LOGIC
It is not easy to summarize in a few
paragraphs the subject matter known as logic. For lawyers and judges, logic is
the science of correct reasoning. They use logic to communicate more
effectively, construct valid arguments, analyze legal contracts, and make
decisions. Law schools consider knowledge of logic to be one of the most
important predictors of future success of their new students. A sizable portion
of the LSAT (Law School Admission Test) considers logical reasoning as part of
their admission process.
Other professions also make extensive
use of logic. Programmers use logic to design computer software, electrical
engineers use logic to design circuits for smart phones, and mathematicians use
logic to solve problems and construct mathematical proofs. In this chapter, you
will encounter several facets of logic. Specifically, logic is use to
·
Analyze
information and the relationship between statements
·
Determine
the validity of arguments
·
Determine
valid conclusions based on given assumptions, and
·
Analyze
electronic circuits.
LOGIC
STATEMENTS AND QUANTIFIERS
Gottfried Wilhelm Leibniz (1646 – 1716) was one of the mathematicians
who make a serious study of symbolic logic. He tried to advance the study of
logic from a merely philosophical subject to a formal mathematical subject.
Leibniz never completely achieve his
goal; however several mathematicians such as Augustus De Morgan (1806 – 1871)
and George Boole (1815 – 1864), contributed to the advancement of symbolic
logic as a mathematical discipline.
Boole published “The Mathematical
Analysis of Logic” in 1848. In 1854 he published the more extensive work, “An Investigation of the Laws of Thought”.
With these documents, the mathematician Bertrand Russell stated, “Pure
mathematics was discovered by Boole in a work which is called The Laws of Thought.”
LOGIC
STATEMENT
Statement or Proposition is a declarative
sentence that is either true or false, but not both true and false. It is
typically expressed as a declarative sentence (as opposed to question or
command). Propositions are the basic building blocks of any theory of logic.
Every language contains different types of
sentences, questions, and commands. For instance,
“Is there a rain today?” is a question.
“Go and set that bird free” is a
command.
“This is a nice design” is an
opinion.
“Pasig City is the capital of the
province of Rizal” is a statement of fact.
The symbolic logic that Boole was
instrumental in creating applies only to declarative sentences.
Examples.
Determine whether each sentence is
statement.
1. Mexico is in Africa.
2. How are your parents ?
3. 99 + 2 is a prime number.
4. x + 2 = 4
Answers:
1. Statement with a truth value
of false because Mexico in Central America.
2. It’s a question and not a
declarative sentence, hence not
statement.
3. You may not know that 99
+ 2 is a prime number, however, you know
that it is a whole number larger than 1, so it is either a prime number or it
is not a prime number. The sentence is either true or false, and it is not both
true and false, so it is a statement.
4. x + 2 =
4 is a statement, known as an
open statement and is true only for x = 2, and false for any other values of x.
Exercises:
Which
of the following are statements or proposition ?
1.
The only positive integer that divide 5 are 1 and 5 itself.
2.
2 + 4 = 7
3. The
earth is an oblate spheroid.
4.
4 – x = 7.
5.
Do you speak Chinese?
6. Oh,
what a beautiful site!
7.
Buy three tickets for the concert on November 25, 2018.
8.
The only positive integer that divide 12 are 3, 4 and itself.
9.
A square is a rectangle having all sides equal.
10. A
circle is round.
SIMPLE STATEMENTS AND COMPOUND STATEMENTS
A simple statement is a statement that
conveys a single idea. A compound statement is
statement that conveys two or more ideas. Connecting simple statements with
words and phrases such as and, or,
not, if...then, and if and only if
creates a compound statement. For
instance, “I will go to school and I will join the debate” is a compound. It is
composed of two simple statements “I will go to school” and “I will join the
debate.” The word and is the
connective for the two simple statements.
George
Boole used the letters p, q, r,
and s
to represent simple statements and the symbols Ù, Ú, ~, ® and « to represent connectives.
Let
us define the meaning of these connectives by showing the relationship between
the truth value (i.e. true or false) of composite propositions and those of
their component propositions.
Let
p and q be propositions.
a) The conjunction of p and q, denoted by p L q, is the proposition p and q.
b) The disjunction of p and q, denoted by p
V q, is the proposition p or q.
c) The negation of p, denoted by ~p, is the proposition not p.
Propositions such as p L q and p V q that result from
combining propositions are called compound propositions. The compound statement
p L
q is true when both p and q are true; otherwise, it is false. The compound
statement p V q is true if at least one
of p or q is true, it is false when both p and q are false.
The
truth value of a simple statement is either true ( T) or false ( F). The truth
value of a compound statement depends on the truth values of its simple statements
and its connectives. The truth values of
propositions such as conjunctions and disjunctions can be described by truth
tables. The truth table shows the truth value of compound statement for all
possible truth values of its simple statements.
The truth table of a proposition
p made up of the individual propositions p1...pn, lists
all possible combinations of truth values for p1...pn ,T
denoting true and F denoting false and for each such combination lists the
truth value of p.
NEGATION
Write
the negation of each statement.
1.
p: Today is not Friday.
~p: Today is Friday.
2.
p: The seventh month is July.
~p: The seventh month is not July.
3. p: Mathematical logic is not an easy
subject.
4.
p: Tacurong City is a city in the
province of Sultan Kudarat.
5.
p: The national flower is
sampaguita.
6.
p: Tarsiers are abundant in the
province of Bohol.
7.
p: Vatican City is a city within
a City.
8.
p: Japan is the land of the
rising sun.
9.
p: Cagayan de Oro City is known
as the City of Golden Friendship.
10. p: Cebu
City is the Queen City of the South.
CONJUNCTION, Ù and DISJUNCTION, Ú
Given p:
Today is Monday.
q:
The weather is cold.
r:
I passed the test.
s: I will be very happy.
1. p Ù q : Today is Monday and the weather is cold.
2. r Ù s : I passed the test and I will be very happy.
3. p Ú q : Today is Monday or the weather is cold.
4. r Ú s :
5. ~p Ù q :
6. r Ù ~s :
7. p Ú ~q :
8. ~r Ú s :
9. q Ú s
10.
p Ù r
11.
q Ú ~s
12.
~p Ù r
Consider the following simple
propositions.
p:
Today is Tuesday.
q: It is a sunny day.
r: I am going to the gym.
s: I am not going to play
basketball.
Write the following compound statements
in symbolic form.
1. Today is Tuesday and it is a
sunny day.
2. It is not a sunny day and I am
going to the gym.
3. I am going to play basketball
or I am not going to the gym.
4. It is not a sunny day and
today is not Tuesday.
5. I am not going to play
basketball or I am not going to the gym.
6. I am going to play basketball
and I am going to the gym.
7. It is not a sunny day and
today is Tuesday.
8. I am going to play basketball
and today is Tuesday.
Answers :
1. p Ù q
2. ~q Ù r
Consider
the following statements
p:
The game will be played in New York.
q:
The game will be shown on ABC 5.
r:
The game will not be shown on ABS-CBN.
s:
The GSW are favored to win.
Write
each of the following symbolic statements in words.
1. p Ù q 6. ~p Ù ~q
2. q Ù p 7. q Ù ~p
3. p Ú ~r 8. ~p Ú ~r
4. q Ù s 9. s Ù q
5. q Ú ~ r 10.
~q Ú r
