1 Introduction
Proof theory of modal logics is a subtle subject, and if a sequent calculus presentation is complex, natural deduction systems are even more difficult. The source of this problem is well highlighted in Dag Prawitz’s foundational book [Prawitz:1965].
One of the most successful prooftheoretical formulations of modal logics are the labelled systems of [Vigano00a, Simpson93, Negri:2011]. which extend ordinary natural deduction by explicitly mirroring in the deductive apparatus the accessibility relation of Kripke models (see also [MVVJANCL2101, MVVJLC2011, MasiniViganoZorzi08, MVVENTCS2010, MVZjmvl, Baratella2019, Baratella2004a, BaMaJANCL13]). In a sense, they may look like a formalization of Kripke semantics in a firstorder deductive fashion (see Section 9.1, below, for a more complete discussion).
Differently from the labelled systems cited above, the aim of our paper is to define natural deduction systems for modal logics that do not explicitly deal with the accessibility relation.
Our leading idea is to extend in a geometrical way the standard natural deductive systems for classical and intuitionistic logic, in order to treat modalities as first order quantifiers are treated in those systems. In doing this we refine and extend to natural deduction some recent proposals by Lellmann and others for sequent calculi for modal logics [Pimentel2019, Lellmann2019] (see later in this introduction).
Our proposal in a nutshell
We add to formulas a kind of spatial coordinates, that we call positions, to adapt to natural deduction the paradigm of 2–Sequents by Masini et al., and Linear–Nested–Sequents by Lellman et al. Main features of our systems are the following:

there is exactly one introduction and one elimination rule for each modal connective;

rules for modal connectives have the same shape as those of first order quantifiers;

no formalization of the first order translation of modal logic formulas is present at the level of deduction rules (hence no formalization of the accessibility relation appears);

a notion of proof reduction is given and a property of normalization is provided, following the standard definitions and techniques for natural deduction systems;

only modal operators can change the spatial positions of formulas.
We underline that, as was the case for 2–Sequents and Linear–Nested–Sequents, a specific goal is not to explicitly embed the notion of accessibility relation, thus equipping the formal systems with adhoc deductive rules (see also Section 9.1).
A short history
To fully understand our proposal is useful to frame it “historically”, and to go back to 2–Sequents, originally formulated in [Mas:TwoSeqInt:93, Mas:TwoSeqProof:92]. The main idea of these works was to add a second dimension to ordinary propositional sequents. Each formula in a 2Sequent lives at a level (that could be seen as a natural number).
Such a proposal was later extended and generalized to a natural deduction setting. Formulas become indexed formulas, i.e. pairs of formulas and natural numbers, where numbers correspond explicitly to levels in 2–Sequents. Such an idea works fine for the negative free fragments of the modal logics K, T, K4 and S4, and for the corresponding MELL (Multiplicative Exponential Linear Logic) subsystems [MM:ComInt:95, MM:ONFineStr:95].
At the time we presented such systems, it was not possible to extend them to full modal logics from K to S4, since the simple notion of level of a formula does not interact well with reduction in presence rules.
The problem, instead, does not show up if, instead of natural deduction, we consider 2–Sequents—see e.g. [GueMarMasPNGC2001, GuMarMasCoher2003, GMM:anexpextseq] where the authors show how 2–Sequents are a suitable framework to deal with full MELL (and other linear systems) both in sequent calculi, and proof nets.
More recently, the approach based on 2–Sequents has been extended in order to deal with linear and branching time time temporal logics [BaMaJANCL13, BarMas:apal]. In particular for temporal logics it was necessary to properly extend the notion of level of formulas, since natural numbers do not suffice.
Finally, the paradigm of 2–Sequents has been reformulated by Lellman and coauthors, under the name of Linear–Nested–Sequents [Lellmann2019, Pimentel2019], in order to deal with an interesting class of modal logics.
Unfortunately, 2–Sequents/Linear–Nested–Sequents cannot be directly translated in a natural deduction setting, since the simple decoration of formulas with natural numbers does not agree with the obvious definition of reduction. To overcome these problems, the simple (simplistic) notion of level has to be generalised to that of position, a kind of spatial coordinate for each formula.
Content of the paper
The paper deals with the normal modal logics varying from K to S4. We give both a classical and intuitionistic formulation, for which we prove soundness and completeness with respect to the axiomatic formulation, passing through a suitable Kripke style semantics. For the intuitionistic versions, we define a notion of reduction and we give a syntactical proof of normalization, along the lines of the analogous proof for standard natural deduction. As usual, consistency is obtained syntactically, as a byproduct of normalization. We conclude with a more detailed discussion of the labelled systems approach to modal natural deduction and some considerations about obtained results and future work.
2 Preliminary Notions
As mentioned in the Introduction, formula occurrences will be labeled with positions—sequences of uninterpreted tokens. We introduce here the notation and operations that will be needed for such notions.
Given a set , is the set of ordered finite sequences on . With we denote the finite non empty sequence s.t. ; is the empty sequence.
The (associative) concatenation of sequences is defined as

