Chapter 3 – Introduction to Logic
Logic is the formal systematic study of the principles of valid
inference and correct reasoning.
It is used in most intellectual activities, but is studied primarily
in the disciplines of philosophy, mathematics, semantics, and
computer science.
Logic examines:
(a) general forms which arguments may take,
(b) which forms are valid, and
(c) which forms are fallacies.
The initial motivation for the study of logic was to learn to
distinguish good arguments from bad arguments.
3.1 – Statements and Quantifiers
Statements
A statement is defined as a declarative sentence that is either
true or false, but not both simultaneously.
Compound Statements
A compound statement may be formed by combining two or
more statements.
The statements making up the compound statement are called
the component statements.
Connectives such as and, or, not, and if…then, can be used in
forming compound statements.
3.1 – Statements and Quantifiers
Determine whether or not the following sentences are
statements, compound statements, or neither.
If Amanda said it, then it must be true.
Compound statement (if, then)
Today is extremely warm.
Statement
The gun is made by Smith and Wesson.
Statement
The gun is a pistol and it is made by Smith and Wesson.
Compound statement (and)
3.1 – Statements and Quantifiers
Negations
A negation is a statement that is a refusal or denial of some
other statement.
Statement: Max has a valuable card.
Negation: Max does not have a valuable card.
The negation of a true statement is false and the negation of a
false statement is true.
Statement: The number 9 is odd.
Negation: The number 9 is not odd.
Statement: The product of 2 negative numbers is not positive.
Negation: The product of 2 negative numbers is positive.
3.1 – Statements and Quantifiers
Negations and Inequality Symbols
Symbolism
ab
a b
ab
ab
Meaning
a is less than b
a is greater than b
a is less than or equal to b
a is greater than or equal to b
Give a negation of each inequality. Do not use a slash symbol.
Statement: p ≥ 3
Negation: p < 3
Statement: 3x – 2y < 12
Negation: 3x – 2y ≥ 12
3.1 – Statements and Quantifiers
Symbols
To simplify work with logic, symbols are used.
Statements are represented with letters, such as p, q, or r,
while several symbols for connectives are shown below.
Connective
Symbol Type of Statement
and
Conjunction
or
not
Disjunction
Negation
3.1 – Statements and Quantifiers
Translating from Symbols to Words
Let: p represent “It is raining,”
q represent “It is March.”
Write each symbolic statement in words.
p˅q
It is raining or it is March.
̴ (p ˄ q)
It is not the case that it is raining and it is March.
3.1 – Statements and Quantifiers
Quantifiers
Universal Quantifiers are the words all, each, every, no, and
none.
Existential Quantifiers are words or phrases such as some,
there exists, for at least one, and at least one.
Quantifiers are used extensively in mathematics to indicate
how many cases of a particular situation exist.
Negations of Quantified Statements
Statement
Negation
All do.
Some do not.
Some do.
None do.
3.1 – Statements and Quantifiers
Forming Negations of Quantified Statements
Statement: Some cats have fleas.
Negation: No cats have fleas.
Statement: Some cats do not have fleas.
Negation: All cats have fleas.
Statement: All dinosaurs are extinct.
Negation: Not all dinosaurs are extinct.
Statement: No horses fly.
Negation: Some horses fly.
3.1 – Statements and Quantifiers
Sets of Numbers
Natural Numbers: {1, 2, 3, 4, …}
Whole Numbers: {0, 1, 2, 3, …}
Integers: {…, -3, -2, -1, 0, 1, 2, 3, 4, …}
Rational Numbers: Any number that can be expressed as a
quotient of two integers (terminating or repeating decimal).
a

 a and b are integers and b  0 
b

Irrational Numbers: Any number that can not be expressed as a
quotient of two integers (non-terminating and non-repeating).
Real Numbers: Any number expressed as a decimal.
3.1 – Statements and Quantifiers
True or False
Every integer is a natural number.
False: – 1 is an integer but not a natural number.
A whole number exists that is not a natural number.
True: 0 is the number.
There exists an irrational number that is not real.
False: All irrational numbers are real numbers.