meaning of (r .⊃. s ⊃ r) [the syntax meaning]

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I'm trying to to determine whether the following is a tautology, contingency, or contradictory:




(p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r)




This is school work. I'm getting that it's a tautology, but only through looking at patterns of solutions of textbook problems and exercises. I would like to know what exactly is meant by the dot prior and after the material implication (.⊃.)?



I understand that material implication - in a truth table - has false only when p is true and q is false. I also understand the or operator where it's true when either both or one of the two, namely: p or q is true. I just need to understand what the dots are supposed to mean. I couldn't find good explanations online.










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    up vote
    1
    down vote

    favorite












    I'm trying to to determine whether the following is a tautology, contingency, or contradictory:




    (p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r)




    This is school work. I'm getting that it's a tautology, but only through looking at patterns of solutions of textbook problems and exercises. I would like to know what exactly is meant by the dot prior and after the material implication (.⊃.)?



    I understand that material implication - in a truth table - has false only when p is true and q is false. I also understand the or operator where it's true when either both or one of the two, namely: p or q is true. I just need to understand what the dots are supposed to mean. I couldn't find good explanations online.










    share|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I'm trying to to determine whether the following is a tautology, contingency, or contradictory:




      (p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r)




      This is school work. I'm getting that it's a tautology, but only through looking at patterns of solutions of textbook problems and exercises. I would like to know what exactly is meant by the dot prior and after the material implication (.⊃.)?



      I understand that material implication - in a truth table - has false only when p is true and q is false. I also understand the or operator where it's true when either both or one of the two, namely: p or q is true. I just need to understand what the dots are supposed to mean. I couldn't find good explanations online.










      share|improve this question















      I'm trying to to determine whether the following is a tautology, contingency, or contradictory:




      (p ⊃ q) ∨ (q ⊃ p) .⊃. (r .⊃. s ⊃ r)




      This is school work. I'm getting that it's a tautology, but only through looking at patterns of solutions of textbook problems and exercises. I would like to know what exactly is meant by the dot prior and after the material implication (.⊃.)?



      I understand that material implication - in a truth table - has false only when p is true and q is false. I also understand the or operator where it's true when either both or one of the two, namely: p or q is true. I just need to understand what the dots are supposed to mean. I couldn't find good explanations online.







      logic philosophy-of-logic symbolic-logic






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      edited 54 mins ago

























      asked 1 hour ago









      wa7d

      305




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          2 Answers
          2






          active

          oldest

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          up vote
          2
          down vote



          accepted










          The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:




          ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))




          Note that without the dots the consequent would be:




          (r ⊃ s ⊃ r)




          Which is ambiguous between the following two:




          ((r ⊃ s) ⊃ r)



          (r ⊃ (s ⊃ r))




          The dots just tell you which connective is the main one.






          share|improve this answer




















          • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
            – wa7d
            5 mins ago

















          up vote
          2
          down vote













          The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).



          To give a simpler example:



          A ∨ B ⊃ C



          is syntactically ambiguous. It could mean



          (A ∨ B) ⊃ C



          or



          A ∨ (B ⊃ C)



          An alternative way to specify the first version would be to write:



          A ∨ B .⊃. C



          It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.






          share|improve this answer




















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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            2
            down vote



            accepted










            The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:




            ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))




            Note that without the dots the consequent would be:




            (r ⊃ s ⊃ r)




            Which is ambiguous between the following two:




            ((r ⊃ s) ⊃ r)



            (r ⊃ (s ⊃ r))




            The dots just tell you which connective is the main one.






            share|improve this answer




















            • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
              – wa7d
              5 mins ago














            up vote
            2
            down vote



            accepted










            The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:




            ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))




            Note that without the dots the consequent would be:




            (r ⊃ s ⊃ r)




            Which is ambiguous between the following two:




            ((r ⊃ s) ⊃ r)



            (r ⊃ (s ⊃ r))




            The dots just tell you which connective is the main one.






            share|improve this answer




















            • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
              – wa7d
              5 mins ago












            up vote
            2
            down vote



            accepted







            up vote
            2
            down vote



            accepted






            The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:




            ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))




            Note that without the dots the consequent would be:




            (r ⊃ s ⊃ r)




            Which is ambiguous between the following two:




            ((r ⊃ s) ⊃ r)



            (r ⊃ (s ⊃ r))




            The dots just tell you which connective is the main one.






            share|improve this answer












            The dots function like parentheses: they disambiguate an otherwise ambiguous expression. In this case the expression is equivalent to:




            ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r))




            Note that without the dots the consequent would be:




            (r ⊃ s ⊃ r)




            Which is ambiguous between the following two:




            ((r ⊃ s) ⊃ r)



            (r ⊃ (s ⊃ r))




            The dots just tell you which connective is the main one.







            share|improve this answer












            share|improve this answer



            share|improve this answer










            answered 21 mins ago









            Eliran H

            3,80621032




            3,80621032











            • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
              – wa7d
              5 mins ago
















            • I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
              – wa7d
              5 mins ago















            I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
            – wa7d
            5 mins ago




            I kind of wish there was a universally agreed upon notation. That would be one way to disambiguate. Thanks!
            – wa7d
            5 mins ago










            up vote
            2
            down vote













            The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).



            To give a simpler example:



            A ∨ B ⊃ C



            is syntactically ambiguous. It could mean



            (A ∨ B) ⊃ C



            or



            A ∨ (B ⊃ C)



            An alternative way to specify the first version would be to write:



            A ∨ B .⊃. C



            It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.






            share|improve this answer
























              up vote
              2
              down vote













              The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).



              To give a simpler example:



              A ∨ B ⊃ C



              is syntactically ambiguous. It could mean



              (A ∨ B) ⊃ C



              or



              A ∨ (B ⊃ C)



              An alternative way to specify the first version would be to write:



              A ∨ B .⊃. C



              It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.






              share|improve this answer






















                up vote
                2
                down vote










                up vote
                2
                down vote









                The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).



                To give a simpler example:



                A ∨ B ⊃ C



                is syntactically ambiguous. It could mean



                (A ∨ B) ⊃ C



                or



                A ∨ (B ⊃ C)



                An alternative way to specify the first version would be to write:



                A ∨ B .⊃. C



                It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.






                share|improve this answer












                The use of dots like that is an alternative to using nested parentheses. Putting dots around a connective indicates that its binding priority is lower than that of another connective in the sentence. Your sentence could also be written ((p ⊃ q) ∨ (q ⊃ p)) ⊃ (r ⊃ (s ⊃ r)).



                To give a simpler example:



                A ∨ B ⊃ C



                is syntactically ambiguous. It could mean



                (A ∨ B) ⊃ C



                or



                A ∨ (B ⊃ C)



                An alternative way to specify the first version would be to write:



                A ∨ B .⊃. C



                It is a matter of style as to which you prefer, but the dot notation is not as common as it used to be.







                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered 16 mins ago









                Bumble

                6,9882830




                6,9882830



























                     

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