Sequent Calculi for the classical fragment of Bochvar and Halldén's Nonsense Logics
DD. Kesner and P. Viana (Eds.): LSFA 2012EPTCS 113, 2013, pp. 125–136, doi:10.4204/EPTCS.113.12 c (cid:13)
M. E. Coniglio & M. I. Corbal´anThis work is licensed under theCreative Commons Attribution License.
Sequent Calculi for the classical fragment of Bochvar andHalld´en’s Nonsense Logics
Marcelo E. Coniglio Mar´ıa I. Corbal´an
Centre for Logic, Epistemology and the History of ScienceandDepartment of PhilosophyState University of Campinas, Brazil [email protected] [email protected]
In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implica-tion-negation fragment) of the nonsense logics B , introduced by Bochvar, and H , introduced byHalld´en, are presented. These calculi are obtained by restricting in an appropriate way the applicationof the rules of a sequent calculus for classical propositional logic CPL . The nice symmetry betweenthe provisos in the rules reveal the semantical relationship between these logics. The Soundnessand Completeness theorems for both calculi are obtained, as well as the respective Cut eliminationtheorems.
Introduction
The study of logical paradoxes from a formal perspective has produced several proposals in the liter-ature. In particular, 3-valued propositional logics were proposed in which, besides the two ‘classical’truth-values, the third one plays the role of a ‘nonsensical’ or ‘meaningless’ truth value. This is whythese logics are known as ‘logics of nonsense’. In 1938 ([3]) A. Bochvar introduced the first logic ofnonsense, by means of 3-valued logical matrices. Since the nonsensical truth value is not distinguished,Bochvar’s logic is paracomplete but it is not paraconsistent: the negation ¬ is explosive (from a contra-diction everything follows) but the third-excluded law does not hold. In 1949 S. Halld´en ([6]) proposeda closely related logic of nonsense by means of 3-valued logical matrices in which the third truth-valueis distinguished, producing a paraconsistent, non-paracomplete logic.Both logics share the same main feature: the nonsensical truth-value is ‘infectious’ in the sense that,given a valuation v , every formula having at least one propositional variable with nonsensical truth-valueunder v also gets the non-sensical truth-value under v . Also, both logics contain, besides the connectives ¬ for negation and ∧ conjunction, an unary connective which allows to recover all the classical inferences(cf. [4, 5]).The respective ‘classical’ fragments of each of these two logics (that is, the {¬ , ∨ , ∧ , →} -fragments)are interesting since they together constitute the only two possibilities for extending the usual matricesof classical logic with a third nonsensical, ‘infectious’ truth-value : either is designated or it is not.The former corresponds to the ‘classical’ fragment of Halld´en’s logic, while the latter corresponds tothe same fragment of Bochvar’s logic. It is not hard to establish, by semantical means, a relationshipbetween these two fragments and classical logic: given a classically valid inference G ⊢ a over thelanguage generated by {¬ , ∨ , ∧ , →} , if the propositional variables ocurring in G also occur in a then G ⊢ a is valid in Halld´en’s logic H . Dually, if the propositional variables ocurring in a also occur in G then such classically valid inference is valid in Bochvar’s logic B . This duality is a direct consequence26 Sequent Calculi for Bochvar and Halld´en’s Nonsense Logicsof the criterion adopted in each logic with respect to the third truth-value (namely, designated vs. non-designated), and the fact that this non-sensical truth-value propagates through any complex formula.Since {∨ , →} and {∧ , →} can be defined as usual from {¬ , ∧} and {¬ , ∨} , respectively, the observationabove can also be applied to the {¬ , ∧} and {¬ , ∨} -fagments of both logics.This paper introduced two cut-free sequent calculi for the {¬ , ∨ , ∧ , →} -fragment of each logic ofnonsense mentioned above. Both systems are obtained by imposing restrictions on the rules of theusual sequent calculus for classical propositional logic CPL . In the calculus for the classical fragmentof Halld´en’s logic, the introduction rules for conjunction, implication and negation on the left side ofthe sequent are restricted. In the calculus for the fragment of Bochvar’s logic the restriction is imposedto the introduction rule for disjunction, implication and negation on the right side. In this manner, therelationship between classical logic and both logics became explicit through restrictions on the rules forthe logical connectives ¬ , ∧ , ∨ and → . Along this paper, we fix a denumerable set prop of propositional variables, as well as three propositionalsignatures: S just containing a negation (unary) connective ¬ and a disjunction (binary) connective ∨ ; S just containing negation ¬ and a conjunction (binary) connective ∧ ; and S , containing ¬ , ∨ , ∧ , and animplication (binary) connective → . The set of formulas generated by S i and prop will be denoted by For i ,for i = , ,
