The article begins and ends with reminiscences and reflections about conversations with Myhill . The topic in the title was never far from these conversations, but remained off stage: questions about the suitability of the focus on logical languages and logical consequence, here meant in contrast to incomparable categories of propositions and proofs. The body of the article goes into cases where this focus has contributed to—and where it has distracted from more rewarding categories for—effective knowledge in mathematics. There are (...) some rules of thumb for recognizing distractions, and for doing better. (shrink)

Intuitionistically. a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights?in particular, the so-called continuity principles?which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer?s reconstruction of mathematics. The predicate ?α (...) is lawless? is not acceptable, the lawless sequences do not form a well-circumscribed intuitionistic set, and quantification over lawless sequences does not make sense. (shrink)