I. Pop-quiz paradox...

(next)

We'll see that the answer to the pop-quiz symbolization question involves saying that ALL remaining class days are IDENTICAL to either a, b, c, or d.

 

II. So: identity! We'll add one specific predicate to PL in order to make PLI, predicate logic with identity.

1. Let's begin by saying that "George is identical to the president but distinct from the winner of the MI vote"
   

   Then use this interpretation

   g: George, m: the winner of the MI vote, p: the president
   
   Ixy: x=y

   So:
   (fill in with just PLI)

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. How would we say that "George and someone else lives in the White House"?
   

   g: George, h: The White House
   Px: x is president, Mx: x is male
   Lxy: x lives in y, Ixy: x=y

   "George lives in the White House" is

   Lgh

   So, we just need to add that someone else does too.

   
   
   
   
   
   
   
   
   
   
   
   
   

   That is, we need to say that someone DISTINCT from George lives in the White House:

   (%x)(~Ixg&Lxh)

   So, the final answer:

   
   
   
   
   
   
   
   
   

   Lgh & (%x)(~Ixg&Lxh)

   
   
   
   
   
   
   
   
   
3. Now, symbolize: "At least two people live in the White House".
   

   (fill in for the question marks)

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Next: "All males except George live outside the White House."
   
   
   These "all except" symbolizations may seem a little ambiguous. (See tutorial 9.1 for details -- remember I'll be holding you doubly responsible for the tutorials now.) But
   
   
   
   
   
   

   ...we'll say that

   All S except a are P

   is a straigtforward universal statement: it's about the subject class including all S execept c.

   
   
   
   
   
   
   
   

   (^x)( (x is in the set S less a)  > Px)

   Then we have

   
   
   
   
   
   

   (^x)( (Sx&~Ixa) > Px)

   So, for "All males except George live outside the White House":
   

   
   
   
   
   
   
   

   (^x)[(Mx&~Ixg)>~Lxh]

   
   
   
   
   
   
   
   
   
   
   
   
5. Finally: "The president lives in the White House"
   
   We can treat "the president" as a new name for George and symbolize this one simply as:
   
   
   
   

   Lgp

   But we can also get at more of what "the" means.

   
   
   
   
   
   
   The idea here is that there is exactly one president and that one person lives in the White House. So,
   
   
   
   

   (%x)(x is president and x is the only person who is president and x live in the white house)

   Finally, then,

   
   
   
   
   
   
   

   (%x)[(Px&(^y)(Py>Ixy))&Lxh]

   or equivalently:
   
   
   
   
   
   
   

   (%x)[(Px&~(%y)(~Iyx&Py))&Lxh]

   
   
   
   
   
   
   
   
6. Problems

III. Homework: The harder rules of PD and PD+

1. Just what is PD? And PD+?
   

   PD is our system of rules including all of SD plus the new rules for introduction and elimination of the two quanitifiers.

   PD+ is PD together with all short-cut rules including the new QN:
   

   QN (Quantifier Negation)
   ~(^x)P (%x)~P or ~(%x)P (^x)~P

    

    

    

    

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Review
   
   
   
   
   
   
   
   
   
   
   
   
3. Homework Problems from chapter 8? 8.4ex II?

IV. Applied Logic (or what we're here for)

1. Analysis and Meaning
   
   We now have the capability to think about statements in a very precise way.
   
   
   
   
   
   
   
   
2. Real Life Argument
   
   We are now good at hypothetical and maybe even indirect reasoning. For a logic example...
   
   
   
   
   Suppose that we have a valid argument A. Then A is sound iff it has no false premise.. Proof:
   
   
   
   

   Suppose A is valid. Then it's sound exactly in case it has no false premise.

   
   
   
   
   
   
   
   
   
   
   
3. Programming is applied logic!
   

   We've seen our lecture notes add something to the end of a grey box.

   We use a "function" to do this. But this is just a tool out of my tool box which appends a bit of text when it's "called". Here's how I write this function in JavaScript...

   
   
   
   
   
   
   
   
   
   

   var newElem, idElem      (These are names)
   function addToEnd(ident,text,time) { (note the logic:)
     if ( (typeof document.all == 'undefined') &&
        ( typeof document.getElementsByTagName !=               'undefined' )) { (then DO the following...)
   document.all = document.getElementsByTagName("*")
   }
   newElem = document.createElement("P") (
   newElem.id = "newP" (here we create a name in the "object language", a name for the HTML page's language)
   newElem.innerHTML=text (we put some text in our "newElem")
   idElem = ident
   setTimeout("document.all[idElem].appendChild(newElem)",time)
   } (and display the text after a delay)

   Exclusive "or":

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   You'll also get our wedge in a standard program language, though it's represented by '||'. For example, one might want to add text to a grey box with ID either "box1" or "box2"

   Finally,
   

   
   
   
   
   
   
   
   

   there are quanitifers in the language.

   One might want to add text to ALL boxes. There is a statement that says roughly:

   For all boxes with ID from 1 to 10, add the word "DONE" to their end.

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Philosophy and Logic I:
   

   Mind is just an automated formal system? The brain represents reality -- coded up somewhat as a computer does -- and processes this information.

   (Well, think about taking a real philosophy class for more on issues of mind.)

   
   
   
   
   
   
   
   
   
   
   
   
   
5. Philosophy and Logic II:
   

   Descartes' Proof of God?

   Descartes was the "I think therefore I am" guy. Also, he's responsible for a lot of physics and analytic geometry. What was his coordinate systemaization of space called?
   

   
   
   
   
   
   
   
   
   Cartesian Coordinates! (the x, y stuff on a plane.)
   
   
   
   
   

   Descartes' Proof of God?

   God must exist because by definition she has all perfections. Perfect things can’t fail to exit!

   Premise? Conclusion?

   Thoughts?
   

   
   
   
   
   
   
   
   
   
   

   We need to be careful about defining God. Descartes just claims that God has all perfections. But this may seem to "beg the question" or assume what he wants to prove: that God exists. Better...

   
   
   
   
   
   

   x is God  if and only if   x has all perfections.

   So there might or might not be any such x. This looks more like the defintions we given in logic. Like

   A is sound if and only if A is both valid and has true premises.

V. Review Sheet

VI. Mock Final Exam

1. Do the new mock exams. See the syllabus for one. Here's another.
2. (Optional:) Review the mock exams for Exam V. Look again at the one from last lecture.
3. (Optional:) Try the next problem for more of a test:

VI. A Harder Problem...