Classical first-order logic assumes a non-empty domain to maintain the validity of standard syllogistic inferences.
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Scholarly and reference literature confirms that classical first-order logic traditionally assumes a non-empty domain of interpretation, distinguishing it from inclusive logics that permit empty domains.
Free logics are a family of first-order logics which came about as a result of examining the existence assumptions of classical logic (Hintikka The Journal of Philosophy , 56 , 125–137 1959 ; Lambert Notre Dame Journal of Formal Logic , 8 , 133–144 1967 , 1997 , 2001 ). 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 reject the claim that names need to denote in (ii). Positive free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, negative free logic treats them as uniformly false, and neutral free logic as taking a third value. There has been a renewed interest in analyzing proof theory of free logic in recent years, based on intuitionistic logic in Maffezioli and Orlandelli ( Bulletin of the Section of Logic , 48 (2), 137–158 2019 ) as well as classical logic in Pavlović and Gratzl ( Journal of Philosophical Logic , 50 , 117–148 2021 ), there for the positive and negative variants. While the latter streamlines the presentation of free logics and offers a more unified approach to the variants under consideration, it does not cover neutral free logic, 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. We discuss extending the results to this third major variant of free logics. We present a series of G3 sequent calculi adapted from Fjellstad ( Studia Logica , 105 (1), 93–119 2017 , Journal of Applied Non-Classical Logics , 30 (3), 272–289 2020 ), which possess all the desired structural properties of a good proof system, including admissibility of contraction and all versions of the cut rule. At the same time, we maintain the unified approach to free logics and moreover argue that greater clarity of intuitions is achieved once neutral free logic is conceptualized as consisting of two sub-varieties.
Appendix:Glossary of logic in Wiktionary, the free dictionary. This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation
This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation.
deviant logic
A term used to describe non-standard or alternative logical systems that deviate from classical logic.
first-order variable
A variable in first-order logic that represents an individual in the domain of discourse.
inclusive first-order logic
A variant of first-order logic that allows for empty domains, in contrast to the standard requirement that domains contain at least one object.
skolemization
A technique in first-order logic for eliminating existential quantifiers by introducing Skolem functions, used in the process of converting formulas to a standard form.
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