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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).<jats:italic>Positive<\/jats:italic>free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true,<jats:italic>negative<\/jats:italic>free logic treats them as uniformly false, and<jats:italic>neutral<\/jats:italic>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 (<jats:italic>Bulletin of the Section of Logic<\/jats:italic>,<jats:italic>48<\/jats:italic>(2), 137\u2013158 2019) as well as classical logic in Pavlovi\u0107 and Gratzl (<jats:italic>Journal of Philosophical Logic<\/jats:italic>,<jats:italic>50<\/jats:italic>, 117\u2013148 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 (<jats:italic>Studia Logica<\/jats:italic>,<jats:italic>105<\/jats:italic>(1), 93\u2013119 2017,<jats:italic>Journal of Applied Non-Classical Logics<\/jats:italic>,<jats:italic>30<\/jats:italic>(3), 272\u2013289 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.<\/jats:p>","DOI":"10.1007\/s10992-022-09679-z","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T13:02:47Z","timestamp":1661346167000},"page":"519-554","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neutral Free Logic: Motivation, Proof Theory and Models"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6331-4256","authenticated-orcid":false,"given":"Edi","family":"Pavlovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0660-6527","authenticated-orcid":false,"given":"Norbert","family":"Gratzl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"issue":"4","key":"9679_CR1","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1007\/BF01471125","volume":"149","author":"AR Anderson","year":"1963","unstructured":"Anderson, A. 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