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.


Introduction
This is a paper in proof theory, specifically in sequent calculi. It has three main goals -first to present a unified approach to the proof theory of free logics (with one caveat to be mentioned shortly), second, to offer a unified, streamlined and above all fruitful semantical approach to those free logics, and finally to extend the results of the first two goals to applications to both inclusive and non-inclusive systems, and all of those to modal logics.
Before we delve deeper let us briefly remind ourselves what a free logic is standardly understood to be. Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic [8,9]. The term is due to Karel Lambert [9][10][11] and is short for first order logic free of existential assumptions, and we take a logic to be free iff (1) it is free of existential presuppositions with respect to its singular terms, (2) it is free of existential presuppositions with respect to its general terms and finally (3) its quantifiers have existential import (the last claim is expressed by the axioms A3 and A4 later in the paper) [6]. The three (families of) free logics we will here distinguish depend on their decisions on the truth values of (atomic) statements containing empty terms. If a free logic holds that those can be true it is called positive, if it holds that they are false it is called negative and if it holds they take a third value it is called neutral.
In the light of recent research on structural proof theory it seems timely to have a fresh look at the family of free logics. Our focus is on positive and negative free logics (according to our definitions above) and not on neutral free logics. Herein lies the caveat mentioned in the beginning. Neutral free logics have been studied (see [12]) but up to now still lack the rigorous systematicity the other two family members enjoy. For us the main reasons for this is that there is some lack of formal intuitions or even a clash of formal intuitions when it comes to, say, the relationship between the semantics stati of sentences/formulas with empty singular terms in the contexts of the existence predicate, and identity statements as well as to a philosophically satisfying account of the conditional in this context. For our proof-theoretic investigations we have chosen the popular G3-style systems. This makes the presentations of the formal systems rather concise and dispenses with structural rules at the core of the systems altogether. Furthermore it has the advantage of putting the insights from [18,19] on the identity predicate to full use.
The approach to semantics we utilize in this paper -which we label generalized semantics -is a purely syntactic representation of semantics, with the main goal being to simplify comparisons between logical systems. Given how predictable (cutfree) derivations are in a G3 systems, the comparison can be achieved simply by noting that the semantics of one system require comparatively higher restrictions (and those are captured in a systematic way in a sequent calculus). So we are able to focus on the presence and absence of individual semantic restrictions. These in turn allow for descriptions of different systems in terms of a series of binary choices, and thus significantly streamline the way in which they can be systematized. In a nutshell, this allows for a comparisons between whole systems in a point-by-point manner.
Plan of the paper: in the following section we present a G3-style sequent calculus for both positive and negative free logic, and then demonstrate a range of desirable structural properties, including being cut-and contraction free. We establish the connection to previous sequent calculi for free logics and moreover we also show that our formalization of positive and negative free logic is adequate by showing that all the axioms of their standard representations are derivable. In Section 3 we present a generalized version of the semantics for positive and negative free logic, which allows for a systematic representation of a wide range of logics and their relations, primarily positive and negative free logic, as well as classical logic, on both inclusive and non-inclusive versions. Moreover, this generalized semantics has further uses in the discussion of modal logic. That, however, is left for a later section and we first utilize the general semantics to prove the soundness and completeness of our calculi in Section 4. After a discussion of some further logics captured by our systematization in Section 5, in Section 6 we examine the modal extension of both the calculi and the generalized semantics, again show they possess the structural and metatheoretic properties, and discuss the different approaches to identity that our formalization allows for. Finally in Section 7 we look at further avenues of research that this paper opens. The appendices contain some useful technical results that have been omitted from the main body of the paper for conciseness.

The Formal Systems G3 pf and G3 nf
In formulating the rules for the free logics discussed here we take as our starting point the sequent calculus for the negative variety first introduced in [6]. We then streamline it by transforming it into a G3-style calculus along the lines of [19], and to a lesser extent, [16,20]. Specifically, we replace multiple types of initial sequents with only a single one, and we simplify the quantifier rules. We defer showing that the new versions of the rules are adequate and deductively equivalent to the previous ones, as well as their comparative advantages, for after the discussion of the structural properties of the new systems.

Language of Free Logics
The language L utilized in this paper is a standard first order language without functions, adapted from [6], with the vocabulary defined as  A formula of our language L is defined as In the rest of the paper, unless otherwise specified we use variables to refer to bound individual variables, names to refer to free individual variables, and formulas to refer to closed formulas (i.e. formulas in which every free occurrence of a term is a name).

