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Wednesday, 25 July 2018

First-order logic

While propositional logic assumes the world contains facts,

first-order logic (like natural language) assumes the world contains
 - Objects: people, house, numbers, colors, baseball games, wars,...
 - Relations: red, round, prime, brother of, bigger than, part of, comes between, ...
 - Functions: father of, best friend, one more than, plus, ...

"Evil King John England in 1200."
  - Object: John, England, 1200; Relation; ruled; Properties: evil, king

Logic in General

Ontological Commitment:
 - What exists in the world - TRUTH
 - PL : facts hold or do not hold.
 - FOL : objects with relations between them that hold or do not hold

Epistemological Commitment:
 - What an agent believes about facts - BELIEEF


Syntax of First Order Logic: Basic elements

Constant Symbols:
 - Stand for objects
 - e.g., KingJohn, 2, UCI, ...

Predicate Symbols
 - Stand for relations
 - E.g., Brother (Richard, John), greater_than(3,2)....

Function Symbols
 - Stand for functions
 - E.g., Sqrt(3), LeftLegOf(John),....

Constant       KingJohn,2, UCI,...
Predicates     Brother,>,....
Function       Sqrt, LeftLegOf,....
Variables       x,y,a,b,....
Connectives  ㄱ,⇒,៱,v,⇔ 
Equality          =
Quantifiers     ∀,∃

Relations

Some relations are properties: they state some fact about a single object: Round (ball), Prime(7).

n-ary relations state facts about two or more objects: Married(John,Mary), LargeThan(3,2).

Some relations are functions: their value is another object: Puls(2,3), Father(Dan).


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