# First-Order Logic (FOL) First-Order [[Logic]], also known as predicate [[Logic]] or first-order predicate calculus, is a formal system that extends [[Truth-Functional Logic]] (TFL) by introducing [[Quantifiers]] and predicates. This allows for more expressive power in representing and analyzing logical statements and arguments. ## Key Features 1. Includes all features of TFL (logical [[Connectives]], truth-functionality). 2. Introduces [[Quantifiers]] to express statements about "all" or "some" objects. 3. Uses predicates to express properties of objects and relationships between objects. 4. Allows for variables and constants to represent objects. ## Basic Components 1. [[Quantifiers]]: - Universal Quantifier (∀): "for all" - Existential Quantifier (∃): "there exists" 2. Predicates: Express properties or relations (e.g., P(x) might mean "x is a person") 3. Variables: Represent unspecified objects in the domain 4. Constants: Represent specific objects in the domain 5. Functions: Represent operations on objects ## Syntax FOL expressions are built using: 1. Terms: Variables, constants, or functions applied to terms 2. Atomic Formulas: Predicates applied to terms 3. Complex Formulas: Built from atomic formulas using logical [[Connectives]] and [[Quantifiers]] ## Semantics [[Interpretation]] in FOL involves: 1. Specifying a domain of discourse (set of objects) 2. Assigning meanings to predicates, constants, and functions 3. Evaluating truth values of formulas based on these assignments ## Advantages over TFL 1. Can express more complex logical relationships 2. Able to represent statements about properties and relations 3. Can handle statements involving "all" or "some" objects ## Limitations 1. Cannot quantify over predicates or functions (this is done in higher-order logics) 2. Some concepts in natural language are still difficult to represent precisely ## Applications 1. Mathematics: Formalizing mathematical theories 2. Computer Science: Database query languages, knowledge representation 3. Linguistics: Formal semantics of natural languages 4. Philosophy: Analyzing arguments and theories with greater precision ## Relation to Other Concepts - [[Truth-Functional Logic]]: FOL is an extension of TFL - [[Formal Language]]: FOL is a more expressive [[Formal Language]] than TFL - [[Validity]]: FOL can be used to assess [[Validity]] of more complex arguments - [[Atomic Sentences]]: In FOL, [[Atomic Sentences]] can have internal structure (predicates applied to terms) First-Order [[Logic]] provides a powerful framework for representing and reasoning about a wide range of statements and arguments, making it a fundamental tool in [[Logic]], mathematics, and computer science.