Positive Free Logic G3 pf
The base from which we will develop the system of positive free logic G3 pf (and shortly also the negative version) is quantified classical calculus G3 c [19] in Fig. 1. The basic unit of a sequent calculus is a sequent ⇒ , where and are finite multisets of (closed) formulas. All the formulas except and are called active formulas of the rule if they occur only in the upper sequent(s) and principal if they occur in the lower sequent of the rule.
The positive free logic G3 pf (Fig. 2) is then obtained by replacing the quantifier rules in G3 c with the rules below, i.e. by adding the condition that E!t. This approach from [16,20] was explicitly used in [14] to simplify the quantifier rules of intuitionistic logic with existence predicate. However, in the more general framework of this paper, E! is for left undefined and open to multiple interpretations (cf. [1,4]).

Negative Free Logic G3 nf
The negative free logic G3 nf (Fig. 3) is obtained by replacing the rule = Ref (reflexivity of identity) in G3 pf by the rules: We can demonstrate a few simple facts about these systems. Full proofs can be found in Appendix A, and here we just note that   Note that this result justifies the decision to keep the free and bound individual variables separate.

Structural Properties
The systems G3 pf and G3 nf possess a range of desirable proof-theoretic properties. Again these proofs are left for Appendix A, where important parts are shown. Here we simply state that   The proofs of all of these become routine, following a general pattern of demonstration from [19], once appropriate modifications are made to account for the specificities of the system discussed here.
From the last theorem it immediately follows that and from there Corollary 2.12 (Consistency) The calculi G3 pf and G3 nf are consistent.

Relation to Previous Systems of Free Logic
The sequent calculi for free logics offered so far, e.g. for either version of free logics in [2] (here simplified to be context-sharing) or specifically for negative free logic in [6] use the following versions of the quantifier rules for L∀ and R∃: Notably, these are both two-premise rules, whereas the rules we use are onepremise. The approaches are deductively equivalent, but ours has multiple advantages. We first demonstrate the former claim:

Lemma 2.13
The rules L∀ † and R∃ † are admissible in G3 pf and G3 nf .
Proof We illustrate on the example of L∀ † , with the other being nearly identical, and make use of the structural properties of G3 pf and G3 nf .
Theorem 2.10, Lemma 2.9 ∀xA, ⇒ Lemma 2.14 The quantifier rules of G3 pf and G3 nf are admissible in a system with L∀ † and R∃ † .
Proof We again illustrate on the example of L∀. Note that the upper left sequent is initial.
The case for the rules E! and = Ref is straightforward as they replace initial sequents.

Advantages of the Present Approach
The first advantage of our approach is that it simplifies both proof-search and metatheory by virtue of not creating a branching when the rules are read bottom-up. The second is that in this formulation, the rules exhibit a perfect symmetry with the modal operator rules in their typical formulation, as in [16,20]. Finally, the rules do not tacitly rely on an implicit conditional and conjunction introduction, respectively for L∀ and R∃, as had been the case with previous iterations. For example, this was the case with the free logic rules in [24, pp. 200-201], where the definitions make this reliance explicit, while here the quantifiers are restored to being independent of other operators (see also [22]).

Adequacy of the Systems G3 pf and G3 nf
In this section we demonstrate the adequacy of the formalization in the systems G3 pf and G3 nf by showing that each proves (one version of) an axiomatization [15,21] of their respective system. In addition to the admissibility of modus ponens, this also serves as an indirect proof of completeness of the systems, but we will not focus on that as a direct proof of it will be presented in a later section.
An axiomatization of positive free logic is the following.

Note that due to the freshness condition and Lemma 2.4, A[t//s][d/x] is the same as A[d/x][t//s].
We can show that it holds for G3 pf that Theorem 2.19 All of the PFL axioms A1-A6 are derivable in G3 pf .
And then for G3 nf that Theorem 2.20 All of the NFL axioms A1-A7 are derivable in G3 nf .
We demonstrate these for each axiom. Since none of the proofs are particularly involved, their presentation is deferred to Appendix B. Instead, we proceed to prove a few interesting results specific to NFL. We start by showing that the axiom A6 of PFL does not hold in G3 nf : Proof By Corollary 2.11, every formula occurring in the derivation of ⇒ t = t is either t = t or atomic. But no application of the rules will remove every instance of those from the antecedent. Therefore, there is no derivation of ⇒ t = t in G3 nf .

