Notes on Ontology Design

1. Symbology

Table of Common Logic Symbols^[1]

Term	Sym	Example	Description
Conjunction	$\wedge$	$p\wedge q$	The conjunction of $p$ and $q$ is the statement $p\wedge q$, which we read “p and q.”
Disjunction	$\vee$	$p\vee q$	The disjunction of $p$ and $q$ is the statement $p\vee q$, which we read “p or q.”
Negation	$~$	$~p$	The negation of $p$ is the statement $~p$, which we read “not p.”
Equivalent	$\equiv$	$s\equiv t$	Two statements are logically equivalent if, for all possible truth values of the variables involved, the two statements always have the same truth values. If $s$ and $t$ are equivalent, we write $s\equiv t$. This is not another logical statement. It is simply the claim that the two statements $s$ and $t$ are logically equivalent.
Conditional	$\to$	$p\to q$	If $p$ is true, then $q$ is true, or, more simply, if $p$, then $q$. We can also phrase this as $p$ implies $q$. $p$ is a sufficient condition for $q$. $q$ is a necessary condition for $p$.
Biconditional	$\leftrightarrow$	$p\leftrightarrow q$	The biconditional, written $p\leftrightarrow q$, is defined to be the statement$\left(p\to q\right)\wedge \left(q\to p\right)$. $p$ is necessary and sufficient for $q$. $p$ is equivalent to $q$.
Subclass Of Subproperty Of	$\subseteq$	$C_1\subseteq C_2$ $P_1\subseteq P_2$	Example: $\mathrm{Human}\subseteq \mathrm{Animal}\cap \mathrm{Biped}$ $\mathrm{hasDaughter}\subseteq \mathrm{hasChild}$
Some	$\exists$	$\exists RC$	Example: $\exists \mathrm{hasColleague}\mathrm{Professor}$, Existential
Only	$\forall$	$\forall RC$	Eample: $\forall \mathrm{hasColleague}\mathrm{Professor}$, Universal
Not	$\neg$	$\neg \left(\exists RC\right)$	Example $\neg \left(\exists \mathrm{hasTopping}\mathrm{MeatTopping}\right)$, Not

Table 1 - Table of Common Logic Symbols

OWL uses Description Logics which starts with propositional calculus, predicate calculus and First Order Logic (FOL)

Interesting Formulas

Term	Description
Disjunctive Normal Form	$p\to q\equiv \left(~p\right)\vee q$ We can convert implication to disjunction. We can convert any logical statement into a disjunction of conjunctions of atoms or their negations. This is called disjunctive normal form, and is essential in the design of the logical circuitry making up digital computers.
Converse	$q\to p$ The statement $q\to p$is called the converse of the statement $p\to q$. A conditional and its converse are not equivalent.
Contrapositive	$\left(~q\right)\to \left(~p\right)$ The statement $\left(~q\right)\to \left(~p\right)$ is called the contrapositive of the statement $p\to q$. A conditional and its contrapositive are equivalent. $p\to q\equiv \left(~p\right)\vee q\equiv q\vee \left(~p\right)\equiv ~\left(~q\right)\vee \left(~p\right))\equiv \left(~q\right)\to \left(~p\right)$
Modus Ponens or Direct Reasoning	$\left[\left(p\to q\right)\wedge p\right]\to q$ If an implication and its premise are both true, then so is its conclusion. Modus ponens is the most direct form of everyday reasoning, hence its alternate name “direct reasoning.” When we know that $p$ implies $q$ and we know that $p$ is true, we canconclude that $q$ is also true. This is sometimes known as affirming the hypothesis.
Modus Tollens or Indirect Reasoning	$\left[\left(p\to q\right)\wedge \left(~q\right)\right]\to \left(~p\right)$ If an implication is true but its conclusion is false, then its premise is false. When we know that $p$ implies $q$ and we know that $q$ is false, we can conclude that $p$ is also false. This is sometimes known as denying the consequent.

