Math 10130 Lecture 32 by wku77463

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									                                    Review
A bit more on the alternation of quantifiers
                          One last example
        Logical Theory for Predicate Logic




                    Math 10130 Lecture 32



                                   9 April 2008




                                              10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


Last time




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


Last time



   Last time:




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Last time



   Last time: We considered quantified formulas in several different
   contexts.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Last time



   Last time: We considered quantified formulas in several different
   contexts.
       Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Last time



   Last time: We considered quantified formulas in several different
   contexts.
       Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
           - true when the domain of discourse is the set of natural
             numbers N




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Last time



   Last time: We considered quantified formulas in several different
   contexts.
       Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
           - true when the domain of discourse is the set of natural
             numbers N
           - false when the domain of discourse is the set of integers Z




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Last time



   Last time: We considered quantified formulas in several different
   contexts.
       Example: ∃x ∀y Mxy , where Mxy is interpreted as “x ≤ y ”
           - true when the domain of discourse is the set of natural
             numbers N
           - false when the domain of discourse is the set of integers Z
       Principle: Truth or falsity of a statement depends on the
       context.




                                                       10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


Alternation of Quantifiers




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Alternation of Quantifiers



   Consider the following structure:




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Alternation of Quantifiers



   Consider the following structure:
   Domain of discourse: people




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Alternation of Quantifiers



   Consider the following structure:
   Domain of discourse: people
   Mxy stands for “x loves y ”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Alternation of Quantifiers



   Consider the following structure:
   Domain of discourse: people
   Mxy stands for “x loves y ”

   Q: Do ∀x ∃y Mxy and ∃y ∀x Mxy express the same proposition?




                                                       10130 Lecture 32
                                           Review
       A bit more on the alternation of quantifiers
                                 One last example
               Logical Theory for Predicate Logic


Comparing the two formulas




                                                     10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Comparing the two formulas


   ∀x ∃y Mxy says, “For each person x, there is a y such that x loves
   y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Comparing the two formulas


   ∀x ∃y Mxy says, “For each person x, there is a y such that x loves
   y .”

   In other words, “Everybody loves somebody.”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Comparing the two formulas


   ∀x ∃y Mxy says, “For each person x, there is a y such that x loves
   y .”

   In other words, “Everybody loves somebody.”

   ∃y ∀x Mxy says, “There is a person y such that for all persons x,
   x loves y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Comparing the two formulas


   ∀x ∃y Mxy says, “For each person x, there is a y such that x loves
   y .”

   In other words, “Everybody loves somebody.”

   ∃y ∀x Mxy says, “There is a person y such that for all persons x,
   x loves y .”

   In other words, “Somebody is loved by everybody.”




                                                       10130 Lecture 32
                                           Review
       A bit more on the alternation of quantifiers
                                 One last example
               Logical Theory for Predicate Logic


A tough example




                                                     10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A tough example



  Domain of discourse: the set of natural numbers, N




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A tough example



  Domain of discourse: the set of natural numbers, N
  Mxy stands for “x ≤ y ”




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A tough example



  Domain of discourse: the set of natural numbers, N
  Mxy stands for “x ≤ y ”

  Let’s discuss this beast of a formula:




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A tough example



  Domain of discourse: the set of natural numbers, N
  Mxy stands for “x ≤ y ”

  Let’s discuss this beast of a formula:

  ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree




   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree




   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

    ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree




   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

    ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

      (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree



   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

    ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

      (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))



                 Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))




                                                      10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree


   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

    ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

      (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))



                 Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

                                                      (Mxz & ¬x = z) &(Mzy & ¬z = y )


                                                         10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

    ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

      (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))



                 Mxy & ¬x = y                 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))

                                                      (Mxz & ¬x = z) &(Mzy & ¬z = y )


                                                       Mxz & ¬x = z         Mzy & ¬z = y

                                                         10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A huge syntactic tree

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))) 6

     ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

       (Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y ))



               Mxy & ¬x = y 1                   ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) 5

                                                      (Mxz & ¬x = z) &(Mzy & ¬z = y ) 4


                                                      Mxz & ¬x = z 2       Mzy & ¬z = y 3
                                                        10130 Lecture 32
                                           Review
       A bit more on the alternation of quantifiers
                                 One last example
               Logical Theory for Predicate Logic


Breaking down the formula




                                                     10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”
   2 Mxz & ¬x = z says “x < z.”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”
   2 Mxz & ¬x = z says “x < z.”
   3 Mzy & ¬z = y says “z < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”
   2 Mxz & ¬x = z says “x < z.”
   3 Mzy & ¬z = y says “z < y .”
   4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
   “x < z < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”
   2 Mxz & ¬x = z says “x < z.”
   3 Mzy & ¬z = y says “z < y .”
   4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
   “x < z < y .”
   5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such
   that x < z < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula

   1 Mxy & ¬x = y says “x < y .”
   2 Mxz & ¬x = z says “x < z.”
   3 Mzy & ¬z = y says “z < y .”
   4 (Mxz & ¬x = z) &(Mzy & ¬z = y ) says “x < z and z < y ” or
   “x < z < y .”
   5 ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )) says “There is a z such
   that x < z < y .”
   6 (the big formula) says “For all x and y , if x < y , then there is a
   z such that x < z < y .”


