Atomic Propositions and Logical Connectives in Symbolic Logic
Learn about atomic propositions and logical connectives in symbolic logic. This guide defines atomic propositions, explains how logical connectives (AND, OR, NOT, etc.) combine propositions to form compound statements, and illustrates their use in building logical expressions.
Atomic Propositions and Logical Connectives
Atomic Propositions
An atomic proposition is a simple statement that has a definite truth value—it's either true or false. It cannot be broken down into smaller, simpler statements.
Examples of Atomic Propositions
- 3 + 3 = 5 (False)
- The Earth is round. (True)
- Water boils at 100°C at sea level. (True)
Non-Atomic Propositions
Some statements are not atomic propositions because their truth value isn't definitively true or false. These often involve variables or are not declarative sentences.
Examples of Non-Atomic Propositions
- x + 2 = 7 (The truth depends on the value of x.)
- Please close the door.
- Is it raining?
Propositional Variables
We use letters (like p, q, r, x, y, z) to represent propositions in symbolic logic. This simplifies working with complex statements.
Logical Connectives
Logical connectives combine propositions to form compound propositions. The main connectives are:
1. AND (∧)
The conjunction "P ∧ Q" (P AND Q) is true only if both P and Q are true.
| P | Q | P ∧ Q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
2. OR (∨)
The disjunction "P ∨ Q" (P OR Q) is true if at least one of P or Q is true.
| P | Q | P ∨ Q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
3. Implication (→)
The implication "P → Q" (If P, then Q) is false only when P is true, and Q is false.
| P | Q | P → Q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
4. If and Only If (⇔)
The biconditional "P ⇔ Q" (P if and only if Q) is true when P and Q have the same truth value (both true or both false).
| P | Q | P ⇔ Q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
5. Negation (¬)
The negation "¬P" (NOT P) is true when P is false and false when P is true.
| P | ¬P |
|---|---|
| F | T |
| T | F |
Truth Table Concepts
Tautologies
A tautology is a statement that's always true, regardless of the truth values of its components.
(An example of a tautology is given in the original text and should be included here.)
Contingencies
A contingency (or consistent formula) is a statement that can be either true or false depending on the truth values of its components.
(An example of a contingency is given in the original text and should be included here.)
Contradictions
A contradiction (or inconsistent formula) is a statement that's always false.
(An example of a contradiction is given in the original text and should be included here.)
Propositional Equivalence
Two propositions are equivalent if they have the same truth value for all possible combinations of truth values of their components. This is often shown using a truth table.
Example: Proving Equivalence
(The example proving the equivalence of ¬(X∨Y) and (¬X∧¬Y) using a truth table is given in the original text and should be included here.)
Conclusion
Atomic propositions and logical connectives are the fundamental building blocks of propositional logic. Understanding these concepts is crucial for constructing and analyzing complex logical statements.