I. Syntax and Symbolization Review

1. Syntax:
   
   Atomic Formula?
   
   

   1. Jj
   2. ~J
   3. Jxd
   4. Vj
   5. Vx                        (answers, next)
   
   
   
   
   
   
   
   
   
   
   
   
   Formulas? Main Connectives? Quantifier Scope?
   
   

   1. A
   2. Lax>JJ
   3. %xBx
   4. (^x)(Lx>Ll)
   5. B%x>x                     (answers, next)
   
   
   
   
   
   
   
   
   
   
   
   
   
   Sentences?
   
   

   1. (^x)Jx>Rx
   2. (^x)(Jx>Rx)
   3. (^x)Jx>(%x)Rx
   4. Jx>(^x)Rx                         (answers)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Symbolization Review
   
   
   
   
   
   
   
   
   
   
   
3. Homework Questions?

(next)

III. Symbolizations with Basic Forms for categorical logic.

1. As symbolizations get a little harder, we'll need to fit them into a couple of basic forms. Here's the first form.
   

   Existential Form

   The first basic form of English is the following.

   existential form:     Some S are P.

   where 'S' (the subject) and 'P' (the predicate of the expression) name groups or classes of individuals. (We will call these the subject class and the predicate class, respectively.)

   So, for example
   

   "Some students are freshman" is of existential form. And it's pretty easy to see how it might be symbolized. Given a natural symbolization key, it could well be rendered as

   '(%x)(Sx&Fx)'.

   Easy! But there are harder cases...
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   (*) There are female logic students who are juniors set to graduate next year.

   We can fit this messy example sentence into the existential form and then symbolize. Here's the mold we need to fit:
   

   
   
   
   
   
   
   
   
   

   (Step I) Some S are P.

   where S is "female logic students".
   and P is "juniors who will graduate next year".

   Now,

   
   
   
   
   
   
   
   
   

   we need to provide a hybrid English, PL symbolization of the form:

   (Step II) (%x)(x is an S  &  x is a P)

   For (*) this should be

   (%x)(x is a female logic student & x is a junior set to graduate next year)

   Finally,

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   we take the hybrid of step II and form it into pure PL:

   (Step III)       (%x)(Sx & Px)

   Take this key:

   universe of discourse: Everything
   
   l: logic, n: next year
   
   Fx: x is female, Jx: x is a junior, Sxy: x is a student of subject y, Gxy: x will graduate in year y

   Then the subject phrase becomes: 'Fx&Sxl' and the predicate phrase becomes 'Jx&Gxn'. So, finally we have:

   (*)'s Symbolization:
   
       (%x)[ (Fx&Sxl) & (Jx&Gxn) ]

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Problems
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. The Second Basic Form
   

   Universal Form

   All such-and-such are so-and-so.

   For example, "All Swedes are Europeans". Again we have a subject class and predicate class:

   universal form:     All S are P.

   Such a universal statement means that anything is such that if it's in the subject class, then it's also in the predicate class. So, our example might be translated as

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   (^x)(Sx>Ex)

   Let's think about fitting this form with a more complex example.
   

   
   
   
   
   
   
   
   
   
   (**) Every female junior will graduate next year.

   This means:

   (Step I) All female juniors are people who will graduate next year.

   Notice that the subject is a conjunction. So, we have the hybrid form:

   (Step II)   (^x)( x is a female and a junior > x is a person who will graduate next year )

   and finally the symbolization:

   (Step III) (^x)( (Fx&Jx) > Gxn )

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. More Problems

(next)

III. Semantics

1. Interpretations
   
   An interpretation is roughly a symbolization key. It tells you what subject matter is under discussion...
   

   The "universe of discourse" (UD) , and
   

   it tells you what the basic symbols indicate, for example:

   The univese of discourse is living U.S. presidents.
   
   b: Bill Clinton, j: Jimmy Carter, g: Gerald Ford
   
   Dx: x is a democrat, viz. Clinton and Carter, Bx: x is a Bush, viz. G.H.W. Bush and G.W. Bush
   
   Bxy: x was in office before y, viz. the ordering of living presidents from Ford to (the last) Bush

   So,

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   for our truth-functional connectives taken over from SL, we can give truth conditions as before.

