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Implications Adriano Joaquim de O Cruz ©2002 NCE/UFRJ

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 2 Implication Logic: –Logical Implication as regarded in mathematical logic. –Material conditional as regarded in philosophical logic. –Semantic entailment between two sets of statements.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 3 Logic Logical implication is a logical relation that holds between a set T of formulas and a formula B when every model (or interpretation or valuation) of T is also a model of B. In symbols, Without using the language of models, the material conditional formed from the conjunction of all the elements of T and B is valid. That is, it is valid that

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 4 Definitions A set of sentences logically implies a sentence B if and only if it is impossible that all the members of the set be true while B false. A peculiar feature of logical implication is that a contradiction implies anything and that anything implies a validity.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 5 Implication If x A then y B. P is a proposition described by the set A Q is a proposition described by the set B P Q: If x A then y B P implies Q

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 6 Implication An implication is a logical construction that tell us if one condition is true, then another condition must also be true. Implication is not if and only if. Implication P -> Q is true even if only Q is true. Elephants can fly, therefore it is hot today. This statement is true if it is hot today.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 7 Implication - uses Kid, if you mown the grass then I will pay you $ Only one consequence is certain. It is not well defined what will happen if the grass is not mowed. If you do not eat all your broccolis then you will not have dessert. Two consequences are understood. The other one is dessert as a consequence of all broccolis eaten.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 8 Implication - uses Kid, if you mown the grass then I will pay you $ Only one consequence is certain. It is not well defined what will happen if the grass is not mowed. This is the mathematical sense of the implication If the proposition A is true then B is too. Nothing can be said about B when A is false.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 9 Implication If I am elected I will decrease taxes I was not elected, I did not decrease taxes I was not elected, I did decrease taxes I was elected, I did not decrease taxes I was elected, I did decrease taxes

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 10 Implication – Truth table TTT FFT TTF TFF A -> BBA A -> B =(not A) or B A->B=(not A) or (A and B)

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 11 Families of Implication Generalization of Material Implication

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 12 Families of Implication Propositional Calculus There are over 40 implication relations reported in the literature

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 13 Interpretations of Implication A -> B as A entails B Entailment is a relation between sets of formulae such that, if A and B are sets of formulae of a formal language, then A entails B if and only if every model (or interpretation) that makes all the members of A true, makes at least one of the members of B true. Entailment differs from implication, where the truth of one (A) suggests the truth of the other (B), but does not require it.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 14 Interpretations of Implication There are fuzzy ways to interpret the fuzzy rule if … then … else First: A -> B as A entails (coupled) with B Where T is a T-norm operator

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 15 A coupled with B Commonly used T-norms are: Mamdani: Larson Bounded Product

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 16 Implication Relation Implication can be described as a relation The relation is defined by the T-norm

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Inference

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 18 Inference Fuzzy inference refers to computational procedures used for evaluating fuzzy rules of the form if x is A then y is B There are two important inferencing procedures –Generalized modus ponens (GMP) - mode that affirms –Generalized modus tollens (GMT) – mode that denies

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 19 Modus Ponens If x is A then y is B We know that x is A then we can infer that y is B All men are mortal All men are mortal (rule) Socrates is a man (this is true) So Socrates is mortal (as a consequence) (A and (A -> B)) -> B

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 20 Fuzzy Modus Ponens If x is A then y is B We know that x is A then we can infer that y is B Tall men are heavy (rule) John is tall (this is true) So John is heavy (as a consequence) (A and (A -> B)) -> B

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 21 Fuzzy Modus Ponens proof - I

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 22 Fuzzy Modus Ponens proof - II

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 23 Modus Tollens If x is A then y is B We know that y is not B then we can infer that x is not A All murderers owns axes (rule) John does not own an axe (this is true) So John is not a murderer (as a consequence) (not B and (A -> B)) -> not A

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 24 Fuzzy Modus Tollens If x is A then y is B We know that y is not B then we can infer that x is not A All rainy days are cloudy (rule) Today is not cloudy (this is true) So Today is not raining (as a consequence) (not B and (A -> B)) -> not A

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 25 Fuzzy Modus Tollens proof ?

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 26 Reasoning Methods Backward Chaining: the reasoning engine is presented with a goal and asked to find all the relevant, supporting processes that lead to this goal. Forward Chaining: data is collected and and a sustainable problem state and, eventually a solution state is built. Fuzzy Reasoning: rules are run in parallel. Every rule contributes to the final shape of the consequent solution. When all rules are evaluated the resulting fuzzy sets are defuzzified.

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 27 How to find the consequent If x is A then y is B This rule is a relation R(x,y) If x is A, we want to know whether y is B B= A R(x,y) B (y)= x [ A (x) R (x,y)]

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 28 Example Consider the fuzzy set A and the fuzzy set B u A (3) uB(y)uB(y)

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 29 Example 1

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 30 Example 2 We will use the Mamdani implication function

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 31 Example 3

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 32 Example 4 Consider the rule if x is A then y is B Consider the statement x is A, what is the conclusion? u A (4)=

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 33 Example u A (3) uB(y)uB(y) u A (4)=

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 34 Example 6

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 35 Example 7 B(y i )=A(x i )oR(x i,y i )

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@2002 Adriano Cruz NCE e IM - UFRJImplicações 36 Example uB(y)uB(y) uB(y)uB(y)

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