Cut--free sequent calculus and natural deduction for the tetravalent modal logic
aa r X i v : . [ m a t h . L O ] J a n Cut–free sequent calculus and natural deductionfor the tetravalent modal logic
Mart´ın Figallo
Departamento de Matem´atica. Universidad Nacional del Sur. Bah´ıa Blanca,Argentina
Abstract
The tetravalent modal logic ( T ML ) is one of the two logics definedby Font and Rius ([13]) (the other is the normal tetravalent modal logic
T ML N ) in connection with Monteiro’s tetravalent modal algebras. Theselogics are expansions of the well–known Belnap–Dunn’s four–valued logic that combine a many-valued character (tetravalence) with a modal char-acter. In fact,
T ML is the logic that preserve degrees of truth withrespect to tetravalent modal algebras. As Font and Rius observed, theconnection between the logic
T ML and the algebras is not so good as in
T ML N , but, as a compensation, it has a better proof-theoretic behavior,since it has a strongly adequate Gentzen calculus (see [13]). In this work,we prove that the sequent calculus given by Font and Rius does not enjoythe cut–elimination property. Then, using a general method proposed byAvron, Ben-Naim and Konikowska ([4]), we provide a sequent calculus for T ML with the cut–elimination property. Finally, inspired by the latter,we present a natural deduction system, sound and complete with respectto the tetravalent modal logic.
The class
TMA of tetravalent modal algebras was first considered by Anto-nio Monteiro (1978), and mainly studied by I. Loureiro, A.V. Figallo, A. Zilianiand P. Landini. Later on, J.M. Font and M. Rius were interested in the log-ics arising from the algebraic and lattice–theoretical aspects of these algebras.From Monteiro’s point of view, in the future these algebras would give rise to afour-valued modal logic with significant applications in Computer Science (see[13]). Although such applications have not yet been developed, the two logicsconsidered in [13] are modal expansions of Belnap-Dunn’s four-valued logic, alogical system that is well–known for the many applications it has found inseveral fields. In these logics, the four non-classical epistemic values emerge:1 (true and not false), 0 (false and not true), n (neither true nor false) and1 (both true and false). We may think of them as the four possible ways inwhich an atomic sentence P can belong to the present state of information : wewere told that (1) P is true (and were not told that P is false); (2) P is false(and were not told that P is true); (3) P is both true and false (perhaps fromdifferent sources, or in different instants of time); (4) we were not told anythingabout the truth value of P . In this interpretation, it makes sense to consider amodal-like unary operator (cid:3) of epistemic character, such that for any sentence P , the sentence (cid:3) P would mean “the available information confirms that P is true”. It is clear that in this setting the sentence (cid:3) P can only be true inthe case where we have some information saying that P is true and we haveno information saying that P is false, while it is simply false in all other cases(i.e., lack of information or at least some information saying that P is false,disregarding whether at the same time some other information says that P istrue); that is, on the set { , n , b , } of epistemic values this operator must bedefined as (cid:3) (cid:3) n = (cid:3) b = (cid:3) { } asdesignated set, that is, ψ follows from ψ , . . . , ψ n in this logic when every inter-pretation that sends all the ψ i to 1 also sends ψ to 1. The other logic, denoted by T ML (the logic we are interested in), is defined by using the preserving degreesof truth scheme, that is, ψ follows from ψ , . . . , ψ n when every interpretationthat assigns to ψ a value that is greater or equal than the value it assigns to theconjunction of the ψ i ’s. These authors proved that T ML is not algebraizablein the sense of Blok and Pigozzi, but it is finitely equivalential and protoalge-braic . However, they confirm that its algebraic counterpart is also the class ofTMAs: but the connection between the logic and the algebras is not so goodas in the first logic. As a compensation, this logic has a better proof-theoreticbehavior, since it has a strongly adequate Gentzen calculus (Theorems 3.6 and3.19 of [13]).In [13], it was proved that
T ML can be characterized as a matrix logicin terms of two logical matrices, but later, in [9], it was proved that
T ML can be determined by a single logical matrix. Besides, taking profit of thecontrapositive implication introduced by A. V. Figallo and P. Landini ([11]), asound and complete Hilbert-style calculus for this logic was presented. Finally,the paraconsistent character of
T ML was also studied from the point of view ofthe
Logics of Formal Inconsistency , introduced by W. Carnielli and J. Marcosin [8] and afterward developed in [7].
Recall that, a
De Morgan algebra is a structure h A, ∧ , ∨ , ¬ , i such that h A, ∧ , ∨ , i is a bounded distributive lattice and ¬ is a De Morgan negation, i.e.,an involution that additionally satisfies De Morgan’s laws: for every a, b ∈ A ¬¬ a = a ( a ∨ b ) = ¬ a ∧ ¬ b. A tetravalent modal algebra (TMA) is an algebra A = h A, ∧ , ∨ , ¬ , (cid:3) , i oftype (2 , , , ,
0) such that its non-modal reduct h A, ∧ , ∨ , ¬ , i is a De Morganalgebra and the unary operation (cid:3) satisfies, for all a ∈ A , the two followingaxioms: (cid:3) a ∧ ¬ a = 0 , ¬ (cid:3) a ∧ a = ¬ a ∧ a. Every TMA A has a top element 1 which is defined as ¬
0. These algebraswere studied mainly by I. Loureiro ([14]), and also by A. V. Figallo, P. Landini([11]) and A. Ziliani, at the suggestion of the late A. Monteiro (see [13]). Theclass of all tetravalent modal algebras constitute a variety which is denoted by
TMA . Let M = { , n , b , } and consider the lattice given by the followingHasse diagram 1 n b L4 (See [1], pg. 516.) Then, TMA isgenerated by the above four–element lattice enriched with two unary operators ¬ and (cid:3) given by ¬ n = n , ¬ b = b , ¬ ¬ (cid:3) is defined as: (cid:3) n = (cid:3) b = (cid:3) (cid:3) M m , has two prime filters, namely, F n = { n , } and F b = { b , } . As we said, M m generates the variety TMA , i.e., an equationholds in every TMA iff it holds in M m . Lemma 2.1 (See [13]) In every TMA A and for all a, b ∈ A the following hold: (i) ¬ (cid:3) a ∨ a = 1 , (viii) (cid:3)(cid:3) a = (cid:3) a , (ii) (cid:3) a ∨ ¬ a = a ∨ ¬ a , (ix) (cid:3) ( a ∧ b ) = (cid:3) a ∧ (cid:3) b , (iii) (cid:3) a ∨ ¬ (cid:3) a = 1 , (x) (cid:3) ( a ∨ (cid:3) b ) = (cid:3) a ∨ (cid:3) b , (iv) (cid:3) a ∧ ¬ (cid:3) a = 0 , (xi) (cid:3) ¬ (cid:3) a = ¬ (cid:3) a (v) (cid:3) a ≤ a , (xii) a ∧ (cid:3) ¬ a = 0 , (vi) (cid:3) , (xiii) (cid:3) ( (cid:3) a ∧ (cid:3) b ) = (cid:3) a ∧ (cid:3) b (vii) (cid:3) , (xiv) (cid:3) ( (cid:3) a ∨ (cid:3) b ) = (cid:3) a ∨ (cid:3) b The next proposition will be needed in what follows.3 roposition 2.2
Let A be a TMA. If x ≤ y ∨ z and x ∧¬ z ≤ y , then x ≤ y ∨ (cid:3) z ,for every x, y, z ∈ A . Proof.