   'Dg=Dj' has what truth value
   

   False

   And for '(^x)Dx'?
   

   
   
   
   
   
   
   
   
   
   
   
   
   

   False again.

   But '(%y)Dy' is true (becuase there is at least one thing in the universe of discourse making 'D' true).

   Hold on, what does this mean? that "something makes D true"? Think about our syntactical definitions, which one means "makes true"? (Hint: "S____ I____".)
   

   Substitution Instance

   '(%y)Dy' is true in our interpretation because there is at least one substitution instance making it true.

   Problem!
   

   This definition depends on having names for all members of the universe of discourse. We don't have a name for any Bush. So, we have to be careful about '(%x)Bx'. Try this...
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   If we need to evaluate the truth value of a sentence of PL with respect to an interpretation lacking names for some elements of its domain, we simply add names to the interpretation until each member has a name and then evaluate.

   '(%x)P' is true on an interpretation I if and only if P(a/x) is true on I for at least one name a (where names for all members of the UD are added if necessary).

   Similarly,

   '(^x)P' is true on an interpretation I if and only if P(a/x) is true on I for all names a (where names for all members of the UD are added if necessary).

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Problems
   
   
   
   
   
   (next)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. Consider the following interpetation.
   

   Universe of Discourse: Positive Integers, 1,2,3,4,...
   
   Gxy: x is greater than y

   Prove that (*) '(^x)(%y)Gyx' is true on the given interpretation.      Fill in...

   
   
   
   
   (next)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

    

    

   L2

   roll

    

    

    

    

4. Semantics Review
   1. Interpretations
      

      These are like truth value assignments in SL.

      An interpretation just tells us enough to know which sentences of PL are true and which false. So,

      an interpretation tells us what we're talking about (the universe of discourse) and
      

      what the language indicates. E.g., 'D' indicates the democrats. We put it this way:
      

      we gave this example:

      The univese of discourse is living U.S. presidents.
      
      b: Bill Clinton, j: Jimmy Carter, g: Gerald Ford
      
      Dx: x is a democrat, viz. Clinton and Carter, Bx: x is a Bush, viz. G.H.W. Bush and G.W. Bush
      
      Bxy: x was in office before y, viz. the ordering of living presidents from Ford to (the last) Bush

      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
   2. So, an interpretation gives a truth value assignment for PL. For (*) '(^x)(Mx>Gx)',
      
      we can apply

      '(^x)P' is true on an interpretation I if and only if P(a/x) is true on I for all names a (where names for all members of the UD are added if necessary).

      But if 'Mx' stands for "x is a martian", and 'Gx' for "x is green, then all substitution instances of (*) are true!
      

      because a substitution instance looks like this,

      Ma>Ga

      and (because there are no Martians) the antecedent is false and the whole conditional true. Is this bad?
      

      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      

      The proponents of traditional logic -- who disagree with the way we've defined PL -- say this is a bad consequence, that universal form statements

      All S are P

      entail that there is something which is S. Hence, (*) '(^x)(Mx>Gx)', is a bad translation of (*). But notice this reductio of their position...
      

      If "All martians are green" is false, then its negation

      "Not all martians are green" is true.

      That's bad. From this we are inclined to conclude "If they are not all green, then there must be some Martians who are blue or red or..." (The Principle)

      "Not all S are P", as our Square of Oppositoin shows, means that "Some S are not P". 

      This is an absurd consequence. So, let's keep our logic the way we've defined it.

      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      (next)
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
5. More Semantical Definitions
   
   It's easy to extend our old deductive concepts from SL to PL. For example...
   

   A sentence of SL in logically true in SL if and only if it is false on no truth value assignment.

   and replace the words "truth value assignment" with "interpretation":

   A sentence of PL in logically true in PL if and only if it is false on no interpretation.

   And, an argument is valid in PL if and only if WHAT???
   

   There is no interpretation making its premises true and conclusion

   false.

