Nathan Sturtevant FOL
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1 Lecture Overview COMP 3501 / COMP Lecture 8 First order logic (FOL) Inference in FOL Prof. JGH 318 First Order Logic FOL is closer to natural languages that prop. logic FOL contains: Objects (Constants): FOL Propositional logic reduces everything to true or false In FOL relations between objects are true or false (do or do not hold) people, locations, etc Relations (between objects): next to(c11, C12), older(a, b), father(a, b) Functions (relation which returns object): father_of(b), plus(one, two), leader(usa)
2 Prop. Logic Syntax Sentence AtomicSentence ComplexSentence AtomicSentence True False P Q R Complex Sentence (Sentence) [Sentence] Sentence Sentence Sentence Sentence Sentence Sentence Sentence FOL syntax Sentence AtomicSentence ComplexSentence Predicate Predicate(Term ) AtomicSentence True False P Q R Term = Term Complex Sentence (Sentence) [Sentence] Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence Sentence QuantifierVariable, Sentence Operator precedence:,,,, Operator precedence:,,,, FOL syntax FOL semantics Term Function(Term, ) Constant Variable Quantifier Constant A P1 Fred Variable a x s Predicate True False After Loves Raining Function Mother LeftLeg Semantics are somewhat flexible In C++ you can overload the + operator to be * Similarly, we can define objects in a way that is nonsensical in the real world Intended Interpretation is when objects represent the names they have in the real world
3 FOL Models crown A model in propositional had the truth value of each variable A model in FOL has: on head king All objects The relations between objects Functions on objects R $ J FOL Models A model contains one set of interpretations What about all models? Not only all interpretations on a fixed set of objects Also all numbers of objects and all interpretations of relations between them Sentences Atomic sentences state facts Married(Father(Richard), Mother(John)) The sentence is true in a model if the relations hold in the model Complex sentences work like prop. logic King(Richard) King(John)
4 Universal Quantifier ( ) Quantifiers let us solve the problem of making general statements about the world All kings are s x King(x) Person(x) Universal Quantifier ( ) Formally, when quantifying a variable we need to look at all the extended interpretations of that variable eg x, what are all things that x can be? x King(x) Person(x) crown For all x, if x is a king, then x is a Lower case letters are variables This is true if it is true for all interpretations of x R $ J on head king Universal Quantifier ( ) Existential quantifier ( ) Richard is a king Richard is a John is a king John is a Richard s is a king Richard s is a John s is a king John s is a The crown is a king The crown is a Note that this works because of our definition of R $ J on head crown king Universal quantifiers make statements about all objects Existential quantifiers claim that at least one object has a given property is the natural connector to use with Why is x King(x) Person(x) wrong?
5 Existential quantifier ( ) x Crown(x) OnHead(x, John) Richard is a crown Richard is on John s head John is a crown John is on John s head Richard s is a crown Richard s is on John s head John s is a crown John s is on John s head The crown is a crown The crown is on John s head Why is x Crown(x) OnHead(x, John) wrong? Nested quantifiers Can nest multiple quantifiers together x, y (x, y) (y, x) With the same quantifier, order doesn t matter What is the difference? x y loves(x, y) x y loves(x, y) Avoid re-using variables in quantifiers Negating quantifiers Negating quantifiers How do we write everyone likes ice cream? De Morgan s rules for quantification: x P x P x P x P x P x P x P x P
6 Equality Equality indicates that two terms refer to the same objects Father(John) = Henry Often used with multiple existential variables: x, y Sibling(x, Nathan) Sibling(y, Nathan) x, y Sibling(x, Nathan) Sibling(y, Nathan) (x = y) First-Order Logic Examples All cows eat grass. x cow(x) eat(x, grass) Some cows don t eat grass x cow(x) eat(x, grass) Every good boy deserves fudge. x good(x) boy(x) deserves(x, fudge) My dog likes popcorn. x dog(x) owns(me, x) likes(x, popcorn) Homework: 8.24
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