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I have the following two English sentences:

  1. Mary is a Person.
  2. Bulldog is a specie of dog. French bulldog is a specie of bulldog.
  3. The only kind of dog that Mary owns is French bulldog.

I would like to know which of the following ways is the correct way to translate the third and the sentences based on the knowledge given.

1st approach

Bulldog ⊆ Dog 
FrenchBulldog ⊆ Bulldog
FrenchBulldog ⊆ Dog

(∀owns.FrenchBulldog ⨅ Person)(MARY)

2nd approach

∀owns.Bulldog ⊆ ∀owns.Dog 
∀owns.FrenchBulldog ⊆ ∀owns.Bulldog

(¬(∀owns.Dog⊔Bulldog) ⨅ ∀owns.FrenchBulldog ⨅ Person)(MARY) (*)

3rd approach

Bulldog ⊆ Dog 
FrenchBulldog ⊆ Bulldog

(Person⨅(∀owns.FrenchBulldog⨅(∀owns.¬Dog⊔∀owns.¬Bulldog)))(MARY) (**)

I know that the first approach is correct. But I would like to re-written the third English sentence as approaches 2-(*), 3-(**).

Thanks in advance for any advice.

CodeEnthusiast
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1 Answers1

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Your approach 1 is correct and approaches 2 and 3 are incorrect.

I assume with

(¬(∀owns.(Dog⊔Bulldog)) ⨅ ∀owns.FrenchBulldog ⨅ Person)(MARY)

by adding (¬(∀owns.(Dog⊔Bulldog)) you trying to ensure Mary only owns FrenchBulldogs, but it is achieving the opposite.

(¬(∀owns.(Dog⊔Bulldog)) ≡ ∃owns.¬(Dog⊔Bulldog) ≡ ∃owns.(¬Dog ⨅ ¬Bulldog)

Thus in essence you are saying that Mary owns only French bulldogs (∀owns.FrenchBulldog) AND you are saying she owns at least 1 thing that is not a dog and not a bulldog (∃owns.(¬Dog ⨅ ¬Bulldog)).

Henriette Harmse
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  • Thanks for the comment Henriette. Is there any way to achieve the same result as approach 1 but with approach 3? ```∀owns.FrenchBulldog⨅(∀owns.¬Dog⨅∀owns.¬Bulldog))``` Doesn't it transale that Mary owns only FrenchBulldong and not any other Dog or Bulldog? – CodeEnthusiast Dec 18 '22 at 21:58
  • No. ∀owns.¬Dog means everything Mary owns is not a Dog. It does NOT mean " not any other dog". – Henriette Harmse Dec 19 '22 at 00:16
  • BTW. To approach 1 you want add `∃owns.FrenchBulldog`. That is you want `(∀owns.FrenchBulldog ⨅ ∃owns.FrenchBulldog ⨅ Person)(MARY). This is to ensure Mary owns at least 1 French bulldog. Otherwise she may have 0 French bulldog. – Henriette Harmse Dec 19 '22 at 00:19
  • Thanks for the note on approach 1. Let's does infer that no more than 1 dog can be owned in this knowledge base. So everyone owns only 1 dog. If I want to achieve as you said "not any other dog" how approach 3 should be written? Is there a way? – CodeEnthusiast Dec 19 '22 at 06:42
  • Also in your answer you have this ```(¬(∀owns.(Dog⊔Bulldog)) ≡ ∃owns.¬(Dog⊔Bulldog) ≡ ∃owns.(¬Dog ⨅ ¬Bulldog)```... if we knew that she could own only 1 Dog (of any specie) and no more (take it as we knew)..could your answer be written as ```(∀owns.FrenchBulldog⨅ ∃owns.(¬Dog ⨅ ¬Bulldog))(MARY)``` – CodeEnthusiast Dec 19 '22 at 06:56
  • No. Approach 3 cannot be rewritten. – Henriette Harmse Dec 19 '22 at 09:14
  • Ok. As far as this (1)```∀owns.FrenchBulldog ⊆ ∀owns.Bulldog``` =>This translates to _For everyone that owns a FrenchBulldog those dogs are Bulldogs_ ... (2)```∃owns.FrenchBulldog ⊆ ∀owns.Bulldog``` =>This translates to _At least one that owns a FrenchBulldog that dog must be Bulldog_ ... (3) ```∀owns.FrenchBulldog ⊆ ∃owns.Bulldog``` => This translates to _For everyone that owns a FrenchBulldog that dog should be Bulldog for at least one of them_ ...It this correct? – CodeEnthusiast Dec 19 '22 at 09:45
  • None of the semantics you given for (1)-(3) is correct. This is too much to deal with in comments. I have answered your original question. You can read more about universal and existential property restrictions on my blog, [here](https://henrietteharmse.com/category/semantic-technologies/owl/universal-property-restrictions/) and [here](https://henrietteharmse.com/category/semantic-technologies/owl/existential-property-restrictions/). – Henriette Harmse Dec 19 '22 at 10:38