Undecidability in Fuzzy Logic

At an elementary level accessible to a general audience of collegestudents, three-valuedtruth tablesprovidean introduction to the history of logic, and a pointer from the mathematical logic to engineering applications. At an intermediate level accessible to students interested in symbolic calculations, three-valued truth tables provide elementary proofs of the non-existence of proofs of selected formulae within specific logical systems. Thus multi-valued logic demonstrates the concept of mathematical impossibility, without the machinery of models needed against a proof of Euclid's fifth postulate, Galois theory against angle trisection, or general recursion against G®odel's incompleteness theorems.

Table of Contents

1. INTRODUCTION

2. TWO-VALUED LOGIC
2.1 Choices of Values for Truth Tables
2.2 Logical Implication
2.3 History
2.4 Tautologies
2.5 Systematization and Mechanization

3. A THREE-VALUED SYSTEM
3.1 Three-Valued Truth Tables
3.2 Tautologies in Three-Valued Logic

4. AXIOMATIC SYSTEMS
4.1 Frege's System
4.2 The Positive Implicational Propositional Calculus
4.3 The Axiom Frege Thought That He Needed-But He Didn't
4.4 Preservation of Tautologies
4.5 A Tautology That Is Not a Tautology
4.6 Independence, Incompleteness, and Impossibility

5. OTHER THREE-VALUED SYSTEMS
5.1 Church's {0, 1, 2} System
5.2 Lukasiewicz's {1, 2, 0} System
5.3 Comparisons
5.4 The Usefulness of Multiple Systems of Multi-Valued Logic

6. FUZZY LOGIC

7. APPLICATIONS
7.1 Finding Oil Fields
7.2 Deciding Where to Drill ProductionWells
7.3 Alleviating the Loss of Drilling Fluids
7.4 Categorization of the Examples

8. CONCLUSIONS

9. ACKNOWLEDGMENTS (Y.N.)

10.COMPUTER SYSTEMS FOR LOGIC
10.1 Computer Systems for Boolean Logic
10.2 Computer Systems for Multiple-Valued Logic

11. SOLUTIONS TO THE EXERCISES

REFERENCES

ABOUT THE AUTHORS