PUCLogic
Abstract
We present a logic for Proximitybased Understanding of Conditionals (PUCLogic) that unifies the Counterfactual and Deontic logics proposed by David Lewis. We also propose a natural deduction system (PUCND) associated to this new logic. This inference system is proven to be sound, complete, normalizing and decidable. The relative completeness for the and logics is shown to emphasize the unified approach over the work of Lewis.
definitionDefinition cds]Ricardo Queiroz de Araujo Fernandes\fnreffn1 di]Edward Hermann Haeusler df]Luiz Carlos Pereira \fntext[fn1]Thanks to PUCRio for the VRac sponsor. Thanks to DAAD (Germany) for the Specialist Literature Programme.
onditionals, Logic, Natural Deduction, Counterfactual Logic, Deontic Logic
1 Counterfactuals

If Oswald did not kill Kennedy, then someone else did.

If Oswald had not killed Kennedy, then someone else would have.Lewis ()
The phrases above are respectively instances of the indicative and the subjunctive conditionals. The indicative conditional is associated to the material implication, whereas the subjunctive construction of the language is traditionally studied by the philosophy as the counterfactual conditionalLewis (); Goodman () or the counterfactual for short.
Conditional propositions involve two components, the antecedent and the consequent. Counterfactual conditionals differ from material implication in a subtle way. The truth of a material implication is based on the actual stateofaffairs. From the knowledge that Kennedy was killed, we can accept the truth of the phrase. On the other hand, a counterfactual conditional should take into account the truth of the antecedent, even if it is not the case. The truth of the antecedent is mandatory in this analysis.
Some approaches to counterfactuals entail belief revision, particularly those based on Ramsey test evaluation Ramsey (). In this analysis, the truth value of a counterfactual is considered within a minimal change generated by admitting the antecedent trueGoodman ().
A possible way to circumvent belief revision mechanisms is to consider alternative (possible) stateofaffairs, considered here as worlds, and, based on some accessibility notion, choose the closest one among the worlds that satisfy the antecedent. If the consequent is true at this considered world, then the counterfactual is also trueLewis ().
Both conditionals have false antecedents and false consequents in the current stateofaffairs. However, the second conditional is clearly false, since we found no reason to accept that, in the closest worlds in which Kennedy is not killed by Oswald, Kennedy is killed by someone else.
We choose the approach of LewisLewis () in our attempt to formalize an inference system for counterfactuals because his accessibility relation leaves out the discussion for a general definition of similitude among worlds, which is considered as given in his analysis.
It also opened the possibility for a contribution in the other way. If we found some general properties in his accessibility relation, considering the evaluation of the formulas in the counterfactual reasoning, we could sketch some details of the concepts of similitude.
2 Lewis analysis
Lewis, in the very first page of his bookLewis ():
”If kangaroos had no tails, they would topple over, seems to me to mean something like this: in any possible state of affairs in which kangaroos have no tails, and which resembles our actual state of affairs as much as kangaroos having no tails permits it to, the kangaroos topple over.”
We can observe that the word “resemble” may be seen as a reference to the concept of similarity between some possible state of affairs in relation to the actual state of affairs. The expression “as much as” here may be understood as a relative comparison of similarities among the possible states of affairs in relation to the actual state of affairs. But Lewis gave no formal definition of similarity in his bookLewis ().
He defined two basic counterfactual conditional operators:

: If it were the case that A, then it would be the case that B;

: If it were the case that A, then it might be the case that B.
And provided also the definition of other counterfactual operators. But, since they are interdefinable, he took as the primitive for the construction of formulas.
In the middle of his book, he introduced the comparative possibility operators and showed that they can serve as the primitive notion for counterfactuals.

