The Theory of Insults


P = George's mother is ugly.

Q = George's mother is more stupid
     than Harry.

R = Harry's mother is ugly.

S = Harry's mother is more stupid 
     than George.


Let (Um x) = x's mother is ugly.

Then P = (Um George) and R = (Um Harry).

Let  (MS x y) = x's mother is
                  more stupid than y.

So Q = (MS George Harry),
   S = (MS Harry George).

 Predicate symbols and Function Symbols

   (ugly x) = x is ugly.

   (mother x) = the mother of x


P = (ugly (mother George))

Q = (ugly (mother Harry))

T = (nice (mother George))


   (more x y) = x is more than y.

   (stupidity x) = the stupidity of x


  (and (ugly (mother George))
       (more (stupidity (mother George))
             (stupidity Harry)))

  (and (ugly (mother Harry))
       (more (stupidity (mother Harry))
             (stupidity George)))

  (nice (mother George))


     Variables and quantifiers


There is no one stupider than George.

  (not
   (Exists (x)
     (more (stupidity x)
           (stupidity George))))



Anyone stupider than Harry is Ugly.

  (All (x)
     (implies (more (stupidity x)
                    (stupidity Harry))
              (ugly x)))

Why do we believe this argument?

  (All (x) (implies (man x) (mortal x)))
  (man Socrates)
  -------------------------------------
  (mortal Socrates)


Method of interpretation!

   Domain

   Predicate symbols map to predicates

   Function symbols map to functions

   Individual Constant symbols map to 
     distinguished members of the domain.


If P is true in an interpretaion we say
that the interpretation is a model of P.

An argument is valid if there is no
interpretation in which the consequent
is false and the antecedents are true.


     (All (x) (Exists (y) P))
     ------------------------
     (Exists (y) (All (x) P))

This is not valid:

  Let the domain be the integers.
  Let P be (<= x y).

  (All (x) (Exists (y) (<= x y))) is
  true, because for any integer you
  choose there is always a bigger one.
  So we have a model for the antecedent.
  In the model the consequent
  (Exists (y) (All (x) (<= x y)))
  is false, because there is no greatest
  integer.


     (Exists (y) (All (x) P))
     ------------------------
     (All (x) (Exists (y) P))

This is a valid rule: Suppose the rule
were invalid, then there is a model of
the antecedent in which the consequent
is false.

If the consequent is false there is an 
x0 for which there is no y that makes
P true.  But the antecedent claims that 
there is a y0 for which P is true, 
independent of the choice of x.  This 
is a contradiction, so the rule is 
valid.

		Remember

1.                    Premise       {1}
    All y [(greek y) --> (human y)]  

2.                    Premise       {2}
    All x [(human x) --> (mortal x)] 

3.                    Premise       {3}
    (greek *G)

4.                    (US 1 *G y)   {1}
    (greek *G) --> (human *G)

5.                     (MP 4 3)    {1 3}
    (human *G)

6.                     (US 2 *G x)   {2}
    (human *G) --> (mortal *G)

7.                     (MP 6 5)  {1 2 3}
    (mortal *G)

8.                     (CP 7 3)    {1 2}
    (greek *G) --> (mortal *G)

9.                     (UG 8 z *G) {1 2}
    All z [(greek z) --> (mortal z)]

   Rule 15. Universal Specification

     Restriction: In this rule t may be a
     term composed of constants or
     arbitrary individuals.

   n     All x P        ---          d
   --------------------------------------
   m     S[t;x;P]    (US n t x)      d


   Rule 16. Universal Generalization

     Restriction: Here t must be some
     arbitrary individual not occurring in
     any line of d, and x may not appear
     in P. 


   n     P                ---        d
   --------------------------------------
   m    All x S[x;t;P]   (UG n x t)  d


BAD! 

10. All w (mortal w)  (UG 7 z *G) {1 2 3}

11. All w (human w)     (UG 5 z *G) {1 3}

12. All w (greek w)       (UG 3 z *G) {3}



GOOD!

13.                       (UG 6 z *G) {2}
    All z [(human z) --> (mortal z)]


   Rule 17. Existential Specification

     Restriction: ???


   n     Exists x P     ---          d
   --------------------------------------
   m     S[t;x;P]    (ES n t x)      d



   Rule 18. Existential Generalization

     Restriction: Here t is a term which
     contains no variables, and x may not
     appear in P.

   n     P                 ---        d
   --------------------------------------
   m    Exists x S[x;t;P] (EG n x t)  d

Consider the following fallacious 
derivation (dependencies are all {1}).

1. (All (x) (Exists (y) (< x y))) Premise
2. (Exists (y) (< *A y))      (US 1 *A x)
3. (< *A a)                    (ES 2 a y)
4. (All (x) (< x a))          (UG 3 x *A)
5. (Exists(y)(All(x)(< x y)))  (EG 4 x a)

Where is the bug?

  
The problem is that a depends on *A.  

Patch: Make it explicit: (a *A).


   Rule 17. Existential Specification

     Restriction: In this rule t is a term
     composed of a new function symbol
     applied to the list of the arbitrary
     individuals occurring in P. 


   n     Exists x P     ---          d
   --------------------------------------
   m     S[t;x;P]    (ES n t x)      d


1. (All (x) (Exists (y) (< x y)))  Premise
2. (Exists (y) (< *A y))       (US 1 *A x)
3. (< *A (a *A))           (ES 2 (a *A) y)
4. (All (x) (< x (a x)))       (UG 3 x *A)
5. STUCK.