,

.
For and , we sometimes write for ; and as a shorthand for . The set is equipped with the following successor relation
We use the following notations:

denotes the reflexive closure of ;

denotes the transitive closure of ;

denotes the reflexive and transitive closure of ;
Given three sequences the prefix replacement is so defined
When and have the same length, the replacement is called renaming of with .
3 Natural deduction calculi
The propositional modal language contains the following symbols:

countably infinite proposition symbols, ;

the propositional connectives

the modal operators

the auxiliary symbols and
As usual, is a shorthand for .
Definition 3.1.
The set of propositional modal formulas of is the least set that contains the propositional symbols and is closed under application of the propositional connectives and the modal operators. A formula is atomic if it is a propositional symbol, or the connective .
In the following denotes a denumerable set of tokens, ranged by metavariables , possibly indexed. Let be the set of the sequences on , called positions; metavariables range on , possibly indexed.
Definition 3.2.
A positionformula (briefly pformula) is an expression of the form , where is a modal formula and . We denote with the set of position formulas.
Given a sequence of pformulas, is the set of prefixes of the positions in :
.
3.1 A class of normal modal systems
We briefly recall the axiomatic (“Hilbertstyle”) presentation of normal modal systems. Let be a set of formulas. The normal modal logic is defined as smallest set of formulas verifying the following properties:
 (i)

 (ii)

contains all instances of the following schemas:
 1.

 2.

 3.

 K.

 MP

if then ;
 NEC

if then .
We write for . If are names of schemas, the sequence denotes the set , where . Figure 1 lists the standard axioms for the wellknown modal systems K, D, T, K4, S4; we use as a generic name for one of these systems.
Axiom schema  Logic 

D T 4  K D T K4 S4 
We will call D, T and S4 total modal logics, since in their well known Kripke semantics the accessibility relation is total. Instead, we will call K and K4 partial modal logics.
4 Natural Deduction Systems
In this section we define natural deduction systems for the class of logics we previously introduced.
4.1 Total logics
We start by defining the system
The set of derivations from a set of assumptions is defined as the least set that contains and is closed under application of the following rules (where, as usual, a formula into square brackets represents a discharged assumption):
Logical rules
In is atomic; moreover, when is we require
In the rule , one has , where is the set of (open) assumptions on which depends.
In the rule , one has , where is the set of (open) assumptions on which depends, with the exception of the discharged assumptions .
It is easy to show the admissibility of the following rule, where the requirement of atomicity of the conclusion is removed:
for .
On the basis of , the natural deduction systems for the logics T and D can be obtained by imposing suitable constraints on the application of and rules, as shown in the following table.
name of the calculus  constraints on the rules and 

