I. Strategy Review (from last week)

II. Warm-up

(next)

III. Exam

(next)

IV. PL again

1. Review
   • This 'PL' means what???
     

     Predicate Language and Predicate Logic.

     (I am conflating the two...but they come to about the same thing. The logic includes a language; and the language implies the logic.)

     
     
     
     
     
     
     
     
     
   • Idea

     We break down sentences like

     George is male.

     into name ('George '...represented perhaps by 'g')

     and

     predicate ("is male"...represented perhaps by 'M')

      

     This gives
     

     Mg

     as our symbolization.

      

      

      

     : Predicates and Names
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • Quantifiers
     

     Universal: ^

     '(^x)Px' means that "every x is such that P".

     and
     

     
     
     
     
     
     
     
     
     
     

     Existential: %

     '(%x)Px' means that "at least one x is such that P".

     
     
     
     
     
     
     
     
     
     
     
     
   • Exercises
     
     
     
     
     
     
     
     
   • Homework Questions? (6.1ex IV)
     
     
     
     
     
     
     
     
     
     
2. Relations
   
   
   • 'Rxy'

     means that x is related to y by R. For example, we may say that Ann is older than Bob. If 'Oxy' means that x is older than y, then we symbolize Ann's age in relation to Bob this way

     Oab

      

      

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • Let's try some.
     
     
     
     
     
     
     
     
     
     
     
     
3. Syntax
   
   
   • Formulas
   
   
   

   We need to define sentences. But the basic parts of sentences are a little different in PL.

   For example, when we conjoin two objects in PL to get

   Px&Rxy

   the two objects need not be sentences exactly. We'll call them
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   

   Formulas.

   'Px' and 'Rxy' are atomic formulas.

   But neither is a sentence: neither expresses a complete idea because 'x' and 'y' are like placeholders or blanks. Let's define formulas now, then define sentences.
   

   
   
   
   
   
   
   
   
   
   
   
   
   
   Start out this way:
   A term is any name ('a', 'b', 'c',...'u') or variable ('w', 'x', 'y', or 'z'). (Subscripts are allowed.)
   
   An expression is an atomic formula in PL if and only if it is a predicate letter followed by 0, 1, or more terms (i.e., names and/or variables).
   And then
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   We can define formulas by our old building definition, except for one new wrinkle. What is the one change?

   I) Each atomic formula of PL is a formula of PL.
   
   II) If P and Q are formulas of PL, then so are
   a) (PvQ), (P&Q), (P>Q), (P=Q), ~P, and
   b) (^x)Px, (%x)Px (provided an x-quantifier does not occur in P)

   Think about some examples:
   

   We start with 'Fx' and 'Oax' as two atomic formulas. then
   

   We could use clause IIa) to build a compound formula:

   (Fx>Oax)

   And then use clause IIb) to add a quantifier.
   

   (^x)(Fx>Oax)          (more)
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   (%x)Pxx (^y)Jy>Ky (%xy)Rsxy
   
   
   
   
   
   
   
   
   
   
   
   • Problems
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • More Definitions
     

     A subformula of a formula P is any formula used or produced in the building of P.

     and
     

     P is an immediate subformula of a formula Q if and only if P is a subformula of Q and is used in the final step of the building process of Q.

     and
     

     The main connective (or main logical operator) of a formula P is the last quantifier or truth functional connective used in P's building process.

     (%x)(Pxx&Jx) (^y)Jy>Ky (%x)(%y)Rsxy

     Most importantly

      

      

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

     to define a sentence of PL, think about

     (*) (%x)(Tx&Lx)

     and

     (**) Tx&(%x)Lx

     First consider the subformulas and main connectives of the above.

     Then consider:
     

     We define a notion that means roughly "the part of the sentence to which the quantifier applies".

     The scope of a quantifier in a PL formula P is the subformula of P for which that quantifier is main connective.

     Notice the difference is scope. (*)'s main connective is '%' and so this quantifier has the whole sentence as scope.

     For (**),
     

     '&' is the main connective, and the first occurrence of 'x' is not within the scope of the existential quantifier.

     Other examples...

     Finally!

      

      

      

      

      

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     

     An instance of a variable x in a formula is bound if and only if it is within the scope of an x-quantifier. Otherwise we say it is free.

     In (*), '(%x)(Tx&Lx)', all instances of 'x' are bound, while in (**), Tx&(%x)Lx', the first instance is free. This is what makes (*) but not (**) a sentence of PL.
     

     A formula of PL is a sentence if and only if it has no free variables.

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
4. Problems
   
   
   
   
   
   
   
   
   
   
   
   
5. One final syntactical definition will come in handy when we discuss the semantical issues and derivations of the next two chapters:
   
   
   

   If a sentence is of the form '(^x)P' or '(%x)P', then the substitution instance P(a/x) is the result of taking P and replacing every occurrence of x with a.

   So, for example, if we have
   

   (^x)(Jx>Kxa)

   then which of the following are substitution instances?

     1. (^x)(Ja>Kaa)
   2. (Ja>Kaa)    
   3. (Jb>Kba)    
     4. (Ja>Kbb)      

   Answer:
   

   The middle two.

   Not the first because it needs to remove the quantifier.

   Not the last because one needs to substitute into the 'x' place with the same name each time and one must not change the 'a' in the original; only change the 'x'.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
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V. More Symbolizations

VI. Derivations! Let's just try some. We'll just try to figure out the rules as we go.

VII. Homework