: It is as possible that A as it is that B.
This operator gave us simpler proofs during this work.
He used possibleworld semantics for intentional logic. For that reason the state of affairs are treated as worlds. To express similarity, he used proximity notions: a world is closer to the actual world in comparison to other worlds if it is more similar to the actual world than other considered worlds.
Lewis called the set of worlds to be considered for an evaluation as the strictness of the conditional. He pointed out that the strictness of the counterfactual conditional is based on the similarity of worlds. He showed that the counterfactual could not be treated by strict conditionals, necessity operators or possibility operators given by modal logics. To do so, he argued that strictness of the conditional can not be given before all evaluations. He constructed sequences of connected counterfactuals in a single English sentence for which the strictness cannot be given for the evaluation:
”If Otto had come, it would have been a lively party; but if both Otto and Anna had come it would have been a dreary party; but if Waldo had come as well, it would have been lively; but…”
to show that the strictness of the counterfactuals cannot be defined by the context, because the sentence provides a single context for the evaluation of all counterfactuals. If we try to fix a strictness that makes a counterfactual true, then the next counterfactual is made false.
Lewis proposed a variably strict conditional, in which different degrees of strictness is given for every world before the evaluation of any counterfactual. To express this concept, the accessibility relation is defined by a system of spheres, which is given for every world by a nesting function that applies over a set of worlds . The nested function attributes a set of nonempty sets of worlds for each world and this set of sets is in total order for the inclusion relation.
A systems of spheres, of any kind, is central in the most traditional analysis of counterfactuals. But the idea behind it is also available to many different logics. So, if we manage to handle them in a satisfactory manner, we will be able to use it in a broader class of logics. The system of neighbourhoods facilitates the development of the model, by leaving open the choice for a proper definition of similarity. And that concept can be used for a broader class of logics, not only the counterfactuals.
From Lewis definitions, the nesting function is a primitive notion:
is true at a world (according to a system of spheres ) if and only if either: no world belongs to any sphere in ^{1}^{1}1 gives the neighbourhoods around the world . They are the available strictness to evaluate counterfactuals at ., or some sphere in does contains at least one world, and holds at every world in .
is true at a world (according to a system of spheres ) if and only if, for every sphere in , if contains any world then contains a world.
LewisLewis () also provided conditions that may be applied to the nesting function . To every condition corresponds a different counterfactual logic:

Normality (N): is normal iff ;

Total reflexivity (T): is totally reflexive iff ;

Weak centering (W): is weakly centered iff ;

Centering (C): is centered iff ;

Limit Assumption (L): satisfies the Limit Assumption iff, for any world and any formula , if there is some world^{2}^{2}2A world is a world in which holds. in , then there is some smallest sphere of that contains a world;

Stalnaker’s Assumption (A): satisfies Stalnaker’s Assumption iff, for any world and any formula , if there is some world in , then there is some sphere of that contains exactly one world;

Local Uniformity (U): is locally uniform iff for any world and any , and are the same;

Uniformity (U): is uniform iff for any worlds and , and are the same;

Local absoluteness (A): is locally absolute iff for any world and any , and are the same;

