 For example, there is the issue raised by the use of the notation above for the operations of L. Such a choice suggests that ∧, →, ¬ will represent conjunction, implication (or the conditional) and negation—that is, be intended to capture some aspect of the behaviour of and, if–then, and not, respectively—and that ⊥ will be some kind of falsity constant. (Similarly, while we are fixing notation, if we had written “∨”, “↔”, or “⊤”, we would be arousing expectations of disjunction, equivalence—or biconditionality—and a truth constant, respectively.) We can think of this as a matter of the inference patterns to be sanctioned by some proof system offered for L, or as a matter of how the formulas of this language are to be interpreted by assigning suitable semantic values to them (truth-values being the simplest candidates here). We refer to the former as a proof-theoretic or syntactical characterization of the logic and the latter as a semantical characterization. The terminology is somewhat unfortunate in that a perfectly respectable view in the philosophy of logic (going back at least to Gentzen) has it that the meanings of items of logical vocabulary, such as the connectives of current concern, for a particular reasoner, are given by a specification of the set of primitive rules, or a favoured subset thereof, governing those items in a proof system representing that reasoner’s (perhaps idealized) inferential  practices.[2]  We take up some of those issues later, beginning with a reprise of the semantical side of the story at the start of the following section.