Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

Atomists explain properties of wholes as compositions of properties of their parts; in particular properties of complex expressions as composed of properties of their parts. Especially, semantic atomists explain meanings of complex expressions as composed of meanings of their parts. Holists deny themselves this way: they insist that at least in some cases properties of wholes are more basic than, or not reducible to, properties of their parts; in particular, semantic holists claim that meanings of (at least some) wholes are (...) more basic than meanings of parts. Now as atomists have composition as the way of getting themselves from the meanings of parts to the meanings of wholes, holists need something to get them from the meanings of wholes to those of their parts. What is usually invoked in this context is the concept of intersubstitutivity, which is, however, not always wholly clear. (shrink)

Accounts of logical independence which coincide when applied in the case of classical logic diverge elsewhere, raising the question of what a satisfactory all-purpose account of logical independence might look like. ‘All-purpose’ here means: working satisfactorily as applied across different logics, taken as consequence relations. Principal candidate characterizations of independence relative to a consequence relation are that there the consequence relation concerned is determined by only by classes of valuations providing for all possible truth-value combinations for the formulas whose independence (...) is at issue, and that the consequence relation ‘says’ nothing special about how those formulas are related that it does not say about arbitrary formulas. Each of these proposals returns counterintuitive verdicts in certain cases—the truth-value inspired approach classifying certain cases one would like to describe as involving failures of independence as being cases of independence, and the de Jongh approach counting some intuitively independent pairs of formulas as not being independent after all. In final section, a modification of the latter approach is tentatively sketched to correct for these misclassifications. The attention is on conceptual clarification throughout, rather than the provision of technical results. Proofs, as well as further elaborations, are lodged in the ‘longer notes’ in a final Appendix. (shrink)

The purpose of this brief note is to prove a limitative theorem for a generalization of the deduction theorem. I discuss the relationship between the deduction theorem and rules of inference. Often when the deduction theorem is claimed to fail, particularly in the case of normal modal logics, it is the result of a confusion over what the deduction theorem is trying to show. The classic deduction theorem is trying to show that all so-called ‘derivable rules’ can be encoded into (...) the object language using the material conditional. The deduction theorem can be generalized in the sense that one can attempt to encode all types of rules into the object language. When a rule is encoded in this way I say that it is reflected in the object language. What I show, however, is that certain logics which reflect a certain kind of rule must be trivial. Therefore, my generalization of the deduction theorem does fail where the classic deduction theorem didn’t. (shrink)

The main result of this paper is the following theorem: each modal logic extendingK4 having the branching property belowm and the effective m-drop point property is decidable with respect to admissibility. A similar result is obtained for intermediate intuitionistic logics with the branching property belowm and the strong effective m-drop point property. Thus, general algorithmic criteria which allow to recognize the admissibility of inference rules for modal and intermediate logics of the above kind are found. These criteria are applicable to (...) most modal logics for which decidability with respect to admissibility is known and to many others, for instance, to the modal logicsK4,K4.1,K4.2,K4.3,S4.1,S4.2,GL.2; to all smallest and greatest counterparts of intermediate Gabbay-De-Jong logicsD n; to all intermediate Gabbay-De-Jong logicsD n; to all finitely axiomatizable modal and intermediate logics of finite depth etc. Semantic criteria for recognizing admissibility for these logics are offered as well. (shrink)

It is well known that no consistent normal modal logic contains (as theorems) both ♦A and ♦¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ♦A, or else no consistent normal modal logic contains ♦¬A.

We explore a relation we call 'anticipation' between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A → B. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as "a", governed by rules which guarantee that for (...) any formula B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context. (shrink)