It is a routine task to check that the assertion holds in M m . The factthat M m generates the variety TMA completes the proof. (cid:4)
Let L = {∨ , ∧ , ¬ , (cid:3) } be a propositional language. From now on, we shall de-note by Fm = h F m, ∧ , ∨ , ¬ , (cid:3) , ⊥i the absolutely free algebra of type (2,2,1,1,0)generated by some denumerable set of variables. We denote by F m the setof sentential formulas, and we shall refer to them by lowercase Greek letters α, β, γ, . . . and so on; and we shall denote finite sets of formulas by uppercaseGreek letters Γ , ∆ , etc. Definition 2.3
The tetravalent modal logic
T ML defined over Fm is the propo-sitional logic h F m, | = T ML i given as follows: for every finite set Γ ∪ { α } ⊆ F m , Γ | = T ML α if and only if, for every A ∈ TMA and for every h ∈ Hom ( Fm , A ) , V { h ( γ ) : γ ∈ Γ } ≤ h ( α ) . In particular, ∅ | = T ML α if and only if h ( α ) = 1 forevery A ∈ TMA and for every h ∈ Hom ( Fm , A ) . Remark 2.4
Observe that, if h ∈ Hom ( Fm , A ) for any A ∈ TMA , we havethat h ( ⊥ ) = 0 . This follows from the fact that ⊥ is the -ary operation in Fm , is the -ary operation in A and the definition of homomorphism (in the senseof universal algebra). Let M = hT , D , Oi be a logical matrix for L , that is, T is a finite, non-empty set of truth values, D is a non-empty proper subset of T , and O includesa k -ary function ˆ f : T k → T for each k -ary connective f ∈ L . Recall that, avaluation in M is a function v : F m → T such that v ( f ( ψ , . . . , ψ k )) = ˆ f ( v ( ψ ) , . . . , v ( ψ k ))for each k -ary connective f and all ψ , . . . , ψ k ∈ F m . A formula α ∈ F m issatisfied by a given valuation v , in symbols v | = α , if v ( α ) ∈ D . Let Γ , ∆ ⊆ F m .We say that the ∆ is consequence of Γ, denoted Γ | = M ∆, iff for every valuation v in M , either v does not satisfy some formula in Γ or v satisfies some formulain ∆.J. M. Font and M. Rius proved in [13] that the tetravalent modal logic T ML isa matrix logic defined in terms of two logical matrices. But later, M. E. Coniglioand M. Figallo proved in [9] that
T ML can be characterized as a matrix logicin terms of a single logical matrix. Indeed, let M = hT , D , Oi be the matrixwhere the set of truth values is T = { , n , b , } , the set of designated values is D = { b , } and O = { ˜ ∨ , ˜ ∧ , ˜ ¬ , ˜ (cid:3) } where ˜ ∨ , ˜ ∧ : T → T and ˜ ¬ , ˜ (cid:3) : T → T aredefined as x ˜ ∨ y = Sup { x, y } , x ˜ ∧ y = Inf { x, y } (here we are assuming that theelements of T are ordered as in the lattice M ).4 ˜ ¬ x ˜ (cid:3) x n n b b
01 0 1then,
Proposition 2.5 ([9])
T ML is sound and complete w.r.t. M . Therefore, given Γ and ∆ sets of formulas, ∆ is consequence of Γ in T ML ,denoted Γ | = T ML ∆, iff for every valuation v in M , either v does not satisfysome formula in Γ or v satisfies some formula in ∆. If ∆ is a set with exactlyone element, we recover the consequence relation given in Definition 2.3.In order to characterize T ML syntactically, that is, by means of a deductivesystem, J. M. Font and M. Rius introduced in [13] the sequent calculus G . Thesequent calculus G is single–conclusion, that is, it deals with sequents of theform ∆ ⇒ α such that ∆ ∪ { α } is a finite subset of F m . The axioms and rulesof G are the following: Axioms (Structural axiom) α ⇒ α (Modal axiom) ⇒ α ∨ ¬ (cid:3) α Structural rules (Weakening) ∆ ⇒ α ∆ , β ⇒ α (Cut) ∆ ⇒ α ∆ , α ⇒ β ∆ ⇒ β Logic rules ( ∧ ⇒ ) ∆ , α, β ⇒ γ ∆ , α ∧ β ⇒ γ ( ⇒ ∧ ) ∆ ⇒ α ∆ ⇒ β ∆ ⇒ α ∧ β ( ∨ ⇒ ) ∆ , α ⇒ γ ∆ , β ⇒ γ ∆ , α ∨ β ⇒ γ ( ⇒ ∨ ) ∆ ⇒ α ∆ ⇒ α ∨ β ( ⇒ ∨ ) ∆ ⇒ β ∆ ⇒ α ∨ β ( ¬ ) α ⇒ β ¬ β ⇒ ¬ α ( ⊥ ) ∆ ⇒ ⊥ ∆ ⇒ α ( ¬¬ ⇒ ) ∆ , α ⇒ β ∆ , ¬¬ α ⇒ β ( ⇒ ¬¬ ) ∆ ⇒ α ∆ ⇒ ¬¬ α ( (cid:3) ⇒ ) ∆ , α, ¬ α ⇒ β ∆ , α, ¬ (cid:3) α ⇒ β ( ⇒ (cid:3) ) ∆ ⇒ α ∧ ¬ α ∆ ⇒ α ∧ ¬ (cid:3) α G is the usual. Besides, forevery finite set Γ ∪ { ϕ } ⊆ F m , we write Γ ⊢ G ϕ iff the sequent Γ ⇒ ϕ has aderivation in G . We say that the sequent Γ ⇒ ϕ is provable iff there exists aderivation for it in G .J. M. Font and M. Rius proved in [13] that G is sound and complete with respectto the tetravalent modal logic T ML . Theorem 2.6 (Soundness and Completeness, [13])
For every finite set Γ ∪{ α } ⊆ F m , Γ | = T ML α if and only if Γ ⊢ G α. Moreover,
Proposition 2.7 ([13])
An arbitrary equation ψ ≈ ϕ holds in every TMA iff ψ ⊣⊢ G ϕ (that is, ψ ⊢ G ϕ and ϕ ⊢ G ψ ). As a consequence of it we have that:
Corollary 2.8 ([13])(i)
The equation ψ ≈ holds in every TMA iff ⊢ G ψ . (ii) For any ψ, ϕ ∈ F m , ψ ⊢ G ϕ iff h ( ψ ) ≤ h ( ϕ ) for every h ∈ Hom ( Fm , A ) , for every A ∈ TMA . G does not admit a cut–elimination theorem Corollary 2.8 is a powerful tool to determine whether a given sequent of G isprovable or not. For instance, Proposition 3.1 In G we have that the sequent ¬ (cid:3) α ⇒ α is provable iff thesequent ⇒ α is provable. Proof.
Indeed, suppose that the sequent ¬ (cid:3) α ⇒ α is provable in G . Then, h ( ¬ (cid:3) α ) ≤ h ( α ), for all h ∈ Hom ( Fm , M m ). But, considering all the cases,we must have that h ( ¬ (cid:3) α ) = 0 and h ( α ) = 1, for all h , and therefore thesequent ⇒ α is provable in G . The converse is straightforward. (cid:4) Recall that a rule of inference is admissible in a formal system if the set oftheorems of the system is closed under the rule; and a rule is said to be derivable in the same formal system if its conclusion can be derived from its premises usingthe other rules of the system. 6 well–known rule for readers familiar with modal logic is the
Rule of Necessi-tation , which states that if ϕ is a theorem, so is (cid:3) ϕ . Formally,(Nec) ⇒ ϕ ⇒ (cid:3) ϕ Then, we have that:
Lemma 3.2
The Rule of Necessitation is admissible in G . Proof.
