(数)公理.ppt

Chapter 2The Representation of Knowledge Propositional Logic Formal logic: the logic is concerned with the form of logical statements rather than their meaning. Example: Algebra, a formal logic of numbers. Algebra A school has 25 computers with a total of 60 memory boards. Some of the computers have two memory boards, and some have four. How many computers of each type are there? Solution: 25 = X + Y 60 = 2X + 4Y Algebra There are 25 animals with a total of 60 legs in a barnyard. Some of the animals have two legs, others have four. How many animals of each type are there? The same algebraic equations apply whether we are talking about computers, animals, or anything else. Algebra Algebraic equations let us concentrate on the mathematical manipulation of symbols without regard to what they represent. Formal logic lets us concentrate on the reasoning without becoming confused by the objects we are reasoning about. Example of formal logic Syllogism Premise: All squeegs are moofs Premise: John is a squeeg Conclusion: John is a moof Although the words squeeg and moof are nonsense and have no meaning, the form of this argument is still correct. The argument is valid no matter which words are used because the syllogism has a valid form. Form In fact any syllogism of the form Premise: All X are Y Premise: Z is a X Conclusion: Z is a Yis valid no matter what is substituted for X, Y, and Z. Meaning do not matter in formal logic. Only the form or appearance is important. Form The concept of separating the form from the meaning or semantics is what makes logic such a powerful tool. This is like algebra, in which the correctness of expressions such as X + X = 2Xholds whether X is an integer, apples, or anything. Axioms A set of Axioms consisting of symbols representing both objects and classes and algebraic operations to manipulate the symbols. Axioms are the fundamental definitions from which logical systems such as mathematics and logic itself are