Ground Terms as Domain of Interpretation
Recall syntax of first-order logic terms in First-Order Logic Syntax.
Ground term is a term without variables, i.e. , i.e. given by grammar:
i.e. built from constants using function symbols.
If has no constants then is empty. In that case, we add a fresh constant into the language and consider that has a non-empty . We call the set Herbrand Universe.
Goal: show that if a formula without equality (for now) has a model, then it has a model whose domain is Herbrand universe, that is, a model of the form .
How to define ?
Term Algebra Interpretation for Function Symbols
Let . Then
This defines . How to define to ensure that elements of a set are true, i.e. that ?
Partition in two sets, one over which is true and the other over which it is false.
- is this possible for arbitrary set? no
Example Consider a set that is not satisfiable :
If , and , we call an Herbrand Atom. HA is the set of all Herbrand atoms:
We order elements of in sequence (e.g. sorted by length) and establish a bijection with propositional variables
We will write .
We define p such that :