From Corollary 2.8 and considering the algebra M m . (cid:4) From the above lemma, we can obtain a proof of ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) in G , forany α ∈ F m . Let Π be a proof of ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) and let ( r ) be the last ruleapplication in Π. Clearly, Π make use of more than one rule since (cid:3) ( α ∨ ¬ (cid:3) α )is not an axiom. Then, we have the following two cases:Case 1: Π is of the form ··· Γ ⇒ ϕ (r) ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) Case 2: Π is of the form ··· Γ ⇒ ϕ Γ ⇒ ϕ (r) ⇒ (cid:3) ( α ∨ ¬ (cid:3) α )In case 1, ( r ) has just one premise, and therefore it can be: ( ⊥ ), weakening,( ∧ ⇒ ), ( ∨ ⇒ ), ( ⇒ ∨ ), ( ¬ ), ( ¬¬ ⇒ ), ( ⇒ ¬¬ ), ( (cid:3) ⇒ ) or ( ⇒ (cid:3) ). In the caseof ( ⊥ ), the only possibility is having Γ = ∅ . But this would imply that thesequent ⇒ ⊥ is provable, which contradicts the soundness of G . Thus, this caseis discarded. On the other hand, none of the other rules above has the structureof ( r ), so they are also discarded.Therefore, π is of the form depicted in Case 2. Then, ( r ) must be one of thefollowing: the cut rule, ( ⇒ ∧ ) or ( ∨ ⇒ ). It is clear that ( r ) cannot be ( ⇒ ∧ )nor ( ∨ ⇒ ). Consequently, ( r ) must be the cut rule.We have just proved, therefore, the following assertion. Proposition 3.3
Every proof of ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) in G uses the cut rule. Moreover, we have that:
Lemma 3.4
For every ϕ ∈ F m such that ⇒ ϕ is provable in G , we have that ⇒ (cid:3) ϕ is provable in G ; and every proof of ⇒ (cid:3) ϕ in G makes use of the cutrule. Consequently,
Theorem 3.5 G does not admit cut–elimination. The general method of Avron, Ben-Naim andKonikowska
In [3], A. Avron and B. Konikowska use the Rasiowa-Sikorski decompositionmethodology to get sound and complete proof systems employing n -sequents forall propositional logics based on non-deterministic matrices. Later, these sameauthors jointly with J. Ben-Naim ([4]) presented a general method to transforma given sound and complete n -sequent proof system into an equivalent soundand complete system of ordinary two-sided sequents (for languages satisfying acertain minimal expressiveness condition). In this section we shall recall bothmethods considering ordinary (deterministic) matrices.In what follows, L is a propositional language and let (in this section) Fm be the absolutely free algebra over L generated by some denumerable set ofvariables, with underlying set (of formulas) F m . Let M = hT , D , Oi be alogical matrix for L . As we said, a valuation v in M satisfies a given formula α if v ( α ) ∈ D . A sequent Γ ⇒ ∆ is satisfied by the valuation v , in symbols v | = Γ ⇒ ∆, if either v does not satisfy some formula in Γ or v satisfies someformula in ∆. A sequent is valid if it is satisfied by all valuations.Now, suppose that T = { t , . . . , t n − } , where n ≥
2, and D = { t d , . . . , t n − } ,where 1 ≤ d ≤ n − Definition 4.1 (see [3]) An n –sequent over L is an expression Γ | · · · | Γ n − where, for each i , Γ i is a finite set of formulas. A valuation v satisfies the n –sequent Γ | · · · | Γ n − iff there exists i , ≤ i ≤ n − and ψ ∈ Γ i such that v ( ψ ) = t i . An n –sequent is valid if it is satisfied by every valuation v . Note that, a valuation v satisfies an ordinary sequent Γ ⇒ ∆ iff v satisfies the n –sequent Γ | · · · | Γ n − where Γ i = Γ for all 0 ≤ i ≤ d − j = ∆ for all d ≤ j ≤ n − n -sequents is by means of sets of signed for-mulas. A signed formula over the language L and T , is an expression of theform t i : ψ where t i ∈ T and ψ ∈ F m . A valuation v satisfies the signed formula t i : ψ iff v ( ψ ) = t i . If U ⊆ T and Γ ⊆ F m , we denote by U : Γ the set U : Γ = { t : α | t ∈ U , α ∈ Γ } If U = { t } , we write t : Γ instead of { t } : Γ. A valuation satisfies the set ofsigned formulas U : Γ if it satisfies some signed formula of U : Γ; and we saythat U : Γ is valid if it is satisfied by every valuation v ∈ V . It is clear that,8he n –sequent Γ | · · · | Γ n − is valid iff the set of signed formulas n − S i =0 t i : Γ i isvalid.A. Avron and B. Konikowska developed in [3] a generic n -sequent systemfor any logic based on an n -valued matrix. Consider the n -valued matrix M = hT , D , Oi and let SF M the system defined as follows: for Ω and Ω ′ sets of signedformulas • Axioms: T : α • Structural rules:
Weakening:ΩΩ ′ in case Ω ⊆ Ω ′ • Logical rules: for each k -ary connective f and every ( a , . . . , a k ) ∈ T k Ω , a : α . . . Ω , a k : α k Ω , ˆ f ( a , . . . , a k ) : f ( α , . . . , α k ) Theorem 4.2 ([3]) The system SF M is sound and complete w.r.t. the matrix M Let
F m p be the set of all formulas of F m that have p as their only proposi-tional variable, i.e., F m p = { α ∈ F m : V ar ( α ) = { p }} . Let M = hT , D , Oi bea logical matrix and denote by N the set T \ D . Definition 4.3 ([4]) The language L is sufficiently expressive for M iff for any i , ≤ i ≤ n − there exist natural numbers l i , m i and formulas α ij , β ik ∈ F m p ,for ≤ j ≤ l i and ≤ k ≤ m i such that for any valuation v , the followingconditions hold:(i) α i = p if t i ∈ N and β i = p if t i ∈ D ,(ii) For ϕ ∈ F m and t i ∈ T v ( ϕ ) = t i ⇔ v ( α i [ p/ϕ ]) , . . . , v ( α il i [ p/ϕ ]) ∈ N and v ( β i [ p/ϕ ]) , . . . , v ( α im i [ p/ϕ ]) ∈ D where α ij [ p/ϕ ] ( β ik [ p/ϕ ] ) is the formula obtained by the substitution of p by ϕ in α ij ( β ik ). Note that, as it is mentioned in [4], condition (i) above is not really limiting, sincegiven α ij , β ik satisfying (ii), we can simply add to them the necessary formula p without violating (ii). Condition (i) will only be used for a backward translationfrom ordinary sequents to n -sequents, and will be disregarded otherwise.If Γ is a set of formulas and α ∈ F m p , we denote by α [ p/ Γ] the set9 [Γ] = { α [ p/γ ] | γ ∈ Γ } The method is based on replacing each n -sequent by a semantically equivalentset of two-sided sequents.Let L be a sufficiently expressive language and for 0 ≤ i ≤ n − l i , m i , α ij and β ik as in Definition 4.3. Consider the n –sequent Σ = Γ | · · · | Γ n − over L . A partition π of the n –sequent Σ is a tuple π = ( π , . . . , π n − ) such that,for every i , π i is a partition of the set Γ i of the form: π i = { Γ ′ ij | ≤ j ≤ l i } ∪ { Γ ′′ ik | ≤ k ≤ m i } Note that π i is not a partition in the usual sense, since its components areallowed to be empty. Besides, observe that the number of sets in this partitionis exactly the number of formulas corresponding to i in Definition 4.3.Then, given a partition π of the n -sequent Σ, we define the two-sided sequentΣ π determined by Σ and the partition π , as follows: l [ j =0 α j [Γ ′ j ] , . . . , l n − [ j = n − α n − j [Γ ′ ( n − j ] ⇒ m [ k =0 β k [Γ ′′ k ] , . . . , m n − [ k = n − β n − k [Γ ′′ ( n − k ]Let Π be the set of all partitions of the n –sequent Σ. Then, the set T W O (Σ) isdefined as follows:
