First-order inference rules, Computer Engineering

First-Order Inference Rules:

Here now we have a clear definition of a first-order model is that we can define soundness for first-order inference rules in the same way such we did for propositional inference rules: in fact the rule is sound if given a model of the sentences above the line that is always a model of the sentence below. 

Justify here to be able for new rules if we must use the notion of substitution than we've already seen substitutions that replace propositions into propositional expressions see in the 7.2 above this and other substitutions that replace variables with terms that represent a given object in the 7.5 above. However in this section we use substitutions that replace variables with ground terms without variables- so to be clearly we will call these ground substitutions. And the another name for a ground substitution is an instantiation, if we considered here example than we see, start with the wonderfully optimistic sentence such everyone likes everyone else:  ∀X, Y (likes(X, Y)), means here we can choose particular values for X and Y. But if we can instantiate this sentence to say: as george, tony. It means that we have chosen a particular value that is the quantification no longer makes sense so that we must drop it.

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