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Classical Logic & Fuzzy Logic

Classical Logic & Fuzzy Logic. Classical predicate logic T: u U  [0,1] U: universe of all propositions. All elements u  U are true for proposition P are called the truth set of P: T(P). Those elements u  U are false for P are called falsity set of P: F(P). T(Y) = 1 T(Ø) = 0.

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Classical Logic & Fuzzy Logic

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  1. Classical Logic & Fuzzy Logic Classical predicate logic T: uU  [0,1] U: universe of all propositions. All elements u  U are true for proposition P are called the truth set of P: T(P). Those elements u  U are false for P are called falsity set of P: F(P). T(Y) = 1 T(Ø) = 0

  2. Classical Logic &Fuzzy Logic Logic connectives Disjunction  Conjunction  Negation – Implication  Equivalence If xA, T(P) =1 otherwise T(P) = 0 Or xA(x)={ 1 if x A, otherwise it is 0 } If T(p)T()=0 implies P true,  false, or  true P false. P and  are mutually exclusive propositions.

  3. Classical Logic &Fuzzy Logic Given a proposition P: xA, P: xA, we have the following logical connectives: Disjunction PQ: x A or x B hence, T(PQ) = max(T(P),T(Q)) Conjunction PQ: xA and xB hence T(P Q)= min(T(P),T(Q)) Negation If T(P) =1, then T(P) = 0 then T(P) =1 Implication (P  Q): xA or xB Hence , T(P  Q)= T(P Q)

  4. Classical Logic &Fuzzy Logic Equivalence 1, for T(P) = T(Q) (P Q): T(PQ)= 0, for T(P)  T(Q) The logical connective implication, i.e.,P  Q (P implies Q) presented here is also known as the classical implication. P is referred to as hypothesis or antecedent Q is referred to as conclusion or consequent.

  5. Classical Logic &Fuzzy Logic T(PQ)=(T(P)T(Q)) Or PQ= (AB is true) T(PQ) = T(PQ is true) = max (T(P),T(Q)) (AB)= (AB)= AB So (AB)= AB Or AB false  AB Truth table for various compound propositions

  6. Classical Logic &Fuzzy Logic PQ: If x A, Then y B, or PQ  AB The shaded regions of the compound Venn diagram in the following figure represent the truth domain of the implication, If A, then B(PQ). X A B Y

  7. Classical Logic &Fuzzy Logic IF A, THEN B, or IF A , THEN C PREDICATE LOGIC (PQ)(PS) Where P: xA, AX Q: yB, BY S: yC, CY SET THEORETIC EQUIVALENT (A X B)(A X C) = R = relation ON X Y Truth domain for the above compound proposition.

  8. Classical Logic &Fuzzy Logic Some common tautologies follow: BB  X AX; A X  X AB (A(AB))B (modeus ponens) (B(AB))A (modus tollens) Proof: (A(AB))  B (A(AB))  B Implication ((AA) (AB))B Distributivity ((AB))B Excluded middle laws (AB)B Identity (AB)B Implication (AB)B Demorgans law A(BB) Associativity AX Excluded middle laws X T(X) =1 Identity; QED

  9. Classical Logic & Fuzzy Logic Proof (B(AB))A (B(AB))A ((BA)(BB)) A ((BA))A (BA)A (BA)A (BA)A B(AA) BX = X T(X) =1 Truth table (modeus ponens)

  10. Classical Logic &Fuzzy Logic Contradictions BB A; A Equivalence PQ is true only when both P and Q are true or when both P and q are false. Example Suppose we consider the universe positive integers X={1 n8}. Let P = “n is an even number “ and let Q =“(3n7)(n6).” then T(P)={2,4,6,8} and T(Q) ={3,4,5,7}. The equivalence PQ has the truth set T(P  Q)=(T(P)T(Q)) (T(P) (T(Q)) ={4} {1} ={1,4} T(A) Venn diagram for equivalence T(B)

  11. Classical Logic &Fuzzy Logic Exclusive or Exclusive Nor Exclusive or P “” Q (AB)  (AB) Exclusive Nor (P “” Q)(PQ) Logical proofs Logic involves the use of inference in everyday life. In natural language if we are given some hypothesis it is often useful to make certain conclusions from them the so called process of inference (P1P2….Pn) Q is true.

  12. Classical Logic &Fuzzy Logic Hypothesis : Engineers are mathematicians. Logical thinkers do not believe in magic. Mathematicians are logical thinkers. Conclusion : Engineers do not believe in magic. Let us decompose this information into individual propositions P: a person is an engineer Q: a person is a mathematician R: a person is a logical thinker S: a person believes in magic The statements can now be expressed as algebraic propositions as ((PQ)(RS)(QR))(PS) It can be shown that the proposition is a tautology. ALTERNATIVE: proof by contradiction.

  13. Classical Logic &Fuzzy Logic Deductive inferences The modus ponens deduction is used as a tool for making inferences in rule based systems. This rule can be translated into a relation between sets A and B. R = (AB)(AY) Now suppose a new antecedent say A’ is known, since A implies B is defined on the cartesian space X  Y, B can be found through the following set theoretic formulation B= AR= A((AB)(AY))  Denotes the composition operation. Modus ponens deduction can also be used for compound rule.

  14. Classical Logic &Fuzzy Logic Whether A is contained only in the complement of A or whether A’ and A overlap to some extent as described next: IF AA, THEN y=B IF AA THEN y =C IF AA , AA, THEN y= BC

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