T W O (Σ) = { Σ π | π ∈ Π } Theorem 4.4 ([4]) Let Σ be an n –sequent over L and v a valuation. Then, v satisfies Σ iff v satisfies Σ ′ , for every Σ ′ ∈ T W O (Σ) . Definition 4.5 ([4]) Let C be an n –sequent calculus over L . Then, let T W O ( C ) the (ordinary) sequent calculus over L given by: Axioms:
T W O ( A ) , for all axiom A of C , Inference rules:
T W O ( S )Σ ′ , where S is a finite set of n -sequents, R isone n -sequent such that SR is a rule in C and Σ ′ ∈ T W O ( R ) . Then,
Theorem 4.6 ([4]) If an n –sequent Σ is provable in C , then each two-sidedsequent Σ ′ ∈ T W O (Σ) is provable in
T W O ( C ) . heorem 4.7 ([4]) Let L be a sufficiently expressive language for M , and let C be a sound and complete sequent calculus w.r.t M . Then, T W O ( C ) is soundand complete w.r.t. M . The analogue of the cut rule for ordinary sequents is the following generalizedcut rule for sets of signed formulas:Ω ∪ { i : α | i ∈ I } Ω ∪ { j : α | j ∈ J } Ω for
I, J ⊆ V , I ∩ J = ∅ Theorem 4.8 ([4]) Under the conditions of Theorem 4.7, the cut rule is ad-missible in
T W O ( C ) . In particular, if C is obtained by the method of [3], thenthe cut rule is admissible in T W O ( C ) . As it was observed in [4], the n -sequent calculi obtained using the abovegeneral method are hardly optimal (the same is true for the two-sided calculi).We can use the three general streamlining principles from [3] to reduce the calculito a more compact form. The three streamlining principles are: Principle 1: deleting a derivable rule,
Principle 2: simplifying a rule by replacing it with onewith weaker premises, and
Principle 3: combining two context–free rules withthe same conclusion into one. Recall that a rule R is context-free if whenever φ ...φ n Σ is a valid application of R , and Σ ′ is a set of signed formulas, then φ ∪ Σ ′ ...φ n ∪ Σ ′ Σ ∪ Σ ′ is also a valid application of R . A rule R of an ordinary two–sided sequent calculus is a context–free if Γ ⇒ ∆ , . . . , Γ k ⇒ ∆ k Γ ⇒ ∆ is a validapplication of R , then Γ , Γ ′ ⇒ ∆ , ∆ ′ , . . . , Γ k , Γ ′ ⇒ ∆ k , ∆ ′ Γ , Γ ′ ⇒ ∆ , ∆ ′ is also a validapplication of R , where Γ ′ and ∆ ′ are finite sets of formulas.Of these three, the first and the third decrease the number of rules, while thesecond simplifies a rule by decreasing the number of its premises.It is worth mentioning that applying Principles 1–3 preserves the cut-eliminationproperty since cut-elimination is obtained via the completeness result and theprinciples are designed to retain completeness. T ML
Now, we shall use the method exhibited in Section 4 to develop a 4-sequent cal-culus for
T ML . In this case, we shall use its alternative presentation providedby sets of 4-signed formulas.Let SF be 4-sequent calculus given by: for α, β ∈ F m , Ω and Ω ′ arbitrary setsof signed formulas Axioms: { α, n : α, b : α, n : α } .11 tructural rules: Weakening.ΩΩ ′ in case Ω ⊆ Ω ′ Logical rules: for i, j ∈ M ( ∨ ij ) Ω , i : α Ω , j : β Ω , Sup { i, j } : α ∨ β ( ∧ ij ) Ω , i : α Ω , j : β Ω , Inf { i, j } : α ∧ β ( ¬ ) Ω , α Ω , ¬ α ( ¬ n ) Ω , n : α Ω , n : ¬ α ( ¬ b ) Ω , b : α Ω , b : ¬ α ( ¬ ) Ω , α Ω , ¬ α ( (cid:3) i ) Ω , i : α Ω , (cid:3) α , for i = 1 ( (cid:3) ) Ω , α Ω , (cid:3) α In rules ( ∨ ij ) (and (( ∧ ij )), the supremum (infimum) is taken on the lattice M . Besides, observe that the system SF has forty logical rules and it is notoptimal. However, in this step we are not going to use the principles mentionedin Section 4 to reduce SF . Proposition 5.1 (i) SF is sound and complete w.r.t. the matrix M ,(ii) the cut rule is admissible in SF . Proof.
From Theorem 4.2. (cid:4)
Now, we shall apply the method described in Section 4 to translate SF to anordinary two-sided sequent calculus. Proposition 5.2
The language L is sufficiently expressive for the semanticsdetermined by the matrix M . Proof.
Let v : F m → M be a valuation and let α ∈ F m an arbitrary formula,then we have that v ( α ) = 0 ⇐⇒ v ( α ) ∈ N and v ( ¬ α ) ∈ D v ( α ) = n ⇐⇒ v ( α ) ∈ N and v ( ¬ α ) ∈ N v ( α ) = b ⇐⇒ v ( α ) ∈ D and v ( ¬ α ) ∈ D v ( α ) = 1 ⇐⇒ v ( α ) ∈ D and v ( ¬ α ) ∈ N where N = M \ D = { , n } . (cid:4) SF to an ordinary one, we have toreplace every axiom A with the equivalent set of ordinary sequents T W O ( A ).In terms of 4-sequents, the only axiom of SF has the form α | α | α | α and it yields the following ordinary two-sided sequents α, ¬ α ⇒ α α, ¬ α ⇒ ¬ α α ⇒ ¬ α, α α, ¬ α ⇒ ¬ α, α ¬ α ⇒ ¬ α, α All of them can be derived from α ⇒ α (or from an instance of it) by the useof weakening.Now, let us focus on rules ( ∨ ij ), i, j ∈ M . First observe that, if ϕ ∈ F m then
T W O ( ϕ | | | ) = { ϕ ⇒ , ⇒ ¬ ϕ } T W O ( | ϕ | | ) = { ϕ ⇒ , ¬ ϕ ⇒} T W O ( | | ϕ | ) = {⇒ ϕ , ⇒ ¬ ϕ } T W O ( | | | ϕ ) = {¬ ϕ ⇒ , ⇒ ϕ } So, after removing the contexts for brevity, the rules ( ∨ ij )’s are translated tothe following thirty-two two-sided sequent rules:( ∨ ) ⇒ α ¬ α ⇒ β ⇒ ⇒ ¬ β ⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) n ⇒ α ¬ α ⇒ β ⇒ ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) b ⇒ α ¬ α ⇒ ⇒ β ⇒ ¬ β ⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) ⇒ α ¬ α ⇒ ⇒ β ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) b ⇒ α ¬ α ⇒ β ⇒ ⇒ ¬ β ⇒ α ∨ β ; ⇒ ¬ ( α ∨ β ) ( ∨ ) bn ⇒ α ¬ α ⇒ β ⇒ ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) bb ⇒ α ¬ α ⇒ ⇒ β ⇒ ¬ β ⇒ α ∨ β ; ⇒ ¬ ( α ∨ β ) ( ∨ ) b ⇒ α ¬ α ⇒ ⇒ β ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) n α ⇒ ¬ α ⇒ β ⇒ ⇒ ¬ βα ∨ β ⇒ ; ¬ ( α ∨ β ) ⇒ ( ∨ ) nn α ⇒ ¬ α ⇒ β ⇒ ¬ β ⇒ α ∨ β ⇒ ; ¬ ( α ∨ β ) ⇒ ( ∨ ) nb α ⇒ ¬ α ⇒ ⇒ β ⇒ ¬ β ⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) n α ⇒ ¬ α ⇒ ⇒ β ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒ ( ∨ ) α ⇒ ¬ α ⇒ β ⇒ ⇒ ¬ βα ∨ β ⇒ ; ⇒ ¬ ( α ∨ β ) ( ∨ ) n α ⇒ ¬ α ⇒ β ⇒ ¬ β ⇒ α ∨ β ⇒ ; ¬ ( α ∨ β ) ⇒ ( ∨ ) b α ⇒ ¬ α ⇒ ⇒ β ⇒ ¬ β ⇒ α ∨ β ; ⇒ ¬ ( α ∨ β ) ( ∨ ) α ⇒ ¬ α ⇒ ⇒ β ¬ β ⇒⇒ α ∨ β ; ¬ ( α ∨ β ) ⇒
13n the above list we use an informal notation by separating the alternate conclu-sion sequents with semicolons. At this point, we shall follow the three principlesmentioned in the above section in order to reduce the number of rules. Our maintool for this job will be the next proposition.
Proposition 5.3
Let SC a sequent calculus in which the cut rule is admissible,let S be a set of sequents and Σ be a sequent such that S ∪ { Γ ⇒ ∆ , ϕ } Σ and S ∪ { Γ , ϕ ⇒ ∆ } Σ are two context-free rules of SC . Then, S Σ is derivable in SC . Proof.
