Logical equivalence of ¬p→q











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Just wondering what other ways $neg p to q$ can be expressed.



I know that $pto q$ is logically equivalent to $neg plor q$, hence I think that $neg pto q$ has the same logical equivalence as $plor q$.










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    up vote
    1
    down vote

    favorite












    Just wondering what other ways $neg p to q$ can be expressed.



    I know that $pto q$ is logically equivalent to $neg plor q$, hence I think that $neg pto q$ has the same logical equivalence as $plor q$.










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Just wondering what other ways $neg p to q$ can be expressed.



      I know that $pto q$ is logically equivalent to $neg plor q$, hence I think that $neg pto q$ has the same logical equivalence as $plor q$.










      share|cite|improve this question















      Just wondering what other ways $neg p to q$ can be expressed.



      I know that $pto q$ is logically equivalent to $neg plor q$, hence I think that $neg pto q$ has the same logical equivalence as $plor q$.







      propositional-calculus






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      edited Nov 11 at 18:41









      Mutantoe

      546411




      546411










      asked Nov 11 at 8:23









      Jordan Solomons

      197




      197






















          3 Answers
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          As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.



          But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.



          Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.






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          • Thanks for confirming!
            – Jordan Solomons
            Nov 11 at 10:55


















          up vote
          2
          down vote













          You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.



          In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.






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            As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
            Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.






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              3 Answers
              3






              active

              oldest

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              3 Answers
              3






              active

              oldest

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              active

              oldest

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              active

              oldest

              votes








              up vote
              4
              down vote













              As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.



              But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.



              Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.






              share|cite|improve this answer





















              • Thanks for confirming!
                – Jordan Solomons
                Nov 11 at 10:55















              up vote
              4
              down vote













              As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.



              But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.



              Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.






              share|cite|improve this answer





















              • Thanks for confirming!
                – Jordan Solomons
                Nov 11 at 10:55













              up vote
              4
              down vote










              up vote
              4
              down vote









              As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.



              But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.



              Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.






              share|cite|improve this answer












              As others have pointed out, truth tables will confirm the equivalence of $neg p to q$ and $p lor q$.



              But also you'll get additional insight by thinking informally. Suppose you are given that either $p$ or $q$. Then, if you rule out $p$, then you are left with $q$. So, in symbols, from $p lor q$ you can infer $neg p to q$.



              Conversely, suppose you are given that if not-$p$ then $q$. Then assuming you accept excluded middle, the principle that either $p$ or not $p$, it follows that either $p$ or (using that conditional) $q$. So, in symbols, from $neg p to q$ you can infer $p lor q$.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Nov 11 at 8:49









              Peter Smith

              40.3k339118




              40.3k339118












              • Thanks for confirming!
                – Jordan Solomons
                Nov 11 at 10:55


















              • Thanks for confirming!
                – Jordan Solomons
                Nov 11 at 10:55
















              Thanks for confirming!
              – Jordan Solomons
              Nov 11 at 10:55




              Thanks for confirming!
              – Jordan Solomons
              Nov 11 at 10:55










              up vote
              2
              down vote













              You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.



              In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.






              share|cite|improve this answer

























                up vote
                2
                down vote













                You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.



                In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.






                share|cite|improve this answer























                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.



                  In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.






                  share|cite|improve this answer












                  You have pretty much given the answer yourself already. Logical equivalence of $p$ and $q$ is given if $p$ is true if and only if $q$ is true (and hence $p$ is false iff $q$ is false). Since $p$ and $q$ can only be true or false, you can use truth tables and check whether logical equivalence is given. From there you can derive permissible manipulations of propositions.



                  In this case, $(lnot p) rightarrow q equiv lnot(lnot p) lor q equiv p lor q$, as you stated correctly.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 11 at 8:34









                  MacRance

                  505




                  505






















                      up vote
                      1
                      down vote













                      As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
                      Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.






                      share|cite|improve this answer

























                        up vote
                        1
                        down vote













                        As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
                        Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.






                        share|cite|improve this answer























                          up vote
                          1
                          down vote










                          up vote
                          1
                          down vote









                          As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
                          Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.






                          share|cite|improve this answer












                          As you said, $pRightarrow q$ is logically equivalent to $neg pvee q$.
                          Then by double negation, $neg p Rightarrow q$ is logically equivalent to $neg(neg p)vee q$, which amounts to $pvee q$.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 11 at 8:32









                          Wuestenfux

                          2,7371410




                          2,7371410






























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