Lemma 2.22
In G3 nf the sentences E!t and t = t are equivalent.
Proof We show this by two simple derivations: Basic case.
Inductive case. Whenever some subformula B does not contain s, an instance of the inductive hypothesis for it becomes an instance of Lemma 2.6. Note for the

Similar for ∃xC.
This, and Lemmas 2.13 and 2.14, suffice to show that our formalization constitutes an adequate representation of the two systems of free logics, both in what is and what is not (in particular in the case of NFL) derivable.
Naturally, since we do not intend to use the axioms as semantics, we will in this section present a system of semantics. The central idea is to separate two distinct elements of semantics, namely the assignment of truth-values to atoms and to complex formulas, and abstract away from the former. In this way, we represent the semantics in purely syntactic terms, which allows us to achieve the required meta-theoretical properties in a streamlined manner.

Generalized Semantics
The usual exposition of semantics consists of two steps -giving a decision procedure for the assignment of truth values to atoms, and inductive rules for assignments of truth values to any other formula. However, since we ever only deal with descriptions of semantic models, i.e. their syntactic superstructure, here we will employ generalized semantics, which will suffice for the meta-theoretic proofs. We simply assume the existence of a method for assigning values to atoms, while focusing on the second step of the exposition (cf. [19, p. 81]).
. . countable list of free individual variables, and ϕ an interpretation function on L: where t ∈ D (to emphasize its dual role we will abuse the notation slightly and write D as . . . , t i , . . . ∈ ϕ(P n ), for any n and any 1 ≤ i ≤ n.
A positive structure is defined as We omit the clause ϕ(E!) ⊆ D.
The reason we can omit ϕ(E!) ⊆ D is that E! is defined like any other unary predicate by the last clause of the definition. This means that E! could here be regarded as a non-logical predicate. As another consequence of this definition, if s = t then E!s iff E!t.
From these two structures we can likewise define a classical structure: A classical structure S c is any structure such that it is both an S p and an S n .

Definition 3.4 (Valuation V)
The truth-value assignment V on the structure D, ϕ is defined as . . , t n ∈ ϕ(P n ), and ⊥ otherwise.
Definition 3.5 (Validity) A formula is valid on a structure iff it is true on that structure. A formula is valid iff it is valid on any structure.

Relationship Between PFL, NFL and Classical Logic
In any of the three definitions above we can further specify that they are non-inclusive by the addition of the condition ϕ(E!) = ∅. In this way, the generalized semantics systematizes the relation between classical, negative and positive structures (marked below as C, N and P respectively) on both inclusive and non-inclusive versions (marked below with subscripts i and n respectively). For the purposes of this schematization we take it that what is usually thought of as the domain is here designated by E! (and not D). This is reinforced by the axioms A3 and A4 (and Theorems 2.19 and 2.20), and some more evidence towards this claim will be provided later in the section on modalities. However, the framework here is also perfectly compatible with the single-domain semantics for positive free logics [1] (note that in Definition 3.2 we can regard E! as just an ordinary predicate).
Each inclusive structure is obtained from its non-inclusive counterpart, by extending the interpretation of E! to empty cases.
A positive structure is obtained from a negative structure when Ref and ϕ(P n ) are extended from members of E! to members of D. Finally, a classical structure is obtained from the positive one by extending the interpretation of E! to all members of D (Fig. 4).

Fig. 4 Relationship between PFL, NFL and CL
Of these, the structure C n marks the classical logic. Here, the domain is not empty, and all the free individual variables refer (since in a negative structure t = t holds only if E!t, and in a positive one t = t holds for every t).
While it is a classical structure, C i is not the classical logic. Even though it is here mostly a theoretical construct, it does correspond to an empty logic (more on that in Section 5.2).

Meta-theoretical Properties
We now proceed to show that the systems G3 pf and G3 nf are sound and complete with respect to their appropriate generalized semantics.

