Interpretations and Models

Mathematical Logic · Axiom Academy

LESSON Interpretations and Models Giving meaning to formulas: how structures assign truth values to first-order logic statements and what it means for a formula to be true in a particular mathematical world. 1. What is a Structure (Interpretation)? A structure (or interpretation ) for a first-order language consists of three components: Domain D : A non-empty set called the universe of discourse Constants: For each constant symbol c in , an element c^ D Functions: For each n -ary function symbol f , a function f^ : D^n D Relations: For each n -ary relation symbol R , a subset R^ D^n 2. The Domain: Universe of Discourse The domain (or universe ) is the set of objects that variables can range over. It determines what "things" exist in your mathematical world. Domain : FALSE (no natural number squared equals 2) Domain : FALSE (no integer squared equals 2) Domain : FALSE (no rational squared equals 2) Domain : TRUE ( exists in reals) 3. Interpreting Constants, Functions, and Relations Once we have a domain, we must specify what each symbol in our language means: 4. Models: When a Structure Makes a Formula True A structure is a model of a formula (written ) if evaluates to TRUE in . R(t_1, , t_n) iff (t_1^ , , t_n^ ) R^ For any n , we can choose y = n+1 Since this works for all n , we have x \, y \, (x < y) A theory T is a set of formulas (called axioms ). A structure is a model of the theory T (written T ) if for every T . ( , +, 0, -) — integers under addition

This is the written version of the interactive lesson above. See the full Mathematical Logic course.