Propositional logic
Logic:
- defines a formal language for representing knowledge and for making logical inferences
- It helps us to understand how to construct a valid argument
Logic defines:
- Syntax of statements
- The meaning of statements
- The rules of logical inference (manipulation)
Propositional logic
- The simplest logic
Definition:
- A proposition is a statement that is either true or false.
Examples:
- Pitt is located in the Oakland section of Pittsburgh.
- (T)
- 5 + 2 = 8.
- (F)
- It is raining today.
- (either T or F)
- How are you?
- a question is not a proposition
- x + 5 = 3
- since x is not specified, neither true nor false
- 2 is a prime number.
- (T)
- She is very talented.
- since she is not specified, neither true nor false
- There are other life forms on other planets in the universe.
- either T or F
Propositional Variables
The lower case letters starting from P onwards are used to represent propositions
Example: p: India is in Asia
q: 2 + 2 = 4
Compound Statements
Statements or propositional variables can be combined by means of logical connectives (operators) to form a single statement called compound statements.
The five logical connectives are:
| Symbol | Connective | Name |
|---|---|---|
| ~ | Not | Negation |
| ∧ | And | Conjunction |
| ∨ | Or | Disjunction |
| ⟶ | Implies or if...then | Implication or conditional |
| ⟷ | If and only if | Equivalence or biconditional |
Basic Logical Operations
1. Negation:
- It means the opposite of the original statement. If p is a statement, then the negation of p is denoted by ~p and read as 'it is not the case that p.' So, if p is true then ~ p is false and vice versa.
Example: If statement p is Paris is in France, then ~ p is 'Paris is not in France'.
| p | ~ p |
| T | F |
| F | T |
2. Conjunction:
- It means Anding of two statements. If p, q are two statements, then "p and q" is a compound statement, denoted by p ∧ q and referred as the conjunction of p and q. The conjunction of p and q is true only when both p and q are true. Otherwise, it is false.
| p | q | p ∧ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
3. Disjunction:
- It means Oring of two statements. If p, q are two statements, then "p or q" is a compound statement, denoted by p ∨ q and referred to as the disjunction of p and q. The disjunction of p and q is true whenever at least one of the two statements is true, and it is false only when both p and q are false.
| p | q | p ∨ q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
4. Implication / if-then (⟶):
- An implication p⟶q is the proposition "if p, then q." It is false if p is true and q is false. The rest cases are true.
| p | q | p ⟶ q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | F |
5. If and Only If (↔):
- p ↔ q is bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true.
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Derived Connectors
1. NAND:
- It means negation after ANDing of two statements. Assume p and q be two propositions. Nanding of pand q to be a proposition which is false when both p and q are true, otherwise true. It is denoted by p ↑ q.
| p | q | p ∨ q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | T |
2. NOR or Joint Denial:
- It means negation after ORing of two statements. Assume p and q be two propositions. NORing of p and q to be a proposition which is true when both p and q are false, otherwise false. It is denoted by p ↑ q.
| p | q | p ↓ q |
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
3. XOR:
- Assume p and q be two propositions. XORing of p and q is true if p is true or q is true but not both and vice-versa. It is denoted by p ⨁ q.
| p | q | p ⨁ q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
- Example1: Prove that X ⨁ Y ≅ (X ∧∼Y)∨(∼X∧Y).
- Solution: Construct the truth table for both the propositions.
| X | Y | X⨁Y | ∼Y | ∼X | X ∧∼Y | ∼X∧Y | (X ∧∼Y)∨(∼X∧Y) |
| T | T | F | F | F | F | F | F |
| T | F | T | T | F | T | F | T |
| F | T | T | F | T | F | T | T |
| F | F | F | T | T | F | F | F |
As the truth table for both the proposition is the same.
- Example2: Show that (p ⨁q) ∨(p↓q) is equivalent to p ↑ q.
- Solution: Construct the truth table for both the propositions.
| p | q | p⨁q | (p↓q) | (p⨁q)∨ (p↓q) | p ↑ q |
| T | T | F | F | F | F |
| T | F | T | F | T | T |
| F | T | T | F | T | T |
| F | F | F | T | T | T |
Conditional and BiConditional Statements
Conditional Statement
Let p and q are two statements then "if p then q" is a compound statement, denoted by p→ q and referred as a conditional statement, or implication. The implication p→ q is false only when p is true, and q is false; otherwise, it is always true. In this implication, p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent).
| p | q | p → q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
For Example: The followings are conditional statements.
- If a = b and b = c, then a = c.
- If I get money, then I will purchase a computer.
Variations in Conditional Statement
- Contrapositive: The proposition ~q→~p is called contrapositive of p →q.
