w4
L1 (L2)
roll
exam return and comments...
Next Exam: two weeks! (through chapter four...so we'll get a jump on
that chapter this week)
I. Table review:
A. The idea...one row at a time like
in 3.1 and...
 review the truth table definitions of the connectives,
 think about
truth functionality
, then...
A connective is used truth functionally
to form a sentence from components if and only if that sentence's
truth value depends only on the truth value of the components.
 do a couple problems, and
 note how these simple tables lead to the full tables.
B. Full Tables
 Do some,
and
 think about table length.
II. Concepts and Tables
A. Possibility
 It's key to our deductive logic concepts.
 On a table, we list possibilities, each row corresponds to a possibility.
 Definition: Truth value assignments and partial tvas.
We need to
think about possibilities
in terms of ways things could be. For this, we need to give meaning
to our language so that sentences can be true or false given a possibility:
we'll call these interpretations.
An interpretation must do at least this:
it assigns a truth value (true or false) to atomic sentences of
SL. In this chapter, we will be interested in one particular kind
of interpretation: a truth value assignment.
A truth value assignment
is an association of a single truth value with every atomic
sentence.
So, a truth value assignment provides an infinite number of truth
values, one for each atomic sentence. You can imagine starting:
(*) 'A' has value true, 'B' has value false, 'C' has
value ...
But of course you can't finish: there is an unending list of atomic
sentences including those with subscripts.
A partial truth value assignment
is an association of a single truth value with some atomic
sentence (usually the ones that are under considerationthe only
ones that are relevant).
B. Logical Truth

First,
think
about a possibility as a way to make sentences true. A truth value
assignment fits the bill.

Next,
We're used to thinking of rows (partial truth value assignments) as
possible assignments to the atomic sentences in question. Now, we'll
get used to them as the possibilities from which we'll define our
deductive concepts.
 Think about:
 Bob will be a student only if he gets a loan, (B>L),
and
 If he doesn't get a loan, he won't be a student. (~L>~B).
 There are 4 possible
ways that things
could go...
 Logical Truth...the definition revisited:

The old definition?
(Chapter One) A sentence is logically
true if and only if it could not possibly be false.
In SL, as we have just seen, we have a better handle on the
possibilities. They are the truth value assignments. Thus, in
the context of SL, we can think about logical truth of a sentence
as meaning "no truth value assignment makes it false".
(Chapter Three) A sentence of SL in logically
true in SL if and only if it is false on no truth
value assignment.
 Let's apply this. To test to see if ~L>~B
is logically true,
what do we do?
Then notice that '~L>~B' is NOT
logically true in SL: it can be false, i.e., in row
three:

L 
B 

~ 
L 
> 
~ 
B 
row one: 
T 
T 

F 
T 
T 
F 
T 
row two: 
T 
F 

F 
T 
T 
T 
F 
row three: 
F 
T 

T 
F 
F 
F 
T 
row four: 
F 
F 

T 
F 
T 
T 
F 
(Just look at the truth values in bold under the main connective
'v' of 'Av~B'. This is where 'Av~B's truth values are found 
under its main connective.)
C. Other concepts...
 Logically False? Indeterminate? Problems!
 Logical Equivalence
For a sentence to be logically equivalent to another
means?
it's not possible for one sentence to be true while the other is false.
Now, let's see
the definition in
SL:
The two members of a pair of SL sentences are logically
equivalent in SL if and only if there is no truth value assignment
on which one of the pair is true while the other is false.
 Here's how to do
one (test to see if 'A=B' is l.equiv. to 'B>(A&B):
 You do:
 ~L>~B vs. B>L
 AvC vs. ~Av~C
(overheads)
roll
exam returns
next exam!
Don't forget:
 The reference manual
 And other printable pages...under results track.
 But you really need to work through the tutorials.
 Things get more complicated each week. We'll see complicated symbolizations
this week and complications of a new way of doing logic very soon (Chapter
5)
 NO VACATIONS!
III Review

Possibility and Logic
 Logical Truth, Falsity, Indeterminacy: A sentence which is...
 Try some: Which of the following are l.t., l.f. or l.i.?
 ~(A>(A&B))
 (C=L)&~(Cv~L)
 J&(K&~U)
(overhead)
 Something New:
Shortcut tables.
If one row, or two, allows you to test for one of our deductive
concepts, then just show that row or those rows.
For example: two rows is enough to show that a sentence can be
true and can be false. Thus, you can just do two rows for the above
example.
 Logical Equivalence...
the definition in
SL:
Two members of a pair of SL sentences are logically
equivalent in SL
if and only if
there is no truth value assignment on which one of the pair is true
while the other is false.
 Are the following l.e.?
WvS , ~S>W
IV. The final concepts

