We define a logic capable of expressing dependence of a variable on designated variables only. Thus has similar goals to the Henkin quantifiers of [4] and the independence friendly logic of [6] that it much resembles. The logic achieves these goals by realizing the desired dependence declarations of variables on the level of atomic formulas. By [3] and [17], ability to limit dependence relations between variables leads to existential second order expressive power. Our avoids some difficulties arising in the original (...) independence friendly logic from coupling the dependence declarations with existential quantifiers. As is the case with independence friendly logic, truth of is definable inside . We give such a definition for in the spirit of [11] and [2] and [1]. (shrink)

Several writers have assumed that when in “Outline of a Theory of Truth” I wrote that “the orthodox approach” – that is, Tarski’s account of the truth definition – admits descending chains, I was relying on a simple compactness theorem argument, and that non-standard models must result. However, I was actually relying on a paper on ‘pseudo-well-orderings’ by Harrison. The descending hierarchy of languages I define is a standard model. Yablo’s Paradox later emerged as a key to interpreting the result.

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)