- Converse: The proposition q→p is called the converse of p →q.
- Inverse: The proposition ~p→~q is called the inverse of p →q.
- Example1: Show that p →q and its contrapositive ~q→~p are logically equivalent.
- Solution: Construct the truth table for both the propositions:
| p | q | ~p | ~q | p →q | ~q→~p |
| T | T | F | F | T | T |
| T | F | F | T | F | F |
| F | T | T | F | T | T |
| F | F | T | T | T | T |
As, the values in both cases are same, hence both propositions are equivalent.
- Example2: Show that proposition q→p, and ~p→~q is not equivalent to p →q.
- Solution: Construct the truth table for all the above propositions:
| p | q | ~p | ~q | p →q | q→p | ~p→~q |
| T | T | F | F | T | T | T |
| T | F | F | T | F | T | T |
| F | T | T | F | T | F | F |
| F | F | T | T | T | T | T |
As, the values of p →q in a table is not equal to q→p and ~p→~q as in fig. So both of them are not equal to p →q, but they are themselves logically equivalent.
BiConditional Statement
- If p and q are two statements then "p if and only if q" is a compound statement, denoted as p ↔ q and referred as a biconditional statement or an equivalence. The equivalence p ↔ q is true only when both p and q are true or when both p and q are false.
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
- For Example: (i) Two lines are parallel if and only if they have the same slope.
- (ii) You will pass the exam if and only if you will work hard.
- Example: Prove that p ↔ q is equivalent to (p →q) ∧(q→p).
- Solution: Construct the truth table for both the propositions:
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
| p | q | p →q | q→p | (p →q)∧(q→p) |
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |
Since, the truth tables are the same, hence they are logically equivalent. Hence Proved.
Principle of Duality
Two formulas A1 and A2 are said to be duals of each other if either one can be obtained from the other by replacing ∧ (AND) by ∨ (OR) by ∧ (AND). Also if the formula contains T (True) or F (False), then we replace T by F and F by T to obtain the dual.
Note1: The two connectives ∧ and ∨ are called dual of each other.
2. Like AND and OR, ↑ (NAND) and ↓ (NOR) are dual of each other.
3. If any formula of the proposition is valid, then it's dual of each other.
Equivalence of Propositions
- Two propositions are said to be logically equivalent if they have exactly the same truth values under all circumstances.
- The table1 contains the fundamental logical equivalent expressions:
Laws of the algebra of propositions
| Idempotent laws | (i) p ∨ p≅p | (ii) p ∧ p≅p |
| Associative laws | (i) (p ∨ q) ∨ r ≅ p∨ (q ∨ r) | (ii) (p ∧ q) ∧ r ≅ p ∧ (q ∧ r) |
| Commutative laws | (i) p ∨ q ≅ q ∨ p | (ii) p ∧ q ≅ q ∧ p |
| Distributive laws | (i) p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r) | (ii) p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) |
| Identity laws | (i)p ∨ F ≅ p (iv) p ∧ F≅F | (ii) p ∧ T≅ p (iii) p ∨ T ≅ T |
| Involution laws | (i) ¬¬p ≅ p | |
| Complement laws | (i) p ∨ ¬p ≅ T | (ii) p ∧ ¬p ≅ T |
| DeMorgan's laws: | (i) ¬(p ∨ q) ≅ ¬p ∧ ¬q | (ii) ¬(p ∧ q) ≅¬p ∨ ¬q |
Example: Consider the following propositions
Are they equivalent?
Solution: Construct the truth table for both
| p | q | ~p | ~q | ~p∨∼q | p∧q | ~(p∧q) |
| T | T | F | F | F | T | F |
| T | F | F | T | T | F | T |
| F | T | T | F | T | F | T |
| F | F | T | T | T | F | T |
Tautologies
A proposition P is a tautology if it is true under all circumstances. It means it contains the only T in the final column of its truth table.
Example: Prove that the statement (p⟶q) ↔(∼q⟶∼p) is a tautology.
Solution: Make the truth table of the above statement:
| p | q | p→q | ~q | ~p | ~q⟶∼p | (p→q)⟷( ~q⟶~p) |
| T | T | T | F | F | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |
As the final column contains all T's, so it is a tautology.
Contradiction:
A statement that is always false is known as a contradiction.
Example: Show that the statement p ∧∼p is a contradiction.
Solution:
| p | ∼p | p ∧∼p |
| T | F | F |
| F | T | F |
Since, the last column contains all F's, so it's a contradiction.
Contingency:
A statement that can be either true or false depending on the truth values of its variables is called a contingency.
| p | q | p →q | p∧q | (p →q)⟶ (p∧q ) |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | F |
| F | F | T | F | F |