Absoluteness (A): is absolute iff for any worlds and , and are the same.
The logic is the most basic counterfactual logic presented by LewisLewis (), where “V” stands for variably strict conditional. If, for example, we accept the centering condition (C), then we have the logic. Lewis showed in his book a chart of 26 nonequivalent logics that arises from the combinations of the conditions.
We prefer to call the spheres as neighbourhoods, because they represent better the concept of proximity, which Lewis used to express similarity. The neighbourhoods provide a relative way to compare distance. The world that is contained in a neighbourhood is closer to the actual world than other world that is not contained in that same neighbourhood.
As far as we know, there is only one natural deduction system for the counterfactuals, which is given by Bonevac Bonevac (). But his system is designed to deals with the logic, since it contains the rule of counterfactual exploitation (E), which encapsulates the weak centering condition. His approach to define rules for the counterfactual operators provides a better intuition of the counterfactual logic. His systems is expressive enough to deal with modalities and strict conditionals. The labelling of world shifts using formulas make easier to capture the counterfactual mechanics.
We also found the work of Sano Sano () which pointed out the advantages of using the hybrid formalism for the counterfactual logic. He presented some axioms and rules for the logic that extends the logic of Lewis.
Another interesting reference is the article of GentGent (), which presents a new sequent or tableauxstyle proof system for V C. His work depends on the operator and the definition of signed formulas.
We recently found a sequent calculus, provided by LellmannJelia2012 (), that treats the logic of Lewis and its extensions. Its language depends on modal operators, specially the counterfactual operators and and the comparative possibility operator .
As far as we know, our deduction system is the only one dealing with Lewis systems in a general form, that is, without using modalities in the syntax and treating the most basic counterfactual logic.
3 Proximitybased Understanding of Conditionals
In LSFA09 (), we presented a sequent calculus for counterfactual logic based on a Local Set Theory Bell (). In this article, we defined the satisfaction relation for worlds, for sets of the worlds and for neighbourhoods, where we encapsulated some quantifications that made it easier to express the operators with fewer quantifiers. But the encapsulation made the the inference system to have no control of the quantifications. Here we propose a logic for Proximitybased Understanding of Conditionals, PUCLogic for short, that take control of the quantifications with labels.
Given a nonempty set (considered the set of worlds), we define a nesting function that assigns to each world of a set of nested sets of . A set of nested sets is a set of sets in which the inclusion relation among sets is a total order.
A frame is a tuple , in which is a truth assignment function for each atomic formula with image on the subsets of . A model is a pair , a frame and a world of , called the reference world of the model. A template is a pair , and is called the reference neighbourhood of the template.
We use the term structure to refer a model or a template.
A structure is finite if its set of worlds is finite.
We now define a relation between structures to represent the pertinence of neighbourhoods in a neighbourhood system of a world and the pertinence of worlds in a given neighbourhood.
Given a model , then, for any , the template is in perspective relation to . We represent this by . Given a template , then, for any , the model is in perspective relation to . We represent this by .
The concatenation of tuples of the perspective relation is called a path of size and is represented by the symbol .
One remark: if the size of a path is even, then a model is related to another model or a template is related to another template.
The transitive closure of the perspective relation is called the projective relation, which is represented by the symbol .
Given a world and the nested neighbourhood function we can build a sequence of sets of worlds:

;

, .
Let and .
We introduce labels in our language, in order to syntactically represent quantifications over two specific domains: neighbourhoods and worlds. So, for that reason, a label may be a neighbourhood label or a world label:

Neighbourhood labels:

Universal quantifier over neighbourhoods of some neighbourhood system;

Existential quantifier over neighbourhoods of some neighbourhood system;

Variables (capital letters) that may denote some neighbourhood of some neighbourhood system.


World labels:

Universal quantifier over worlds of some neighbourhood;

Existential quantifier over worlds of some neighbourhood;

Variables (lower case letters) that denote some world of some neighbourhood.

We denote the set of neighbourhood labels by and the set of world labels by .
The language of PUCLogic consists of:

countably neighbourhood variables: ;

countably world variables: ;

countably proposition symbols: ;

countably proposition constants: ;

connectives: ;

neighbourhood labels: ;

world labels: ;

auxiliary symbols: .
As in the case of labels, we want to separate the sets of wellformed formulas into two disjoint sets, according to sort of label that labels the formula. We denote the set of neighbourhood formulas by and the set of world formulas by .
The sets and of wellformed formulas^{3}^{3}3We use the term wff to denote both the singular and the plural form of the expression wellformed formula. are constructed the following rules:

;

;

, for every neighbourhood variable ;

, for every atomic formula , except and ;

if , then ;

if , then ;

if , then ;

if , then ;

if and , then ;

if and , then .
We introduced the two formulas for true and false, in order to make the sets of formulas disjoint. The formula is introduced to represent that a neighbourhood contains the neighbourhood and the formula represent a neighbourhood is contained in .
The last two rules of definition 3 introduces the labelling the formulas. Moreover, since we can label a labelled formula, every formula has a stack of labels that represent nested labels. We call it the attribute of the formula. The top label of the stack is the index of the formula. We represent the attribute of a formula as a letter that appear to the right of the formula. If the attribute is empty, we may omit it and the formula has no index. The attribute of some formula will always be empty if the last rule, used to build the formula, is not one of the labelling rules, as in the case of .
To read a labelled formula, it is necessary to read its index first and then the rest of the formula. For example, should be read as: there is some world, in all neighbourhoods of the considered neighbourhood system, in which it is the case that .
We may concatenate stacks of labels and labels, using commas, to produce a stack of labels that is obtained by respecting the order of the labels in the stacks and the order of the concatenation, like , where is a formula and and are stacks of labels. But we admit no nesting of attributes, which means that is the same as .
Given a stack of labels , we define as the stack of labels that is obtained from by reversing the order of the labels in the stack.{definition} Given a stack of labels , the size is its number of labels.{definition} Given a set of worlds , a set of world variables and a set of neighbourhood variables, we define a variable assignment function , that assigns a world of to each world variable and a nonempty set of to each neighbourhood variable.{definition} Given a variable assignment function , the relation of satisfaction between formulas, models and templates is given by:

