4.4 Semantics of First-Order Logic

Structures and interpretations form the backbone of semantic meaning in first-order logic. They provide a concrete framework for understanding abstract logical concepts, linking symbols to real-world elements and relationships within a specific domain.

Truth values of formulas are determined by evaluating atomic components and building up to complex expressions. This process allows us to assess the validity and satisfiability of statements, crucial for logical reasoning and proof construction.

Structures and Interpretations

Structures and interpretations in logic

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• Structure (model) in first-order logic forms foundation for semantic interpretation
  • Non-empty domain (universe) of discourse contains objects under consideration (natural numbers, people, colors)
  • Interpretation function maps symbols to elements, relations, functions in domain
• Components of interpretation assign meaning to logical symbols
  • Constants linked to specific domain elements (0 to zero, Eiffel Tower to actual structure)
  • Predicates mapped to relations over domain (less than, taller than)
  • Function symbols associated with domain functions (addition, father of)
• Structures exemplify abstract concepts in concrete settings
  • Natural numbers with arithmetic operations model basic mathematics
  • Graphs with vertices and edges represent networks or relationships
  • Sets with membership relation capture grouping and inclusion

Truth values of formulas

• Atomic formulas evaluated based on interpretation
  • Predicates checked against assigned relations (is 3 less than 5?)
  • Equality determined by comparing interpreted elements
• Compound formulas build on atomic truth values
  • Logical connectives combine subformula results ($P \land Q$, $P \lor Q$, $\neg P$, $P \rightarrow Q$)
  • Quantifiers consider all ($\forall$) or some ($\exists$) domain elements
• Variable assignments temporarily bind free variables to domain elements
• Complex formulas recursively evaluated by breaking down into simpler components

Satisfiability and validity concepts

• Satisfiability indicates formula true for at least one interpretation
  • $\exists x (P(x) \land Q(x))$ satisfiable if some element has both properties
• Validity means formula true under all possible interpretations
  • $\forall x (P(x) \lor \neg P(x))$ valid as it's a tautology
• Satisfiability and validity interconnected
  • Valid formula's negation always unsatisfiable
• Proving satisfiability or validity involves:
  1. Constructing models to show satisfiability
  2. Finding counterexamples to disprove validity
  3. Using formal proof techniques for validity

First-order vs propositional semantics

• Propositional logic deals with simple true/false statements
  • Truth tables exhaustively list all possibilities
  • Atomic propositions represent indivisible statements
• First-order logic introduces domain, quantifiers, relations
  • Structures and interpretations provide rich semantic framework
  • Variables and quantifiers allow expressing general statements
• First-order logic significantly more expressive
  • Can represent complex relationships and generalities
  • Propositional logic limited to combinations of atomic facts
• Satisfiability and validity differ in complexity
  • Propositional logic decidable through systematic methods
  • First-order logic undecidable in general cases
• First-order logic builds upon propositional foundation
  • Incorporates propositional connectives
  • Propositional formulas translatable to first-order logic