First-Order Models - Artificial intelligence:
We proposed first-order logic like good knowledge representation language rather than propositional logic because it is more expressive, so we may write more of our sentences in logic. So the sentences we want to apply inference andrewrites rules will include quantification. All of the rewrite rules we have seen so far may be used in propositional logic (and hence first-order logic). Now we consider rules in which rely on information regarding the quantifiers, so there are not available to an agent working with a propositional logic representation scheme.
Before we look at first-order inference rules we have to pause to consider what it means for such an inference rule to be sound. Previously we defined it as meaning the top entails the bottom: that any model of the former was a model of the latter. But first order logic introduces new syntactic elements (quantifiers, functions, constants, variables, and predicates) alongside the propositional connectives. It means we have to fully revise our definition of model, a notion of a 'possible world' that defines whether a sentence is false ortrue in that world.
A propositional model was only an assignment of truth values to propositions. A first-order model is a pair (Δ, Θ) in contrast where
- Δ is a domain, a non-empty set of 'objects', for example things which our first-order sentences are referring to.
- Θ is an interpretation, a process for calculating the truth of sentences relative to Δ.
This appears very different from propositional logic. Luckily, everything we have discussed so far regarding deduction carries over into first-order logic when we use this new definition of model.
First-order logic lets us to talks regarding properties of objects, so the first task for our model (Δ, Θ) is to assign a meaning to the terms that represent objects. A ground term is any arrangement of constant and function symbols, and Θ maps each specific ground term to a specific object in Δ. This means that a ground term refers to a single particular object. The meaning of sub terms is always not dependent of the term they appear in.
The specific way that terms are mapped to objects depends on the model. Different models may define terms as referring to different things. Note down that although father (john) and jack are separate terms, they must both be mapped to the same object (say Jack) in Δ. That is, the 2 terms are syntactically different but (in this model) they are semantically the similar, for example, they both refer to the same thing!
Terms may also contain variables (for example father(X)) - these are non-ground terms. They do not refer to any particular object, and so our model cannot directly assign any single meaning to them. We will come back to what variables mean.