2. The disjunction ∨ and the implication → are defined in For as a ∨ b = de f ¬ ( ¬ a ∧ ¬ b ) and a → b = de f ¬ ( a ∧ ¬ b ) , respectively. By its turn, the conjunction ∧ and the implication → aredefined in For as a ∧ b = de f ¬ ( ¬ a ∨ ¬ b ) and a → b = de f ¬ a ∨ b , respectively.For i = , ,
2, the function var : For i → ˆ ( prop ) which assigns to each formula the set of proposi-tional variables appearing in it is defined recursively as usual. When G ⊆ For i is a set of formulas then var ( G ) = S g ∈ G var ( g ) .The next step is to recall a well-known cut-free sequent calculus for classical propositional logic CPL defined over the signature S . Definition 1
By a sequent S over S i (i = , , ) we shall mean an ordered pair h G , D i of (non-simultaneouslyempty) finite sets of formulas in For i . We shall use the more suggestive notation G ⇒ D for the sequent h G , D i . Sequents of the form h G , /0 i , h /0 , D i , h G , { a }i and h{ a } , D i will be denoted by G ⇒ , ⇒ D , G ⇒ a and a ⇒ D , respectively. As usual,we write a , G (or G , a ) and a , b , G (or G , a , b ) instead of G ∪ { a } and G ∪ { a , b } , respectively. Definition 2
The sequent calculus C over S is defined as follows: Axioms Ax a ⇒ a Structural rules W ⇒ G ⇒ Da , G ⇒ D ⇒ W G ⇒ DG ⇒ D , a Cut G ⇒ D , a a , G ⇒ DG ⇒ D .E.Coniglio &M.I.Corbal´an 127 Operational rules ¬ ⇒ G ⇒ D , a ¬ a , G ⇒ D ⇒ ¬ a , G ⇒ DG ⇒ D , ¬ a ∧ ⇒ a , a , G ⇒ Da ∧ a , G ⇒ D ⇒ ∧ G ⇒ D , a G ⇒ D , a G ⇒ D , a ∧ a ∨ ⇒ a , G ⇒ D a , G ⇒ Da ∨ a , G ⇒ D ⇒ ∨ G ⇒ D , a , a G ⇒ D , a ∨ a →⇒ G ⇒ D , a a , G ⇒ Da → a , G ⇒ D ⇒→ a , G ⇒ D , a G ⇒ D , a → a For i = , ,
2, consider the usual classical valuations from
For i over the set V CPL = { , } of classicaltruth-values , where 1 denotes the “true” value and 0 denotes the “false” value. Let (cid:15) CPL be the semanticalconsequence relation of
CPL over
For , that is: G (cid:15) CPL a iff, for every classical valuation v : if v ( g ) = g ∈ G then v ( a ) =
1. The following theorems are well known:
Theorem 3 (Soundness and Completeness of C)
Let G ∪ D be a finite set of formulas in For . Then:the sequent G ⇒ D is provable in C iff G (cid:15) CPL W a ∈ D a . In particular: the sequent G ⇒ a is provable in C iff G (cid:15) CPL a . The same holds for the {¬ , ∨} and the {¬ , ∧} -fragments of C . Theorem 4 (Cut elimination for C)
Let G ∪ D be a finite nonempty set of formulas in For . If the se-quent G ⇒ D is provable in C then there is a cut-free derivation of it in C , that is, a derivation withoutusing the Cut rule. The same holds for the {¬ , ∨} and the {¬ , ∧} -fragments of C . and H B of Bochvar and H of Halld´en are three-valued logics. Their set of truth-valuesis V = (cid:8) , , (cid:9) where the third non-classical truth-value is interpreted as a nonsensical truth-value.In H this third truth-value is designated; on the other hand, is undesignated in B . So, D B = { } isthe set of designated values of B and D H = (cid:8) , (cid:9) is the set of designated values of Halld´en’s logic H . The logic B is defined over the signature S B obtained from the signature S by adding an unary‘meaningful’ connective B . By its turn, H is defined over the signature S H obtained from S by addingan unary ‘meaningful’ connective H . By means of the connectives B and H it is possible to express themeaninglessness of a formula at the object-language level of each logic. The abbreviations for definingthe other classical connectives in each signature are the same as in CPL (recall Section 1). The truth-tables for negation, conjunction, disjunction, implication and meaningful connectives in B and H areas follows: ¬
12 12 ∧
01 1
12 12 12 12 ∨
01 1
12 12 12 12 →
01 1
12 12 12 12 Here, W a ∈ D a denotes the formula a ∨ ( a ∨··· ( a n − ∨ a n ) ... ) , if D = { a ,..., a n } . If D = { a } or D = /0 then W a ∈ D a = a and W a ∈ D a = p ∧ ¬ p , respectively, where p is the first propositional variable.