Soundness
Definition 4.1 A sequent ⇒ is valid on a structure S iff whenever all formulas in are valid on S, some formula in also is.
We prove that Proof By induction on the height of the derivation of ⇒ . Straightforward for the basic case, and simple for identity, E!, and connectives. We illustrate on an example.
If the last step of the derivation is obtained by L∧, then it has the form Assume C ∧D and all the formulas in are valid. Then by Definition 3.4, C, D, are all valid. Then by inductive hypothesis, some formula in is valid.
If the last step of the derivation is obtained by R∧, then it has the form Assume all the formulas in are valid. Then by inductive hypothesis, either some formula in is valid, in which case so is some formula in , C ∧ D, or both C and D are, in which case by Definition 3.4 so is C ∧ D.
If the last step of the derivation is obtained by L∀, then it has the form Assume E!t, ∀xC and all the formulas in are valid. Then, since ∀xC and E!t are valid, by Definition 3.4 so is C[t/x], therefore all of C[t/x], E!t, ∀xC, are valid, and by the inductive hypothesis so is some formula in .
If the last step of the derivation is obtained by R∀, then it has the form Assume all the formulas in are valid and neither any formula in nor ∀xC are. So by Definition 3.4 there is a free individual variable t such that E!t is valid, but C[t/x] isn't. Let that variable be t. So, none of the formulas in , C[t/x] are valid, and therefore by contraposition of the inductive hypothesis, either some formula in or E!t isn't. Contradiction either way, and so either some formula in or ∀xC is valid.
Parallel for R∃ and L∃ (for the latter the proof is direct).

Completeness
We start by the definition of a reduction tree. Intuitively, this represents a bottom-up proof search.

Definition 4.3 (Reduction tree)
A reduction tree for a sequent ⇒ is built in steps. At step 0, the tree is just ⇒ . Any sequent that does not contain the same atomic formula in both the antecedent and the consequent is called active.
Each subsequent step consists of stages. At each stage and for each sequent i ⇒ i active at the beginning of it, we apply to any eligible (pair of) formulas in the sequent the rule of the stage once (thereby extending the height of the tree by n, for n such formulas in i ⇒ i , and creating at most 2 n branches, before proceeding to the next stage). We call an application of a rule to the formula(s) their reduction.
The order of stages is: E!, for G3 nf only For each active sequent to which no rule can be applied, we just copy it.
We now show that Lemma 4.4 For any sequent ⇒ its reduction tree either produces a proof or it produces a structure and a valuation that validates all the formulas in and none of the formulas in .
Proof It is clear that a reduction tree with no active sequents will produce, read top down (and thus beginning with initial sequents and ending with ⇒ ), a finite derivation of that sequent. The second part is more involved and goes through several lemmas below.
We now build an invalidating structure and a valuation from a reduction tree to prove the second part. The existence of an infinite branch is guaranteed by the König's lemma in the usual way [19, p. 82].

Definition 4.5 (Refutation structure C) Take an infinite branch
of a reduction tree for a sequent ⇒ (where 0 ⇒ 0 is ⇒ ) and consider sets * ≡ i and * ≡ i for 0 ≤ i. A refutation structure C for a sequent ⇒ is built by assigning to all atomic formulas in * and ⊥ to all other atomic formulas (therefore including all atomic formulas in * ), and otherwise the same as V.
We first show that this produces the correct type of structure. Since the proof for PFL is very similar, we will here focus on NFL.

Lemma 4.6
The refutation structure C for a sequent ⇒ in G3 nf is a structure S n . Specifically, (ii) By stage (14), for every atomic formula E!t in * , t = t is in * . Therefore by Definition 4.5, t = t is likewise true. (iii) By stage (15), for any atomic formula in * , including s = t, the formulas E!s and E!t are likewise in * and therefore true by Definition 4.5. (iv) Similar to (iii).

Now we show that
Lemma 4.7 Any formula A occurring in * is assigned by the refutation structure C and any formula B occurring in * is assigned ⊥ by the refutation structure C.
Proof By simultaneous induction on the weight of A and B. Basic step. Immediate from Definition 4.5 and noting that * and * share no atoms.
Inductive step. We illustrate for one example of a connective, as all are straightforward.
(∧) If A is a formula C ∧ D, then by stage (1) of Definition 4.3, C and D are also in * , and by inductive hypothesis assigned , and so C ∧ D is . If B is a formula C ∧ D, then by stage (2) of Definition 4.3, either C or D are also in * , and by inductive hypothesis assigned ⊥, and so C ∧ D is ⊥.
Similar for other connectives. (∀) If A is a formula ∀xC, then by stage (9)

of Definition 4.3, for every t ∈ D,
if E!t is in * (and by inductive hypothesis ) then C[t/x] is in * (and by inductive hypothesis ), and so ∀xC is . If B is a formula ∀xD, then by stage (10)

of Definition 4.3, for some t ∈ D, E!t is in * (and by inductive hypothesis ) and D[t/x] is in * (and by inductive hypothesis ⊥), and so ∀xD is ⊥.
Similar for ∃.
Finally, we have that

