Propositional Logic Introduction

Boolean Algebra

Share

This is an introduction to Propositional Logic tutorial.

What is a Proposition?

A Proposition is an atomic sentence that can either be TRUE or FALSE
and nothing else.

Following sentences are example of proposition.

Proposition: India is a country
Result: TRUE

Proposition: 100 is greater than 200
Result: FALSE

Whereas the sentence How are you? is not a proposition as the answer can’t be TRUE or FALSE.

Simple and Compound Proposition

A simple proposition is one that does not contain
any other propositions as its part.

A compound proposition is one that is made up of
two or more simple propositions.

We use lower case letters a,b,c… to represent proposition.

Example of simple proposition

It is raining.

This statement can either be TRUE or FALSE.

Example of compound proposition

Today is Sunday and Sunday is a holiday.

This statement contain two simple propositions "Today is Sunday" and "Sunday is a holiday" both the statement can be either TRUE or FALSE.

Operator or Logical Connective

Operator or logical connective are the things that joins simple propositions into compound propositions and joins compound propositions into larger compound propositions.

What is Propositional Logic?

Propositional Logic is a way to represent logic through propositions and logical connectives.

Types of Logical Connectives (Operators)

Following are the types of logical connectives (operators) used in propositional logic.

  • Disjunctive (also called OR)
  • Conjunctive (also called AND)
  • Conditional (also called Implication)
  • Bi-conditional (also called Equivalence)
  • Negation (also called NOT)

Disjunctive operator

Disjunctive (also called OR) means one of the two arguments is true or
both of them are true.

We use the word OR and + and ∨ symbols to represent disjunctive.

Example
p + q
p ∨ q
p OR q
They all mean
either p is true, or q is true, or both are true.

Consider two arguments (proposition)
p = Oct 21, 2012 was Sunday
q = Sunday is a holiday
then,
p + q
p ∨ q
p OR q

They all means either p is true, or q is true, or both are true.

i.e., either Oct 21, 2012 was Sunday is true or Sunday is a holiday is true or both are true.

Conjunctive operator

Conjunctive (also called AND) means both the arguments are true. We use the word AND and . & and ∧ symbols to represent conjunctive.

Example
p . q
p & q
p ∧ q
p AND q

They all means both p and q are true.

Consider two arguments (proposition)
p = Oct 21, 2012 was Sunday
q = Sunday is a holiday
then,
p . q
p & q
p ∧ q
p AND q

They all means both p and q are true.

i.e., both Oct 21, 2012 was Sunday and Sunday is a holiday are true.

Conditional (Implication) operator

Conditional also called Implication (If...Then).

Implication means if one argument is true then the other argument is true.

We use the ⇒ symbol to represent conditional operator.

Example
p ⇒ q
this means if p is true, then q is true.

Consider two arguments (proposition)
p = 10 is greater than 0
q = 10 is positive
then,
p ⇒ q
this means if p is true, then q is true.
i.e., if 10 is greater than 0 then 10 is positive.

Bi-conditional (Equivalence) operator

Bi-conditional also called Equivalence (If and only If).

Equivalence means either both arguments are true or both are false.

We use the ⇔ symbol to represent bi-conditional.

Example
p ⇔ q
this means
p and q either both are true or both are false.

Consider two arguments (proposition)
p = 10 is greater than 0
q = 10 is positive
then,
p ⇔ q
this means
p and q either both are true or both are false.

i.e., 10 is greater than 0 and 10 is positive
either both are true or both are false.

Negation (also called NOT)

Negation is an operator that affects only one statement and does not join two statements.

We use the ~ and ' symbol to represent negation

~p
p'
this means
if p is true then,
~p is false.

Example

Consider the argument (proposition)
p = It is raining
if p is true i.e., it is raining
then,
~p is false i.e., it is not raining.