                                                       10130 Lecture 32
                                           Review
       A bit more on the alternation of quantifiers
                                 One last example
               Logical Theory for Predicate Logic


Breaking down the formula, continued




                                                     10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula, continued



   “For all x and y , if x < y , then there is a z such that x < z < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula, continued



   “For all x and y , if x < y , then there is a z such that x < z < y .”
   Is this true of all pairs of natural numbers?




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula, continued



   “For all x and y , if x < y , then there is a z such that x < z < y .”
   Is this true of all pairs of natural numbers?
   Consider the pair (0, 1).




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula, continued



   “For all x and y , if x < y , then there is a z such that x < z < y .”
   Is this true of all pairs of natural numbers?
   Consider the pair (0, 1).
   0 < 1, but there is no z such that 0 < z < 1.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Breaking down the formula, continued



   “For all x and y , if x < y , then there is a z such that x < z < y .”
   Is this true of all pairs of natural numbers?
   Consider the pair (0, 1).
   0 < 1, but there is no z such that 0 < z < 1.
   Thus, the formula is false.




                                                       10130 Lecture 32
                                           Review
       A bit more on the alternation of quantifiers
                                 One last example
               Logical Theory for Predicate Logic


Changing the context




                                                     10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?
   Remember, the formula is translated “For all x and y , if x < y ,
   then there is a z such that x < z < y .”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?
   Remember, the formula is translated “For all x and y , if x < y ,
   then there is a z such that x < z < y .”
   Suppose we have rational numbers x and y such that x < y .




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?
   Remember, the formula is translated “For all x and y , if x < y ,
   then there is a z such that x < z < y .”
   Suppose we have rational numbers x and y such that x < y .

   Q: Is there a z such that x < z < y ?


                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?
   Remember, the formula is translated “For all x and y , if x < y ,
   then there is a z such that x < z < y .”
   Suppose we have rational numbers x and y such that x < y .

   Q: Is there a z such that x < z < y ?
                             x+y
   A: Yes. Let z =            2 .
                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


Changing the context
   Let’s change the domain of discourse to the set of rational
   numbers Q, i.e. the set of postive and negative fractions.
   Is the formula

   ∀x ∀y ((Mxy & ¬x = y ) ⇒ ∃z ((Mxz & ¬x = z) &(Mzy & ¬z = y )))

   true or false in this new context?
   Remember, the formula is translated “For all x and y , if x < y ,
   then there is a z such that x < z < y .”
   Suppose we have rational numbers x and y such that x < y .

   Q: Is there a z such that x < z < y ?
                             x+y
   A: Yes. Let z =            2 .      Thus, the formula is true.
                                                       10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation

    Now, we have the following new definitions:




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation

    Now, we have the following new definitions:
    A closed predicate formula is




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation

    Now, we have the following new definitions:
    A closed predicate formula is
        logically true if it is true in every structure,




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation

    Now, we have the following new definitions:
    A closed predicate formula is
        logically true if it is true in every structure,
        consistent if it is true in some structure, and




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


Recall. . .


    “logically true” means true in every situation
    “consistent” means true in some situation
    “inconsistent” means true in no situation

    Now, we have the following new definitions:
    A closed predicate formula is
        logically true if it is true in every structure,
        consistent if it is true in some structure, and
        inconsistent if it is true in no structure.


                                                        10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A consistent formula




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula



   We will work with a vocabulary consisting of a 1-place predicate
   Gx and a 2-place predicate Mxy .




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula



   We will work with a vocabulary consisting of a 1-place predicate
   Gx and a 2-place predicate Mxy .
   To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula



   We will work with a vocabulary consisting of a 1-place predicate
   Gx and a 2-place predicate Mxy .
   To show: ∀x (¬Gx ⇒ ∃y Mxy ) is consistent.
   That is, we need to find a structure in which ∀x (¬Gx ⇒ ∃y Mxy )
   is true.




