Mathematics in the Modern World Rosalinda A. Obias
HED, USI-Naga City
Lesson 3
Elementary Logic and Connectives
Overview
Mathematics works according to the laws of logic, which specify how to make valid
deductions. In order to apply the laws of logic to mathematical statements, you
need to understand their logical forms. This lesson is an introduction to first-order
symbolic logic. Our goal is to understand these logical systems. We will focus
primarily on mastering these formal systems of symbolic logic and to learn how to
reason with greater clarity and rigor, by applying the principles of logic.
Learning Outcomes
At the end of this module, you will:
1. Represent English sentences in symbolic logic.
2. Construct truth tables for compound propositions.
3. Apply principles of logic to arguments contained in ordinary language.
Learning Content
Can you solve the logic problem below? If you solved correctly, congratulations!
You have high cognitive skills. If not, you just need to practice more on improving
your logical skills.
Which person enters last? ______________
What is Logic?
Logic is the science of reasoning. Reasoning is a thought process in which
inference takes place. Logic tells us how we ought to reason if we want to
reason correctly.
The term ’logic’ came from Greek word ’logos’, which means thought. There
are many thought processes such as ’reasoning’, ’remembering’, ’imagining’.
Critical thinking is a process of evaluation which uses logic to separate truth
from falsehood, reasonable from unreasonable beliefs. Logic helps us to
understand how to construct a valid argument.
1
, Mathematics in the Modern World Rosalinda A. Obias
HED, USI-Naga City
Propositional Logic
Propositional logic is the most basic branch of mathematical logic. A proposition is
a declarative statement that is either true or false. If a proposition is true, its truth
value is true, denoted by T. If it is false, its truth value is false, denoted by F.
Examples:
1) Manila is the capital of the Philippines. Proposition True (T)
2) 5 + 2 = 8 Proposition False (F)
3) It is raining today. Proposition Either T or
F
4) How are you? Not a proposition because
not a declarative statement
5) x + 5 = 3 Not a proposition, since x is not specified
Connectives and Compound Propositions
A propositional connective is an operation that combines two propositions to
yield a new one whose truth value depends only on the truth values of the two
original propositions. Propositions built up by combining propositions using
propositional connectives are called compound propositions. Example:
Proposition p: “It rains outside.”
Proposition q: “We will see a movie.”
A combined proposition p → q : “If it rains outside, then we will see a
movie.”
Logical connectives
The different logical connectives are:
- Conjunction - Implication
- Disjunction - Biconditional
- Exclusive or - Negation
Conjunction
Definition: Let p and q be propositions. The proposition "p and q"
denoted by p ˄ q, is true when both p and q are true and is false otherwise.
The proposition p ˄ q is called the conjunction of p and q.
The conjunction connective is more defined by the following truth table:
p q p˄q
All possible truth T T T …TRUE when
value
T F F both p and q
combinations of
F T F are true.
two or more
F F F
propositions.
2
HED, USI-Naga City
Lesson 3
Elementary Logic and Connectives
Overview
Mathematics works according to the laws of logic, which specify how to make valid
deductions. In order to apply the laws of logic to mathematical statements, you
need to understand their logical forms. This lesson is an introduction to first-order
symbolic logic. Our goal is to understand these logical systems. We will focus
primarily on mastering these formal systems of symbolic logic and to learn how to
reason with greater clarity and rigor, by applying the principles of logic.
Learning Outcomes
At the end of this module, you will:
1. Represent English sentences in symbolic logic.
2. Construct truth tables for compound propositions.
3. Apply principles of logic to arguments contained in ordinary language.
Learning Content
Can you solve the logic problem below? If you solved correctly, congratulations!
You have high cognitive skills. If not, you just need to practice more on improving
your logical skills.
Which person enters last? ______________
What is Logic?
Logic is the science of reasoning. Reasoning is a thought process in which
inference takes place. Logic tells us how we ought to reason if we want to
reason correctly.
The term ’logic’ came from Greek word ’logos’, which means thought. There
are many thought processes such as ’reasoning’, ’remembering’, ’imagining’.
Critical thinking is a process of evaluation which uses logic to separate truth
from falsehood, reasonable from unreasonable beliefs. Logic helps us to
understand how to construct a valid argument.
1
, Mathematics in the Modern World Rosalinda A. Obias
HED, USI-Naga City
Propositional Logic
Propositional logic is the most basic branch of mathematical logic. A proposition is
a declarative statement that is either true or false. If a proposition is true, its truth
value is true, denoted by T. If it is false, its truth value is false, denoted by F.
Examples:
1) Manila is the capital of the Philippines. Proposition True (T)
2) 5 + 2 = 8 Proposition False (F)
3) It is raining today. Proposition Either T or
F
4) How are you? Not a proposition because
not a declarative statement
5) x + 5 = 3 Not a proposition, since x is not specified
Connectives and Compound Propositions
A propositional connective is an operation that combines two propositions to
yield a new one whose truth value depends only on the truth values of the two
original propositions. Propositions built up by combining propositions using
propositional connectives are called compound propositions. Example:
Proposition p: “It rains outside.”
Proposition q: “We will see a movie.”
A combined proposition p → q : “If it rains outside, then we will see a
movie.”
Logical connectives
The different logical connectives are:
- Conjunction - Implication
- Disjunction - Biconditional
- Exclusive or - Negation
Conjunction
Definition: Let p and q be propositions. The proposition "p and q"
denoted by p ˄ q, is true when both p and q are true and is false otherwise.
The proposition p ˄ q is called the conjunction of p and q.
The conjunction connective is more defined by the following truth table:
p q p˄q
All possible truth T T T …TRUE when
value
T F F both p and q
combinations of
F T F are true.
two or more
F F F
propositions.
2
