Abstract
Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (namely self-identity) are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject (i), are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics (free and classical). The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework.
Highlights
This is a paper in proof theory, in sequent calculi
Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic [8, 9]
The term is due to Karel Lambert [9,10,11] and is short for first order logic free of existential assumptions, and we take a logic to be free iff (1) it is free of existential presuppositions with respect to its singular terms, (2) it is free of existential presuppositions with respect to its general terms and (3) its quantifiers have existential import [6]
Summary
This is a paper in proof theory, in sequent calculi. It has three main goals – first to present a unified approach to the proof theory of free logics (with one caveat to be mentioned shortly), second, to offer a unified, streamlined and above all fruitful semantical approach to those free logics, and to extend the results of the first two goals to applications to both inclusive and non-inclusive systems, and all of those to modal logics. We are able to focus on the presence and absence of individual semantic restrictions These in turn allow for descriptions of different systems in terms of a series of binary choices, and significantly streamline the way in which they can be systematized. After a discussion of some further logics captured by our systematization, in Section 6 we examine the modal extension of both the calculi and the generalized semantics, again show they possess the structural and metatheoretic properties, and discuss the different approaches to identity that our formalization allows for. The appendices contain some useful technical results that have been omitted from the main body of the paper for conciseness
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