Do we really need two syntaxes to represent intensional and extensional operators?

[Clément Michaud]

Hey there,

I am playing with the NAL theory lately and I wonder whether it is really necessary to have two syntaxes to represent extensional and intensional operators in general instead of one (I am mainly refering to the compound image and the difference operators).

From what I understand, the intensional or extensional property of a compound term depends on whether the term is subject or predicate in an inheritance statement. Therefore, I tend to think that the property of the operator can be made implicit by using only one syntax in the grammar and deducing whether it is the extensional or intensional variant by looking at its context, ie, the statement.

The above reasoning might be a bit abstract so let me argue with a concrete example. The image of a compound term can be intensional or extensional and, according to the book and papers, there is one representation for each:

1. Alice --> /(married_with _ Bob)    |  Extensional

2. \(married_with _ Bob) --> Alice    |  Intensional

Obviously, the two sentences have two different meaning. The first sentence makes sense, but the second one much less even though we can expect to observe it anyway after following some inference rules since it is totally valid both syntactically and in theory.

In my opinion though, the semantic of the compound term in both sentences is exactly the same and refers to ""anything" married with Bob".

It came to my mind because I observed an inference in one of my experiences where an image is transformed with the conversion rule and ended up with a weird conclusion. If we follow the rule

S --> P |-  P --> S <Fconv>

with the example above the system actually does the following inference:

Alice --> /(married_with _ Bob)  |-   /(married_with _ Bob) --> Alice <Fconv>

However if / represents the extensional image, the conclusion shoud therefore be considered invalid in theory I guess because we end up with an extensional image as a subject. Is my reasoning correct, or am I missing something?

Cannot we simply keep one operator in the grammar that would be used for both interpretations of the image but where the interpretation is disambiguated with the context? Doing so would also reduce the number of concepts in the system and facilitate concept matching improving inference efficiency as a side effect.

What do you think?

Thanks,

Clément

[Clément Michaud]

I figured that the word "operator" was not the right one in my last comment because it might be confused with the concept of operations from the theory. "term connector" would be more appropriate so please replace any occurence of "operator" with "term connector" when reading my previous comment.
Sorry for that.

[Pei Wang]

Hi Clément,

At the beginning I also thought "image" can be defined directly as the reverse of "product" (it was called "quotient" in my PhD dissertation and only has the extensional version). Later, I realized that intuition mainly came from the traditional definition of sets, which is extensional. The needs for two "images" mainly come from two aspects of a "relational term". Though in set theory a "relation" R is defined as a set of pairs, so (A * B) --> R is the usual usage, R --> (A * B) is also valid when the "domain" or "range" of R is specified, though it is less often used and unique for normal relations. Anyway, here (A * B) --> R and R --> (A * B) have different meanings as the two (A * B)s have different relations to R. Since the function of image is to "peel out" a component from a product or relation, the two cases need to be distinguished from each other, so the two become A --> (R / _ B) and (R \ _ B) --> A, respectively. You are right that because of conversion (and other inference) it is possible to also have (R / _ B) --> A in the system, but it is still different from (R \ _ B) --> A. Here "extensional" (or "intensional") indicates how the image is related to the relation R in it, not the position of this image in inheritance statements, especially the derived ones.

Extensional/intensional differences have a similar story. My dissertation only mentioned extensional difference (which is obviously needed), and the introduction of the intensional version was mostly caused by my obsession on the symmetry/duality between extension and intension. ;-) . It took me a long time to find a proper interpretation for this term connective. Now I'd say that while (animal - human) is for "non-human animals", (human ~ animal) is for something like "humanness beyond animalness".

You are correct that very often only one of the two term connectives is meaningful for the given components, as (animal ~ human),  (human - animal), and  \(married_with _ Bob) are not very useful, so to distinguish the two by context is possible. However, it is not always the case. Furthermore, since the two concepts have different meanings anyway, to use the same syntax will cause troubles, both in theory and in implementation.

Regards,

Pei

--
You received this message because you are subscribed to the Google Groups "open-nars" group.
To unsubscribe from this group and stop receiving emails from it, send an email to open-nars+...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/open-nars/58dfb30a-daa3-40d1-ac7a-7a40ccebf4afn%40googlegroups.com.

[Clément Michaud]

Hi Pei,

Thanks for your explanation. After wrapping my head around the problem further, I think I got it.

Now, I actually even think that the judgements produced by the inferences applied on compound images (like conversion) and producing judgements with extensional image on the left or intensional image on the right are actually valid too.

Despite the statement <(R / _ B) --> A> has no equivalence compared to <(R \ _ B) --> A> which has two equivalences (<R --> (A * B)> and <(R \ A _) --> B), the extensional image on the left of the first statement can still be selected as premise for an inference and draw new valid conclusions.

For instance:

C --> (R / _ B),  (R / _ B) --> A  |-- C --> A.

Also, with the single syntax, as I initially suggested, we would not be able to distinguish whether the concept has the equivalences or not.

That now makes perfect sense. Thanks.

Regarding your example with human and animal, I think it would be fairer to compare the interpretation of "(animal - human)" and "(animal ~ human)" but I got what you mean.

Clément

[Clément Michaud]

Hey Pei,

Another question related to the difference between intension and extension.

I am wondering how you think the conversion rule should apply to a sentence involving intensional or extensional sets.

The reasoning is done on intensional set but is equivalent when using extensional set.

Since by definition of an intensional set:

[S] -> P  <==>  [S] <-> P

and since the similarity is symmetric we can conclude that

P <-> [S] <==> P -> [S]

Assuming the above, and let say that  [S] -> P <1, 0.9>, are we supposed to allow a conversion with a different truth value than if we applied the standard conversion rule?

In that case it would mean P -> [S] <1, 0.9> when following the above reasoning, instead of P -> [S] <1, 0.47> when applying the conversion rule.

Have you thought about such a case and what assumption should we take to prefer one truth value over the other?

Thanks,

Clément

[Clément Michaud]

I think I have my answer now. Basically, both judgements can be derived and may exist in the memory (with different truth values and evidential bases) even though one is more likely to be selected in the future because it has a higher confidence and/or expected value.

It did not make sense to me initially because while trying to implement my own NAL for learning purposes, I tried to squash conflicting judgements as soon as possible to keep the number of them as low as possible. However in the original theory, it is totally fair to have both and it is even necessary in the case we cannot confidently select one of the conflicting judgements yet because the truth values are still too similar somehow.

[Pei Wang]

Clément,

Sorry I somehow missed the previous email.

You are right that in this case the conclusion can be higher than that provided by conversion. In general, it is possible for the same statement to get different truth-values when derived from different inference paths, even if the original premises are the same. In that case the choice rule will pick the "strongest".

In general, there is a balance to keep between the expense of keeping redundant (though not identical) data and the expense of checking for special cases in the inference rules.

Regards,

Pei

Virus-free.www.avast.com

To view this discussion on the web visit https://groups.google.com/d/msgid/open-nars/e7ab6f21-2f40-4d5b-b329-fa2a5be3848cn%40googlegroups.com.