First Order Logic

First Order Logic Chapter 7

PL is a Weak Representational Language • Propositional 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 Language • Consider 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 Language • In 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 Advantages • Concise Representations • Recursive Relations • Arbitrary structures (non-fixed size)

Propositional logic v. FOL • PL 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 PL • Example in FOL

Alphabet Non-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)

Alphabet Logical Symbols (fixed meaning): • Connectives: not () and () or () implies () equivalent () • Punctuation Symbols: ( , ) • Truth symbols: TRUE, FALSE

Alphabet Logical Symbols (fixed meaning): • Variable symbols- strings beginning with an uppercase letter near the end of the alphabet: X, Y, Z • Quantifier symbols: Universal:  Existential:  • (Optional) Equality predicate: =

Terms • Terms 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 predicatesof 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  B A  B A 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

First Order Logic