Inverses for Normal Modal Operators

Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right.     

Heterogeneous logic

This paper considers the question: what becomes of the notion of a logic as a way of codifying valid arguments when the customary assumption is dropped that the premisses and conclusions of these arguments are statements from some single language? An elegant treatment of the notion of a logic, when this assumption is in force, is that provided by Dana Scott's theory of consequence relations; this treatment is appropriately generalized in the present paper to the case where we do not make this assumption of linguistic homogeneity. Several applications of the resulting concept of a heterogeneous logic are suggested, but the main emphasis is on the formal development. One topic touched on is a certain contrast between the boolean and the intensional sentence-connectives in this more general setting.     

Replacement in Logic

We study a range of issues connected with the idea of replacing one formula by another in a fixed (linguistic) context. The replacement core of a consequence relation ? is the relation holding between a set of formulas {A ₁, ..., A _m , ...} and a formula B when for every context C(·), we have C(A ₁), ..., C(A _m ), ...???C(B). Section 1 looks at some differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we find that while the inference from A and B to $A \land B$ , sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two (and some other) logics occupies Sections 3 and 4. Section 2 looks at the m?=?1 case, describing A as replaceable by B according to ? when B is a consequence of A by the replacement core of ?, and inquiring as to which choices of ? render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop (and a comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3.     

For Want of an ‘And’: A Puzzle about Non-Conservative Extension

The paper analyses Frege's approach to the identity conditions for the entity labelled by him as Sinn. It starts with a brief characterization of the main principles of Frege's semantics and lists his remarks on the identity conditions for Sinn. They are subject to a detailed scrutiny, and it is shown that, with the exception of the criterion of intersubstitutability in oratio obliqua, all other criteria have to be discarded. Finally, by comparing Frege's views on Sinn with Carnap's method of extension and intension and the method of intensional isomorphism, it is proved that these methods do not provide a criterion for the identity of Frege's Sinn, even for extensional contexts, that the concept of intension does not coincide, as stated by Carnap, in these contexts, with Frege's concept of Sinn, and that Carnap's claim that in oratio obliqua Frege's semantics leads to an infinite hierarchy of Sinn entities can be questioned at least hypothetically in the light of certain new historical facts.     

The Relevant Equivalence Property

The pure implicational and the multiplicative fragments of arange of propositional relevant (and other) logics are shownto have the property that any two formulas equivalent in sucha logic are constructed from exactly the same propositionalvariables – as opposed to merely having (as the definitionof relevance itself would require) some propositional variablein common.     

Valuational semantics of rule derivability

If a certain semantic relation (which we call local consequence) is allowed to guide expectations about which rules are derivable from other rules, these expectations will not always be fulfilled, as we illustrate. An alternative semantic criterion (based on a relation we call global consequence), suggested by work of J.W. Garson, turns out to provide a much better — indeed a perfectly accurate — guide to derivability.