28 Sequent Calculi for Bochvar and Halld´en’s Nonsense Logics B
00 0 H
00 1Additionally, H can be defined in terms of the connectives of B but the same relationship between B and H is not true, and so the expressive power of the matrices of B is strictly stronger than that ofthe matrices of H (cf. [5]).The key feature of both logics is the following, which can be easily proved by induction on thecomplexity of the formula a : Proposition 5
Let a be a formula of L without and let v be a valuation of L , where L = B or L = H .Then: v ( a ) = iff v ( p ) = for some propositional variable p ∈ var ( a ) . This means that in the ‘classical’ fragment of B and H the non-classical truth-value is ‘infec-tious’: an atomic formula ‘infects’ complex formulas with the nonsensical truth-value. It is easy to provethat, over the respective S i , both B and H are deductive fragments of classical logic: every valid infer-ence in B or in H written in the classical signature S i is valid in CPL . In fact, the following proposition(whose proof is immediate) holds in B and H . Proposition 6
Let a be a formula of L without , let v CPL be a classical valuation and let v L be avaluation of L , where L = B or L = H . If v L ( p ) = v CPL ( p ) for every propositional variable p ∈ var ( a ) then v L ( a ) = v CPL ( a ) (and so v L ( a ) ∈ { , } ). Despite these similarities, there are important differences between B and H with respect to classicallogic as a consequence of choosing different sets of designed truth-valued: • There are no tautological formulas over S in B ; H contain every classical tautology over S . • No contradiction written over S is a trivializing formula in H ; every contradiction over S is atrivializing formula in B . • The Deduction Theorem is not valid in B and modus ponens is not valid in H . So, the followingmetaproperty does not hold in B : if G , a (cid:15) b , then G (cid:15) a → b ; on the other hand, the followingmetaproperty does not hold in H : if G (cid:15) a → b , then G , a (cid:15) b . • The inference a (cid:15) a ∨ b does not hold in B ; in H the inference a ∧ b (cid:15) a does not hold. • In B the Principle of Excluded Middle: (cid:15) a ∨ ¬ a (PEM)does not hold; in H the Principle of Explosion: a , ¬ a (cid:15) b (PE)does not hold. Thus, B is paracomplete w.r.t. the negation ¬ , while H is paraconsistent w.r.t. ¬ .These differences between Bochvar and Halld´en’s connectives with respect to classical connectivesare not independent from each other, and their connections are expressed in the following theorems,which constitute the basis of our proposal. Theorem 7
Let G ∪ { a } be a set of formulas in For such that G (cid:15) CPL a . Then:if var ( a ) ⊆ var ( G ) or G (cid:15) CPL p ∧ ¬ p then G (cid:15) B a . .E.Coniglio &M.I.Corbal´an 129 Proof.
Cf. [3, 9, 7, 5].
Theorem 8
Let G ∪ { a } be a set of formulas in For such that G (cid:15) CPL a . Then:if var ( G ) ⊆ var ( a ) or (cid:15) CPL a then G (cid:15) H a . Proof.
Assume that G (cid:15) CPL a . If G H a then there is a valuation v H for H such that v H ( G ) ⊆ (cid:8) , (cid:9) and v H ( a ) =
0. Suppose that var ( G ) ⊆ var ( a ) . Since v H ( a ) = v H ( p ) ∈ { , } for every propositional variable p ∈ var ( a ) . Thus v H ( G ) ⊆ { } . Let v CPL be a classicalvaluation such that v CPL ( p ) = v H ( p ) for every p ∈ var ( a ) = var ( a ) ∪ var ( G ) . Then, by Proposition 6, v CPL ( G ) ⊆ { } but v CPL ( a ) =
0, a contradiction. Then, var ( G ) * var ( a ) . Thus, if var ( G ) ⊆ var ( a ) then G (cid:15) H a .Finally, if a is a classical tautology, let v H be a valuation for H . If v H ( p ) = for some p ∈ var ( a ) then v H ( a ) = , by Proposition 5. On the other hand, if v H ( var ( a )) ⊆ { , } then, by Proposition 6, v H ( a ) =
1. Then, (cid:15) H a and so G (cid:15) H a .So, by Theorem 7, we have that if a valid classical inference G (cid:15) a is invalid in Bochvar’s nonsenselogic then G is a consistent set of formulas of CPL such that var ( a ) var ( G ) . On the other hand,Theorem 8 expresses that if a valid classical inference G (cid:15) a is invalid in Halld´en’s nonsense logic then a is not a tautological formula in CPL and var ( G ) var ( a ) . Therefore, it is clear that (cid:15) H a but B a ,for every a such that (cid:15) CPL a .By Theorems 7 and 8 we obtain a sufficient condition in order to determine whether a valid classicalinference is also valid in both B and H . Corollary 9
Let G ∪ { a } be a set of formulas in For such that G (cid:15) CPL a . Then: if var ( G ) = var ( a ) , then G (cid:15) B a and G (cid:15) H a . We will introduce cut-free sequent calculi for the {¬ , ∨} -fragment of H and for the {¬ , ∧} -fragmentof B , where ∧ and → ( ∨ and → , respectively) are derived connectives. The strategy adopted is tomodify the classical sequent rules for classical connectives by adding suitable provisos. As we shall see,the provisos are applied to symmetrical rules: in the fragment of Halld´en’s logic, the provisos apply tothe introduction rules for conjunction, implication and negation on the left side of the sequent while, inthe case of Bochvar’s logic, the proviso applies to the introduction rules for disjunction, implication andnegation on the the right side. This reflects the relationship between these logics and classical logic, asdepicted in theorems 7 and 8. {¬ , ∨} -fragment of Halld´en’s logic H C should be blocked in any sequent calculus for H . Wepresent now a cut-free sequent calculus H for the fragment of H over S by adding provisos on theapplication of (classical) rules such that the construction of complex formulas in the antecedent of thesequents is blocked in some cases. By symmetry, a sequent calculus B for B will be also introduced byadding provisos on the application of (classical) rules such that the construction of complex formulas inthe succedents of the sequents is blocked under certain circumstances. Obviously we are identifying here a primitive connective of S with its abbreviation in S i , for i = ,
30 Sequent Calculi for Bochvar and Halld´en’s Nonsense Logics
Definition 10
The sequent calculus H is obtained from the {¬ , ∨} -fragment of C by replacing the rule ¬ ⇒ by the following one: ¬ H ⇒ G ⇒ D , a ¬ a , G ⇒ D provided that var ( a ) ⊆ var ( D ) Proposition 11
The following rules are derivable in H : ∧ H ⇒ a , a , G ⇒ Da ∧ a , G ⇒ D ⇒ ∧ G ⇒ D , a G ⇒ D , a G ⇒ D , a ∧ a with the following proviso: var ( { a , a } ) ⊆ var ( D ) in ∧ H ⇒ . Proof.
