Difference between intended interpretation and extended interpretation in first-order logic

Question

I am currently reading "Artificial Intelligence - A modern approach" and I really do not get the difference between intended interpretation and extended interpretation in first-order logic.

Are extended interpretations all possible assignments from a symbol in my knowledge base to an object in the world? And is the intended interpretation the interpretation the "author" of the sentences wanted to have?

So when there is a symbol like "John" in my sentences does that mean that it really refers to the human being John in the real world (if I have intended interpretation).

And the extended interpretation could be: "John" in my sentences refers to the human being Mary in the real world.

Are the extended interpretations a set of all possible interpretations and is the intended interpretation one specific interpretation in this set?

So do I interpret my sentences either intended or extended?

And if I say in my interpretation that John and Mary (as constants) refer to the same object in my model can I unify them then, i.e. UNIFY(John,Mary) = {John/Mary}?

Answer 1

You should only care about intended interpretation. I don't like the concept of the extended interpretation. The latter is only included to make precise the concept of quantification, but I think it makes things more confusing instead.

Before understanding what an intended interpretation is, you should really know what a model is, and what a language or signature is. To follow their example: A language can be consisting of the constants Richard and John, the predicates (the arity in backets) Brother (2), OnHead (2), Person (1), King (1) and Crown (1). Finally you have the unary function LeftLeg (1).

That was the language, and though we use words as Brother, this should really impose no meaning, we could just as well have written it A.

A model, on the other hand, or what they call an intended interpretation is a domain, together with *interpretations of the above predicates, constants and functions. Here's an example:

• Domain = { Richard, John, Crown, Dog }
• Brother = { (Richard, Crown), (Dog, John) }
• OnHead = { (Dog, John), (Richard, Dog), (Crown, Richard) } (the crown' on Richard's head, Richard's on Dog's head, Dog's on John's head)
• Person = {John, Dog}
• Richard = Richard (oops, heavy overloading!)
• John = Dog (oops, heavy overloading!)

I didn't complete the definition, but I hope you get the point. If you put all those above together as a KB, (Domain, Richard, John, Brother, OnHead, Person, etc...), you get your inteded interpretation.

The extended interpretation you should ignore. Completely. At least if you understand what $\forall x P(x)$ means for some first order sentence $P$. And $\exists x P(x)$.