no constraints  
is a singleton sequence 
Let be one of , , ; as usual we write if there is a deduction in with conclusion , whose non discharged assumptions appear in .
Definition 4.1 (Proper Token).
We refer to the token that explicitly appears in any of the rules , as to the proper token of the corresponding rule. We say that a token is proper in a derivation if it is the proper token of some , rule in the derivation.
By token renaming we can we can prove the following (see [tvd1988, Vol. 2, pag. 529] for the analogous proof for proper variables):
Proposition 4.2.
Let . Then there exists a deduction of from in the system such that

each proper token is the proper token of exactly one instance of or rule;

the proper token of any instance of rule occurs only in the subderivation above that instance of the rule;

the proper token of any instance of rule occurs only in the subderivation above the minor premiss of that instance of the rule.
Definition 4.3 (Token condition).
A deduction satisfying conditions 1–3 of Proposition 4.2 is said to satisfy the token condition.
By Proposition 4.2 we can always assume that all deductions satisfy the token condition.
We denote by the tree obtained by replacing each position in a deduction with .
Remark 4.1.
If the following holds:

is a deduction satisfying the token condition;

is a token that is not a proper token of

is a position not containing any proper token of
then is a deduction satisfying the token condition.
Notice that if the last rule of is , and the last formula is for some , it might be the case that, after the token substitution, the side condition of this application is no longer satisfied (that is, its premise and conclusion are both , for the same ). In such a case by we mean the deduction obtained by deleting, after the substitution, the last—incorrect—application of
We want to make sense of the deduction even when the conditions of Remark 4.1 are not satisfied. Notice that if is a deduction satisfying the token condition, we can replace all proper tokens in by new tokens, to obtain a deduction of the same formula from the same assumptions, and such that and satisfy all the conditions of Remark 4.1. Hence we define as this . In the sequel we will implicitly assume that by we actually mean for some as above.
4.2 Weak Completeness
We prove a Weak Completeness theorem passing through some auxiliary results.
Proposition 4.4.

Let be one of the systems , , : ;

Let be one of the systems , , : ;

Let be one of the systems , : ;

Let be one of the systems , : ;

;
Proof.
∎
Closure under NEC is obtained by showing that all positions in a provable sequent may be “lifted” by any prefix. Observe first that, for , we have .
Proposition 4.5 (lift).
Let be one of the systems , , , and let be a position. If , then .
Proof.
Standard induction on derivation (with suitable renaming of proper tokens). It is easily verified that the constraints on the modal rules remain satisfied. ∎
Corollary 4.6.
Let be one of the systems , , .
If , then
.
Finally, closure under MP is trivially ensured by rule .
Theorem 4.7 (weak completeness).
Let be one of the modal systems D, T, S4. If , then .
5 Partial logics
The treatment of partial logics K and K4 is delicate and requires the introduction of auxiliary notions to soundly define their formal system and prove prooftheoretic results. To motivate the formal systems for K and K4, it could be useful to anticipate that, in the semantics we will define in Section 6, positions will be mapped into nodes of a Kripke structure.
Recall that both K and K4 are complete with respect to the class of models where the accessibility relation is not always defined. This means that the correspondence between positions and nodes could be undefined at some position, a situation reminiscent of the case of first order logic with undefined terms^{1}^{1}1The formal analogy between variables/terms and tokens/positions (and hence between quantifiers and modalities) is one of the leitmotive of the 2sequents approach, as it is evident from the shape of the modal rules.. In fact, we will treat this case with an existence predicate for positions, a tool introduced by D. Scott in the late seventies [Scott1979] to deal with empty domains, and therefore with partially defined terms. For a first order logic term , the predicate has the following intuitive meaning: is defined^{2}^{2}2For an extensive treatment of existence predicates for first order natural deduction, see the two volumes [tvd1988], or the review [BaazIemhoff2005]..
The natural deduction systems introduced in the previous section are now expanded with formulas of the form , where is any position and which we informally read as: denotes an existing node/object. Such formulas may be used only as premises in deductions. The only modified rules w.r.t. the previously introduced formal system are the modal ones.
Rules for are the following:
where in the rule , , where is the set of (open) assumptions on which depends.
Rules for are the following:
where in rule , , where is the set of (open) assumptions on which depends, with the exception of the discharged assumptions .
These “generic” rules are further constrained to take into account the specifics of the systems K4 and K. The following table gives such constraints for the systems and .
name of the calculus  constraints on the rules and 