Assume that var ( { a , a } ) ⊆ var ( D ) . Then var ( ¬ a ∨ ¬ a ) ⊆ var ( D ) and so the followingderivation can be done in H : a , a , G ⇒ DG ⇒ D , ¬ a , ¬ a (by ⇒ ¬ ) G ⇒ D , ¬ a ∨ ¬ a (by ⇒ ∨ ) ¬ ( ¬ a ∨ ¬ a ) , G ⇒ D (by ¬ H ⇒ )In order to obtain ⇒ ∧ , the following derivation can be done in H : G ⇒ D , a G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) , a ¬ a , G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) ¬ H ⇒ ⇒ W G ⇒ D , a G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) , a ¬ a , G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) ¬ H ⇒ ⇒ W ¬ a ∨ ¬ a , G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) G ⇒ D , ¬ ( ¬ a ∨ ¬ a ) ⇒ ¬ ∨ ⇒ Proposition 12
The following implicational rules are derivable in H : → H ⇒ G ⇒ D , a a , G ⇒ Da → a , G ⇒ D ⇒ → a , G ⇒ D , a G ⇒ D , a → a with the following proviso: var ( { a , a } ) ⊆ var ( D ) in → H ⇒ . Proof.
Straightforward, by considering that a → a stands for ¬ a ∨ a in H . In this subsection we shall prove the soundness of sequent calculus H . Firstly, some semantical notionswill be extended from formulas to sequents. Definition 13
Let L be a matrix logic over a signature S . A valuation v of L is a model of a sequent G ⇒ D over S iff, if v ( G ) ⊆ D L , then v ( d ) ∈ D L for some d ∈ D . When v is a model of the sequent G ⇒ D ,we will write v (cid:15) L G ⇒ D . .E.Coniglio &M.I.Corbal´an 131 Definition 14 A sequent G ⇒ D is valid in L if, for every valuation v of L , v is a model of the sequent G ⇒ D . When the sequent is valid, we will write (cid:15) L G ⇒ D . It is worth noting that (cid:15) L G ⇒ a iff G (cid:15) L a . Additionally, (cid:15) CPL G ⇒ D iff G (cid:15) CPL W a ∈ D a Definition 15
A sequent rule R preserves validity in L if, for every instance ¡ S of R and for everyvaluation v of L , if v (cid:15) L S ′ for every S ′ ∈ ¡ then v (cid:15) L S. Lemma 16
Every sequent rule of the calculus H preserves validity. Proof.