Further Varieties
In this section we observe some further varieties of G3 pf and G3 nf mentioned in the Section 3.2. We first note that the two systems described so far are compatible with an empty domain. We then add a rule preventing that, and briefly discuss the resulting (non-inclusive) logics. Furthermore, we also take a look at the logic corresponding to the structure C i .

Non-inclusive Logics
Note that for the sequent ∀xA ⇒ ∃xA the refutation structure C will be generated by an infinite branch containing only that sequent, and the truth condition for ∀ will be vacuously fulfilled as no E!t appears as true in * . So nothing we have said so far actually excludes the possibility of ϕ(E!) being empty in G3 pf and G3 nf , i.e. of them being inclusive logics. We can make them explicitly non-inclusive by the addition of the rule: This rule makes "something exists" a theorem: This rule is not necessary in the classical system G3 pf +nf , which contains all the rules of both G3 pf and G3 nf (and by inspection of their proofs also all the structural properties), since it can be easily shown that Theorem 5.2 In a system G3 pf +nf , the rule NI is admissible.
This derivation mirrors the semantical explanation of the interaction of the two systems from the Section 3.2. Of course, nothing we have said so far excludes the list of free individual variables D being empty either. Without that assumption, we obtain the logic G3C i which corresponds to an empty logic.

Empty Logic G3C i
If we assume D is empty, ∀xA becomes an abbreviation for (see [23]). To represent this semantics in terms of a sequent calculus any formula P n (t 1 , . . . , t n ) can be seen simply as an abbreviation for ⊥ (for the sake of simplicity we exclude 0ary predicates from the language). Therefore, the rules E!, = Ref and = Repl reduce to contraction, which can be shown admissible. Furthermore, every initial sequent is simply an instance of the rule L⊥. The system then has the following form: Equivalently, this logic can be obtained by the same procedure from G3 pf , (with the rule E! vacuously removed).
Structural rules for this logic are demonstrated in a routine manner, so we will limit ourselves here to just showing the admissibility of contraction, as it was used in the preceding paragraph.

Lemma 5.4 (Contraction in G3C i ) Contraction is height-preserving admissible in
Proof Simultaneous for (i) and (ii) by induction on the height of the derivation.
In the initial case (for zero-premise rules since there are no initial sequents), if C, C, ⇒ ( n ⇒ , C, C) is a conclusion of a zero-premise rule, then so is C, ⇒ ( ⇒ , C).
Standard for propositional rules.
For L∀ and R∃ we apply the inductive hypothesis to the upper sequent of the rule and then the rule in a routine way.
Like this case, all other properties of the system are straightforward and routine, and will be skipped.

Adequacy of G3C i
One axiomatization by which we can measure the adequacy of our formalization is that in [7]. The notation is adjusted to be more in line with the current standards and the present paper.  If A and A → B then B. Theorem 5.6 All of the Axioms 1-7 of QE are derivable in G3C i .

If A is tautologous, then
Proof Note that the propositional part of G3C i is simply standard G3 classical propositional logic, so (1) follows from completeness of it. (2)-(6) are easily derivable from R∀ and R→. Finally, (7) follows from Lemma 2.17 (again noting that the propositional part of G3C i is standard).
It would be interesting to check whether this axiomatization is complete with respect to the intended semantics, as it is not obvious to us the axiomatization can be used to prove anything other than universal closures of tautologies (as opposed to any universally quantified formula). However, such a discussion would take us too far off topic since, as stated previously, we are here to a large extent dealing with an artifact of the systematization.