Table 1 - Interesting Formulas

These formulas form the basis of many of the other formuli in this document.

Class Constructors^[2]

Constructor	DL Syntax	Manchester	Example
intersectionOf	$C\cap D$	C AND D	Person $\cap$ EmployedThings Person AND EmployedThings The class EmployedPerson can be described as the intersectionOf Person and EmployedThings.
unionOf	$C\cup D$	C OR D	USCitizens $\cup$ DutchCitizens A class contains things that are either USCitizens or DutchCitizens.
complementOf	$\neg C$		$\neg$Male
oneOf	$\{i_1...i_n\}$		$\{\mathrm{John},\mathrm{Mary}\}$
hasClass	$\exists P.C$		$\exists$hasChild.Lawyer
toClass	$\forall P.C$		$\forall$hasChild.Doctor
hasValue	$\exists P.\left\{i\right\}$		$\exists$citizenOf.$\left\{\mathrm{USA}\right\}$
minCardinalityQ	$\ge nP.C$		$\ge 2$hasChild.Lawyer
maxCardinalityQ	$\le nP.C$		$\le 1$hasChild.Male
cardinalityQ	$=nP.C$		$=1$hasParent.Female

Table X - Class Constructors

• C is a concept (class)
• P is a role (property)
• $x_i$ is an individual/nominal
• XMLS datatypes as well as classes in$\forall P.C$and $\exists P.C$

2. OntoClean

2.1. Assumptions

If the time argument is omitted for a certain predicate ϕ, then the predicate is assumed to be time invariant, that is $\exists t\varphi \left(x,t\right)\to \forall t\varphi \left(x,t\right)$. Some Thing implicates All Thing

• □ϕ - Means ϕ is necessarily true in all possible worlds
• ◇ϕ - Means ϕ is possibly true in at least one possible world
• Property variables are restricted to discriminating properties, properties ϕ such that$◇\exists x\varphi \left(x\right)\wedge ◇\exists x\neg \varphi \left(x\right)$. There is possibly something which has this property, and possibly something that does not have this property.

2.2. Rigidity

Rigid Property

A property that is essential to all its instances

A property ϕ such that $\square (\forall \mathrm{xt}\varphi (x,t)\to \square \forall t\text{'}\varphi (x,t\text{'}\left)\right)$

Non-Rigid Property

A property that is not essential to some of its instances

A property ϕ such that $◇(\exists x,t\varphi (x,t)\wedge ◇\exists t\text{'}\neg \varphi (x,t\text{'}\left)\right)$

Anti-Rigid Property

A property that is not essential to all of its instances

2.3. Formal Ontological Analysis

The following theories for the basis for a common ontology vocabulary:

• Theory of Essence and Identity
• Theory of Parts (Mereology)
• Theory of Wholes
• Theory of Dependence
• Theory of Composition and Constitution
• Theory of Properties and Qualities

2.3.1. Mereology

Mereology is the calculus of individuals, "nominalism". The word is derived from the Greek meros, a part - one of the constituent parts of a whole. Tarski linked the theory of Mereology with boolean algebra in 1935. The theory is expressed in First Order Logic (FOL). The theory places constraints on what can be described:

• No null individuals
• No abstract entities
• No hierarchical distinction between individuals
• A single relation of parthood

All ontologies use a parthood relation, in the best cases fully specified with respect to Peter Simon's work in 1986.