From the fact that the rules are context-free and using the cut rule. (cid:4)
Then, from ( ∨ ) , ( ∨ ) n and Proposition 5.3 we get ⇒ α ¬ α ⇒ β ⇒⇒ α ∨ β | ¬ ( α ∨ β ) ⇒ .From ( ∨ ) b , ( ∨ ) and Proposition 5.3 we get ⇒ α ¬ α ⇒ ⇒ β ⇒ α ∨ β | ¬ ( α ∨ β ) ⇒ .From these rules and Proposition 5.3 we obtain (1) ⇒ α ¬ α ⇒⇒ α ∨ β and (1’) ⇒ α ¬ α ⇒¬ ( α ∨ β ) ⇒ . Analogously, from ( ∨ ) b , ( ∨ ) bn , ( ∨ ) bb , ( ∨ ) b we obtain(2) ⇒ α ⇒ ¬ α ⇒ α ∨ β . Finally, from (1), (2) and Proposition 5.3 we get that ⇒ α ⇒ α ∨ β (3)is derivable. On the other hand, following an analogous reasoning we can provethat ⇒ β ⇒ α ∨ β (4)is derivable. Then, after combining rules (3) and (4) and restoring the contextwe get the rule ( ⇒ ∨ ) Γ ⇒ ∆ , α, β Γ ⇒ ∆ , α ∨ β From ( ∨ ) n , ( ∨ ) nn , ( ∨ ) nb and ( ∨ ) n we obtain (5) ⇒ α ¬ α ⇒⇒ α ∨ β ; then using(1’) and restoring the context we get (5) Γ , ¬ α ⇒ ∆Γ , ¬ ( α ∨ β ) ⇒ ∆ . In a similar way, itcan be proved that (6) Γ , ¬ β ⇒ ∆Γ , ¬ ( α ∨ β ) ⇒ ∆ is derivable. Then, combining (5) and(6) and restoring the context we get( ¬∨ ⇒ ) Γ , ¬ α, ¬ β ⇒ ∆Γ , ¬ ( α ∨ β ) ⇒ ∆From ( ∨ ) n , ( ∨ ) nn , ( ∨ ) and ( ∨ ) n and restoring context we obtain the rule( ∨ ⇒ ) Γ , α ⇒ ∆ Γ , β ⇒ ∆Γ , α ∨ β ⇒ ∆14nd, from ( ∨ ) , ( ∨ ) b , ( ∨ ) b and ( ∨ ) bb we get( ⇒ ¬∨ ) Γ ⇒ ∆ , ¬ α Γ ⇒ ∆ , ¬ β Γ ⇒ ∆ , ¬ ( α ∨ β )In the same way, we obtain the following rules for the connective ∧ :( ∧ ⇒ ) Γ , α, β ⇒ ∆Γ , α ∧ β ⇒ ∆ ( ⇒ ∧ ) Γ ⇒ ∆ , α Γ ⇒ ∆ , β Γ ⇒ ∆ , α ∧ β ( ¬∧ ⇒ ) Γ , ¬ α ⇒ ∆ Γ , ¬ β ⇒ ∆Γ , ¬ ( α ∧ β ) ⇒ ∆ ( ⇒ ¬∧ ) Γ ⇒ ∆ , ¬ α, ¬ β Γ ⇒ ∆ , ¬ ( α ∧ β )On the other hand, rules ( ¬ ) i with i ∈ M are translated to (after eliminatingthe trivial rules)( ¬ ) α ⇒ ⇒ ¬ α ¬¬ α ⇒ ( ¬ ) n α ⇒ ¬ α ⇒¬¬ α ⇒ ( ¬ ) b ⇒ α ⇒ ¬ α ⇒ ¬¬ α ( ¬ ) ⇒ α ¬ α ⇒⇒ ¬¬ α From ( ¬ ) , ( ¬ ) n and Proposition 5.3 on the one hand; and ( ¬ ) b , ( ¬ ) andProposition 5.3 on the other, we obtain( ¬¬ ⇒ ) Γ , α ⇒ ∆Γ , ¬¬ α ⇒ ∆ ( ⇒ ¬¬ ) Γ ⇒ ∆ , α Γ ⇒ ∆ , ¬¬ α Finally, rules ( (cid:3) ) i are translated to( (cid:3) ) α ⇒ ⇒ ¬ α (cid:3) α ⇒ ; ⇒ ¬ (cid:3) α ( (cid:3) ) n α ⇒ ¬ α ⇒ (cid:3) α ⇒ ; ⇒ ¬ (cid:3) α ( (cid:3) ) b ⇒ α ⇒ ¬ α (cid:3) α ⇒ ; ⇒ ¬ (cid:3) α ( (cid:3) ) ⇒ α ¬ α ⇒⇒ (cid:3) α ; ¬ (cid:3) α ⇒ and, from these rules and Proposition 5.3, we obtain( (cid:3) ⇒ ) Γ , α ⇒ ∆Γ , (cid:3) α ⇒ ∆ ( (cid:3) ⇒ ) Γ ⇒ ∆ , ¬ α Γ , (cid:3) α ⇒ ∆ ( ⇒ (cid:3) ) Γ ⇒ ∆ , α Γ , ¬ α ⇒ ∆Γ ⇒ ∆ , (cid:3) α ( ¬ (cid:3) ⇒ ) Γ ⇒ ∆ , α Γ , ¬ α ⇒ ∆Γ , ¬ (cid:3) α ⇒ ∆ ( ⇒ ¬ (cid:3) ) Γ , α ⇒ ∆Γ ⇒ ∆ , ¬ (cid:3) α ( ⇒ ¬ (cid:3) ) Γ ⇒ ∆ , ¬ α Γ ⇒ ∆ , ¬ (cid:3) α Definition 5.4
Let SC T ML be the sequent calculus given by the axiom α ⇒ α the structural rules of cut and left and right weakening w ⇒ ) Γ ⇒ ∆Γ , α ⇒ ∆ ( ⇒ w ) Γ ⇒ ∆Γ ⇒ ∆ , α and the logical rules ( ∨ ⇒ ), ( ⇒ ∨ ), ( ¬∨ ⇒ ), ( ⇒ ¬∨ ), ( ∧ ⇒ ), ( ⇒ ∧ ), ( ¬∧ ⇒ ),( ⇒ ¬∧ ), ( ¬¬ ⇒ ), ( ⇒ ¬¬ ), ( (cid:3) ⇒ ) i , ( ⇒ (cid:3) ), ( ¬ (cid:3) ⇒ ), ( ⇒ ¬ (cid:3) ) i i = 1 , . We shall write Γ ⇔ ∆ to indicate that both the sequents Γ ⇒ ∆ and ∆ ⇒ Γ areprovable. Then, it is not difficult to verify that α ∧ ¬ α ⇔ α ∧ ¬ (cid:3) α , for everyformula α . Besides, the modal axiom ⇒ α ∨ ¬ (cid:3) α of G is derivable in SC T ML .Indeed, α ⇒ α ( ⇒ ¬ (cid:3) ) ⇒ α, ¬ (cid:3) α ( ⇒ ∨ ) ⇒ α ∨ ¬ (cid:3) α Moreover, the sequent ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) is derivable in SC T ML without the cutrule: α ⇒ α ( ⇒ ¬ (cid:3) ) ⇒ α, ¬ (cid:3) α ( ⇒ ∨ ) ⇒ α ∨ ¬ (cid:3) α ¬ α ⇒ ¬ α ( (cid:3) ⇒ ) ¬ α, (cid:3) α ⇒ ( ¬¬ ⇒ ) ¬ α, ¬¬ (cid:3) α ⇒ ( ¬∨ ⇒ ) ¬ ( α ∨ ¬ (cid:3) α ) ⇒ ( ⇒ (cid:3) ) ⇒ (cid:3) ( α ∨ ¬ (cid:3) α ) Remark 5.5
In Font and Rius’ system G , the propositional constant ⊥ isused. By following Avron, Ben-Naim and Konikowska’s method, we obtained asystem in which ⊥ does not appear. However, it is easy to check that the sequent ¬ α ∧ (cid:3) α ⇒ is provable in SC T ML , for any formula α . Then, if we denote by ⊥ the formula ¬ α ∧ (cid:3) α , for any formula α , we have that the rule ( ⊥ ) of G isderivable in SC T ML . Theorem 5.6 (i) SC T ML is sound and complete w.r.t. M .(ii) The cut rule is admissible in SC T ML , Proof.
