Logical Consequence

Introduction Whenever one asserts a claim of any kind, one engages in a commitment not just to that claim itself, but to a variety of other claims that follow in its wake, claims that, as we tend to say, follow logically from the original claim. To say that Smith and Jones are both great basketball players is to say something from which it follows that Smith is a great basketball player, that someone is a great basketball player, that there is something at which Smith is great, and so on. This general fact, that certain claims follow logically from others, is the central concern of a theory of logical consequence. Logical consequence is just the relation that connects a given claim or set of claims with those things that follow logically from it: to say that B is a logical consequence of A is simply to say that B follows logically from A. All of our ordinary reasoning turns on the recognition of this relation. When we notice, for example, that a certain prediction follows from a given theory, that a particular view is a consequence of some initial commitments, that a collection of premises entails a given conclusion, and so on, we are engaged in reasoning about logical consequence. The other logical properties and relations whose recognition is central to ordinary reasoning are closely related to, and can be defined in terms of, logical consequence. We say that an argument is valid iff its conclusion is a logical consequence of its premises; a set of claims G entails a claim a iff a is a logical consequence of G; a set of claims is consistent iff no contradiction is a logical consequence of it, and a claim a is independent of a set of claims G iff a is not a logical consequence of G. Finally, a claim is a logical truth iff it is a logical consequence of the empty set of claims. The investigation of logical consequence and related notions consists largely in the attempt (a) to give a systematic treatment of the extension of this relation, i.e. of the issue of which claims do in fact follow logically from which 2 others; and (b) to give an informative account of the nature of the relation. There is much room for debate about both of these issues. Though there is no doubt about the fact that some …