logic2008[1]

logic2008[1] - Logic The discipline that deals with the...

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Logic The discipline that deals with the methods of reasoning 04/01/10 1

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The example of reason If I am your teacher then I should give you lessons I am your teacher So I should give you lessons Is it right? Maybe just now! 04/01/10 2
Another example of reason If you invest in the stock market then you will get rich If you get rich then you will be happy so, if you invest in the stock market then you will be happy Is it right? Maybe just now! 04/01/10 3

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Another example of reason if taxes are lowered, then income rises income rises so, taxes are lowered Is it right? Maybe just now! 04/01/10 4
Proposition Proposition (statement) A declarative sentence that is either true or false, but not both. Examples 1+1=2 John is a student. Today is Tuesday. 04/01/10 5

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More Examples of Proposition Do you speak English? This is a question, not a statement. Let’s go! It is not a statement, but a command. 3-x=5 x is a variable, so the truth value of this sentence is open. The 4-color guess is true. We believe that it is true or false, but not both even before it was solved. The sun will come out tomorrow It is a statement What a good man he is. It is not a statement I am lying. It is a paradox. 04/01/10 6
Proposition Truth value: The value (result) of the proposition One of the element from set {True, False} 2 is an odd number I am a teacher. 04/01/10 7

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Propositional Variable A proposition can be represented by a proposition variable A propositional variable is often denoted as p , q , r , etc. e.g. p : Today is Tuesday. q : 2+2=4 r: 2+3=6 04/01/10 8
Logical Connectives A simple statement can be represented by an atom proposition. More than one atom propositions can be combined into a compound statement. The combination is achieved using “connectives”. Usually, the connective roughly corresponds some conjunctive in the natural language. 04/01/10 9

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Logical Connectives What is the truth value of a compound statement? Depends on the truth value of each atom proposition Depends on the connectives The connective is defined exactly using truth table ”. 04/01/10 10
04/01/10 11 Negation ~ p : it is not the case that p p ~ p T F F T Truth table for ~ All possible value of p

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04/01/10 12 Conjunction p and q ” is denoted as p q p q T T T F F T F F p q T F F F All possible value of < p , q > p q=true iff both p and q
04/01/10 13 Disjunction p or q ” is denoted as p q, but not exactly p q T T T F F T F F p q T T T F All possible value of < p , q > p q= false iff both ~ p and ~ q

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How to make a truth table? 1,Make the heads of truth table 2,list the 2 n n-tuples (when n variables) 3,compute the remain values For examples: (p q) (~p) 04/01/10 14
Exclusive Disjunction Sentence: I will leave for beijing tomorrow or I will leave for shanghai tomorrow Proposition: p: I will leave for beijing tomorrow q: I will leave for shanghai tomorrow p q?

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This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

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logic2008[1] - Logic The discipline that deals with the...

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