E
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
P
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
R
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
Limits of sl
Objects and properties
Quantication
Homework
Warm-up
Translate this argument into sl form and then determine whether
its valid:
Every place in Illinois gets cold in the winter. DeKalb is a place in
Illinois. So, DeKalb gets cold in the winter
Four Types of General L-Statements
A
All S are P.
All surgeons are doctors.
x(Sx
Dx)
E
No S are P.
No crabs are crows.
x(Cx
~Rx)
I
Some S are P.
Some apples are pippins.
x(Ax & Px)
O
Some S are not P.
Some apples are not pippins.
x(Ax & ~Px)
Some red-hair
Limits of sl
Objects and properties
Quantication
Homework
Something
Now consider the following statement:
1. Something is hilarious.
Let H name the property of being hilarious. You might think we
should symbolize 1 using s as a name for something:
Hs
But
Limits of sl
Objects and properties
Quantication
Homework
Predicates as truth-functions
From objects to truth values
You can think of a predicate letter as determining a function from
objects to truth values: you put an object in, and you get a truth
valu
1. Assume the opposite of the conclusion if the conclusion has a quantifier.
2. Double negation
3. Quantifier exchange
4. Existential Generalization (EG)
OR
Existential instantiation (EI)
5. Universal Instantiation (UI)
OR
Universal Generalization (UG)
6.
Limits of sl
Objects and properties
Quantication
Homework
Predicate letters and individual constants
In q, we will use predicate letters to name properties, and
individual constants to name particular objects.
Predicate letters
F , G , H , . . ., M
Indivi
PRACTICE QUIZ 6
PHILOSOPHY 230
FALL 2011
NAME: _
D: animals
Cx: x is a cat
Dx: x is a dog
Mx: x is a mouse
j: Jerry
Fx: x is female
Bx: x is male
Sxy: x is the sidekick of y
Using the above interpretation, symbolize each of the following English statement
Limits of sl
Objects and properties
Quantication
Homework
Objects and properties
Consider the following descriptive statements:
1. DeKalb is a city.
2. Brett Favre is a grandfather.
3. This shoe ts comfortably.
Each statement describes a certain thing as
Limits of sl
Objects and properties
Quantication
The universal quantier
But we also have a symbol we can for statements about
everything: . This symbol is called the universal quantier,
and when used with a variable, it means roughly for everything
that e
SOLUTIONS TO SELECTED EXERCISES
EXERCISES FOR S ECTION 2.1
PART I
1.
Given the form p. Case (a): Let p be the statement Salt Lake City is in Utah.
Then p is true. Case (b): Let p be the statement Salt Lake City is in Nevada. Then p is
false.
2.
Given the
16.
Truth-functionally compound. The simple statements are The man walked up to
the stand and The man ordered an ice cream cone.
17.
Truth-functionally simple.
18.
Truth-functionally simple.
20.
Truth-functionally compound. The simple statement is Ellen i
58. M: My computer makes mistakes; C: My computer has been correctly programmed; P:
There is a power failure.
~(~C P) ~M
~P & M
~C
The following truth table shows that this argument is valid:
C
T
T
T
T
F
F
F
F
M
T
T
F
F
T
T
F
F
P
T
F
T
F
T
F
T
F
~
F
T
F
50. This argument is valid.
A
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
B
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
C
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
D
T
T
F
F
T
T
F
F
T
T
F
F
T
T
40. This argument is valid.
A
T
T
T
T
F
F
F
F
C
T
T
F
F
T
T
F
F
D
T
F
T
F
T
F
T
F
~
F
F
F
F
T
T
T
T
A
T
T
T
T
F
F
F
F
(C
T
T
F
F
T
T
F
F
T
T
T
F
T
T
T
F
D)