Validity
(Chapter One) An argument is valid
just in case it is not possible that its conclusion be false and
its premises all be true.
An argument is invalid
if and only if it is not valid.
We can now restate this for SL sentences as:
An argument is valid in SL
just in case there is no truth value assignment on which its conclusion
is false and its premises are all true.
An argument is invalid
in SL if and only if it is not valid in SL, i.e., if and
only if there is a truth value assignment on which its premises
are true and its conclusion is false.
We do...
You do...
(overheads)

Consistency
A set of SL sentences is logically
consistent in SL if and only if there is some truth value
assignment on which all members of the set are true.
next...
A set of SL sentences is logically
inconsistent in SL if and only it it is not logically
consistent, i.e., there is no truth value assignment on which
all members of the set are true.

Let's work some...
(note this row...)
You do...
Check for consistency:
{L=R, R=~S}
(overheads)
V. Symbolizations
 "or" revisited.


P 
Q 
PvQ 
row one: 
T 
T 
??? 
row two: 
T 
F 
T 
row three: 
F 
T 
T 
row four: 
F 
F 
F 
 inclusive 'or' vs. exclusive 'or'
 Some people say that English 'or' is usually exclusive (witness
"she's ten or eleven)
 But,
here's a reductio...
Assume for contradiction that Halpin is wrong and English 'or'
is normally exclusive.
Consider this example: "Either George or Laura is human".
Then, it
follows from our assumption that ...
this
disjunction is false. But if the disjunction is false, it's
negation
is...
true.
Is that bad? Well, if "Either George or Laura is human"
is false then it's
true
negation is
Neither George nor Laura is human.
Oops. This isn't true. So, apparently we cannot so readily
suppose that English "or" is exclusive.
Complex Symbolizations:
(From the reference manual)
In chapter two, we became used to symbolizing fairly simple
English sentences involving only a few connectives. But there
are lots of everyday examples of more complicated compounding
in English. We should be able to translate these into SL. For
example,
Private schools implementing a voucher program
will fail to provide equal educational opportunities
across a community if they either skim off the best students
and leave the poorer ones behind or they charge parents fees
beyond what is paid for in private funds and so exclude children
of poorer families.
Is this a conditional sentence? Or a disjunction? There are
no parentheses to help us. But there are "grouping"
words in the English. For instance, what goes between "either"
and "or" will be a first disjunction and the word
either works like a left hand parenthesis. Here's a reformulation.
Private schools implementing a voucher program
will fail to provide equal educational opportunities across
a community if [either(they skim off the best students
and leave the problem students behind) or they
exclude children of poorer families by charging parents fees
beyond what is paid for in private funds].
This sentence is now more pretty clearly a conditional with
consequent "Private schools implementing a voucher program
will fail to provide equal educational opportunities".
The antecedent is more complicated.
A hybrid formulation may help:
~P if [(S&L) or E]
(Here we've used P: "Private schools implementing a voucher
program provide equal educational opportunities", S: "Private
schools implementing a voucher program skim off the best students",
L: "Private schools implementing a voucher program leave
the problem students behind", E: "Private schools
implementing a voucher program exclude...") But we recall
that 'P if Q'
is symbolized as 'Q>P',
so '~P' is the consequent and we should symbolize the whole
English sentence as:
[(S&L)vE]>~P
The following table summarizes some of the mechanisms English
uses to group.
When one has a sentence of form: 
Then: 
Example (Hybrid): 
Symbolization: 
If P, then Q

Antecedent is P 
If A and B, then C. 
(A&B)>C 
Either P or Q 
First disjunct is P 
Either both A and B, or C. 
(A&B)vC 
Both P and Q 
First conjunct is P 
Both if A then B, and C. 
(A>B)&C 
One other good indicator of grouping in sentences is a comma
(or semicolon). These often mark a major structural division
in the sentence corresponding to the main connective. So...
If Private schools implementing a voucher program
succeed, then their students will benefit directly and traditional
public schools will benefit from the competition.
Which might have the hybrid form...
If S, then D and C.
...and, because of the comma marking the main connective, would
have symbolization as follows:
S>(D&C)
Now, make sure you have carefully read T4.2 and do lots of
exercises. Experience is the key to symbolization.
 Let's do some
symbolizations for review
W5?
VI. Informal Proofs: Do each of the following using deductive concepts for SL. (That is, when thinkng about validity, think about validinSL; there is no tva making premises true and conclusion false.)
A. If P entails Q
(i.e., P/Q
is valid) then P>Q
is logically true (l.t.).
C. PvQ can
be logically true while both P and Q are logically indeterminate.
