We define and examine a notion of logical friendliness, which is a broadening of the familiar notion of classical consequence. The concept is studied first in its simplest form, and then in a syntax-independent version, which we call sympathy. We also draw attention to the surprising number of familiar notions and operations with which it makes contact, providing a new light in which they may be seen. Friendliness Rationale, Definition, Notation Recall the definition of classical consequence in

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... propositional logic. Let A be any set of formulae, and x any individual formula. Then x is said to be a classical consequence of A, written A x, iff for every valuation v on all letters of the language, if v Trivially, the only letters that count here are those occurring in A or in x. So the definition may be rephrased as: A x iff for every partial valuation v on E(A,x), if v(A) = 1 then v(x) = 1. Equivalently again, A x iff for every partial valuation v on E(A), if v(A) = 1 then v + (x) = 1 for every extension v + to E(A, x). Expressed in this last way, classical consequence is a ∀∀ concept. It is natural to ask: what does the corresponding ∀∃ concept look like, and how does it behave? This simple question, born of no more than curiosity, is the starting point of our investigation. The definition is straightforward: † This paper revises and extends the version that appeared in the first edition of Logica Universalis. Specifically, it adds several new sections (1.8-1.10, 3.5-3.6, and all of part 2) as well as additional material in other sections (notably the axiomatization of friendliness in 1.5, a much stronger version of compactness in 1.6, more information about interpolant formulae in 1.7 and 3.5, and counterexamples to proof by exhaustion and to compactness for sympathy in 3.2). The present version also appeared as part of a festschrift for Dov Gabbay, see [Makinson 2005a ].