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

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}?

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}

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 $\exists x P(x)$.

An interpretation creates a link between symbols, on one hand, and objects, relations or functions, on the other hand. Examples of interpretations are the links between names and people in the real world or the links between Numbers (which are objects in the world of mathematics), and the corresponding symbols (1, 2, …).

To discuss intended/extended interpretations in this context of an example, let's assume a finite domain, containing only the 9 first natural numbers (1, 2, …, 9).

Some of the possible interpretations are as follows:

1. One ← 1 (symbol 1 refers to number one), Two ← 2, Three ← 3, …
2. One ← 9, Two ← 8, …
3. One ← 2, Two ← 3, …

In this case the intended interpretation is the first one, which we are all used to and the rest are unintended interpretations. And as you said, is the interpretation that the author would have.

An extended interpretation comes into play only when we are talking about quantifiers.

Consider this sentence: $$\forall x: x \ge 1$$

Here x is a variable which can refer to any object in the domain. That means we can link it to any of the objects in the domain:

• x linked to number one: $$1 \ge 1$$
• x linked to number two: $$2 \ge 1$$

An extended interpretation is created as we link a variable to an object in the domain. So, basically a universal/existential quantification is the conjunction/disjunction of all the extended interpretations.