Mereology Formulas

Term	Label	Description	Hasse Diagram
Partial Order	$M_0$	$P$ Parthood partially orders the universe. Partial order is an antisymmetric relation that is also transitive and reflexive.
Reflexivity	$M_1$	$\forall xP\left(x,x\right)$ Every whole is part (non proper part) of itself. Reflexivity is largely mathematical.
Transitivity	$M_2$	$\forall \mathrm{xyz}\left(\left(P\left(x,y\right)\wedge P\left(y,z\right)\right)\to P\left(x,z\right)\right)$ An important and common requirement for the basic relation from a part to its whole is that it is transitive, i.e. if x is part of y, and y is part of z, then x is part of z.
Antisymmetry	$M_3$	$\forall \mathrm{xy}\left(\left(P\left(x,y\right)\wedge P\left(y,x\right)\right)\to x=y\right)$ If Pxy and Pyx both hold, then x and y are the same object.
Proper Part		$\mathrm{PP}\left(x,y\right){\equiv }_{\mathrm{df}}P\left(x,y\right)\wedge \neg P\left(y,x\right)$ "x is a proper part of y," written PPxy, which holds if Pxy is true and Pyx is false.
Overlap		${O\left(x,y\right)\equiv }_{\mathrm{df}}\exists z\left(P\left(z,x\right)\wedge P\left(z,y\right)\right)$ x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold. The parts of z, the "overlap" or "product" of x and y, are precisely those objects that are parts of both x and y.
Weak Supplementation	$M_4$	$\forall xy\left(\mathrm{PP}\left(x,y\right)\to \exists z\left(P\left(z,y\right)\wedge \neg P\left(z,x\right)\right)\right)$ Every proper part must be supplemented by another, disjoint part -- a remainder. If PPxy holds, there exists a z such that Pzy holds but Ozx does not.

Table X - Mereology Definitions

Mereology formulas can be described using Hasse diagrams which are graphs on finite domains with the convention that all arcs are vertical or oblique and are implicitly oriented from top to bottom.

3. Conjunction

Conjunction is a truth-functional connective similar to "and" in English and is represented in symbolic logic with the symbol $\wedge$." The symbol$\wedge$is a connective forming compound propositions which are true only in the case when both of the propositions joined by it are true. "John left and Carol arrived" can be symbolized as $J\wedge C$, so long as we remember that the statement does not mean "Carol arrived after John left" which is a simple proposition. Other ordinary language conjoiners besides "and" include some uses of "but," "although," "however "yet," and "nevertheless."

Conjuction Truth Table

$p$	$q$	$p\wedge q$
T	T	T
T	F	F
F	T	F
F	F	F

Table X - Truth Table for Conjunction

Conjunction is a truth functional connective. It is represented by the symbol $\wedge$.

The symbol $\wedge$as a truth functional connective does not do everything that the "and" does in English. It might be thought of in terms of a "minimum common logical meaning" to conjoined statements. The temporal or causal sequence "Bill tripped and fell" cannot be transposed as "Bill fell and tripped." The clauses cannot be interchanged. Truth functional connectives are more limited than their corresponding English connectives: the whole meaning of the truth functional connective is given in its truth table. So long as we do not expect more from truth-functional connectives, there should be few difficulties in translation.

3.1. Associative Law

Let $p$: “This document is well formed,” $q$: “This document validates” and $r$: “Valid documents are well formed.” We can express the statement “Not only is this document well formed, but this document validates as well and valid documents are well formed.” in logical form. The statement is asserting that all three statements $p$, $q$ and $r$ are true. Note that “but” is simply an emphatic form of “and.” Now we can combine all three in two steps:

1. Combine $p$ and $q$ to get $p\wedge q$, meaning “This document is well formed and this document validates.”
2. Conjoin this with $r$ to get: $\left(p\wedge q\right)\wedge r$. This says: “This document is well formed and this document validates and valid documents are well formed.”

On the other hand, we could equally well have done it the other way around:

1. Conjoin $q$ and $r$ to get $q\wedge r$ meaning “This document validates and valid XML documents are well formed.”
2. Conjoin this with $p$ to get $p\wedge \left(q\wedge r\right)$, which again says: “This document is well formed and this document validates and valid documents are well formed.”

We say that $\left(p\wedge q\right)\wedge r$ is logically the same as $p\wedge \left(q\wedge r\right)$, a fact called the associative law for conjunction.