Observe that the axiom Ax and the structural rules preserve validity, since they correspond toproperties which are valid in every Tarskian logic (and H is Tarskian since it is a matrix logic). ⇒ ¬ Let v be a valuation of H such that v (cid:15) H a , G ⇒ D , and suppose that v ( G ) ⊆ (cid:8) , (cid:9) . If v ( ¬ a ) = v ( a ) =
1. Then, by hypothesis, we infer that v ( d ) ∈ (cid:8) , (cid:9) , for some d ∈ D . If v ( ¬ a ) = v ( ¬ a ) ∈ (cid:8) , (cid:9) . This shows that v (cid:15) H G ⇒ D , ¬ a . ¬ H ⇒ Let v be a valuation of H such that v (cid:15) H G ⇒ D , a and assume that var ( a ) ⊆ var ( D ) . Supposethat v ( ¬ a ) ∈ (cid:8) , (cid:9) and v ( G ) ⊆ (cid:8) , (cid:9) . Then, by hypothesis, v ( d ) ∈ (cid:8) , (cid:9) , for some d ∈ D , or v ( a ) ∈ (cid:8) , (cid:9) . Since v ( ¬ a ) ∈ (cid:8) , (cid:9) , then v ( a ) ∈ (cid:8) , (cid:9) . If v ( a ) = v ( d ) ∈ (cid:8) , (cid:9) , forsome d ∈ D . And if v ( a ) = , then, by Proposition 5, we infer that v ( p ) = for some atomicformula p ∈ var ( a ) . Since var ( a ) ⊆ var ( D ) then p ∈ var ( d ) for some d ∈ D and so, again byProposition 5, we infer that v ( d ) = (cid:8) (cid:9) . Therefore, we conclude that v (cid:15) H ¬ a , G ⇒ D . ⇒ ∨ Let v be a valuation of H such that v (cid:15) H G ⇒ D , a , a and assume that v ( G ) ⊆ (cid:8) , (cid:9) . If v ( a ) = v ( a ) = v ( d ) ∈ (cid:8) , (cid:9) , for some d ∈ D . Therefore v (cid:15) H G ⇒ D , a ∨ a . Otherwise, if v ( a ) ∈ (cid:8) , (cid:9) or v ( a ) ∈ (cid:8) , (cid:9) then v ( a ∨ a ) ∈ (cid:8) , (cid:9) and so v (cid:15) H G ⇒ D , a ∨ a . ∨ ⇒ Let v be a valuation of H such that v (cid:15) H a , G ⇒ D and v (cid:15) H a , G ⇒ D . Suppose that v ( a ∨ a ) ∈ (cid:8) , (cid:9) and v ( G ) ⊆ (cid:8) , (cid:9) . Then, either v ( a ) ∈ (cid:8) , (cid:9) or v ( a ) ∈ (cid:8) , (cid:9) . By hypothesis, it fol-lows that v ( d ) ∈ (cid:8) , (cid:9) , for some d ∈ D and so v (cid:15) H a ∨ a , G ⇒ D . Theorem 17 (Soundness of H)
Let G ∪ D be a set of formulas in For . Then: if G ⇒ D is provable in H then (cid:15) H G ⇒ D . In particular, if G ⇒ a is provable in H then G (cid:15) H a . Proof.
If the sequent G ⇒ D is an instance of axiom Ax, then G ⇒ D is valid in H . By induction onthe depth of a derivation of G ⇒ D in H it follows, by the previous Lemma 16, that the sequent G ⇒ D isvalid in H . Proposition 18
Let G be a nonempty set of formulas in For . Then the sequent G ⇒ is not provablein H . Proof.
Let v be a H -valuation such that v ( p ) = for every p ∈ var ( G ) . Then v H G ⇒ and so H G ⇒ . By contraposition of Theorem 17, we conclude that the sequent G ⇒ is not provable in H .32 Sequent Calculi for Bochvar and Halld´en’s Nonsense Logics The following result follows straightforwardly:
Proposition 19
Let G ∪ D be a finite nonempty set of formulas in For . Then:if (cid:15) H G ⇒ D , then (cid:15) CPL G ⇒ D . Proof.
Assume that (cid:15) H G ⇒ D and let v be a classical valuation such that v ( G ) ⊆ { } . By Proposition 6, v can be seen as a H -valuation such that v ( G ) ⊆ { , } . By hypothesis, v ( d ) ∈ { , } for some d ∈ D .Since v is classical, it follows that v ( d ) = d ∈ D , therefore (cid:15) CPL G ⇒ D . Proposition 20
Let G ∪ D be a finite nonempty set of formulas in For . Then: if G ⇒ D is provable in H then G ⇒ D is provable in the {¬ , ∨} -fragment of C . Proof.
This is obvious, since H is a restricted version of the {¬ , ∨} -fragment of C . Lemma 21
Let G ∪ D be a finite nonempty set of formulas in For . Then: if G ⇒ D is provable in the {¬ , ∨} -fragment of C and var ( G ) ⊆ var ( D ) then G ⇒ D is provable in H without using the Cut rule. Proof.