Modal Logic
When we move to the modal context, the principle of unrestricted specification, ∀xA → A[t/x], fails to hold in general [20, pp. 223-224]. Therefore, it seems prima facie that free and modal logics might be a good fit (cf. [5,12]). In fact, as we will see in this section, not only is there structural correspondence between modal and quantifier rules (this is by design), but the extension of one into the other is exceedingly straightforward. The modal logic system presented here will be an extension of the base systems G3 pf and G3 nf into labeled sequent calculi, following [16,20], with the addition of the modal operator rules. Definition 6.1 (G3K pf , G3K nf ) We add a countable set of labels w 1 , . . . , w n , . . . to our language, extend it with relational formulas of the form wRo and replace every formula of the form A with a labeled formula of the form w : A. Moreover, we replace every side and principal formula A in the rules of G3 pf and G3 nf by a labeled formula w : A (Fig. 6). Finally, we add the modal operator rules (Fig. 7). Note that the labeled formula w : A is a formula of the new language iff A was a formula of the old one. Therefore, sentence formation rules do not apply to relational formulas (i.e. they are always atomic, and never active in logical rules).
Should we wish to also extend the systems with the explicit rule for noninclusiveness NI, the resulting rule would correspond to the non-emptiness rule from [20].
The interesting case here are the new quantifier rules, which will now have the following form: Note that the usual extension of the quantifier rules into a modal context requires us to specify that t is an element of the domain of w [20]. No such provision is required here due to the presence of the formula E!t. Therefore, all that is required to transform the rules is adding the same label to all the active and principal formulas of the rules. The modal operator rules are:

Structural Properties
Given that the non-modal rules have been minimally modified, it is easy to see that Lemmas 2.6 -2.9, as well as Theorem 2.10 will still hold, so we just need to extend those with the case of modal rules. Moreover, since the modal rules closely resemble Fig. 7 Modal rules for G3K pf , G3K nf the quantifier ones, the proofs there will be quite similar. Therefore, we simply note that Theorem 6.2 Axiom generalization, height-preserving weakening and contraction, as well as cut are admissible in G3K pf and G3K nf . Moreover, the rules of the systems are (height-preserving) invertible.

Generalized Semantics
In developing the generalized semantics for G3K pf and G3K nf we adopt standard Kripke semantics, and define two different types of frames, depending on whether identity is taken as invariant (marked with a superscript I ) or varying (marked with V ). We start off with a more general type: where W is a countable multiset (we note the i-th member of W as w i ) of negative structures S n = D, ϕ (positive structures S p ), and R ⊆ W × W .
This frame allows for lists of names that vary between worlds. If we want to keep them fixed between worlds, we define the frame . . . , b 1 , . . . as before, W is a countable multiset of negative (positive) interpretation functions ϕ, and R ⊆ W × W . Now the list of names is fixed between worlds, but all predicates, including identity, are interpreted separately in each. Since identity can vary, we designate the frames with V . To get an identity predicate that is invariant, we deflate W further and obtain a frame Definition 6.5 A negative (positive) Kripke frame F I n (F I p ) is a tuple F * n , W, R , ( F * p , W, R ) where F * n is some negative structure without ϕ(P n ) (F * p a positive structure without ϕ(P n )), W is a countable multiset of negative interpretation functions over n-ary predicates ϕ(P n ) (positive functions ϕ(P n )), and R ⊆ W × W . For each ϕ i ∈ W , ϕ i (=) and ϕ i (E!) (ϕ i (=)) are identical to ϕ of F * n (F * p ).
Given that the unrestricted instantiation fails in models with varying domains regardless of the limitation to E!, the first type of frames will not be of particular interest to us, so we focus on the other two. We call the logics defined by these successive frames K nf , K pf , K inf and K ipf .
In each of these cases, a model is Definition 6.6 (Model M) A model M is a pair F, V , where F is some frame from Definitions 6.4 -6.5, and V is valuation such that -M ϕ i P n (t 1 , . . . , t n ) iff t 1 , . . . , t n ∈ ϕ i (P n ), -Standard for connectives, A in is valid, then some w : A in is valid.

Invariance of Identity
In the sequent calculus case, to make identity invariant across worlds/ labels (whence it follows it is necessary), we add the rule of invariance of identity to G3K pf and G3K nf thereby obtaining, respectively, G3K ipf and G3K inf . The rule is adopted from [17], and we follow the authors' naming of it as the rule for rigidity.
From this rule the necessity of identity easily follows: Lemma 6.7 Necessity of identity: w :

