Classical Logic

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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
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.
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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)
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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.
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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
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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
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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.
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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
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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)
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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 X1 ?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)
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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.
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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.
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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.
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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 yB IF A??A THEN y
C IF A??A ??, A??A??, THEN y B?C