Both answers $\left(p\wedge q\right)\wedge r$ and $p\wedge \left(q\wedge r\right)$ are equally valid. This is like saying that $\left(1+2\right)+3$ is the same as$1+\left(2+3\right)$. As with addition, we often drop the parentheses and write $p\wedge q\wedge r$.

4. Disjunction

Disjunction

A B

Figure 1 - Sets A and B

$A\vee B$ includes all individuals of class A and all individuals from class B and all individuals in the overlap (if A and B are not disjoint).

Disjunction (or as it is sometimes called, alternation) is a connective which forms compound propositions which are false only if both statements (disjuncts) are false. The connective "or" in English is quite different from disjunction. "Or" in English has two quite distinctly different senses. The exclusive sense of "or" is "Either A or B (but not both)" as in "You may go to the left or to the right." In Latin, the word is "aut."

The inclusive sense of "or" is "Either A or B {or both)." as om "John is at the library or John is studying." In Latin, the word is vel." It is the second sense that we use the "vel" or "wedge" symbol: " or " Consider the statement, "John is at the Library or he is Studying." If, in this example, John is not at the library and John is not studying, then the truth value of the complex statement is false^[2]:

5. Class vs Individual

One of the decision making processes I go through involves deciding what I want to "query" against. In my particular case, I like to query using SPARQL and to a lesser extent computed inferred individuals (using Pellet). For SPARQL, I tend to model my ontologies in such a way that makes it easy to use SPARQL, but I'm not sure if I could recommend this overall because you run into logical problems very quickly.

Now I am (re)learning ontological engineering and I see how I can use defined classes (necessary & sufficient) and Compute Inferred Individuals as a sort of query mechanism through carefully thought out restrictions. This is the next logical step after consistency checking and classifying the ontology. That's why I believe there are three buttons on the tool bar for this.

From a personal philosophical perspective, I believe that the goal of a well crafted OWL model should describe in precise detail 'HOW" the entities and concepts in a domain are related and the Individuals stand as place holders for things that really exist - the "WHAT" of the domain.

Yes, you probably could define a class John - but keep in mind that class John describes how things called John in your domain works. OWL individuals represent actual entities. They are an instantiations of Classes. So, conceivably you may have individuals John1 and John2 that may or may not be the same person depending on your domain.

With all this in mind, I refer to http://www.w3.org/TR/swbp-classes-as-values/ for assistance in figuring out what to do when the lines between class and individual get blurry.

6. n-ary

If I understand your model, you have three concepts that are in a relationship together. OWL (and by definition Protege-OWL) use properties to link two concepts together - so what do you do when you have three, four, or many concepts linked together? Such a relationship is called an n-ary relationship and an excellent tutorial is available on how to deal with n-ary relationships at [1]. That document was prepared by the now defunct Semantic Web Best Practices and Deployment Working Group whose web site is located at [2].

If I understand their model, what you do is considered the relationship among the three concepts to also be a concept. In other words, create a class in Protege that represents the link among all three other classes. In databases, we would create an associate-entity that can be used in joins to get the information you're looking for.

There is an article at the Protege Wiki that describes a way to display n-ary relations in a form.

[1] http://www.w3.org/TR/swbp-n-aryRelations/

[2] http://www.w3.org/2001/sw/BestPractices/

7. Open World Assumption

In a closed world (like DBs), the information we have is everything. In an open world, we assume there is always more information than is stated. Where a database, for example, returns a negative if it cannot find some data, the reasoner makes no assumption about the completeness of the information it is given. The reasoner cannot determine something does not hold unless it is explicitly stated in the model.

With closed-world reasoning, allValuesFrom can be trivially satisfied if there are no fillers at all, which comes as a surprise to many users. With open world reasoning (like in OWL), it is typically very difficult to satisfy allValuesFrom, because (in the absence of other assertions that restrict the set of fillers), there could be additional unknown fillers that do not satisfy the restiction.