, atomic, iff: . For every world , and ;

iff: and ;

iff: and
; 
iff: and
; 
iff: and
; 
iff: ;

iff: ;

iff: ;

iff: and ;

iff: and ;

iff: ;

iff: ;

iff: and ;

iff: and ;

iff: and
; 
iff: and
; 
iff: and
; 
and , for every template.
The relation of logical consequence is defined iff and for all model , we have . The relation is also defined iff and for all template , we have . Given , the relation of logical consequence is defined iff for all model that satisfies every formula of , . Given , the relation is defined iff for all templates that satisfies every formula of , .
() is a ntautology (wtautology) iff for every model (template) ().
Lemma 1.
is a ntautology iff is a ntautology.
Proof.
If is a ntautology, , . In particular, given a world , and, by definition, for every world of and is also a ntautology. Conversely, if is a ntautology, then for every choice of , , and . So, given , and , we can choose to be the constant function . So, , and must also be a ntautology.∎
The relation defined below is motivated by the fact that, if a model satisfies a formula like , then for every template , such that , satisfies by definition. And also for every model , such that , satisfies by definition.
Given a model , called the reference model, the relation of referential consequence is defined iff:

and ( implies , for any structure );

and (if implies ).
Given , iff:

and (, for any structure that satisfies every formula of );

and ( if satisfies every formula of ).
\AxiomC \RightLabel \UnaryInfC \LeftLabel1: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel2: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel3: \RightLabel \BinaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \LeftLabel4: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \alwaysNoLine\UnaryInfC[] \RightLabel \alwaysSingleLine\UnaryInfC \RightLabel \UnaryInfC \AxiomC \alwaysNoLine\UnaryInfC[] \RightLabel \alwaysSingleLine\UnaryInfC \RightLabel \UnaryInfC \LeftLabel5: \RightLabel \alwaysSingleLine\TrinaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel6: \RightLabel \UnaryInfC \DisplayProof 
\AxiomC \alwaysNoLine\UnaryInfC[] \RightLabel \alwaysSingleLine\UnaryInfC \RightLabel \UnaryInfC \LeftLabel7: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel8: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel9: \UnaryInfC \DisplayProof 
\AxiomC \LeftLabel10: \RightLabel \UnaryInfC \DisplayProof  \AxiomC \alwaysNoLine\UnaryInfC[] \RightLabel \alwaysSingleLine\UnaryInfC \RightLabel \UnaryInfC \alwaysSingleLine\LeftLabel11: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel12: \RightLabel \BinaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \LeftLabel13: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel14: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel15: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \LeftLabel16: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel17: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \LeftLabel18: \RightLabel \BinaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel19: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \LeftLabel20: \RightLabel \BinaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \LeftLabel21: \RightLabel \UnaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel22: \RightLabel \BinaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel23: \RightLabel \BinaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel24: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof 
\AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel25: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \LeftLabel26: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \LeftLabel27: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof 
\AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \LeftLabel28: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \LeftLabel29: \RightLabel \BinaryInfC \alwaysNoLine\UnaryInfC \DisplayProof  \AxiomC \LeftLabel30: \RightLabel \UnaryInfC \alwaysNoLine\UnaryInfC \DisplayProof 
Every rule of PUCND has a stack of labels, called its context. The scope is represented by a capital Greek letter at the right of each rule. The scope of a rule is the top label of its context. Given a context , we denote its scope by . If the context is empty, then there is no scope. As in the case of labels and formulas, we want to separate the contexts into two disjoint sets: if ; if is empty or .
We say that a wff fits into a context iff .
The wff and fit into the context , because and . The wff and do not fit into the context , because and are not wff and, therefore, cannot be in . There is no wff that fits into the context , because the label and the rule of labelling can only include the resulting formula into . We can conclude that if a wff is in , then the context must be in and the same for and . The fitting restriction ensures that the conclusion of a rule is always a wff.
Moreover, the definition of fitting resembles the attribute grammar approach for context free languages Knuth (). This is the main reason to name the stack of labels of a formula as the attribute of the formula.
Here it follows the names and restrictions of the rules of PUCND:

