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Non-classic logics / Bergmann. Introduction to Many-Valued and Fuzzy Logic CUP, 2007.pdf
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298

Derivation Systems for Fuzzy First-Order Logic

Note that if a formula P has any value less than 1, P will have the value 0. But we don’t bother to include this as part of the rule EA because we can derive any formula with graded value 0.

We’ll show that if Ta has at least the value .5, in FuzzyL augmented with the external assertion operator then ( x)¬Tx is clearly not clearly true (recall the reading of the external assertion operator as clearly):

1

[Ta, .5]

 

Assumption

2

[1/2 Ta, 1]

1, TCI

3

[¬Ta → ¬1/2, 1]

2, GCON

4

[¬1/2 1/2, 1]

FL P6.1

5

[¬Ta 1/2, 1]

3,4 HS

6

[( x)¬Tx → ¬Ta, 1]

FL P9, with ( x)¬Tx / ( x)Px, a / a

7

[( x)¬Tx 1/2, 1]

5,6 HS

8

[ ( x)¬Tx ( x)¬Tx, 1]

3, with ( x)¬Tx / P

9

[ ( x)¬Tx 1/2, 1]

7,8 HS

10

[¬1/2 → ¬ ( x)¬Tx, 1]

9, GCON

11

[1/2 → ¬1/2, 1]

FL P6.2

12

[1/2 → ¬ ( x)¬Tx, 1]

10,11 HS

13

[ ( x)¬Tx → ¬ ( x)¬Tx, 1]

9,12 HS

14

[( ( x)¬Tx → ¬ ( x)¬Tx) → ¬ ( x)¬Tx, 1]

1, with ( x)¬Tx / P

15

[¬ ( x)¬Tx, 1]

13,14 MP

16

[ ¬ ( x)¬Tx, 1]

15, EA

15.5 Exercises

SECTION 15.2

1 Construct a derivation that justifies the derived axiom schema

FL PD21. [P(a/x) ( x)P, 1] in FL PA.

2Present a derivation in FL PA based on the method of reasoning in the corresponding “chain” derivation in Section 13.3 of Chapter 13, that shows that if the Principle of Charity premise in the weak conjunction version of the Sorites argument has at most the value 191/192 and the other premises have at most the value 1, then the conclusion has at most the value 0.

3Present a derivation in FL PA that shows that if the Principle of Charity premise in the bold conjunction version of the Sorites argument has at most the value

191/192 and the other premises have at most the value 1, then the conclusion has at most the value 0.

4Present a derivation that shows that if the premises of the Kleenean version of the Sorites argument all have at most the value .6, then so does the conclusion.

15.5 Exercises

299

5Consider the non-Pavelka axiomatic system FL A that consists of axiom schemata FL P1–FL P4, FL P8 and FL P9, and the rules MP and UG, all with grades removed from the formulas. We know that this system must be incomplete, as explained in Section 15.1. Nevertheless, the Lukasiewicz weak conjunction Sorites argument, the Lukasiewicz strong conjunction Sorites argument, and the Kleene conditional Sorites arguments are all valid in this system. Show this.

SECTION 15.3

6Explain why we cannot define the existential quantifier in Godel¨ fuzzy firstorder logic as we did in classical logic; that is, explain why ( x)P and ¬ ( x)¬GP are not in general equivalent in FuzzyG . You may do this by giving an instance of these formulas and an interpretation on which these instances have different truth-values.

7Construct a derivation of the modified conclusion of the Godel¨ conjunctive version of the Sorites paradox from its premises in BLG A (remember that ¬P is defined as (P 0) 0):

Ts1

Es2s1

Es3s2

. . .

Es193s192

( x)( y)¬G((Tx &G Eyx) &G ¬GTy)

¬G¬GTs193

SECTION 15.4

8Construct a derivation of the graded formula [ ( x)Tx ( x)Tx, 1] in FL PA.

9Construct a derivation of the graded formula [ ( x)Tx ( x) Tx, 1] in FL PA.

10Construct a derivation that shows that [¬ ( x)Tx, 1] follows from [ ¬ ( x)Tx, 1] in FL PA.

11Construct a derivation that shows that [ Ra, 1] follows from [¬Pa, .5] and [( x) (Px Rx), .5] in FL PA.