   ....
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
6. Problems
   

   First think about this problem from SL:

   How do we show that

   AvB
      A  
   B

   is NOT valid?

   We'd do...
   

   a truth table and give a truth value assignment showing how the premises can be true and the conclusion false.

   No derivation will work for this case! (We can only show an argument valid by derivation. Not invalid.) Then...
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   the same holds for showing

   (^x)(AxvBx)
    (%x)Ax 
   (%x)Bx  

   is not valid, you don't give a truth value assignment but an in_______
   

   interpretation making the premises true and the conclusion false. Like:

   Universe of discourse: all living U.S. presidents. Ax: x is male, Bx: x is ???

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   (next)

IV. Categorical Logic

1. Traditional categorical logic (the logic which relates categories or one-place predicates) involves two further basic forms
   

   No S are P       (^x)(Sx>~Px)   or   ~(%x)(Sx&Px)

   and

   Some S are not P     (%x)(Sx&~Px)

   Notice...
   

   that these two forms are just variants on our universal and existentail forms. But it's sometimes useful to keep the four forms of cateorical logic in mind.
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   Type	English Form	PL Form
   A-form:	All S are P	(^x)(Sx>Px)
   E-form:	No S are P	(^x)(Sx>~Px) or ~(%x)(Sx&Px)
   I-form:	Some S are P	(%x)(Sx&Px)
   O-form:	Some S are not-P	(%x)(Sx&~Px)

   
   
   
   
   
   
   
   
   
   Note
   

   The Modern "Square of Opposition"

   (Pairs of sentences connected by diagonal lines are contradictory.)

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   (next)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Problems
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

V. Complex Symbolizations

Here are but a few of the many quantifier complications you will in natural language. Read about more in the tutorials and reference!

1. Quantifier Type Complications
   
   
   
   • Each
     

     1) Each of Tom's male friends is a friend of Jenny.

     Here too we need to fit the universal quantification mold:

     (^x)( x is a male friend of Tom > x is a friend of Jenny )

     To say "x is a male friend of Tom's" is to say that x is male and a friend of Tom's: '(Mx&Fxt)'. So, 2)'s symbolization is
     

     (^x)( (Mx&Fxt) > Fxj )

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • A
     

     2) A male friend of Tom is a friend of Jenny.

     This is like 1) but is of existential form. So, to fit the mold, we have:

     (%x)( x is a male friend of Toms & x is a friend of Jenny )

     Which is symbolized as
     

     (%x)( (Mx&Fxt) & Fxj )

     However,

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

     notice that this one

     A whale is a mammal.

     Seems to be about ALL whales:

     (^x)(Wx>Mx)

     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • Any
     

     3) If anyone joins me, I'll be happy for the help.

     Which means that if even one person joins me I'll be happy:

     (%z)Jzm > Hm

     Here 'm' stands for me.

     But:
     

     4) If anyone joins me, he or she will be happy.

     5) If someone joins me, he or she will be happy.

     are both

     (^x)(Jxm > Hx)

     because the "anyone" and the "someone" are both arbitrary: the person could be you, someone from another class, or someone form anywhere.

      

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

2. Multiple Quantification
   

   6) For every number there exists a greater one.

   I.e.,

   For every number x there exists another y which is greater than x.

   or
   

   i)        (^x)(%y)Gyx

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   But,
   
   This is different from

   ii)        (%y)(^x)Gyx

   How so?
   

   Because this second one, with main connective %, means that there is one number greater than all numbers. False.

   The correct symbolization, i), says that for any number you pick, there is a greater one that I can pick. BUT there is no one number I can pick each time.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. Complex Subjects and Predicates
   

   7) Everyone who has a friend is a friend of Jenny.

   This is best fit slowly into the universal form.

   (^x)( x has a friend > x is a friend of Jenny )

   The subject clause...

   involves a quantifier. The indefinite article "a" becomes '%' in the symbolization:

   (^x)( (%y)Fyx > Fxj )

    

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Problems...
   
   
   
   
   
   
   
   
   
   
   
   
   
5. Something more fun...
   
   
   
   
   
   
   
   
   
   
   
   
   
6. Homework