CSE579_U2_M1_FOL-intro-recorded

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Oct 30, 2023

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First Order Logic and KR Introduction to First Order Logic
Objectives Objective Explain the limitation of propositional logic Objective Explain the basic idea of first-order logic
Limitations of Propositional Logic and KR
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| Study of declarative sentences, statement about the world which can be given a truth value | Dealt very well with sentence components like: not , and , or , if…then… | Propositional logic is compositional: - meaning of F G is derived from meaning of F and of G ∧ | Limitations: - Cannot express individuals and relations between them - Cannot deal with modifiers like there exists, all, among, only Propositional Logic
Representation in Propositional Logic (1 of 4) Knowledge from a medical domain: | A juvenile disease affects only children or teenagers | Children and teenagers are not adults | Juvenile arthritis is a kind of arthritis and a juvenile disease | Arthritis affects some adults In propositional logic: | ?????????? → ????????ℎ??? ∨ ??????????????? | ?ℎ??? ∨ ???????? → ¬????? | ??????ℎ????? → ?????????? ∧ ???ℎ????? | ???ℎ????? → ????????????
| Some intuitive consequences of our statements: - Juvenile arthritis does not affect adults - Arthritis is not a juvenile disease | We expect the following formulas to be entailed: - → ?????????? ¬???????????? - ???ℎ????? → ¬?????????? | However, neither of them is entailed. Limitations of Propositional Logic (2 of 4) In propositional logic: | ?????????? → ????????ℎ??? ∨ ??????????????? | ?ℎ??? ∨ ???????? → ¬????? | ??????ℎ????? → ?????????? ∧ ???ℎ????? | ???ℎ????? → ???????????? These knowledge base doesn’t entails these two formulas (we found an interpretation make these logics true but doesn’t make theses formulas true)
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| Even worse, if we add to our initial formulas the following ones, we obtain an unsatisfiable set of formulas - → ?????????? ¬???????????? - ??????ℎ????? Limitations of Propositional Logic (3 of 4) In propositional logic: | ?????????? → ????????ℎ??? ∨ ??????????????? | ?ℎ??? ∨ ???????? → ¬????? | ??????ℎ????? → ?????????? ∧ ???ℎ????? | ???ℎ????? → ????????????
| What is going wrong? - A juvenile disease affects only children or teenagers - Children and teenagers are not adults - Juvenile arthritis is a kind of arthritis and a juvenile disease - Arthritis affects some adults | Intuitively… - Green color represents sets of objects - Blue color represents relationships between objects - Red color indicates whether a statement holds for “all” or for “some” objects. Limitations of Propositional Logic (4 of 4) Can’t be expressed in propositional logic
| We need a language that allows us to - Represent sets of objects - Represent relationships between objects - Write statements that are true for some or all objects satisfying certain conditions - Express everything we can express in propositional logic (and, or, implies, not, …) Need for a Richer Language
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| Examples of conditions we want to express: - For all objects c, • if c belongs to the set of juvenile diseases and it affects object d, • then d belongs to the set of children or to the set of teenagers . | There exist objects c, d, such that c belongs to the set of arthritis and d belongs to the set of adults and c affects d. Need for a Richer Language, cont’d
Introduction to First Order Logic (FOL)
| Express: “Every student is younger than some instructor” | We could identify the entire phrase with the propositional symbol p | However, the phrase has a finer logical structure. It is a statement about the following properties: - Being a student - Being an instructor - Being younger than somebody else Example
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| Individuals are expressed by object/function constants: andy, paul, father(andy) | Properties are expressed by predicates. S, I, Y are predicates. - S(andy) : Andy is a student. - I(paul) : Paul is an instructor. - Y(andy, paul) : Andy is younger than Paul. Predicates
| Variables are placeholders for concrete values. - S(x) : x is a student. - I(x) : x is an instructor. - Y(x, y) : x is younger than y. | Quantifiers make possible encoding the phrase: - “ Every student is younger than some instructor. ” - Encoding x (S x ∀ → ( y ∃ I y Y x, y ∧ ) Variables and Quantifiers Indicate the range of x, for all x, There exits
No books are gaseous. Dictionaries are books. Therefore, no dictionary is gaseous. | We denote: B(x) : x is a book G(x) : x is gaseous D(x) : x is a dictionary More Examples (1 of 3)
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No books are gaseous. Dictionaries are books. Therefore, no dictionary is gaseous. | We denote: B(x) : x is a book G(x) : x is gaseous D(x) : x is a dictionary ¬ ∃ ∃? ? ? ∧ ? ? , ∀( ∀?(? ? → ) ? ? ? ¬ ∃ ( ∃?(? ? ) ∧ ? ? More Examples (2 of 3)
More Examples (3 of 3)
| Whereas propositional logic assumes world contains facts, first-order logic (like natural language) assumes the world contains - Objects : people, houses, numbers, colors, cities, nba player, … - Functions : father of, best friend, successor, one more than, end of, . . . - Relations : red, round, bogus, prime, brother of, bigger than, inside, part of, has color, occurred after, owns, comes between, . . . First-Order Logic
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Wrap-Up

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