First-Order Semantics

Mathematical Logic · Axiom Academy

Understanding the meaning of first-order logic formulas through structures, interpretations, and the satisfaction relation Mathematical Logic • Unit 4 - First-Order Logic While syntax tells us which strings are valid formulas, semantics tells us what those formulas mean . In first-order logic, we give meaning to formulas by interpreting them in mathematical structures. The Central Question of Semantics Given a formula and a structure , is true or false in ? We write this as: (read: "M models phi" or "phi is true in M") A structure (or interpretation ) gives concrete meaning to the abstract symbols in our language. It consists of a domain and interpretations for all function and relation symbols. A structure for a language consists of: Domain: A non-empty set of objects Constant interpretation: Each constant maps to an element Function interpretation: Each n-ary function maps to a function Relation interpretation: Each n-ary relation maps to a relation Constant : interpreted as the number 0 Function : interpreted as addition Relation : interpreted as "less than" Variables are placeholders that can refer to any element in the domain. A variable assignment tells us which domain element each variable refers to. Formal Definition: Variable Assignment A variable assignment is a function that maps each variable to an element of the domain. We often write assignments as: Example: Assignment in Natural Numbers

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