# the semantic web a gide to the future of XML, Web Services and Knowledge Management - Daconta M,C.

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Table 8.7 Propositional Logic Example

PROPOSITIONS IN

PROPOSITIONS IN ENGLISH PROPOSITIONAL LOGIC

If John is a management employee, then John manages an organization. John is a management employee. p " q p

John manages an organization. q Modus ponens

Modus ponens

228

Chapter 8

One limitation of propositional logic is that you cannot speak about individuals (instances like John, who is an instance of a management employee) because the granularity is not fine enough. The basic unit is the proposition, which is either true or false. More complicated propositions use compositions of propositions, composed by using the logical connectives such as and, or, and as earlier, implication. One cannot "get inside" the proposition and pull out instances or classes or properties. For these, one needs first-order predicate logic.

First-Order Predicate Logic

In first-order predicate logic, finer semantic distinctions can be made. In Table 8.8, distinct predicates p and q can refer to the same individual x. A predicate is a feature of language (and logic) that can be used to make a statement or attribute a property to something, in this case the properties of being a management employee and managing an organization. So both properties and individuals can be represented in predicate logic. We also note that an instantiated predicate is a proposition, for instance, management_employee(john) = true. An uninstantiated predicate—for example, management_employee(x)—is not a proposition because the statement does not have a truth value (and only propositions have truth values); in other words, we don't know what x refers to and so cannot tell if "x is a management_employee" is true or not. In this example, we have only two predicates, management employee and managing an organization; we have not yet teased apart the statement into three parts: a management employee part, a managing an organization part, and a manages part. But in Table 8.9, we will do just that.

Table 8.8 Predicate Logic Example

PROPOSITIONS AND PREDICATES IN ENGLISH PROPOSITIONS AND PREDICATES IN FIRST-ORDER PREDICATE LOGIC

If John is a management employee, then John manages an organization. John is a management employee. p(x) " q(x) p(john)

John manages an organization. q(john) Modus ponens

Modus ponens

Understanding Ontologies

229

In addition to predicates, predicate logic also has quantifiers. Quantifiers come in many flavors, but we are only interested in two simple kinds: the universal quantifier and the existential quantifier. A quantifier is a logical symbol that enables you to quantify over instances or individuals (most modeling languages use the term instance; usually logic uses the term individual). The universal quantifer means All; the existential quantifier means Some.

In fact, this is why ordinary predicate logic is called first-order: It only quantifies over instances. If you use a logic to quantify over both instances and predicates, then that logic is called second-order logic. The universal quantifier binds a designated instance variable in the expression so that wherever that variable occurs (in whatever predicate), every possible substitution of that variable by an instance must make the complex expression true. In Table 8.9, everyone and anyone who is a management employee also manages an organization (we don't know yet if the person is a manager or a director or a vice president or president, but in any case, we know that person manages some organization).

This final example may seem a bit complicated, but it demonstrates that fine logical (and semantic) distinctions can be made and formalized in predicate logic. High-end ontologies (ontologies that are logical theories in our Ontology Spectrum) are modeled in semantic languages such as DAML+OIL and OWL that have a logic behind them, a logic that is almost but not quite as complicated as first-order predicate logic (description logics explicitly try to achieve a good trade-off between semantic richness and machine tractability). This is the reason that ontologies modeled in those languages can be machine-interpretable: The machine knows exactly what the model means and how the model works logically, and can infer in a step-by-step fashion those inferences a human would make. But you need not worry about the formal logic behind those languages. You just use the languages like OWL to create your ontologies, and then the OWL interpreter will do the right thing. That is the power of using ontologies, especially those developed in a semantically rich language that expresses what you want to express.

Table 8.9 Example of Quantifiers in Predicate Logic

PROPOSITIONS AND PREDICATES IN ENGLISH PROPOSITIONS AND PREDICATES IN FIRST-ORDER PREDICATE LOGIC

Everyone who is a management employee manages some organization. Or: For everyone who is a management employee, there is some organization that that person manages. 6x. [p(x) " 3y. [q(y) / r(x,y)] ] “for all x, if x is a p, then there is some y such that y is a q and x is in the r relation to y"

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