Difference between propositional logic and first-order logic

[quantumadmin]

Propositional Logic:
Propositional logic, also known as sentential logic, deals with propositions or statements that are either true or false. It focuses on the relationships between propositions using logical connectives (such as AND, OR, NOT, IMPLIES) and truth-functional operators. However, it doesn't deal with the internal structure of propositions or the quantification over variables.

Example propositions:

P: It is raining.
Q: The ground is wet.

Example expressions in propositional logic:

P AND Q (It is raining and the ground is wet.)
NOT P (It is not raining.)
P IMPLIES Q (If it is raining, then the ground is wet.)

First-Order Logic:

First-order logic (FOL), also known as first-order predicate logic or first-order predicate calculus, is a more expressive logic that allows for quantification over variables, relationships between objects, and internal structure within propositions. It includes predicates (relations), functions, quantifiers (such as ∀ for "for all" and ∃ for "exists"), and variables.

Example predicates and quantified expressions in first-order logic:

Loves(x, y) (Person x loves person y.)
∀x ∃y Loves(x, y) (Everyone loves someone.)

Key Differences:

Expressiveness: Propositional logic deals with simple true/false propositions and their combinations using logical operators. First-order logic goes beyond this by allowing the representation of complex relationships, quantification over variables, and functions that operate on objects.

Quantification: Propositional logic lacks quantifiers (like "for all" and "exists") which are essential for expressing general statements and relationships involving variables. First-order logic includes quantifiers to make statements about entire classes of objects.

Structure: In propositional logic, propositions are atomic and not further decomposed. In first-order logic, propositions can contain variables, predicates, and functions, allowing for more detailed representation of relationships and properties.

Scope: Propositional logic is often used for simple reasoning tasks and truth tables. First-order logic is more suitable for representing complex relationships, making inferences, and expressing higher-level concepts.

In total, propositional logic deals with true/false propositions and their logical relationships, while first-order logic extends this by allowing quantification, variable binding, and representation of more intricate relationships between objects.