Answer :
First-order logic is differentiated from higher-order logic by the restriction on the types of quantifiers that can be used. In first-order logic, only individual variables can be quantified over, whereas in higher-order logic, quantification over predicates or functions is allowed.
1. First-order logic:
- In first-order logic, quantifiers such as "for all" (∀) and "there exists" (∃) are used to quantify over individual variables.
- Example: , which can be read as "For every x, there exists a y such that x is less than y."
2. Higher-order logic:
- In higher-order logic, quantifiers can be used to range over not only individual variables but also predicates or functions.
- Example: , where P and Q are predicates and x is an individual variable. This statement asserts that for every predicate P, there exists a predicate Q such that for all x, if P(x) is true, then Q(x) is also true.
Therefore, the adjective "first-order" in first-order logic distinguishes it from higher-order logic, where quantification over predicates or functions is permitted.