Meta-theoretical Properties
In this section each of the systems G3K pf , G3K nf , G3K ipf and G3K inf will be shown sound and complete with respect to the appropriate semantics.
Proof Routine by induction on the height of the derivation with relational atoms wRo valid iff w, o ∈ R and validity of a sequent mutatis mutandis as in Definition 3.5. Fig. 8 Invariance of identity rule Theorem 6.9 If a sequent ⇒ is derivable in G3K ipf (G3K inf ), it is valid on any Kripke frame F I p (F I n ).
Proof By induction on the height of the derivation. A new case to check is the rule = Rig . If the last step of the derivation was obtained by it then it has the form: is valid. Then for any model such that whenever for any r : A in it holds that M r A and moreover that M w s = t and M o s = t, it holds for some r : B in that M r B. Now assume w : s = t, is valid. But M w s = t iff M o s = t by Definition 6.5, so w : s = t, o : s = t, is likewise valid. Therefore, so is some r : B in .
We begin the proof of completeness by extending the definition of the reduction tree: Definition 6. 10 We extend Definition 4.3 by first noting that the appropriate applications of the rules now include labeled formulas (this is straightforward) and then adding the following steps: We build a refutation structure same as in Definition 4.5. Note that the results of the Lemma 4.6 are extended. Specifically, since relational formulas occur only in the antecedent, for each wRo we add w, o ∈ R to generate an appropriate Kripke frame. Moreover, in the case of G3K ipf and G3K inf , for any w and o, It is easy to see that

Lemma 6.11
The results of the Lemma 4.7 hold mutatis mutandis, and are extended for the cases of modal formulas.
Proof The first clause is straightforward (replacing A with w : A). For modal formulas: ( ) If A is a formula w : C then by stage (16) for every o such that wRo is in * (and therefore w, o ∈ R), o : C is in * , and therefore by inductive hypothesis M o C. Therefore, M w C.
If B is a formula w : D then by stage (17) for some o such that wRo is in * (and therefore w, o ∈ R), o : C is in * , and therefore by inductive hypothesis M o C. Therefore, M w D.
Similar for ♦.
Finally, we have that Theorem 6.12 If ⇒ is valid in the K pf , K nf , K ipf and K inf , then it is derivable in G3K pf , G3K nf , G3K ipf and G3K inf , respectively.

Applications
We make use of this framework to illustrate the point made earlier about the role the predicate E! plays in free logics. We first note that in frames F V n the Barcan formulas fail. As an illustration, consider the converse Barcan formula, ∀xAx → ∀x Ax, and a model where Here M ϕ 2 ∀xAx and therefore M ϕ 1 ∀xAx, but M ϕ 2 At 1 , so M ϕ 1 At 1 and therefore M ϕ 1 ∀x Ax. Note that this counterexample works for both inclusive and non-inclusive versions.
To obtain Barcan formulas we would need to add standard restrictions, applied to the predicate E! instead of the domain. Interestingly, that is not the case in frames F I n . In the modal negative free logic with the invariance of identity rule added, G3K inf , and corresponding to frames F I n , the predicate E! behaves in the same way as identity (as shown in the Lemma that immediately follows), and therefore Barcan formulas hold: Such a situation does not occur in the positive case, since there is no close connection between the existence and identity predicates. This is, to the best of our knowledge, the first quantified modal logic based on the negative free logic, and it is interesting to note that in G3K inf the Barcan formulas come about as the result of the rigidity of names. However, it should be clear that it is really the invariance of the E! predicate that establishes said formulas. So, on any variant we can simply add a rule for invariance of E! and obtain the same results.
Moreover, while adding such a rule will yield both the Barcan formula and its converse, we can make the result more fine-grained by adding two separate rules: It is straightforward to show that these will allow the derivation of the Barcan formula and the Converse Barcan formula, respectively.

