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
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
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
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
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
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
FORMAL LOGIC
PL SYMBOLIZATION AND SD AND SD+
EXAM #3
PROFESSOR JULIE YOO
Name: _
I. PL Translations: (2 points each; 50 points total 2 points free)
Symbolization Key
UD: Unrestricted
Px = x is a penguin
Dx = x is a dolphin
Sx = x can swim
Fx = x can fly
L
5.15. Valid; the conclusion that is desired states that b is smaller than c. The first premise states that b is
a tetrahedron and the second says that c is a cube. Since the last premise is a disjunction, it states that
two scenarios in which the premise
I. Pre-Test Extra Credit: Prove the following using ONLY the rules of PD. (5 points each; 25
points total)
a. Prove validity: cfw_(x)(Fx Gx), Fb Hb (y)(Fy Hy)
b. Prove validity: cfw_(x)(Fx), (y)Gy) (z)(Fz Gz)
c. Prove validity: cfw_(x)(Fx Gx), (x)Fx (x)(F
FORMAL LOGIC
SYMBOLIZATION WITH POLYADIC PREDICATES
Monadic Predicates
Hx = x is a house
Dx = x is a day
Px = x is a person
LESSON PLAN
PROFESSOR JULIE YOO
Polyadic Predicates Predicates
Lxy = x likes y; y is liked by x
Bxy = x is born on y
Vxy = x lives
FORMAL LOGIC
PL BASIC CONCEPTS
LESSON PLAN
PROFESSOR JULIE YOO
Predicate Logic (PL) v Sentential Logic (SL)
Inadequacy of SL: SL can only represent logical relations among whole sentences, not
logical relations within sentences. Recall the lecture on the
FORMAL LOGIC
PD INTRO AND ELIM RULES
Universal Elimination
Universal Elimination (E)
(x) P
P (a / x)
Existential Introduction
Existential Introduction (I)
P (a / x)
(x) P
Universal Introduction
Universal Introduction (I)
P (a / x)
(x) P
provided that:
i.
lL Ml361:1" 'TION AND SD AND SD+ ' PROFESSOR JULIE YOO
Name: : ;Z2 Lid/cfw_l0 K?
I. PL Translations: (2 points each; 50 points total 2 points free)
Symbolization Key UD: Unrestricted Sx = x can swim
PX = x is a penguin Fx = x can y
Dx = x is a dolphin Lx
FORMAL LOGIC
SYNTAX OF PL
LESSON PLAN
PROFESSOR JULIE YOO
A Quantifier of PL
A quantifier of PL is an expression of PL of the form (x) the universal quantifier or (x)
the existential quantifier.
Atomic Formulas PL
An atomic formula of PL is every express
FORMAL LOGIC
SYMBOLIZATION WITH MONADIC PREDICATES
LESSON PLAN
PROFESSOR JULIE YOO
Basic Universal and Existential Statements A, E, I O
Subjects/Constants
(none)
Predicates
Sx = x is a swan
Wx = x is white
A-sentence
All swans are white.
(x)(Sx Wx)
UD: un
FORMAL LOGIC
EXPLANATION OF PD RULES
LESSON PLAN
PROFESSOR JULIE YOO
FOUR NEW RULES OF PD
Inferences in PD Require Replacement of Variables with the RIGHT Constants
The derivation system, PD, adds four new rules to all the rules of SD an Intro and Elim ru
PHILOSOPHY 230
INTRODUCTION TO FORMAL LOGIC
PROFESSOR JULIE YOO
DEPARTMENT OF PHILOSOPHY
CALIFORNIA STATE UNIVERSITY AT NORTHRIDGE
Tu and Th 3:30 4:45
Juniper Hall 1206
Class (Ticket) No: 14556
email: [email protected]
Office Hours: Wednesdays 1:00 4:00
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
Interpretations
Homework
Interpretations
Quantier domains
Consider the following statement and its q translation:
1. All of the beers are cold.
x (Bx Cx )
Given how we dened , 1s q translation says that for anything
that is a beer, that thing is cold. But