Elkan's theoretical argument, reconsidered

  title={Elkan's theoretical argument, reconsidered},
  author={Enric Trillas and Claudi Alsina},
  journal={Int. J. Approx. Reason.},
On Elkan's theorems: Clarifying their meaning via simple proofs
This article deals with the claims that “a standard version of fuzzy logic collapses mathematically to two‐valued logic” made by Charles Elkan in two papers and presents alternative, considerably simpler proofs of Elkan's theorems and uses these proofs to argue thatElkan's claims are unwarranted.
Twenty years later: Remarks on a polemic
  • E. Trillas
  • Mathematics
    2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS)
  • 2013
This paper just deals with the starting point in the twenty years old paper [1] by Charles Elkan, with which he tried to puzzle fuzzy logic: That the formula 1-min (a,1-b) = max (b, min (1-a, 1-a))
Searching for the roots of non-contradiction and excluded-middle
This paper tries to obtain frameworks in which one can prove as theorems, and with few assumptions, the laws of non-contradiction (NC) and excluded-middle (EM), for a large class of very general
Comments to "Paradoxes of fuzzy logic, revisited"
On contra-symmetry and MPT conditionality in fuzzy logic
The contra-positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor is studied.
Clarifying Elkan's theoretical result
This paper is devoted to clarify the only theoretical result included in the controversial work of C. Elkan “The paradoxical success of fuzzy logic” (1994), by offering both a short proof of it and a
On contra‐symmetry and MPT conditionality in fuzzy logic
The contra‐positive of S implications, R implications, Q implications, and Mamdani–Larsen operators, verifying either Modus Ponens or Modus Tollens inequalities or both, the conditionality's aspect on which lies the complementarity with Fodor is studied.
Some Comments on Ordinary Reasoning with Fuzzy Sets
This chapter proposes a general framework to model a part of commonsense reasoning based on its ability to adequately represent noncontradiction, the minimum condition for considering a reasoning as valid.
Fuzzy logic and self-referential reasoning: a comparative study with some new concepts
It is shown that the reasoning based on the truth qualification principle requires some modifications so that it naturally gives rise to type-2 fuzzy sets, in order to capture the uncertainty about selection of a solution for a set of self-referential statements.
How I would like to foresee the future of theoretic fuzzy logic
By arguing on some aspects fuzzy sets and fuzzy logic research could follow to not only being a theoretic ground for Zadeh’s Computing with Words, but also for going towards a new experimental science dealing with the imprecision and non-random uncertainty permeating language and reasoning.


On the power of fuzzy logic
An investigation of Elkan's theorem revealed that its conclusion is not correct, and it is shown that one does not need to change the definition of logical equivalence in Theorem 1 in order to prove that Fuzzy Logic does not collapse to a two‐valued logic.
Elkan's Reply: The Paradoxical Controversy over Fuzzy Logic
The responses to my article provide an that I have is whether the distinction is re-knowledge becomes implicit background exceptionally wide range of perspectives ally well defined and vagueness is the domain-independent im-items of explicit shallow knowledge.
On non-contradictory input/output couples in Zadeh's CRI
  • E. Trillas, S. Cubillo
  • Computer Science
    18th International Conference of the North American Fuzzy Information Processing Society - NAFIPS (Cat. No.99TH8397)
  • 1999
It is shown that, under some conditions on the values of the fuzzy conditional relation, the output given by the CRI is not contradictory with the input, provided that this is a normal fuzzy set.
The paradoxical success of fuzzy logic
This work hopes to resolve paradoxes in fuzzy logic by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential.
On Some Logical Connectives for Fuzzy Sets Theory
When QM‐operators are implication functions and conditional fuzzy relations
This paper focuses on the QM‐implication operator both as an implication function and also as a T‐conditional function, giving useful tools to characterize them.
A first course in fuzzy logic
Fuzzy rule bases Design methodologies Some mathematical background Approximation capability Exercises POSSIBILITY THEORY Probability and uncertainty Random sets Possibility measures Exercised.
On (S, N)-implications in fuzzy logic consistent with T-conjunctions
An outline of a Naõve loose-set theory
  • in: Proceedings of the Eight IPMU
  • 2000