A More Unified Approach to Free Logics

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.

[1]  Paolo Maffezioli,et al.  Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate , 2019, Bulletin of the Section of Logic.

[2]  Gian Aldo Antonelli,et al.  Proto-Semantics for Positive Free Logic , 2000, J. Philos. Log..

[3]  Sara Negri,et al.  Proof Analysis: Index of names , 2011 .

[4]  Jaako Hintikka,et al.  Existential Presuppositions and Existential Commitments , 1959 .

[5]  Karel Lambert,et al.  Free logics: Their foundations, character, and some applications thereof , 1997 .

[6]  W. V. Quine,et al.  Quantification and the empty domain , 1954, Journal of Symbolic Logic.

[7]  Francesca Poggiolesi.  Proof Analysis. A Contribution to Hilbert's Last Problem , 2013 .

[8]  Norbert Gratzl,et al.  A Sequent Calculus for a Negative Free Logic , 2010, Stud Logica.

[9]  Sara Negri,et al.  Cut Elimination in the Presence of Axioms , 1998, Bulletin of Symbolic Logic.

[10]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[11]  Sara Negri,et al.  Proof Analysis: Index of names , 2011 .

[12]  Hannes Leitgeb,et al.  HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes) , 2018, J. Philos. Log..

[13]  Norbert Gratzl,et al.  PROOF-THEORETIC ANALYSIS OF THE QUANTIFIED ARGUMENT CALCULUS , 2019, The Review of Symbolic Logic.

[14]  L. Goble.  The Blackwell guide to philosophical logic , 2001 .

[15]  George D. W. Berry,et al.  On Quine's axioms of quantification , 1941, Journal of Symbolic Logic.

[16]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[17]  Carl J. Posy.  Free logics , 2007, The Many Valued and Nonmonotonic Turn in Logic.

[18]  Scott Lehmann,et al.  More Free Logic , 2002 .

[19]  Karel Lambert,et al.  Free logic and the concept of existence , 1967, Notre Dame J. Formal Log..

[20]  Theodore Hailperin,et al.  Quantification theory and empty individual-domains , 1953, Journal of Symbolic Logic.

[21]  Sara Negri,et al.  Structural proof theory , 2001 .

[22]  Sara Negri,et al.  Proof theory for quantified monotone modal logics , 2019, Log. J. IGPL.

[23]  Karel Lambert,et al.  New Essays in Free Logic in Honour of Karel Lambert , 2001 .