The system SC T ML was constructed according to the method dis-played in Section 4. (cid:4)
Corollary 5.7 SC
T ML is a cut-free sequent calculus that provides a syntacticalcounterpart for
T ML . Some applications of the cut elimination the-orem
In this section, we shall use the cut-free system SC T ML to show independentproofs of some (known) interesting properties of the logic
T ML . In what followsΓ, ∆ are sets of formulas and α , β , ψ are formulas.In the first place, we shall present a new independent proof of Proposition 2.5.To do this, we need the following technical result. Proposition 6.1 If ⊢ SC T ML Γ ⇒ ∆ then, for every A ∈ TMA and for every h ∈ Hom ( Fm , A ) , V γ ∈ Γ h ( γ ) ≤ W δ ∈ ∆ h ( δ ) . Proof.
Suppose that ⊢ SC T ML Γ ⇒ ∆ and let P be a cut–free proof of thesequent Γ ⇒ ∆ in SC T ML . Let A ∈ TMA and let h ∈ Hom ( Fm , A ). Weuse induction on the number n of inferences in P . If n = 0 the propositionis obviously valid. (I.H.) Suppose that the proposition holds for n < k , k >
0. Let n = k and let ( r ) be the last inference in P . Ir ( r ) is the right/leftweakening rule, the proposition holds since A is, in particular, a lattice. If ( r )is one of the rules ( ∨ ⇒ ), ( ⇒ ∨ ), ( ¬∨ ⇒ ), ( ⇒ ¬∨ ), ( ∧ ⇒ ), ( ⇒ ∧ ), ( ¬∧ ⇒ ),( ⇒ ¬∧ ), ( ¬¬ ⇒ ), ( ⇒ ¬¬ ), the proposition holds since A is, in particular, a DeMorgan algebra. Finally, if ( r ) is one of the rules , ( (cid:3) ⇒ ) i , ( ⇒ (cid:3) ), ( ¬ (cid:3) ⇒ ),( ⇒ ¬ (cid:3) ) i i = 1 , A is a tetravalent modalalgebra. For instance, suppose that ( r ) is ( ⇒ (cid:3) ) and the last inference of P isΓ ⇒ ∆ , α Γ , ¬ α ⇒ ∆Γ ⇒ ∆ , (cid:3) α . By (I.H.), we have (1) V γ ∈ Γ h ( γ ) ≤ W δ ∈ ∆ h ( δ ) ∨ h ( α )and (2) V γ ∈ Γ h ( γ ) ∧ h ( ¬ α ) ≤ W δ ∈ ∆ h ( δ ). Then, from (1), (2) and Proposition2.2 we have V γ ∈ Γ h ( γ ) ≤ W δ ∈ ∆ h ( δ ) ∨ h ( (cid:3) α ). (cid:4) Proposition 6.2
The following conditions are equivalent.(i) Γ | = T ML ψ ,(ii) Γ | = M ψ . Proof. (i) imples (ii) : immediate. (ii) implies (i) : It is consequence of Theorem 5.6 (i) and Proposition 6.1. (cid:4)
Next, we shall prove that the rule ( ¬ ) of Font and Rius’ system is addmissiblein SC T ML . Let X a set of formulas, we shall denote by ¬ X the set ¬ X = {¬ γ : γ ∈ X } . Theorem 6.3 If ⊢ SC T ML Γ ⇒ ∆ , then ⊢ SC T ML ¬ ∆ ⇒ ¬ Γ . Proof.
Suppose that ⊢ SC T ML Γ ⇒ ∆ and let P a cut–free proof of the sequentΓ ⇒ ∆. We use induction on the number n of inferences in P . If n = 0, then17 ⇒ ∆ is α ⇒ α , for some α , and ¬ ∆ ⇒ ¬ Γ is ¬ α ⇒ ¬ α which is provablein SC T ML . (I.H.) Suppose that the lemma holds for n < k , with k >
0. Let n = k and let ( r ) be the last inference in P . If ( r ) is left weakening, then thelast inference of P is Γ ⇒ ∆Γ , α ⇒ ∆ . By (I.H.), ¬ ∆ ⇒ ¬ Γ is provable in SC T ML and using right weakening we have ⊢ SC T ML ¬ ∆ ⇒ ¬ Γ , ¬ α . If ( r ) is an instanceof the right weakening the treatment is analogous.Suppose now that ( r ) is (an instance of) a logic rule. If ( r ) is ( ⇒ ∨ ) and the lastinference of P is Γ ⇒ ∆ , α, β Γ ⇒ ∆ , α ∨ β . By (I.H.), ¬ α, ¬ β, ¬ ∆ ⇒ ¬ Γ is provable, andusing ( ¬∨ ⇒ ) we have that ¬ ( α ∨ β ) , ¬ ∆ ⇒ ¬ Γ is provable. The cases where( r ) is one of the rules ( ∨ ⇒ ), ( ⇒ ¬∨ ), ( ¬∨ ⇒ ), ( ⇒ ∧ ) ( ∧ ⇒ ), ( ⇒ ¬∧ ), ( ¬∧ ⇒ )are left to the reader.If ( r ) is ( ⇒ ¬¬ ) and the last inference of P is Γ ⇒ ∆ , α Γ ⇒ ∆ , ¬¬ α . By (I.H.), ¬ α, ¬ ∆ ⇒¬ Γ is provable in SC T ML and using ( ¬¬ ⇒ ) we have that ¬¬¬ α, ¬ ∆ ⇒ ¬ Γ isprovable. If ( r ) is ( ¬¬ ⇒ ) the proof is analogous.If ( r ) is ( (cid:3) ⇒ ) and the last inference of P is Γ , α ⇒ ∆Γ , (cid:3) α ⇒ ∆ . By (I.H.), we havethat ¬ ∆ ⇒ ¬ Γ , ¬ α is provable in SC T ML . Then, using ( ⇒ ¬ (cid:3) ) we have that ¬ ∆ ⇒ ¬ Γ , ¬ (cid:3) α is provable.If ( r ) is ( (cid:3) ⇒ ) and the last inference of P is Γ ⇒ ∆ , ¬ α Γ , (cid:3) α ⇒ ∆ . By (I.H.), wehave that ¬ ∆ , ¬¬ α ⇒ ¬ Γ is provable in SC T ML and using left weakeningwe have (1) ⊢ SC T ML α, ¬ ∆ , ¬¬ α ⇒ ¬ Γ. On the other hand, one can easilycheck that ⊢ SC T ML α ⇒ ¬¬ α and by means of (right/left) weakening(s) wehave (2) ⊢ SC T ML α, ¬ ∆ ⇒ ¬¬ α, ¬ Γ. From (1), (2) and the cut rule, we have ⊢ SC T ML α, ¬ ∆ ⇒ ¬ Γ (the cut rule is admissible in SC T ML ). Then, using( ⇒ ¬ (cid:3) ) we have ⊢ SC T ML ¬ ∆ ⇒ ¬ Γ , ¬ (cid:3) α .If ( r ) is ( ⇒ (cid:3) ) and the last inference of P is Γ ⇒ ∆ , α Γ , ¬ α ⇒ ∆Γ ⇒ ∆ , (cid:3) α . By (I.H.)we have that (3) ⊢ SC T ML ¬ α, ¬ ∆ ⇒ ¬ Γ and (4) ⊢ SC T ML ¬ ∆ ⇒ ¬¬ α, ¬ Γ. From(4) and a similar reasoning to the above, we have that (5) ⊢ SC T ML ¬ ∆ ⇒ α, ¬ Γ.From (3), (5) and ( ¬ (cid:3) ⇒ ) we get ⊢ SC T ML ¬ ∆ , ¬ (cid:3) α ⇒ ¬ Γ.The cases where ( r ) is one of the rules ( ⇒ ¬ (cid:3) ) , ( ⇒ ¬ (cid:3) ) and ( ¬ (cid:3) ⇒ ) aretreated similarly. (cid:4) Corollary 6.4 ( ¬ ) is admissible in SC T ML . Finally,
Theorem 6.5 ⊢ T ML (cid:3) ψ iff ⊢ T ML ψ . Proof. (= ⇒ ) Suppose that ⊢ T ML (cid:3) ψ . By Theorem 5.6, Proposition 2.5 weknow that the sequent ⇒ (cid:3) ψ has a cut-free proof P in SC T ML . Let ( r ) be thelast inference of P . By inspecting the rules of SC T ML we may assert that ( r )18as to be an instance of the rule ( ⇒ (cid:3) ). So, ( r ) is ⇒ ψ ¬ ψ ⇒⇒ (cid:3) ψ and clearlythe sequent ⇒ ψ is provable in SC T ML . Therefore ⊢ T ML ψ .( ⇐ =) Suppose that ⊢ T ML ψ . By Theorem 5.6 (i), we have: (1) ⇒ ψ isprovable in SC T ML . From (1) and Theorem 6.3, we have that: (2) ¬ ψ ⇒ isalso provable in SC T ML . From (1), (2) and the rule ( ⇒ (cid:3) ), we may assert that ⇒ (cid:3) ψ is provable in SC T ML . Therefore, ⊢ T ML (cid:3) ψ . (cid:4) T ML
In this section, we shall present a natural deduction system for
T ML . Wetake our inspiration from the construction made before. In particular, it threwsome light on how the connective (cid:3) behaves. We think that this system showsan interesting example of a rule (different from the usual ones), namely theintroduction rule of the connective (cid:3) , that needs to produce a discharge ofhypothesis; and this is related to the intrinsic meaning of the connective.The proof system ND T ML will be defined following the notational conventionsgiven in [15].
