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Mathematical Logical Connectives
A logical connective is a symbol used to connect two or more propositional or predicate logics in such a manner that the resultant logic depends only on the input logics and the meaning of the connective used.
There are five fundamental connectives in mathematical logic −
- OR (∨) − Disjunction
- AND (∧) − Conjunction
- NOT (¬) − Negation
- IF-THEN (→) − Implication
- IF AND ONLY IF (⇔) − Biconditional
OR (∨) − Disjunction
The OR operation of two propositions A and B (written as A ∨ B) is true if at least one of the propositional variables A or B is true. It is false only when both are false.
| A | B | A ∨ B |
|---|---|---|
| True | True | True |
| True | False | True |
| False | True | True |
| False | False | False |
AND (∧) − Conjunction
The AND operation of two propositions A and B (written as A ∧ B) is true only if both propositional variables A and B are true.
| A | B | A ∧ B |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
Negation (¬) − NOT
The negation of a proposition A (written as ¬A) flips the truth value − it is false when A is true and true when A is false.
| A | ¬A |
|---|---|
| True | False |
| False | True |
Implication (→) − If-Then
An implication A → B is the proposition "if A, then B". It is false only when A is true and B is false. In all other cases, it is true.
| A | B | A → B |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |
Biconditional (⇔) − If and Only If
A ⇔ B is a biconditional logical connective which is true when A and B have the same truth value − both true or both false.
| A | B | A ⇔ B |
|---|---|---|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | True |
Summary of All Connectives
| Connective | Symbol | Name | True When |
|---|---|---|---|
| OR | ∨ | Disjunction | At least one operand is true |
| AND | ∧ | Conjunction | Both operands are true |
| NOT | ¬ | Negation | The operand is false |
| IF-THEN | → | Implication | A is false, or both A and B are true |
| IFF | ⇔ | Biconditional | Both operands have the same value |
Conclusion
The five logical connectives − OR, AND, NOT, implication, and biconditional − form the foundation of propositional logic. Each connective combines propositions in a specific way defined by its truth table, and understanding these is essential for constructing and evaluating logical expressions.
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