Every universal assertion has a correlative double-negative assertion and both assertions are true. That is, given

x=Pizza

T=hasPizzaTopping

we assert

forallX(Tx)

which also implies

not(forallX(notTx)

Not quite correct. When you introduce the negation, the quantifier needs to change from universal to existential. The correct equivalent statement would be

not(existsX(notTx))

Here I'll assume that you have an equivalence between:

Pizza <=> hasTopping allValuesFrom PizzaTopping

Well, it isn't quite so simple, since the restriction ranges over quantifiers of the fillers. It is certainly allowed that non-Pizzas have PizzaToppings. It is just required that they have some other toppings as well, so that the universal quantification is preserved.

It is certainly true that one does not need to know the specific toppings for an individual asserted to be a pizza. But it is definitely not the case that all instances will classify as pizzas.

In first order logic

(forall x (p x)) implies (exists x (p x))

is alway true. The proof is a simple consequence of the definition of a model. It should appear in any elementary logic book. This of course is the "open world" assumption. A consequence of this is that normal set theoretic models cannot act as the semantics of the "closed world" assumption or the "closed world" logic will simply be inconsistent.

In any event, In my opinion, the "closed world" assumption is a bad way to model anything except in restricted domains where it can be proved to be true on the basis of the "open world" assumption. The "closed world" assumption means, for example, that if your mother is not in the "database" she doesn't and has never existed! A very strange way to model the world. Maybe mom will become a person tomorrow?

Here's a line relevant to this from the OWL...

"To see why this is so, observe that the owl:allValuesFrom constraint demands that all values of P should be of type T, and if no such values exist, the constraint is trivially true."

3.1.2.1.1 owl:allValuesFrom

http://www.w3.org/TR/2004/REC-owl-ref-20040210/#allValuesFrom-def

Every time I read this sentence I think to myself, "There's some predicate logic facts behind this that make sense of it, but I'm just missing them" because otherwise it's as non-sensical to me as anything could be.

Is there a loose usage of "trivial satisfied" in the OWL community that contributes to the confusion, because I thought satisfying something trivally was a characteristic of open world reasoning and not closed world reasoning?

(forall x (p x))

can be satisfied along with

(not (exists x (p x)))

precisely in the trivial case where there are no individuals in the model. An empty model will satisfy the universal quantification, since all zero of the bindings for X satisfy the case. Now, these are not generally interesting models, but the principle applies to the more restricted form of allValuesFrom.

The relation of allValuesFrom to the classical predicate calculus is

p allValuesFrom c <=> (forall (x y) (=> (p x y) (c y)))

It is the implication in the forall that makes this satisfiable with empty fillers. If there are no values satisfying (p x y) then the implication is trivially satisfied.

the use of CWA reasoning should be restricted to cases where it holds. Certain restricted areas of knowledge are amenable to CWA reasoning. And since it is an assumption, it is typically treated in a non-monotonic fashion so that later information can invalidate the inferences based on it.

CWA, of course, should not be applied to properly open world tasks. But some things can presumably be known. In a model of the people in a particular room at a particular time, the enumeration of the guests can be properly and usefully treated as closed world. Registration for college classes is another example. Open ended relationships such as ancestry are not proper subjects for CWA.

> "To see why this is so, observe that the

> owl:allValuesFrom constraint demands that all values

> of P should be of type T, and if no such values exist,

> the constraint is trivially true."

This has to do with the definition of universal quantification in the degenerate case, which is linked to the following equivalence:

forall X: p(x) <=> not exists X: not p(x)

Restating the right hand side a bit, it means that there is no individual who is do not satisfy p. So if there are no individuals at all, then there are (trivially), no individuals of satisfying p. There is often a tendency to assume that universal quantification implies a cardinality of at least one, but that is not a requirement.

If I were to have a paper bag and say that "There are no balls in this paper bag which are not red", then it would still be a true statement if there were no balls (or anything else for that matter) in the bag. But that statement is the same (logically) as saying "All balls in this paper bag are red".