                                                       10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A consistent formula, continued




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:
       Domain: people




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:
       Domain: people
       Gx: “x is male”




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:
       Domain: people
       Gx: “x is male”
       Mxy : “x is married to y ”


                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:
       Domain: people
       Gx: “x is male”
       Mxy : “x is married to y ”
   Then the formula says, “For every person x, if x is not male, then
   x is married.”
                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A consistent formula, continued
   Here’s the desired structure:
       Domain: people
       Gx: “x is single”
       Mxy : “x is married to y ”
   In this structure, ∀x (¬Gx ⇒ ∃y Mxy ) says “For every person x, if
   x is not single, then x is married.” True.
   Is ∀x (¬Gx ⇒ ∃y Mxy ) logically true, i.e. true in all structures?
   No. It is false in the following structure:
       Domain: people
       Gx: “x is male”
       Mxy : “x is married to y ”
   Then the formula says, “For every person x, if x is not male, then
   x is married.” False.
                                                       10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A logically true formula




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
       ∀x Fx is true in S: Every object in the domain of S has the
       property F .




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
       ∀x Fx is true in S: Every object in the domain of S has the
       property F .
       ∃x Fx is false in S: There is no object in the domain of S has
       the property F .



                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
       ∀x Fx is true in S: Every object in the domain of S has the
       property F .
       ∃x Fx is false in S: There is no object in the domain of S has
       the property F .
       Impossible!


                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
       ∀x Fx is true in S: Every object in the domain of S has the
       property F .
       ∃x Fx is false in S: There is no object in the domain of S has
       the property F .
       Impossible!
       Thus, ∀x Fx ⇒ ∃x Fx is true in S.

                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula
   Vocabulary: 1-place predicate Fx.
   To show: ∀x Fx ⇒ ∃x Fx is logically true.
   Method 1: Check that ∀x Fx ⇒ ∃x Fx is true in every structure by
   abstract reasoning about structures.
       Let S be any structure.
       Assume that ∀x Fx ⇒ ∃x Fx is false in S.
       Thus, ∀x Fx is true in S, but ∃x Fx is false in S.
       ∀x Fx is true in S: Every object in the domain of S has the
       property F .
       ∃x Fx is false in S: There is no object in the domain of S has
       the property F .
       Impossible!
       Thus, ∀x Fx ⇒ ∃x Fx is true in S.
       But S is an arbitrary structure, so this holds for all structures.
                                                       10130 Lecture 32
                                            Review
        A bit more on the alternation of quantifiers
                                  One last example
                Logical Theory for Predicate Logic


A logically true formula, continued




                                                      10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.




                                                       10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

   1.                                                  ∀x Fx               (assume)




                                                        10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

   1.                                                  ∀x Fx               (assume)
   2.                                                  ???                   (∀E 1)




                                                        10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

   1.                                                  ∀x Fx               (assume)
   2.                                                  ???                   (∀E 1)
   .
   .                                                     .
                                                         .
   .                                                     .
   .
   .                                                     .
                                                         .
   .                                                     .




                                                        10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

   1.                                                  ∀x Fx                (assume)
   2.                                                  ???                    (∀E 1)
   .
   .                                                     .
                                                         .
   .                                                     .
   .
   .                                                     .
                                                         .
   .                                                     .
   ?                                                   ∃x Fx                 (∃I, ??)




                                                         10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

   1.                                                  ∀x Fx                 (assume)
   2.                                                  ???                     (∀E 1)
   .
   .                                                     .
                                                         .
   .                                                     .
   .
   .                                                     .
                                                         .
   .                                                     .
   ?                                                   ∃x Fx                  (∃I, ??)
   ? ∀x Fx ⇒ ∃x Fx                                                          (⇒I, 1-??)




                                                         10130 Lecture 32
                                             Review
         A bit more on the alternation of quantifiers
                                   One last example
                 Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

    1.                                                 ∀x Fx                (assume)
    2.                                                 ???                    (∀E 1)
    .
    .                                                    .
                                                         .
    .                                                    .
    .
    .                                                    .
                                                         .
    .                                                    .
    ?                                                  ∃x Fx                 (∃I, ??)
   ? ∀x Fx ⇒ ∃x Fx                                                         (⇒I, 1-??)




                                                        10130 Lecture 32
                                              Review
          A bit more on the alternation of quantifiers
                                    One last example
                  Logical Theory for Predicate Logic


A logically true formula, continued


   Method 2: Write a proof of the validity of ∴ ∀x Fx ⇒ ∃x Fx.

    1.                                                  ∀x Fx                (assume)
    2.                                                  ???                    (∀E 1)
    .
    .                                                     .
                                                          .
    .                                                     .
    .
    .                                                     .
                                                          .
    .                                                     .
    ?                                                   ∃x Fx                 (∃I, ??)
   ? ∀x Fx ⇒ ∃x Fx                                                          (⇒I, 1-??)

   We’ll do this later. . .



                                                         10130 Lecture 32

								
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