Combinatory interpretation of lambda calculus

cs.stackexchange https://cs.stackexchange.com/questions/28648

Question

According to Peter Selinger, The Lambda Calculus is Algebraic (PDF). Early in this article he says:

The combinatory interpretation of the lambda calculus is known to be imperfect, because it does not satisfy the $ξ$-rule: under the interpretation, $M = N$ does not imply $\lambda x.M = \lambda x.N$ (Barendregt, 1984).

Questions:

  • What kind of equivalence is meant here?
  • Given this definition of equivalence, what is a counter-example of the implication?

No correct solution

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