A More Unified Approach to Free Logics

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. Keywords Positive free logic • Negative free logic • Sequent calculus • G3 • Modal logic An anonymous reviewer has offered numerous suggestions which have, combined, amounted to a significant improvement of the paper, and for that we extend our gratitude. Special thanks goes to O. Foisch.

Neutral Free Logic: Motivation, Proof Theory and Models

“…Proof theory, and especially sequent calculi, of various systems of free logic are being extensively investigated at the moment, and recently. In [31] the authors develop a sequent calculus for an intuitionistic free logic with existence, [40] offered a unified approach to free logics on a classical base, while [20] further extended them with functions, and also added a treatment of quasi-free logics.…”

“…Of these, the present paper follows [40] most directly. The (somewhat) unified approach there comes with a caveat, namely the treatment focused only on positive and negative free logics and omitted the third major variety, neutral free logics, since there is some lack of both clear formal intuitions on the semantic status of formulas with empty names, as well as a satisfying account of the conditional in this context.…”

A Completeness Proof for A Regular Predicate Logic with Undefined Truth Value

“…Perhaps the next idea is negative free logic: every atomic formula that contains an undefined term yields F [8,20,22,23]. It suffers from a problem illustrated by the following example.…”

“…The important issue regarding related work is how to say that a term or formula is undefined. An obvious idea is to introduce a new atomic predicate * t or connective * ϕ that yields F if its input is undefined and T otherwise [2,6,9,13,18,20,23]. Alternatively, the idea of ⌊t⌉ discussed above can be used and naturally extended to formulas [3,5], at the cost of needing a new component in addition to the traditional signature and set of axioms.…”

Tableaux for Free Logics with Descriptions

“…Recently, cut-free sequent calculi for several free logics, yet without DD, have been presented by Pavlović and Gratzl [25] and by Indrzejczak [15]. In particular, in the latter work it has been shown that if we restrict instantiation in quantifier rules only to variables, we do not lose completeness, provided that some special rules are added.…”