elimination: (a) and must fit into the context; (b) has no existential quantifier;
The existential quantifier is excluded to make it possible to distribute the context over the operator, what is shown in lemma 8. 
elimination: (a) and must fit into the context; (b) has no existential quantifier;
The existential quantifier is excluded to make it possible to distribute the context over the operator, what is shown in lemma 8. 
introduction: (a) and must fit into the context; (b) has no existential quantifier;
The existential quantifier is excluded because the existence of some world (or neighbourhood) in which some wff holds and the existence of some world in which holds do not implies that there is some world in which and holds. 
introduction: (a) and must fit into the context; (b) has no universal quantifier;
The universal quantifier is excluded to make it possible to distribute the context over the operator, what is shown in lemma 8. 
elimination: (a) and must fit into the context ; (b) has no universal quantifier; The universal quantifier is excluded because the fact that for all worlds (or neighbourhoods) holds does not implies that for all worlds holds or for all worlds holds.

introduction: (a) and must fit into the context; (b) has no universal quantifier;
The universal quantifier is excluded to make it possible to distribute the context over the operator, what is shown in lemma 8. 
classical: (a) and must fit into the context;

intuitionistic: (a) and must fit into the context;

absurd expansion: (a) must have no occurrence of ; (b) must fit into the context; (c) must be non empty.
The symbol is used to denote a formula that may only be or . In the occurrence of , we admit the possibility of an empty system of neighbourhoods. In that context, the absurd does not mean that we actually reach an absurd in our world. must be non empty to avoid unnecessary detours, like the conclusion of from in the empty context; 
hypothesisinjection: (a) must fit into the context.
This rule permits an scope change before any formula change. It also avoids combinatorial definitions of rules with hypothesis and formulas inside a given context; 
introduction: (a) and must fit into the context;

elimination (modus ponens): (a) and must fit into the context; (b) has no existential quantifier; (c) the premises may be in reverse order;
The existential quantifier is excluded because the existence of some world (or neighbourhood) in which some wff holds and the existence of some world in which holds do not implies that there is some world in which holds. 
contextintroduction: (a) and must fit into their contexts;

contextelimination: (a) and must fit into their contexts;

world universal introduction: (a) must fit into the context; (b) must not occur in any hypothesis on which depends; (c) must not occur in the context of any hypothesis on which depends;

world universal elimination: (a) must fit into the context; (b) must not occur in or ;

world existential introduction: (a) must fit into the context;

world existential elimination: (a) the formula must fit into the context; (b) must not occur in , , or any open hypothesis on which depends; (c) must not occur in the context of any open hypothesis on which depends; (d) the premises may be in reverse order;

neighbourhood existential introduction: (a) must fit into the context; (b) the premises may be in reverse order;

neighbourhood existential elimination: (a) the formula must fit into the context; (b) must not occur in , , or any open hypothesis on which depends; (c) must not occur in the context of any open hypothesis on which depends; (d) the premises may be in reverse order;

neighbourhood universal introduction: (a) the formula must fit into the contexts; (b) must not occur in any open hypothesis on which depends; (c) must not occur in the context of any open hypothesis on which depends;

neighbourhood universal wildcard: (a) the formulas and must fit into their contexts; (b) the premises may be in reverse order;
This rule is necessary, because a system of neighbourhood may be empty and every variable must denote some neighbourhood because of the variable assignment function . The wildcard rule may be seen as a permition to use some available variable as an instantiation, by making explicit the choice of the variable. 
world existential propagation: (a) and fit into their contexts; (b) the premises may be in reverse order;

world universal propagation: (a) and fit into their contexts; (b) the premises may be in reverse order;

transitive neighbourhood inclusion: (a) and fit into their contexts; (b) the premises may be in reverse order;

transitive neighbourhood inclusion: (a) and fit into their contexts; (b) the premises may be in reverse order;

neighbourhood total order: (a) , and fit into their contexts; (b) the premises may be in reverse order;

