Propositional logic

Propositional logic is the simplest form of logic, where all the statements are made by propositions. A proposition is either True statement or False statement but can’t be both. A propositional formula which is always True is called Tautology and a propositional formula which is always false is called contradiction. It is a technique of knowledge representation in logical and mathematical form.

Examples of propositional logic:

  • a) The Sun rises from east. (True proposition)
  • b) The Sun rises from west. (False proposition)
  • c) 2 + 2 is 5. (False proposition)
  • d) 2 + 2 is 4. (True proposition)

Propositional logic is also called Boolean logic as it works on 0 and 1. Symbolic variables are used to represent the logic and we can use any symbolic variables to represent a logical statement such as A, B, C, P, Q, R, X, Y, Z etc. The statements which are question, command or opinion are not propositions. The propositional logic consists of object, relations or function and logical connectives and the connectives are also called logical operators. The prepositions and connectives are basic elements of propositional logic. Connectives can be said as a logical operator which connects two sentences.

There are 2 types of propositions:

1. Atomic proposition: These are the simple propositions; it consists of a single proposition symbol. These are the sentences which must be either true or False. For example: 2+2 is 4, is an atomic proposition and it is True. 2+2 is 5, is an atomic proposition but it is False.

2. Compound proposition: Compound propositions are the combination of two or more Atomic-propositions using parenthesis and logical connectives. For example: it is raining today and the street is wet. Here “and” is the logical connective which connects these 2 statements.

Logical connectives:

Logical connectives are used to connect two or more atomic propositions to make them a compound proposition and represents them logically. There are mainly 5 connectives which we can use.

1. Negation: A sentence such as ¬P is called the negation of P. Negation of a statement makes its meaning opposite of original statement.

2. Conjunction: A sentence having connective such as P ∧ Q is called a conjunction operator. Which is used like AND. For example,
P: John is intelligent.
Q: John is hardworking.
p ∧ q: John is intelligent and hardworking.

3. Disjunction: A sentence having connective such as P ∨ Q is called a disjunction operator. Which is used like OR. For example,
P: John is a doctor.
Q: John is an engineer.
p ∨ q: John is a doctor or an engineer.

4. Implication: Implications are like if – then rules, for example,
P: It is raining today
Q: The street is wet
P → Q: If it is raining today, then the street is wet.

5. Biconditional: A sentence which is like P ⇔ Q i.e., it is similar to implication but bidirectionally. For example,
P: He is breathing.
Q: He is alive
P ⇔ Q: If he is breathing then he is alive.

Connective Symbols Word Technical Term Example
¬ or ~ Not Negation ¬A or ~A
AND Conjunction A ∧ B
OR Disjunction A ∨ B
Implies Implication A → B
If and only if (iff) Biconditional A ⇔ B

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