Definition 7.1
Deductions in ND T ML are inductively defined as follows:Basis: The proof tree with a single occurrence of an assumption φ with a markeris a deduction with conclusion φ from open assumption φ .Inductive step: Let D , D , D , D be deductions. Then, they can be extendedby one of the following rules below. The classes [ ¬ φ ] u , [ ¬ ψ ] v , [ φ ] u , [ ψ ] v belowcontain open assumptions of the deductions of the premises of the final inference,but are closed in the whole deduction. MA (modal axioma) φ ∨ ¬ (cid:3) φ D φ D ψ ∧ I φ ∧ ψ D φ ∧ ψ ∧ E φ D φ ∧ ψ ∧ E ψ D¬ φ ¬∧ I ¬ ( φ ∧ ψ ) D¬ ψ ¬∧ I ¬ ( φ ∧ ψ ) D ¬ ( φ ∧ ψ ) [ ¬ φ ] u D χ [ ¬ ψ ] v D χ ¬∧ E, u , vχ D φ ∨ I φ ∨ ψ D ψ ∨ I φ ∨ ψ D φ ∨ ψ [ φ ] u D χ [ ψ ] v D χ ∨ E, u , vχ ¬ φ D ¬ ψ ¬∨ I ¬ ( φ ∨ ψ ) D¬ ( φ ∨ ψ ) ¬∨ E ¬ φ D¬ ( φ ∨ ψ ) ¬∨ E ¬ ψ D φ ¬¬ I ¬¬ φ D¬¬ φ ¬¬ E φ D ψ ∨ φ [ ¬ φ ] u D ψ (cid:3) I ∗ , uψ ∨ (cid:3) φ D (cid:3) φ (cid:3) E φ D¬ φ ¬ (cid:3) I ¬ (cid:3) φ D ¬ (cid:3) φ D φ ¬ (cid:3) E ¬ φ D¬ φ ∧ (cid:3) φ ⊥ I ⊥ D⊥ ⊥ E α Remark 7.2
If we take ψ as ⊥ in (cid:3) I ∗ we get D φ [ ¬ φ ] u D ⊥ (cid:3) I, u (cid:3) φ Formally, (cid:3)
I is derivable in ND T ML . The intuition behind this rule is thefollowing:“if we have a deduction for α and ¬ α is not provable, then we have adeduction for (cid:3) α ”. As usual, by application of the rule ¬∧ E a new proof-tree is formed from D , D , and D by adding at the bottom the conclusion χ while closing the sets[ ¬ φ ] u and [ ¬ ψ ] u of open assumptions marked by u and v , respectively. Idem forthe rules ∧ E and (cid:3)
I. Note that we have introduced the symbol ⊥ , it behaveshere as an arbitrary unprovable propositional constant.Let Γ ∪ { α } ⊆ F m . We say that the conclusion α is derivable from a set Γ ofpremises, noted Γ ⊢ α , if and only if there is a deduction in ND T ML of α fromΓ. 20 heorem 7.3 (Soundness and Completeness) Let Γ , ∆ ⊆ F m , Γ finite. Thefollowing conditions are equivalent:(i) the sequent Γ ⇒ ∆ is derivable in SC T ML ,(ii) there is a deduction of the disjunction of the sentences in ∆ from Γ in ND T ML . Proof. (i) implies (ii): Suppose that the sequent Γ ⇒ ∆ is derivable in SC T ML , that is, there is a formal proof P of Γ ⇒ ∆ in SC T ML which doesnot use the cut rule. We shall show that there is a deduction of the disjunctionof the formulas in ∆ (denoted by W ∆) from Γ in ND T ML , using induction onthe number n of rule applications in P , n ≥ n = 0, then Γ ⇒ ∆ is α ⇒ α and it is clear that α ⊢ α . Now, (I.H.) supposethat “(i) implies (ii)” holds for n < k , with k > n = k , that is P is a derivation in SC T ML with last rule (r) of the formΓ ⇒ ∆ . . . . . . ... Γ t ⇒ ∆ t ... ( r ) Γ ⇒ ∆If (r) is left weakening, then the last rule of P has the form (r) Γ ′ ⇒ ∆Γ ′ , β ⇒ ∆ . By(I.H.), there exists a deduction D of ∆ from Γ ′ , then D W ∆ β ∧ I W ∆ ∧ β ∧ E W ∆is a deduction of W ∆ from Γ ′ ∪ { β } . If (r) is right weakening, then (r) has theform (r) Γ ⇒ ∆ ′ Γ ⇒ ∆ ′ , β , then by (I.H.) there is a deduction D of W ∆ ′ from Γ. D W ∆ ′ ∨ I W ∆ ′ ∨ β Now, suppose that (r) is a logical rule, we shall prove it just for ( ⇒ ∨ ), ( ∨ ⇒ ),( ⇒ ¬∨ ), ( ¬∨ ⇒ ). If (r) is ( ∨ ⇒ ), then we may assume that the last inferenceof P has the form ( ⇒ ∨ ) Γ ⇒ ∆ ′ , α, β Γ ⇒ ∆ ′ , α ∨ β . Then, by (I.H.) we have a deduction D of W ∆ ′ ∨ α ∨ β from Γ and the proof is complete.If (r) is ( ∨ ⇒ ) and last inference of P has the from ( ∨ ⇒ ) Γ , γ ⇒ ∆ Γ , γ ⇒ ∆Γ , γ ∨ γ ⇒ ∆ ,then by (I.H.) there are deductions D i , i = 1 ,
2, of α from Γ ∪ { γ i } . Then, thefollowing 21 ∨ γ [ γ ] u D W ∆ [ γ ] u D W ∆ ∨ E, u , u W ∆is a deduction of W ∆ from Γ ∪ { γ ∨ γ } . Note that in this last deduction wehave made every assumption γ i in D i an open assumption with label u i .If (r) is ( ¬∨ ⇒ ) then we may assume that the last instance of P has the form( ¬∨ ⇒ ) Γ , ¬ γ ⇒ ∆Γ , ¬ ( γ ∨ γ ) ⇒ ∆ . By (I.H.), there is a deduction D of α from Γ ∪ { γ } and the following ¬ ( γ ∨ γ ) ¬∨ E ¬ γ D W ∆is a deduction of α from Γ ∪{¬ ( γ ∨ γ ) } . If (r) is ( ⇒ ¬∨ ) we proceed analogously.For (r) being any of the rules ( (cid:3) ⇒ ) i , ( ⇒ (cid:3) ), ( ¬ (cid:3) ⇒ ), ( ⇒ ¬ (cid:3) ) i i = 1 ,