neighbourhood total order: (a) , and fit into their contexts; (b) the premises may be in reverse order;

neighbourhood total order: (a) , and fit into their contexts. (b) the premises may be in reverse order;

truth acceptance: (a) must have no occurrence of ; (b) must fit into the context. The symbol is used to denote a formula that may only be or . If we accepted the occurrence of , the existence of some neighbourhood in every system of neighbourhoods would be necessary and the logic of PUCND should be normal according to Lewis classification Lewis (). must be non empty to avoid unnecessary detours, like the conclusion of from in the empty context.
We present here, as an example of the PUCND inference calculus, a proof of a tautology. Considering Lewis definitions, we understand that if there is some neighbourhood that has some world but no world, then, for all neighbourhoods, having some world implies having some world. The reason is the total order for the inclusion relation among neighbourhoods.
\AxiomC[] \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \UnaryInfC \LeftLabel3 \BinaryInfC \LeftLabel4 \UnaryInfC \alwaysNoLine\UnaryInfC \UnaryInfC \DisplayProof
\AxiomC\RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \BinaryInfC \RightLabel \UnaryInfC \AxiomC \RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \BinaryInfC \RightLabel \UnaryInfC \RightLabel \UnaryInfC \AxiomC[] \RightLabel \UnaryInfC \RightLabel \UnaryInfC \RightLabel \UnaryInfC \RightLabel \BinaryInfC \RightLabel \UnaryInfC \RightLabel \UnaryInfC \RightLabel \UnaryInfC \RightLabel \LeftLabel1 \UnaryInfC \LeftLabel2 \RightLabel \BinaryInfC \RightLabel \UnaryInfC \LeftLabel \UnaryInfC \DisplayProof
Lemma 2.
If , then is odd. If , then is even.
Proof.
By definition, if is empty, then and is even. According to the rules of the PUCND, if is empty, it can only accept an additional label , then and is odd. We conclude that changing the context from to and viceversa always involves adding one to the size of the label and the even sizes are only and always for contexts in .∎
4 PUC Soundness and Completeness
For the proof of soundness of PUCLogic, we prove that the PUCND derivations preserves the relation of resolution, which is a relation that generalizes the satisfability relation. To do so, we need to prove some lemmas. In many cases we use the definition 3 of the referential consequence relation.
Given a model , a context and a wff , the relation of resolution is defined iff fits into the context and . If or , then if the resolution relation holds for every formula of .
Lemma 3.
Given a model , if and , then .
Proof.
If is empty (), the resolution gives us . From we know that if and, by the definition of resolution, ;
If (), then, by definition, means and for every template , such that , .
represent all neighbourhoods of . From , we know that and, by definition, we can change by in all endpoints of the directed graph and conclude and ;
If (), then .
represent all neighbourhoods of such that holds. We know that there is at least one of such neighbourhoods. From , we can change by in all endpoints and conclude because we know that there is at least one of such downward paths. By definition, ;
If (), then .
From , we change by in the endpoint and conclude . By definition, ;
If (), then .
represent all neighbourhoods of . represent all worlds of . From , we can change by in all endpoints and conclude . By definition, ;
If (), then .
represent all neighbourhoods of . represent all worlds of in which holds. We know that there is at least one of these worlds. From , we can change by in all endpoints and conclude and ;
If (), then .
represent all neighbourhoods of . From , we can change by in the endpoint and conclude . So, by definition, ;
Any combination of labels follows, by analogy, the same arguments for each label presented above.∎
Lemma 4.
Given a model , if and , then .
Proof.
We follow the argument of lemma 3, by changing by in all endpoints, what is possible by the definition of logical consequence. ∎
Lemma 5.
Given without universal quantifiers, if is wff, then .
Proof.
We proceed by induction on the size of :
If is empty, then equivalence is true;
(base) If contains only one label, it must be a neighbourhood label:

may be read as or . But implies, by definition, . Then we have or . Since the neighbourhood variables are bound, we have , which is represented whit labels as . Then implies . On the other hand, may be read as , which means, by definition, . In the first case, , which may be read as . In the second case, , which may be read as . Since we have one or the other case, we have . So, ;

may be read as and or and . Then we have and ( or ), which is, by definition,