Blackwell Reference Online

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The goal of a logic is to define a consequence relation between a set of formulas Γ and, in most cases, an individual formula A. This definition generally takes one of two forms. From a proof theoretic standpoint, A is said to be a consequence of Γ whenever there is a deduction of A from the set Γ, viewed as a set of premises; from a model theoretic standpoint, A is said to be a consequence of Γ whenever A holds in every model that satisfies each formula in Γ. Although the detailed inferences sanctioned by particular logics vary widely depending on the connectives present and the properties attributed to them, certain abstract features of the consequence relation are remarkably stable across logics. Among these is the property of monotonicity: if A is a consequence of Γ, then A is a consequence of Γ ∪ { B }. What this means is that any conclusion drawn from a set of premises will be preserved as a conclusion even if the premise set is supplemented with additional information-that the set of conclusions grows monotonically as the premise set grows. The monotonicity property flows from assumptions that are deeply rooted in both the proof theory and the semantics, not only of classical logic, but of most philosophical logics as well. From the proof theoretic standpoint, monotonicity follows from the fact that any derivation of the formula A from the premise set Γ ... log in or subscribe to read full text