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.

中文翻译：

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一种更统一的自由逻辑方法

自由逻辑是一阶逻辑家族，它是通过检查经典逻辑的存在假设而产生的。这些假设各不相同，但核心假设是（i）解释域不是空的，（ii）每个名称都表示域中的一个对象，以及（iii）量词具有存在意义。自由逻辑通常拒绝在（ii）中名称需要表示的主张，并且在本文考虑的系统中，正自由逻辑承认一些包含非表示名称（即自我同一性）的原子公式为真，而否定自由逻辑甚至拒绝后一种主张。同样考虑拒绝 (i) 的包含逻辑。这些逻辑具有复杂多样的公理化和语义，本文的目的是对各种系统及其相互关系进行有序的检查。这是通过首先提供形式化来完成的，使用具有良好证明系统所有所需结构特性的后续演算，包括收缩和切割的可接受性，同时以其他方法没有的方式简化自由逻辑。然后，我们提出了一个简单而统一的抽象语义系统，它允许直接演示元理论属性，并提供对不同逻辑（自由和经典）之间关系的见解。本文的最后一部分致力于通过使用标记的序列演算来扩展具有模态的系统，在这里我们再次能够使用相同的框架绘制出不同的方法及其相互关系。