L1 W12 (L2)
roll
exam: one week 5.8  8.1; see mock exam
DO: all tutorials; exams will take some problems from these.
Questions: (1) Which of the following styles is least distracting?  I worry that some are too jumbled. (2) Do we need lots more inclass work and less listening?
Lecture 1...
I. Complex Symbolization Overview (7.3)
 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,
(%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":

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))
 Next:
"only"
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 nonS are nonP. (All nonF are nonP).
(answers)
 Problems
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II. Derivations in PL

Because of what '(^x)Bx' means......
^E 
input:
output: 
(^x)P
P(a/x) 

And because of what '(%y)Sy' means...
%I 
input:
output: 
P(a/x)
(%x)Px 
 Let's try some!
(next)
 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?
(next)
L2
roll
EXAM! (there could be some extra credit)
NOTE: PL as of 8.1 includes all of SD+
Office hours?
(start)
IV. Pigs Feet...
 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.
(next)

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 shortcut rule
 This one's easy:
For example...
About bats, if NOT all are blind, then
some are
not blind.
...
 Our new rule formalizes
this kind of reasoning:

~(^x)P (%x)~P 
or 
~(%x)P (^x)~P 

 Problems...
(skip to PopQuiz Paradox)

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 Irules appropriate for the main connective of your goal sentence.
 Consider the Erules 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. 
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VII. Popquiz paradox...
(skip to Mock Exam: VIII on Identity is repeated in Lectures for W13.)
VIII. And now for identity!
The word "is" is
ambiguous.
Here we use "is" to attribute a property or relation and apply a predicate.
We call this the "is" of predication. Our symbolizatons might be something like:
Pg
and
Sge
However...
... for
George is George W. Bush
we are saying that the objects named are the same; they are
identical.
When we use the English word "is" to indicate identity, as in (*) we call this the "is" of identity.
Here we need to symbolize with a new symbol for identity, "I"; we might write this as:
Igb
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IX. Mock Exam