Concluding Remarks
In this paper we have established and systematized a range of sequent calculi for free logics. Utilizing the similarity between modal operators and quantifiers, but working from the former to the latter, we formulated the quantifier rules for free logic in a way that treats E! as a relational atom. This proved to carry several benefits -in addition to simplification of the meta-theoretical and structural proofs, it also allowed us to construct systems for free logic where the rules do not rely on implicit implication and conjunction for the rules L∀ and R∃, respectively.
Moreover, by generalizing the semantic restrictions in a way that is still adequate for the sequent calculus rules, we have obtained a system of semantics that enables us to deal with a multitude of different systems of free logic. These approaches tend to be mutually incompatible as well as difficult to extricate from their philosophical motivations. The approach taken in this paper allows, on the other hand, for a prooftheoretic examination of free logics while avoiding such pitfalls.
The possibilities of application of our results have not been exhausted in this paper -specifically, even more unification is possible. In the introduction to this paper we mentioned and described briefly the core idea of neutral free logics. It is on our future research agenda to see if one (or more versions) of this family of free logic can be integrated in our new framework. Having system(s) for a neutral free logic seems timely and interesting also when viewed from a different perspective: hyperintensional logic, e.g. [13]. In [13] the hyperintensional logic is construed as an extension of intuitionistic logic with the constant domain axiom. Viewed from this angle it is also natural to construct free logics (including neutral ones) within the ever growing new field of hyperintensional logics.
Moreover, [14] demonstrates the usefulness of the single-premise approach to quantification in obtaining the Craig's interpolation theorem. We also plan to extend those results to the systems developed here. And finally, we can also see a further avenue for unification in bringing systems with intutionistic base into our framework.
Funding Information Open access funding provided by University of Helsinki including Helsinki University Central Hospital. This paper was written with support of the Academy of Finland research project no. 1308664. This paper was written with support of the Deutsche Forschungsgemeinschaft (DFG) research project no. 390218268 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommonshorg/licenses/by/4.0/.
If ⇒ is an initial sequent, then so is ⇒ . If the last rule is any of the propositional or identity rules, the namings remain unchanged from the premis(es). If the last rule applied is R∀, then it is derived from some sequent E!t, ⇒ , A [ Propositional rules do not alter the free and bound individual variables between their premis(es) and conclusion. The rules = Ref and E! in G3 nf also do not alter the free and bound individual variables between their premise and conclusion, and the step is straightforward for = Ref in G3 pf . The rule = Repl does not alter the free and bound individual variables.
If the last rule applied is R∀ and t is an eigenvariable of that application of the rule (otherwise we skip the first application of the inductive hypothesis), the premise of the application of the rule is some E!t, ⇒ , A[t/x]. We use the inductive hypothesis to replace it with some d that has not so far occurred anywhere above that application of the rule to obtain E!d, Proof By induction on the weight of A. Since the two systems do not differ in their language, the proofs proceed the same. The interesting cases here are for quantified formulas: Lemma 2.7 (Weakening) Weakening is height-preserving admissible in G3 pf and G3 nf : i) If n ⇒ then n C, ⇒ . ii) If n ⇒ then n ⇒ , C.
Proof Routine by induction on the height of the derivation, using Lemma 2.5 when necessary. We demonstrate (i) by induction on the height of the derivation, n. If n = 0, then ⇒ , ∀xA is an initial sequent, but then so is E!t, ⇒ , A[t/x]. If n > 0, then if ∀xA is not principal in the last step, ⇒ , ∀xA follows from some sequents E!t, ⇒ , ∀xA (E!t, ⇒ , ∀xA) with height ≤ n − 1. Applying the Lemma 2.5 if t is an eigenvariable of the application of the rule and then the inductive hypothesis, we obtain E!t, ⇒ , A[t/x] (E!t, ⇒ , A[t/x]), and then applying the rule again we get E!t, ⇒ , A[t/x].
If on the other hand ∀xA is principal, then the upper sequent of the last application of the rule is already of the required form and with height ≤ n.

Lemma 2.9 (Contraction)
Contraction is height-preserving admissible in G3 pf and G3 nf : i) If n C, C, ⇒ then n C, ⇒ . ii) If n ⇒ , C, C then n ⇒ , C.
Proof Simultaneous for (i) and (ii) by induction on the height of the derivation. The interesting part is when the formula C is principal in R∀ or L∃. So, assume that C is ∀xA and principal in R∀ in the last step. Then the last step of the derivation is ⇒ , ∀xA, ∀xA, derived by R∀ from E!t, ⇒ , ∀xA, A[t/x]. Applying the Lemma 2.8 to that sequent we get E!t, E!t, ⇒ , A[t/x], A[t/x] with the same height of ≤ n − 1. We then apply the inductive hypothesis to obtain E!t, ⇒ , A[t/x] and then R∀ to finally obtain ⇒ , ∀xA. Similar for L∃. Theorem 2.10 Cut is admissible in G3 pf and G3 nf .
Proof By induction on the weight of a formula and subinduction on the sum of heights of the two upper sequents of a cut. Standard for propositional rules, identity rules in G3 pf and the rule = Repl in G3 nf . We check the remaining cases.
When the cut formula is principal in the rule E! the cut has the following form: is an initial sequent we distinguish two cases. First, if 1 ⇒