Recall that derivations in C and H are rooted binary trees such that the root is the sequent beingproved, and the leaves are always instances of the axiom Ax of the form a ⇒ a for some formula a .Assume that P is a cut-free derivation in the {¬ , ∨} -fragment of C of a sequent G ⇒ D such that var ( G ) ⊆ var ( D ) (we can assume this by Theorem 4). If P is also a derivation in H then the resultfollows automatically. Otherwise, there are in P , by force, applications of the rule ¬ ⇒ , namely ¬ ⇒ G ′ ⇒ D ′ , a ¬ a , G ′ ⇒ D ′ such that the proviso required by this rule in H is not satisfied. Since P is cut-free then the set of variablesoccurring in the root sequent G ⇒ D contains all the propositional variables occurring in P . Then, byhypothesis, all the propositional variables occurring in P belong to the set var ( D ) . Consider now thederivation P ′ in C obtained from P in two steps: firstly, the right-hand side of each sequent (that is, ofeach node) of P is enlarged by adding simultaneously all the formulas in D . This generates a rootedbinary tree P whose leafs are sequents of the form a ⇒ a , D . Each of such leaves of P correspondsto the original occurrence of an axiom (that is, a leaf) a ⇒ a in the derivation P . In the second step, wereplace each leaf a ⇒ a , D of P by a branch started by a ⇒ a and followed by iterated applicationsof the weakening rule ⇒ W until obtaining the sequent a ⇒ a , D . The resulting rooted binary tree P ′ is clearly a (cut-free) derivation in the {¬ , ∨} -fragment of C of the sequent G ⇒ D . But the criticalapplications of the rule ¬ ⇒ mentioned above have in P ′ the form ¬ ⇒ G ′ ⇒ D ′ , D , a ¬ a , G ′ ⇒ D ′ , D .Being so, these applications are allowed in H (since all the propositional variables occurring in P ′ belongto the set var ( D ) ) and so P ′ is in fact a cut-free derivation in H of the sequent G ⇒ D . That is, G ⇒ D isprovable in H without using the Cut rule. Corollary 22
Let D be a finite nonempty set of formulas in For . Then: ⇒ D is provable in the {¬ , ∨} -fragment of C if and only if ⇒ D is provable in H . Observe that some applications of the weakening rule ⇒ W in P may be innocuous in P ′ . .E.Coniglio &M.I.Corbal´an 133 Corollary 23 (Modus Ponens)
Let a , b ∈ For . Then: if ⇒ a and ⇒ a → b are provable in H then ⇒ b is provable in H . Lemma 24
Let G ∪ D be a finite nonempty set of formulas in For . If (cid:15) H G ⇒ D but var ( G ) var ( D ) then there exists G ′ ⊂ G such that (cid:15) H G ′ ⇒ D , where var ( G ′ ) ⊆ var ( D ) . Proof.
Observe that if (cid:15) H G ⇒ D then D = /0.Assume that (cid:15) H G ⇒ D such that var ( G ) var ( D ) . So, given a valuation v for H , if v ( G ) ⊆ (cid:8) , (cid:9) then v ( d ) ∈ (cid:8) , (cid:9) for some formula d ∈ D . Given that var ( G ) var ( D ) consider the set G ′ = G \{ g ∈ G : var ( g ) var ( D ) } . Then, G ′ ⊂ G and var ( G ′ ) ⊆ var ( D ) . Let v be a valuation for H suchthat v ( G ’ ) ⊆ (cid:8) , (cid:9) . If ∈ v ( G ’ ) then v ( p ) = for some propositional variable p ∈ var ( G ′ ) . Since var ( G ′ ) ⊆ var ( D ) , then ∈ v ( D ) . If v ( G ′ ) ⊆ { } , suppose that v ( D ) = { } . Then v ( p ) ∈ { , } forevery propositional variable p ∈ var ( D ) , by Proposition 5. Since var ( G ′ ) ⊆ var ( D ) then, for everypropositional variable p ∈ var ( G ′ ) , v ( p ) ∈ { , } . Consider now a valuation v ′ for H such that v ′ ( p ) = for every p ∈ var ( G ) \ var ( D ) , and v ′ ( p ) = v ( p ) for every p ∈ var ( D ) . Then, v ′ ( G ) ⊆ (cid:8) , (cid:9) . But then,by hypothesis, v ′ ( d ) ∈ (cid:8) , (cid:9) , for some d ∈ D . That is, v ( d ) ∈ (cid:8) , (cid:9) for some d ∈ D , a contradiction.Therefore, if v ( G ′ ) ⊆ { } then v ( d ) =
0, for some d ∈ D . So, (cid:15) H G ’ ⇒ D . Theorem 25 (Completeness of H)
Let G ∪ D be a finite nonempty set of formulas in For . If (cid:15) H G ⇒ D then G ⇒ D is provable in H without using the Cut rule. In particular, if G (cid:15) H a then the sequent G ⇒ a is provable in H , for every finite set G ∪ { a } . Proof.
Assume that (cid:15) H G ⇒ D . Then, by Proposition 19, (cid:15) CPL G ⇒ D . By Theorem 3, G ⇒ D is provablein the {¬ , ∨} -fragment of C . If var ( G ) ⊆ var ( D ) then, by Lemma 21, G ⇒ D is provable in H withoutusing the Cut rule. If var ( G ) var ( D ) then, by Lemma 24, there exist a set G ′ ⊂ G such that (cid:15) H G ′ ⇒ D ,where var ( G ′ ) ⊆ var ( D ) . Then, using Proposition 19 and Theorem 3 again, we obtain that G ′ ⇒ D isprovable in the {¬ , ∨} -fragment of C . Since var ( G ′ ) ⊆ var ( D ) then, by using Lemma 21, it follows that G ′ ⇒ D is provable in H without using the Cut rule. By applying the structural rule W ⇒ several timeswe obtain a derivation of G ⇒ D in H without using the Cut rule, as desired. Corollary 26 (Cut elimination for H)
Let G ∪ D be a finite nonempty set of formulas in For. If thesequent G ⇒ D is provable in H then there is a cut-free derivation of it in H . Proof.
