First-Order Inference Rules -artificial intelligence:

Now we have a perfect definition of a first-order model is,in the same way, we may define soundness for first-order inference rules as we did for propositional inference rules: the rule is sound if given a model of the sentences above the line, this is always a model of the sentence below.

To be able to specify these new rules, we might use the notion of substitution. We've already seen substitutions which replace propositions with propositional expressions (7.2 above) and other substitutions which replace variables with terms that represent a given object (7.5 pt. above). In this section we use substitutions which replace variables with ground terms (terms without variables) so being clear we will call these ground substitutions. Another name for the ground substitution is an instantiation,

I.e. If we begin with the superbly optimistic sentence that everyone likes everyone else:X, Y (likes(X, Y)), then we can select specific values for X and Y. So, we may instantiate this sentence to say: likes(tony,George). Because we have selected a specific value, the quantification not makes sense longer, so we might drop it.

The activity of performing an instantiation is a function, as there is just one possible outcome, so we may write it as a function. The notation

Subst({X/george, Y/tony}, likes(X,Y)) = likes(george, tony)

Show that we have made a ground substitution.

We also need to recognize that we are working with sentences which form part of a knowledge base of various such like sentences. More to the point, there will be constants which seem throughout the knowledge base and some which are local to a specific sentence.