chapter 3 – introduction to logic
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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. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 3 – Introduction to Logic
The initial motivation for the study of logic was to learn to distinguish good arguments from bad arguments.
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.
StatementsA statement is defined as a declarative sentence that is either true or false, but not both simultaneously.
3.1 – Statements and Quantifiers
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.
Determine whether or not the following sentences are statements, compound statements, or neither.
3.1 – Statements and Quantifiers
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
Compound statement (and)
The gun is a pistol and it is made by Smith and Wesson.
Negations
3.1 – Statements and Quantifiers
A negation is a statement that is a refusal or denial of some other statement.
Max has a valuable card.
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:
Negation:
The number 9 is odd.
The number 9 is not odd.
Statement:
Negation:
The product of 2 negative numbers is not positive.
The product of 2 negative numbers is positive.
Statement:
Negation:
Negations and Inequality Symbols
Symbolism 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
a ba b
a ba b
3.1 – Statements and Quantifiers
Give a negation of each inequality. Do not use a slash symbol.
p ≥ 3
p < 3
Statement:
Negation:
3x – 2y < 12
3x – 2y ≥ 12
Statement:
Negation:
Symbols
To simplify work with logic, symbols are used.
Connective Symbol Type of Statement
3.1 – Statements and Quantifiers
Statements are represented with letters, such as p, q, or r, while several symbols for connectives are shown below.
and
or
not
Conjunction
Negation
Disjunction
Translating from Symbols to Words
Write each symbolic statement in words.
3.1 – Statements and Quantifiers
q represent “It is March.”
Let: p represent “It is raining,”
p ˅ q
It is raining or it is March.
�̴ (p ˄ q)
it is raining and it is March. It is not the case that
Quantifiers
Universal Quantifiers are the words all, each, every, no, and none.
Quantifiers are used extensively in mathematics to indicate how many cases of a particular situation exist.
3.1 – Statements and Quantifiers
Existential Quantifiers are words or phrases such as some, there exists, for at least one, and at least one.
Negations of Quantified Statements
Statement Negation
All do. Some do not.
Some do. None do.
Forming Negations of Quantified Statements
3.1 – Statements and Quantifiers
Some cats have fleas.
No cats have fleas.
Statement:
Negation:
Some cats do not have fleas.
All cats have fleas.
Statement:
Negation:
All dinosaurs are extinct.
Not all dinosaurs are extinct.
Statement:
Negation:
No horses fly.
Some horses fly.
Statement:
Negation:
Sets of Numbers
3.1 – Statements and Quantifiers
Natural Numbers: {1, 2, 3, 4, …}
Whole Numbers: {0, 1, 2, 3, …}
Rational Numbers: Any number that can be expressed as a quotient of two integers (terminating or repeating decimal).
and are integers and 0a
a b bb
Irrational Numbers: Any number that can not be expressed as a quotient of two integers (non-terminating and non-repeating).
Integers: {…, -3, -2, -1, 0, 1, 2, 3, 4, …}
Real Numbers: Any number expressed as a decimal.
True or False
3.1 – Statements and Quantifiers
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.