Now what makes this a bit more useful is that the universally quantified statement in the allValuesFrom constraint is an implication rather than a ground formula. As in my recent reply to Richard Weyhrauch, I noted that the meaning of

p allValuesFrom c

forall x: [forall y: (p x y) => c(y)]

That means that if there are no individuals satisfying (p x y), then the implication is true regardless of whether there are any C's in the world or not. This follows from the fundamental definition of logical implication. The implication is true whenever the antecedent is false. (It is also true if the consequent is true, but that's not relevant to the trivial case here.)

It's not so much nonsensical, as being an edge case. Just like the algebraic definition of empty summation and multiplication. (+) = 0, (*) = 1. Although the degenerate cases may not have an intrinsic or intuitive meaning, there are often practical factors which lead one to choose a particular convention, so as not to have to treat the degenerate cases at each step of the calculation.

Another way to think about some of these logical reasoning steps is as a game where the point is to come up with a disproof of the item in question. So, for example, if I say "all the balls in this bag are red", how would you go about disproving it? You would need to find a ball in the bag that is not red. But if there are no balls in the bag at all, you won't be able to find the non-red ball you need to disprove the statement. I may have been devious and tricky to use the degenerate case, but it is true that "all [zero] of the balls in the bag are red". This can be something that it is difficult to get one's head around, but it is consistent with the entire framework of logic, and is simpler than other solutions would be.

Invent a new example, if it helps. "All the humans who lived on the Moon in the year 2000 CE were under the age of 55." There's no counterexample (namely, a resident of the Moon during 2000 who was age 55 or older). Is it thus not true?

A less contrived and perhaps more relevant example is the convention about the truth value of empty conjunctions and disjunctions. To wit: A conjunction is false if any of its conjuncts are false; A disjunction is true if any of its disjuncts are true. Thus empty conjunctions are considered true while empty disjunctions (including empty clauses) are considered false.

If the logic is two-valued (i.e., the are two truth values: "true" and "false"), then "meaningless" is not a truth value that any statement in that logic can have. If you want a truth value such as "meaningless" (or "maybe" or "possibly" or "sometimes", etc.), then you need to use (or invent) such a logic.

a non-empty domain really only applies to the general, unrestricted case. The case of the implication (as derived from the allValuesFrom clause) can be trivially satisfied, since there is no requirement that every predicate in the model have a realization. So, if instead I say

(forall x (implies (p x) (q x)))

that doesn't mean that every valid model for this theory requires that there be at least one (p x). There could easily be models where there is no constant satisfying P, which leaves the model still valid.

8. Defined Classes

Unless your classes have definitions, a reasoning system had nothing to work with. You will need to define, using necessary and sufficient (N&S) conditions, what is required to belong to a class. Actually, all you really need are sufficient conditions, but there is not any direct way to enter those in the Protege interface. The closest method is to create fully defined (possibly anonymous?) subclasses with N&S conditions, so that satisfaction of any subclass will enable recognition of class membership.

Assuming that the classes you care about have appropriate definitions, then you will not need the "convert individual to class" function. That would only be necessary if you defined your classes as instances in the first place, and now you realize that you really need them to be classes.

You will have to make sure that the restrictions in the class definitions are in the N&S position in order for recognition to take place. But the "compute individuals belonging to class" or "compute types" seems exactly like the reasoning that you would want for a particular case. They will use the sufficient conditions on the class definition to identify those individuals that meet the criteria for the class. This may mean that you will need to move restrictions which are only under necessary conditions under N&S, though. There isn't really any way around that, since necessary conditions do not allow inference about membership. They only tell you what must be true of a particular individual.

For example, it could be that you define having the property "weight" is a necessary condition for a person. But there are lots of other things, like cows, automobiles, bricks, etc. that also have a "weight" property, so clearly that is not a sufficient condition for being a person.

9. References