A Dynamic Characterization of the Pure Logic of Relevant Implication

This paper spells out a dynamic proof format for the pure logic of relevant implication. (A proof is dynamic if a formula derived at some stage need not be derived at a later stage.) The paper illustrates three interesting points. (i) A set of properties that characterizes an inference relation on the (very natural) dynamic proof interpretation, need not characterize the same inference relation (or even any inference relation) on the usual set-theoretical interpretation. (ii) A proof format may display an internal dynamics (defeasible conclusions) in the absence of an external dynamics (non-monotonicity). (iii) A monotonic logic may have a non-monotonic characterization.     

Relevant implication and the weak deduction theorem

It is shown that the implicational fragment of Anderson and Belnap's R, i.e. Church's weak implicational calculus, is not uniquely characterized by MP (modus ponens), US (uniform substitution), and WDT (Church's weak deduction theorem). It is also shown that no unique logic is characterized by these, but that the addition of further rules results in the implicational fragment of R. A similar result for E is mentioned.I am grateful to the referees; their comments enabled me to correct a mistake and to clarify several passages.     

The Adaptive Logic of Compatibility

This paper describes the adaptive logic of compatibility and its dynamic proof theory. The results derive from insights in inconsistency-adaptive logic, but are themselves very simple and philosophically unobjectionable. In the absence of a positive test, dynamic proof theories lead, in the long run, to correct results and, in the short run, sometimes to final decisions but always to sensible estimates. The paper contains a new and natural kind of semantics for S5from which it follows that a specific subset of the standard worlds-models is characteristic for S5.     

Logics for Qualitative Inductive Generalization

The paper contains a survey of (mainly unpublished) adaptive logics of inductive generalization. These defeasible logics are precise formulations of certain methods. Some attention is also paid to ways of handling background knowledge, introducing mere conjectures, and the research guiding capabilities of the logics.     

THE THEORY OF THE PROCESS OF EXPLANATION GENERALIZED TO INCLUDE THE INCONSISTENT CASE

This paper proposes a generalization of the theory of the process of explanation to include consistent as well as inconsistent situations. The generalization is strong, for example in the sense that, if the background theory and the initial conditions are consistent, it leads to precisely the same results as the theory from the lead paper (Halonen and Hintikka 2004). The paper presupposes (and refers to arguments for the view that) inconsistencies constitute problems and that scientists try to resolve them.     

A procedural criterion for final derivability in inconsistency-adaptive logics

This paper concerns a (prospective) goal directed proof procedure for the propositional fragment of the inconsistency-adaptive logic ACLuN1. At the propositional level, the procedure forms an algorithm for final derivability. If extended to the predicative level, it provides a criterion for final derivability. This is essential in view of the absence of a positive test. The procedure may be generalized to all flat adaptive logics.     

Nice Embedding in Classical Logic

It is shown that a set of semi-recursive logics, including many fragments of CL (Classical Logic), can be embedded within CL in an interesting way. A logic belongs to the set iff it has a certain type of semantics, called nice semantics. The set includes many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for CL that are goal directed with respect to CL are turned into partial decision methods that are goal directed with respect to the embedded logics.