Juxtaposition: A New Way to Combine Logics

Abstract

This paper develops a new framework for combining propositional logics, called "juxtaposition." Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefly investigates the question of which rules, when added to these logics, lead to collapse. §1. Introduction. Methods of combining logics are of great interest. 1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multiple kinds of modal operators-epistemic, temporal, and deontic.There are also purely philosophical reasons to be interested in the combination of logics. One illustration of this comes from so-called collapse theorems.Suppose there is a language with two stocks of the usual logical connectives (for propositional or for first-order logic)-∧ 1 and ∧ 2 , ∨ 1 and ∨ 2 , and so on. There is a well-known result which states that given any logic for this language such that each logical constant obeys the usual natural deduction rules (for classical or even for intuitionist logic), sentences that differ only in some or all of their subscripts are intersubstitutable. 2 Corresponding connectives, such as ∧ 1 and ∧ 2 , behave as mere notational variants. In such a logic, if one stock of constants obeys the classical natural deduction rules, so does the other. This result has been used to argue for several striking philosophical theses. For instance, it has been used to argue that the logical constants in our language have unique and determinate extensions. The argument goes as follows: Suppose that one of our logical constants had an indeterminate extension. Then we could precisify the term. We could introduce two (or more) precise terms into our language with distinct extensions. These terms would presumably obey the usual natural deduction rules (understood to apply to the expanded language). But since the terms would have distinct extensions, they wouldnot be Received: February 17, 2011. 1 Methods of combining logics include fibring and its variants, as developed by Gabbay, as well as algebraic fibring and its variants, as developed by A. Sernadas and his collaborators. See Gabbay (1998) and Carnielli et al. (2008), respectively, for comprehensive overviews of this work. See Caleiro et al. (2005) for a summary of the central results concerning algebraic fibring. 2 See McG...

On the characterization of fibred logics, with applications to conservativity and finite-valuedness

“…Namely, we use it to close, in the disjoint case, a long standing question of the theory of fibring: conservativity. Namely, we prove a full conservativity characterization result for disjoint fibring, extending the partial result obtained in [19]. We start by reviewing the conservativity problem for fibring, by means of a series of examples.…”

“…However, getting rid of such pathological cases is still not completely satisfactory, as the next examples will help illustrate. [7], and also as a consequence of [19].…”

“…It is also worth noting that the conservativity result for disjoint fibring provided by [19] covers only four entries in our table, which we marked with , corresponding to the combination of two non-trivial logics without quasi-theorems.…”

“…However, if one considers the problem in full generality, it is immediate that a complete characterization of conservativity for fibred logics is far from obvious, even more so when the logics at hand share some of their connectives (which is nevertheless the case of fusion). More recently, a partial result about the conservativity of logics obtained by disjoint fibring was obtained in [19], covering only non-trivial component logics without quasi-theorems (strongly determined logics, in the author's terminology). Using our main result, we shall obtain a complete characterization of the conservativity problem for disjoint fibring, thus extending the result of [19].…”

“…More recently, a partial result about the conservativity of logics obtained by disjoint fibring was obtained in [19], covering only non-trivial component logics without quasi-theorems (strongly determined logics, in the author's terminology). Using our main result, we shall obtain a complete characterization of the conservativity problem for disjoint fibring, thus extending the result of [19].…”