Lecture 1...

I. Complex Symbolization Overview (7.3)

1. The idea is to fit our two basic forms :
   
   

   For a sentence like

   (*)   A small dog is whining at the door.

   one uses existential form:

   (Step I) Some S are P.

   Then...
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   (*) is about "small dogs": this is S.

   And the predicate P is "being at the door". So,

   
   
   
   
   

   we provide a hybrid symbolization of the form:

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

   For (*), this should be

   
   
   
   
   
   
   
   

   (%x)(x is a small dog & x is whining at the door.)

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

   (Step III)       (%x)(Sx & Px)

   Then we just have to translate "x is a small dog":

   
   
   
   
   
   
   
   
   
   
   
   

   Sx&Dx

   and "x is whining at the door":

   
   
   
   
   
   

   Wx&Axd

   To get our final result

   
   
   
   
   
   

   (%x)[(Sx&Dx)&(Wx&Axd)]

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
2. Similarly for Universal Form:
   
   

   Take a sentence like

   (**)     The only small dogs (we own) are whining at the door.

   The idea is that each of our small dogs is just outside being a nuisnance.

   So, can fit this into WHAT form?

   
   
   
   
   
   
   
   
   
   
   
   

   We can fit (**) into universal form.

   (Step I) All S are P

   where S is "small dogs" and P is "whining at the door". Next we have:

   (Step II)   (^x)( x is an S > x is a P )

   Finally, we translate S and P into PL as before to move into...

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   (Step III)    (^x)(Sx>Px)

   which, with universe of discourse including only things we own, is

   (^x)((Sx&Dx)>(Wx&Axd))

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. Next: "only"
   

   What about

   Only S are P

   (Think about "Only females become pregnant.") This is equivalent to which of the following?

   
   
   
   
   
   
   
   
   

   All S are P.   (All F are P)
   No S are P.  (No F are P)
   All P are S.  (All P are F)
   All non-S are non-P.  (All non-F are non-P).    

   (answers)

   
   
   
   
   
   
   
   
   
   
4. Problems
   
   
   
   
   
   
   
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II. Derivations in PL

1. Because of what '(^x)Bx' means......
   

   ...we can deduce, for example, 'Bn' from it, or 'Br', etc.:

   Premise		1		(^x)Bx
   1 ^E		2		Bn
   1 ^E		3		Br

   This rule, ^E, is represented this way:

   
   
   
   
   
   
   
   
   
   
   
   

   ^E
   input: output:	(^x)P  P(a/x)

   
   
   
   
   
   
   
   
   
   
   
   
2. And because of what '(%y)Sy' means...
   

   We can deduce it from, for example, 'St' or 'Sc', etc.:

   Premise		1		St
   1 %I		2		(%y)Sy

   This rule, %I, is represented this way:

   
   
   
   
   
   
   
   
   
   

   %I
   input: output:	P(a/x)  (%x)Px

   
   
   
   
   
   
   
   
   
   
3. Let's try some!
   
   
   
   
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4. The new rule, ^I, has provisos!
   
   
   • One can't just argue that...
     
     Because George is president, 'Pg', that everyone is president. That would be:
     
     
     
     
     
     

     Premise		1		Pg
     1 ^I		2		(^x)Px???

     We can only make the jump from the substitution instance to the universally quantified sentence when the name is "arbitrary".

     Let's see what this amounts to by example...

     
     
     
     
     
     
     
     
     
     
     
     

     P1: If a city is in Oakland County, then it is in MI.
     P2: If a city is in MI, then it's in the USA.
     Therefore, if a city is in Oakland County, then it is in the USA.

     How would we prove this? (Think informal proofs...)
     
     
     
     
     
     
     
     
     

     Suppose that c is any arbitrary city in Oakand County. Then by P1, c is in MI, and so, by P2, c is in the USA. Hence, because c is arbitrary...
     

     
     
     
     
     
     

     ANY city that is in Oakland County is in the USA.

     Here's the rule: ^I

     
     
     
     
     
     

     ^I input: output: P(a/x) (^x)P Provided 'a' is arbitrary in this sense:  'a' does not occur in any premise or undischarged assumption.  'a' does not occur in P.

     "c"  in our example is arbitrary because it is neither mentioned in the premises nor the conclusion we draw: we picked it "out of the blue".

     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
   • So, let's apply ^I. And don't forget the "*" tool for assumptions.
     
     
     
     
     
     
     
     
     
     
     
     

III. Semantics Homework? Other Homework?

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(start)

IV. Pigs Feet...

1. It's even harder to think about %E.
   
   Just because you know that someone has property P does not mean that you know who that person is.
   
   Maybe you've been in the neigborhood delis in Chicago. And seen the pickled pigs feet for sale by the gallon! But you have no idea who does the buying.
   
   Now, what do you do with a premise that someone likes pigs' feet?
   
   

   Suppose we have this argument:

   Someone in Chicago likes pigs' feet.
   Everyone from Chicago is from Illinois.          
   Therefore someone from Illinois likes pigs' feet.

   
   
   
   
   
   
   
   
   
   
   

   We would naturally argue like this:

   I don't know who she (he) is, but there is (at least one) person out in Chicago who likes pigs feet.
   And, she, this lover of pigs' feet, must be from Illinois.

   We use the word "she" to "illustrate"...

   
   
   
   
   
   
   
   
   
   
   

   ...put differently, we don't have an English proper name for her, whoever she is, but we temporarily use "she".

   In our system PD, we'll use a temporary name to illustrate "someone". So, let's see how this reasoning and %E works formally.

   
   
   
   
   
   
   
   
   
   
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2. Oh, and how do we write %I?
   

   %E input: input: output:   (%x)Px P(a/x) Q Q   Provided 'a' is illustrative in this sense:  'a' does not occur in any premise or undischarged assumption.  'a' does not occur in P.  'a' does not occur in Q.

V. Homework for 8.2

VI. QN...our one short-cut rule

1. This one's easy: For example...
   
   About bats, if NOT all are blind, then some are

   not blind.

   ...
   
   
   
   
   
   
   
   
   Thinking about fish, if all are NOT whales, then NO fish is a

   whale.

   ...
   
   
   
   
   
   
   
   
2. Our new rule formalizes this kind of reasoning:
   

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

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
3. Problems...
   
   
   
   
   
   
   
   
   (skip to Pop-Quiz Paradox)
   
   
   
   
   
   
   
   
   
   
   
   
   
4. Strategy
   

   General Strategy

   Consider goals that may plausibly allow you to complete your derivation. Usually you will be given an ultimate goal: what you are asked to derive. Often during the course of a derivation, strategy will require intermediate goals -- other sentences required to complete the derivation. Do what is obvious to you if that will clearly help you toward your goal(s). Use introduction and elimination rules: Consider the I-rules appropriate for the main connective of your goal sentence. Consider the E-rules appropriate for the main connectives of any accessible sentence you may have already derived (this includes assumptions). When all else fails (in the attempt to derive your goal P), assume ~P and attempt to subderive a contradiction so you can justify P by ~E. Now Recycle: After you have applied a rule go back to 1 and start the goal analysis anew. Continue the process until you are finished.

(next)

VII. Pop-quiz paradox...

(skip to Mock Exam: VIII on Identity is repeated in Lectures for W13.)

VIII. And now for identity!

The word "is" is ambiguous.

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IX. Mock Exam