First Order Logic

Chapter 7

PL is a Weak Representational LanguagePropositional Logic (PL) is not a very

expressive language because: Hard to identify objects : mary, 3 Can't directly talk about properties of

individuals or relations between individuals:

Employee(Name, SoSec, Salary, Dept)Generalizations, patterns, regularities can't

easily be represented:

All persons are mortals

PL is a Weak Representational LanguageConsider the problem of representing the

following information:

Every person is mortal.

Bill is a person.

Bill is mortal.

How can these sentences be represented so that we can infer the third sentence from the first two?

PL is a Weak Representational LanguageIn PL we have to create propositional symbols to stand for all or part of each sentence. For example, we might do:

Person => Mortal

Person-Bill

Mortal-Bill

That is, we have used four symbols to represent the three given sentences. But, given this representation, the third sentence is not entailed by the first two.

A different representation would be to use three symbols to represent the three sentences as

Person => Mortal

Bill => Person

Bill => Mortal The third sentence is entailed by the first two, but we needed an explicit symbol, Bill, to represent an individual who is a member of the classes "person" and "mortal." So, to represent other individuals we must introduce separate symbols for each one, with means for representing the fact that all individuals who are ”person" are also "mortal." FOL is expressive enough to concisely represent this kind of situation.

Representational AdvantagesConcise Representations

Recursive Relations

Arbitrary structures (non-fixed size)

Propositional logic v. FOLPL deals only with propositions that have no

internal structure P : I-love-all-dogs

FOL makes a stronger commitment to what there is (ontology) objects: things in the world properties & relations of & between objects

FOL is as an extension of PLExample in FOL

AlphabetNon-Logical Symbols

(meaning given by interpretation) (infinite) set of object constants - strings of

alphanumeric characters beginning with a lowercase letter or a numeral: john, mary

(infinite) set of function constants of all arities - strings of alphanumeric characters beginning with a lowercase letter: fatherOf, distanceBetween, times

(infinite) set of predicates (relation constants) of all arities - strings of alphanumeric characters beginning with an uppercase letter: MarriedTo(john, mary)

AlphabetLogical Symbols (fixed meaning):Connectives:

not ()

and ()or ()implies ()

equivalent ()Punctuation Symbols: ( , )Truth symbols:

TRUE, FALSE

AlphabetLogical Symbols (fixed meaning):Variable symbols- strings beginning with an

uppercase letter near the end of the alphabet:

X, Y, ZQuantifier symbols:

Universal:

Existential: (Optional) Equality predicate: =

TermsTerms refer to objects in the world

(do not have a truth value assigned to them): object constants function constants of arity n, followed by n terms in

parentheses separated by commas:

fatherof(john, bill)

times(4, plus(3, 6)) variable symbols

Syntax rules: Wffs

atomic formulas are predicates of arity n, followed by n terms in parentheses separated by commas:

Loves(john, mary)

=(john, john)

(= usually in infix form: john = john)

(A proposition in PL is an atomic formula of arity 0)

Truth symbols (TRUE, FALSE) are wffs

WFFs

Complex wffs (just as in PL, we can make more complicated sentences by combining predicates using connectives):If A is a wff, so is A If A and B are wffs, so are

A BA BA B

A B

WFFs

If A is a wff and X is a variable then both X(A) and X(A) are wffs;

X (Person(X) Mortal(X))

X (Student(X) Smart(X))

X y Loves(X, Y)

There are no other wffs.

Language: set of all wffs