Suppose that G ⇒ D is provable in H . By Theorem 17, (cid:15) H G ⇒ D . Then, by Theorem 25, thereis a cut-free derivation of G ⇒ D in H . {¬ , ∧} -fragment of Bochvar’s logic B B which will result cut-free, sound and complete for theconjunction-negation fragment of the nonsense logic B , where ∨ and → are derived connectives. As weshall see, there exists a symmetry between the provisos imposed in the rules of B and those imposed in H , as long as the language ¬ , ∧ , ∨ , → is considered. Definition 27
The sequent calculus B is obtained from the {¬ , ∧} -fragment of C by replacing the rule ⇒ ¬ by the following one: ⇒ ¬ B a , G ⇒ DG ⇒ D , ¬ a provided that var ( a ) ⊆ var ( G )
34 Sequent Calculi for Bochvar and Halld´en’s Nonsense Logics
Proposition 28
The following rules are derivable in B : ∨ ⇒ a , G ⇒ D a , G ⇒ Da ∨ a , G ⇒ D ⇒ ∨ B G ⇒ D , a , a G ⇒ D , a ∨ a with the following proviso: var ( { a , a } ) ⊆ var ( G ) in ⇒ ∨ B . Proof.
We leave the easy proof as an exercise to the reader.
Proposition 29
The following implicational rules are derivable in B : → ⇒ G ⇒ D , a a , G ⇒ Da → a , G ⇒ D ⇒ → B a , G ⇒ D , a G ⇒ D , a → a with the following proviso: var ( { a , a } ) ⊆ var ( G ) in ⇒ → B . Proof.
The proof is also left to the reader.
In order to prove the Soundness Theorem for B , we will prove that every sequent rule of the calculus B preserves validity. Lemma 30
Every sequent rule of the calculus B preserves validity. Proof.
As in the case of H , it is enough to analyze the rules for connectives. ⇒ ¬ B Assume that v | = B a , G ⇒ D for some valuation v in B , where var ( a ) ⊆ var ( G ) . Suppose that v ( G ) ⊆ { } . Then, by Proposition 5, v ( p ) ∈ { , } , for every propositional variable p such that p ∈ var ( G ) . Since var ( a ) ⊆ var ( G ) , then v ( p ) ∈ { , } , for every propositional variable p ∈ var ( a ) .By Proposition 5 again, we obtain that v ( a ) ∈ { , } . If v ( a ) =
1, then by hypothesis, we obtainthat { } ⊆ v ( D ) . If v ( a ) = v ( ¬ a ) =
1. In both cases it follows that { } ⊆ v ( D ∪ {¬ a } ) .Therefore v | = B G ⇒ D , ¬ a . ¬ ⇒ Assume that v | = B G ⇒ D , a for some valuation v in B . Suppose that v ( ¬ a ) = v ( G ) ⊆ { } .So, { } ⊆ v ( D ) or v ( a ) =
1, by hypothesis. But, since v ( ¬ a ) =
1, then v ( a ) =
0. Thus, { } ⊆ v ( D ) and so v | = B ¬ a , G ⇒ D . ⇒ ∧ Assume that v | = B G ⇒ D , a and v | = B G ⇒ D , a for some valuation v in B . Suppose that v ( G ) ⊆ { } . By hypothesis, we obtain that either { } ⊆ v ( D ) or both v ( a ) = v ( a ) =
1. Inboth cases it follows that { } ⊆ v ( D ∪ { ( a ∧ a ) } ) . Then v | = B G ⇒ D , a ∧ a . ∧ ⇒ Assume that v | = B a , a , G ⇒ D for some valuation v in B . Suppose that v ( a ∧ a ) = v ( G ) ⊆ { } . So, v ( a ) = v ( a ) = v ( G ) ⊆ { } . By hypothesis, { } ⊆ v ( D ) . Therefore, v | = B a ∧ a , G ⇒ D .As a consequence of this it follows the soundness theorem for B : Theorem 31 (Soundness of B)
Let G ∪ D be a finite nonempty subset of For . Then: if G ⇒ D is provablein B then | = B G ⇒ D . In particular, if G ⇒ a is provable in B then G | = B a . Corollary 32
Let D ⊆ For be a nonempty set of formulas. Then the sequent ⇒ D is not provable in B . Proof.
Consider a valuation v for B such that v ( p ) = for every p ∈ var ( D ) . Then v = B ⇒ D and so = B ⇒ D . By Theorem 31, the sequent ⇒ D is not provable in B ..E.Coniglio &M.I.Corbal´an 135 The proof of completeness of B is similar to that of H and so we will omit some proofs. Proposition 33
Let G ∪ D be a finite nonempty subset of For . Then:if | = B G ⇒ D , then | = CPL G ⇒ D . Proposition 34
Let G ∪ D be a finite nonempty subset of For . Then: if G ⇒ D is provable in B then it isprovable in the {¬ , ∧} -fragment of C . Lemma 35
Let G ∪ D be a finite nonempty subset of For . Then: if G ⇒ D is provable in the {¬ , ∧} -fragment of C and var ( D ) ⊆ var ( G ) then G ⇒ D is provable in B without using the Cut rule. Proof.
