Predicate Logic Motivations Propositional logic can only express

Predicate Logic

Motivations • Propositional logic can only express whole sentences. • It doesn’t allow you to access the components within a sentence, e. g. , subject, object, adjectives, etc. • Predicate logic has that capability.

Objectives • • • Syllogism Predicates Universal quantifiers Existential quantifiers Define legal predicate sentences recursively

Many names • • • Predicate logic Predicate calculus First-order logic First-order predicate calculus Logic of quantifiers

All humans are mortal • Syllogism in Aristotle's Prior Analytics, – Major premise: All humans are mortal. – Minor premise: Socrates is human. – Conclusion: Socrates is mortal. • Propositional Logic – – P: All humans are mortal. Q: Socrates is human. R: Socrates is mortal. But R cannot be deduced from P and Q. • First Order Logic – – – The for-all symbol is need to qualify the variable x to indicate that the IF-conditional is true not just for some x. • SWI-Prolog – – mortal(X) : - man(X). man(socrates). ? - mortal(socrates). Yes Prolog uses the variable X and the ONLY-IF-conditional to show that the clause is true for all X.

Subject and predicate • Access the subject of a sentence: Tony likes BBQ ribs. Lynda likes BBQ ribs. • We need a special function called a predicate, e. g, like_ribs() like_ribs(tony) like_ribs(lynda) • The predicate like_ribs/1 predicates (depends) on one argument which is the subject of a sentence.

Examples • Given a world of discourse, or a domain D = {tony, lynda, samuel, daniel, ezekiel, jonathan, joy} • D contains 7 constant symbols or atoms. We can say that the whole family in the House of Chan like BBQ ribs: – X like_ribs (X), X in D. X is a variable, not a constant. – Compare with the proposition “Everyone in the House of Chan likes BBQ ribs. ” • Some computers have mouse connected on the USB port. Y computer (Y) USB_conn (Y, mouse_of(Y))

Predicate logic • Subject / Predicate John / went to the store. The sky / is blue. • Propositional logic uses propositions • Predicate logic uses predicates – predicates must be applied to a subject in order to be true or false – In general, a logical predicate can be applied to the objects, adjectives, or any other component of a sentence. • P(X) – means this predicate represented by P – applied to the entity (item) represented by X

Family relationships 1. 2. 3. 4. 5. mother(eve, abel). mother(eve, cain). father(adam, abel). father(adam, cain). X Y father(X, Y) mother(X, Y) parent(X, Y). 6. X Y Z parent(X, Y) parent(X, Z) sibling(Y, Z). 7. sibling(cain, abel)?

Connection between the quantifiers Change of variable X to Y. X q(X) Y q(Y) X p(X) Y p(Y) It is false that for all X, such that p(X) is true. E. g. : Not every student has a textbook. Or: There is at least one student who does not have the textbook. X p(X) Not even one student has the textbook. X p(X)

Types of symbols 1. Truth symbols true and false These are reserved symbols. 2. Constant symbols are symbols having the first character lowercase. – E. g. , today, fisher 3. Variable symbols are symbols beginning with an uppercase character or underscore. – E. g. , X, Y, Z, Building, _building 4. Function symbols are symbols having the first character lowercase. Arity: /number of arguments – E. g. , mother_of/1; maximum/2

Mutually recursive definitions • A function expression consists of a function symbol of arity n, followed by n terms, t 1 , t 2 , …, tn, enclosed in parentheses and separated by commas. – mother_of(sam) – maximum(5, 38) – maximum(7, 18), add_one(18)) – Non-example: maximum(7, 18) add_one(18) • A FOL term is either a constant, variable, or function expression. – sam – Sam – house_of(X) – color_of(house_of(neighbor(joe)))

Atomic sentences • Predicate symbols are symbols beginning with a lowercase letter. Predicates are special functions with true/false as the range. Arity: number of arguments • An atomic sentence is a predicate constant of arity n, followed by n terms, t 1 , t 2 , …, tn, enclosed in parentheses and separated by commas. • The truth values, true and false, are also atomic sentences. • Atomic sentences do not contain logical connectives symbols.

3 examples of atomic sentences 1. greater_than(2, 3) Predicate symbol term, constant 2. mother_of(joe, susan) 3. mother_of(sister_of(joe), susan)

Predicate logic sentences • Every atomic sentence is a sentence. • If s is a sentence, then so is its negation, s. • If s 1 and s 2 are sentences, then so is their – Conjunction, s 1 s 2. – Disjunction, s 1 s 2. – Implication, s 1 s 2. – Equivalence, s 1 s 2.

Predicate logic sentences (cont’d) • If X is a variable and s is a sentence, then so are – X s. • Remember that logical sentences evaluate to true or false, therefore only such objects are atomic sentences. • Functions are not atomic sentences.

Checking a sentence in predicate logic 1. 2. 3. 4. 5.

Summary • Unlike natural languages such as English, the semantics of predicate logic sentences are precise and unambiguous. • Predicate logic is more expressive than propositional logic. • It extends propositional logic by allowing functions, universal, and existential quantifiers in its sentences.