T
F
T
F
T
F
T
F
F
F
F
T
T
T
T
T
~
F
F
F
F
T
T
T
T
A
T
T
T
T
F
F
F
F
A
T
T
T
T
F
F
F
F
F
T
F
T
T
T
T
20. This statement is a tautology.
X
T
T
T
T
F
F
F
F
Y
T
T
F
F
T
T
F
F
Z
T
F
T
F
T
F
T
F
[(X
T
T
T
T
F
F
F
F
&
T
T
F
F
F
F
F
F
T
F
T
T
T
T
T
T
Y)
T
T
F
F
T
T
F
F
Z]
T
F
T
F
T
F
T
F
[(X
T
TT
T
TF
T
TT
T
TF
T
FT
T
FT
T
FT
T
FT
Z)
T
F
T
F
T
F
T
F
T
F
T
T
T
EXERCISES FOR SECTION 3.4
PART I: TAUTOLOGIES
(Note: You can find solutions to the odd-numbered problems in the authors solutions manual,
which you can find at the following web site: http:/blue.butler.edu/~sglennan/SSM.pdf.)
2. This statement is not a ta
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
B C.
This is an SL statement. It has the logical form of a conjunction; & is its
main connective. (The parentheses around B are OK (since B is a
statement letter and therefore an SL statement), but we might just hav
25.
N: There is a nuclear war; U: It will be started by the United States; R: It will be
started by Russia.
N (U R )
~U & ~R
~N
26.
G: The game will be won; S: Samson is allowed to play; T: Samson is tied up for
a while.
GS
T & (T ~S)
~G
27.
O: The conc
33.
R: I will allow you to read the comics; G: You give me the front page. (R G)
EXERCISES FOR S ECTION 2.4
PART I
2.
L: Luc will stay at the party; A: Andy will stay at the party. (L A) & ~(L & A)
3.
L: Luc will stay at the party; A: Andy will stay at th
All astronomers are stargazers.
Hypatia is an astronomer.
Therefore, Hypatia is a stargazer.
1
The Language of Predicate Logic (L)
INDIVIDUAL CONSTANT: A lower-case letter (e.g. h) used to stand for a particular individual
(e.g. Hypatia)
PREDICATE SYMBOL:
Answer Key for Exam 3 Extra Practice Problems
1. Either Lloyd is male or Sheila is female.
Ml
Fs
2. Lloyd makes fun of Diane only if he has a crush on her.
Uld
Rld
3. Any boisterous teenager is the child of Lloyd and Diane.
x[(Bx & Tx)
Cxld]
4. If Lloyd p
Relations
Syntax of q
Homework
Syntax of q
Scope and binding
The scope of a quantier is the formula directly to its right. E.g.:
x (Gx & Hx )
scope of x : x (Gx & Hx )
xGx Ha
scope of x : x Gx Ha
A quantier binds every occurrence of its associated variabl
Relations
Syntax of q
Syntax of q
Vocabulary
The vocabulary of q consists of all and only the following:
Statement letters: p , q , r , s
Predicate letters: F n , G n , H n , . . . , M n
Individual constants: a, b , c , . . . , o
Individual variables: t ,
Relations
Syntax of q
Homework
Multiple quantiers
Consider the following statement:
1. Everybody loves somebody.
We symbolize statement 1 using two quantiers and two variables:
x yLxy
This says, roughly: for everybody that exists (x ), there is
somebody (
Relations
Syntax of q
Homework
Quantication and relations
We can use quantiers to express things like:
1. Alex loves somebody.
Statement 1 can be symbolized as:
xLax
Note that the x goes in the second place after the L. What would
it mean if we instead wr
Relations
Syntax of q
Homework
Love
Here are two statements:
1. Alex is loving.
2. Alex is happy.
1 and 2 attribute dierent properties to Alex. Now consider:
3. Alex loves Bertie.
4. Alex loves Chris.
Do statements 3 and 4 attribute dierent properties to
Relations
Syntax of q
Homework
Warm-up
Symbolize the following statements.
1. Omar is hungry.
6. Someone is hungry.
2. Omar is not hungry.
7. No one is hungry.
3. Either Omar is hungry or he
isnt.
8. Someone is not hungry.
4. If Omar is hungry, he eats.
5