W(F) → (S(F) ∨ P(F))$ i.e. $lnot W(F) ∨ (S(F) ∨ P(F))$



          6) $lnot W(T) ∨ (S(T) ∨ P(T))$




          and




          7) $lnot W(M) ∨ (S(M) ∨ P(M))$.




          Then, using Resolution, we get :



          8) $lnot W(T) ∨ P(T)$ --- from 3) and 6)



          9) $lnot W(F) ∨ S(F)$ --- from 4) and 5)



          10) $W(F) lor S(M) ∨ P(M)$ --- from 2) and 7)



          11) $W(M) lor S(F)$ --- from 2) and 9).



          All this is not very useful...



          8) is $W(T) to P(T)$ and from it : $exists x (W(x) to P(x))$.



          In the same way, from 9) : $exists x (W(x) to S(x))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
            $endgroup$
            – Elliott
            Jan 22 at 4:04
















          1












          $begingroup$

          Hint



          The premises refer to three days : Mon, Tue, Fri.



          Thus, it can be useful to use all of them in Universal instantiation of 1) to get :




          5) $W(F) → (S(F) ∨ P(F))$ i.e. $lnot W(F) ∨ (S(F) ∨ P(F))$



          6) $lnot W(T) ∨ (S(T) ∨ P(T))$




          and




          7) $lnot W(M) ∨ (S(M) ∨ P(M))$.




          Then, using Resolution, we get :



          8) $lnot W(T) ∨ P(T)$ --- from 3) and 6)



          9) $lnot W(F) ∨ S(F)$ --- from 4) and 5)



          10) $W(F) lor S(M) ∨ P(M)$ --- from 2) and 7)



          11) $W(M) lor S(F)$ --- from 2) and 9).



          All this is not very useful...



          8) is $W(T) to P(T)$ and from it : $exists x (W(x) to P(x))$.



          In the same way, from 9) : $exists x (W(x) to S(x))$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
            $endgroup$
            – Elliott
            Jan 22 at 4:04














          1












          1








          1





          $begingroup$

          Hint



          The premises refer to three days : Mon, Tue, Fri.



          Thus, it can be useful to use all of them in Universal instantiation of 1) to get :




          5) $W(F) → (S(F) ∨ P(F))$ i.e. $lnot W(F) ∨ (S(F) ∨ P(F))$



          6) $lnot W(T) ∨ (S(T) ∨ P(T))$




          and




          7) $lnot W(M) ∨ (S(M) ∨ P(M))$.




          Then, using Resolution, we get :



          8) $lnot W(T) ∨ P(T)$ --- from 3) and 6)



          9) $lnot W(F) ∨ S(F)$ --- from 4) and 5)



          10) $W(F) lor S(M) ∨ P(M)$ --- from 2) and 7)



          11) $W(M) lor S(F)$ --- from 2) and 9).



          All this is not very useful...



          8) is $W(T) to P(T)$ and from it : $exists x (W(x) to P(x))$.



          In the same way, from 9) : $exists x (W(x) to S(x))$.






          share|cite|improve this answer











          $endgroup$



          Hint



          The premises refer to three days : Mon, Tue, Fri.



          Thus, it can be useful to use all of them in Universal instantiation of 1) to get :




          5) $W(F) → (S(F) ∨ P(F))$ i.e. $lnot W(F) ∨ (S(F) ∨ P(F))$



          6) $lnot W(T) ∨ (S(T) ∨ P(T))$




          and




          7) $lnot W(M) ∨ (S(M) ∨ P(M))$.




          Then, using Resolution, we get :



          8) $lnot W(T) ∨ P(T)$ --- from 3) and 6)



          9) $lnot W(F) ∨ S(F)$ --- from 4) and 5)



          10) $W(F) lor S(M) ∨ P(M)$ --- from 2) and 7)



          11) $W(M) lor S(F)$ --- from 2) and 9).



          All this is not very useful...



          8) is $W(T) to P(T)$ and from it : $exists x (W(x) to P(x))$.



          In the same way, from 9) : $exists x (W(x) to S(x))$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 22 at 9:15

























          answered Jan 20 at 8:19









          Mauro ALLEGRANZAMauro ALLEGRANZA

          66.3k449115




          66.3k449115












          • $begingroup$
            So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
            $endgroup$
            – Elliott
            Jan 22 at 4:04


















          • $begingroup$
            So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
            $endgroup$
            – Elliott
            Jan 22 at 4:04
















          $begingroup$
          So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
          $endgroup$
          – Elliott
          Jan 22 at 4:04




          $begingroup$
          So if translated back to english than we can conclude that it was Sunny on Friday, and Sunny on Monday, or Partly Sunny on Monday?
          $endgroup$
          – Elliott
          Jan 22 at 4:04


















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