is a non empty sequence  
is a singleton sequence 
5.1 Weak Completeness
We prove a Weak Completeness Theorem also for partial logics.
Proposition 5.1.

Let be one of the systems , ;

Let be one of the systems , ;

;
Proof.
In the following derivations, observe the interplay between modal introduction and elimination, which allows to discharge all existence predicates.
∎
Closure under NEC and under MP is shown in the same manner as for the total systems. Therefore:
Theorem 5.2 (weak completeness).
Let be one of the modal systems K and K4. If then .
6 Semantics
We introduce in this section a treebased Kripke semantics for our modal systems, in order to prove their completeness with respect to the standard axiomatic presentations.
6.1 Trees and Treesemantics
Let be the set of finite sequences of natural numbers with the partial order as defined in Section 2.
Definition 6.1.
A tree is a subset of s.t. ; and if and , then where is the restriction of to .
The elements of are called nodes; a leaf is a node with no successors. Given a tree and , we define (the subtree of rooted at ) to be the tree defined as: . Observe that . In this section, and will range over the generic elements (nodes) of .
If is the set of proposition symbols of our modal language, a Kripke model is a triple , where is a tree, is an assignment of proposition symbols to nodes, and . Given a modal system , a model is a Kripke model s.t.
modal system  conditions on  conditions on 

K  no condition  
D  does not have leaves  
T  no condition  
K4  no condition  
S4  no condition 
The satisfiability relation of formulas on a Kripke model is standard; e.g., for a model and node , . As usual, we write , when for all nodes of .
Theorem 6.2 (standard completeness).
For each modal system in K, D, T, K4, S4, and for every formula , for all –model , we have .
6.2 Semantics: Total logics
Definition 6.3 (Structures).
Let be a modal system. A structure is a pair where:

is an –model

is a map from positions to nodes (the evaluation).

Moreover for , and for a fixed , with we denote an evaluation .
Depending on the specific modal system, has to satisfy the following, additional constraints:
modal system  conditions on 

D  
T  
S4 
The satisfiability relation between a 2structure and a position formula is defined in the following way:
where is the standard satisfiability relation w.r.t. modal Kripke semantics.
Finally, given a modal system , we define the notion of logical consequence for positions formulas. Let be one of the systems :
We now introduce some notation for the semantical substitution of values into the evaluation functions , in correspondence of specific subtrees. For and , define
We define the following set of elements:

;

;

.
As for other notations, we will write for any of these sets.
Let us now fix a specific structure ; we have the following.
Lemma 6.4.
Let .
Proof.
∎
Let in a tree , we define the subtraction operation between nodes as:
Lemma 6.5.
Let be an evaluation s.t. , then
Proof.
Observe that
;
therefore
∎
We are finally in the position to prove the soundness theorem, by an easy induction on proofs which—we remark once again—strictly mimics the standard proof of soundness for first order natural deduction. In the rest of the paper with we denote the set of undischarged hypoteses of the deduction . We write
for is a deduction of formula whose last rule is .
Theorem 6.6 (soundness 1).
Let be a modal system.
If then .
Proof sketch..
Let and assume that in
We prove by induction on the length of , for each such that , that . We discuss only the cases where is or .
 ()

Let be
We observe first that the rule is the same for all the systems under consideration, and that , with .
By IH we have:
(by the genericity of )
(since