The proof is analogous to that of Lemma 21, but now using the rule W ⇒ . Lemma 36
Let G ∪ D be a finite nonempty subset of For . If | = B G ⇒ D but var ( D ) var ( G ) then thereexist a set D ′ ⊂ D such that | = B G ⇒ D ′ , where var ( D ′ ) ⊆ var ( G ) . Proof.
Let D ′ = D \ { d ∈ D : var ( d ) var ( G ) } . Suppose that there is a B -valuation v such that v ( G ) ⊆{ } but v ( D ′ ) ⊆ (cid:8) , (cid:9) . Thus, the B -valuation v ′ such that v ′ ( p ) = v ( p ) for every p ∈ var ( G ) and v ′ ( p ′ ) = for every p ′ ∈ var ( D ) \ var ( G ) is such that v ′ ( G ) ⊆ { } but v ′ ( D ) ⊆ (cid:8) , (cid:9) , a contradiction.Therefore | = B G ⇒ D ′ , where var ( D ′ ) ⊆ var ( G ) . Theorem 37 (Completeness of B)
Let G ∪ D be a finite nonempty subset of For . If | = B G ⇒ D then G ⇒ D is provable in B without using the Cut rule. In particular, if G | = B a then G ⇒ a is provable in B . Proof.
Assume that | = B G ⇒ D . Then, by Proposition 33 and Theorem 3, it follows that G ⇒ D isprovable in the {¬ , ∧} -fragment of C . If var ( D ) ⊆ var ( G ) then, by Lemma 35, the sequent G ⇒ D isprovable in B without using the Cut rule. On the other hand, if var ( D ) var ( G ) , then by Lemma 36, (cid:15) B G ⇒ D ′ , for some set D ′ ⊂ D such that var ( D ′ ) ⊆ var ( G ) . By Proposition 33 and Theorem 3 again, itfollows that G ⇒ D ′ is provable in the {¬ , ∧} -fragment of C . Using again Lemma 35, the sequent G ⇒ D ′ is provable in B without using the Cut rule. Finally, by applying the structural rule ⇒ W several timeswe obtain a derivation of G ⇒ D in B without using the Cut rule. Corollary 38 (Cut elimination for B)
Let G ∪ D be a finite nonempty set of formulas in For . If thesequent G ⇒ D is provable in B then there is a cut-free derivation of it in B . Proof.
Suppose that G ⇒ D is provable in B . By Theorem 31, | = B G ⇒ D . Then, by Theorem 37, thereis a cut-free derivation of G ⇒ D in B as desired. In this paper a cut-free sequent calculi for the {¬ , ∨} -fragment of Bochvar’s logic, as well as a cut-free sequent calculi for the {¬ , ∧} -fragment of Halld´en’s logic, were proposed. In the former calculus,conjunction and implication are derived connectives, while disjunction and implication are derived con-nectives in the latter. The main feature of both calculi is that they are obtained by imposing provisos tothe rules of the respective fragments of a well-known sequent calculus for classical propositional logic.The signature for each calculus was choosen in order to keep as close as possible to the respective frag-ment of classical logic. Observe that both {¬ , ∨} and {¬ , ∧} -fragments are adequate, that is, they canexpress all the other (classical) connectives.36 Sequent Calculi for Bochvar and Halld´en’s Nonsense LogicsThus, concerning the calculus for the {¬ , ∨} -fragment of Halld´en’s logic, the only change requiredwith respect to the calculus for the respective fragment of classical logic was the inclusion of a provisoin the introduction rule for negation on the left side of the sequent. As a consequence of this, a provisoappear in the (derived) introduction rules for conjunction and implication on the left side of the sequent.In the calculus for the {¬ , ∧} -fragment of Bochvar’s logic, the situation is entirely symmetrical: therestriction was imposed to the introduction rule for negation on the right side, and so this restriction alsoapplies to the introduction rules for disjunction and implication on the right side (both are derived rules).In this manner, the existing relationship between classical logic and both logics became explicit throughrestrictions on the rules for the logical connectives.Since these two logic of nonsense are related to classical logic in such particular way, the ad hoc definition of sequent calculi presented here, which exploit these particularities, seems to be justified.However, it would be interesting to compare the cut-free sequent calculi introduced here with the oneswhich could be obtained by applying general techniques such as those proposed in [2, 1, 8].As a future research, we plan to extend the calculi to the full language of both logics. Clearly theresulting calculi will not be so simple and symmetrical because of the subtleties of the ‘meaningful’connectives and their relationship with the other connectives. Acknowledgements:
We would like to thank the anonymous referees for their extremely useful com-ments on an earlier draft, which have helped to improve the paper. The first author was financed byFAPESP (Brazil), Thematic Project LogCons 2010/51038-0 and by an individual research grant fromThe National Council for Scientific and Technological Development (CNPq), Brazil.
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