Knowledge engineering in first order logic

• A sentence (also called a formula or well-formed formula or wff) is defined as:

1. A symbol

2. If S is a sentence, then ~S is a sentence, where "~" is the "not" logical

operator

3. If S and T are sentences, then (S v T), (S ^ T), (S => T), and (S <=> T) are

sentences, where the four logical connectives correspond to "or," "and,"

"implies," and "if and only if," respectively

4. A finite number of applications of (1)-(3)

• Examples of PL sentences:

o (P ^ Q) => R (here meaning "If it is hot and humid, then it is raining")

o Q => P (here meaning "If it is humid, then it is hot")

o Q (here meaning "It is humid.")

• Given the truth values of all of the constituent symbols in a sentence, that

sentence can be "evaluated" to determine its truth value (True or False). This is

called an interpretation of the sentence.

• A model is an interpretation (i.e., an assignment of truth values to symbols) of a

set of sentences such that each sentence is True. A model is just a formal

mathematical structure that "stands in" for the world.

• A valid sentence (also called a tautology) is a sentence that is True under all

interpretations. Hence, no matter what the world is actually like or what the

semantics is, the sentence is True. For example "It's raining or it's not raining."

• An inconsistent sentence (also called unsatisfiable or a contradiction) is a

sentence that is False under all interpretations. Hence the world is never like what

it describes. For example, "It's raining and it's not raining."

• Sentence P entails sentence Q, written P |= Q, means that whenever P is True, so

is Q. In other words, all models of P are also models of Q