2, wepresent the next table showing the deduction corresponding to the premise(s)of (r) and the deduction corresponding to the consequence of (r).22 ule (r) Upper sequent(s)’s Lower sequent’sdeduction(s) deductionΓ , γ ⇒ ∆ ( (cid:3) ⇒ )1 Γ , (cid:3) γ ⇒ ∆ γ D W ∆ (cid:3) γ (cid:3) E γ D W ∆ Γ ⇒ ∆ , ¬ γ ( (cid:3) ⇒ )2 Γ , (cid:3) γ ⇒ ∆ D W ∆ ∨ ¬ γ D W ∆ ∨ ¬ γ [ W ∆] u W ∆ [ ¬ γ ] u (cid:3) γ ∧ I ¬ γ ∧ (cid:3) γ ⊥ I ⊥ ⊥ E W ∆ ∨ E, u , v W ∆ Γ ⇒ ∆ , γ Γ , ¬ γ ⇒ ∆ ( ⇒ (cid:3) ) Γ ⇒ ∆ , (cid:3) γ D W ∆ ∨ γ ¬ γ D W ∆ D W ∆ ∨ γ ¬ γu D W ∆ (cid:3) I ∗ , u W ∆ ∨ (cid:3) γ Γ ⇒ ∆ , γ Γ , ¬ γ ⇒ ∆ ( ¬ (cid:3) ⇒ ) Γ , ¬ (cid:3) γ ⇒ ∆ D W ∆ ∨ γ ¬ γ D W ∆ D W ∆ ∨ γ [ ¬ γ ] u D W ∆ (cid:3) I, u W ∆ ∨ (cid:3) γ ¬ (cid:3) γ ∨ I2 W ∆ ∨ ¬ (cid:3) γ ∧ I( W ∆ ∨ (cid:3) γ ) ∧ ( W ∆ ∨ ¬ (cid:3) γ ) 1 W ∆ ∨ ( (cid:3) γ ∧ ¬ (cid:3) γ ) 2 W ∆ ∨ ⊥ W ∆ Γ , γ ⇒ ∆ ( ⇒ ¬ (cid:3) )1 Γ ⇒ ∆ , ¬ (cid:3) γ γ D W ∆ (MA) γ ∨ ¬ (cid:3) γ γv D W ∆ W ∆ ∨ ¬ (cid:3) γ ¬ (cid:3) γu W ∆ ∨ ¬ (cid:3) γ ∨ E, u , v W ∆ ∨ ¬ (cid:3) γ Γ ⇒ ∆ , ¬ γ ( ⇒ ¬ (cid:3) )2 Γ ⇒ ∆ , ¬ (cid:3) γ D W ∆ ∨ ¬ γ D W ∆ ∨ ¬ γ W ∆ u W ∆ ∨ ¬ (cid:3) γ ¬ γv ¬ (cid:3) I ¬ (cid:3) γ W ∆ ∨ ¬ (cid:3) γ ∨ E, u , v W ∆ ∨ ¬ (cid:3) γ (ii) implies (i): Let D be a deduction of the disjunction of the sentences in∆ from Γ in ND T ML . As before, we use induction on the number n of ruleinstances in the deduction D . If r = 0 the proof is trivial. (I.H.) Suppose that“(ii) implies (i)” holds for n < k , k >
0; and let ( r ) the last rule instance in D . If ( r ) is one of the introduction/elimination rule of ∧ I, ∧ E, ¬∧ I, ¬∧ E, ∨ I, ∨ E , ¬∨ I, ¬∨ E, ¬¬ I and ¬¬ E; the proof is immediate since these rules are justtranslations of the corresponding rules of SC T ML . Suppose that ( r ) is (cid:3) I ∗ ,23hen D is D ψ ∨ φ [ ¬ φ ] u D ψ (cid:3) I ∗ , uψ ∨ (cid:3) φ Then, by (I.H), we have that the sequents Γ ⇒ ψ ∨ φ and Γ , ¬ φ ⇒ ψ areprovable in SC T ML , where Γ ∪ Γ = Γ. By using weakening(s) and the cutrule we obtain Γ ⇒ ψ, φ and Γ ¬ φ ⇒ ψ are provable. Then, using ( (cid:3) ⇒ ), wehave that ⊢ SC T ML Γ ⇒ ψ, (cid:3) φ . If ( r ) is (cid:3) E, then D is D (cid:3) φ (cid:3) E φ By (I.H.), we have ⊢ SC T ML Γ ⇒ (cid:3) φ . From the fact that ⊢ SC T ML (cid:3) φ ⇒ φ andthe cut rule the proof is completed. If ( r ) is ¬ (cid:3) I, then D is D¬ φ ¬ (cid:3) I ¬ (cid:3) φ By (I.H.), we have ⊢ SC T ML Γ ⇒ ¬ φ . By Theorem 6.3, ⊢ SC T ML ¬¬ φ ⇒ ¬ Γand from ⊢ SC T ML φ ⇒ ¬¬ φ and the cut rule, we have ⊢ SC T ML φ ⇒ ¬ Γ. Using( (cid:3) ⇒ ) we obtain ⊢ SC T ML (cid:3) φ ⇒ ¬ Γ and by Theorem 6.3 ⊢ SC T ML ¬¬ Γ ⇒ ¬ (cid:3) φ .Finally, from ⊢ SC T ML Γ ⇒ ¬¬ Γ and cut(s) (and weakening(s) if necessary) weobtain ⊢ SC T ML Γ ⇒ ¬ (cid:3) φ . If ( r ) is ¬ (cid:3) E, then D is D ¬ (cid:3) φ D φ ¬ (cid:3) E ¬ φ By (I.H) and using weakening(s) we have that the sequents Γ ⇒ ¬ (cid:3) φ and Γ ⇒ φ are provable in SC T ML . Using ( ⇒ ∧ ), we obtain ⊢ SC T ML Γ ⇒ φ ∧ ¬ (cid:3) φ andsince φ ∧¬ (cid:3) φ ⇔ φ ∧¬ φ and the cut rule we obtain ⊢ SC T ML Γ ⇒ φ ∧¬ φ . Finally,taking into account that ⊢ SC T ML φ ∧ ¬ φ ⇒ ¬ φ we have ⊢ SC T ML Γ ⇒ ¬ φ .The cases in which ( r ) is ⊥ I or ⊥ E are immediate (see Remark 5.5). (cid:4)
Since our natural deduction system is strongly inspired by the cut-free se-quent calculus SC T ML , one can likely expect normalization to hold for SC T ML . ( γ ∨ α ) ∧ ( γ ∨ β ) ⊣⊢ γ ∨ ( α ∧ β ) α ∨ ( (cid:3) γ ∧ ¬ (cid:3) γ ) ⊣⊢ α ∨ ⊥ Conclusions
In the present paper we focused on the proof-theoretic aspects of the tetrava-lent modal logic
T ML . In the first place, we showed that the strongly ade-quate Gentzen calculus given by Font and Rius for
T ML does not enjoy thecut–elimination property. Then, by applying a method due to Avron, Ben-Naim and Konikowska, we developed a sequent calculus for
T ML with thecut–elimination property. This allowed us to provide new independent proof ofsome known interesting properties of
T ML . Finally, strongly inspired by thiscut–free sequent calculus, we presented a natural deduction system, sound andcomplete with respect to the
T ML .Despite the fact that
T ML was originally defined as the logic that preservesdegrees of truth w.r.t. tetravalent modal algebras, we could use Avron, Ben-Naim and Konikowska’s method; and this is because
T ML is also a matrixlogic. An interesting task to be done is to extend this method to logics to logicsthat preserves degrees of truth w.r.t. some ordered structure but which do nothave a matrix semantics.
I would like to thank the anonymous referees for their extremely careful reading,helpful suggestions and constructive comments on this paper.
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