sequent-systems and groupoid models. ikosta/dosen radovi/[p][21... · groupoid models will be based...

KOSTA

DO~EN Sequent-Systems and Groupoid Models. I

Abstract. The purpose of this paper is to connect the proof theory and the model theory of a family of propositional logics weaker than Heyting's. This family includes systems analogous to the Lambek calculus of syntactic categories, systems of relevant logic, systems related to BCK algebras, and, finally, Johansson's and Heyting's logic. First, sequent-systems are given for these logics, and cut-elimination results are proved. In these sequent-systems the rules for the logical operations are never changed: all changes are made in the structural rules. Next, Hilbert-style formulations are given for these logics, and algebraic completeness results are demonstrated with respect to residuated lattice-ordered groupoids. Finally, model structures related to relevant model structures (of Urquhart, Fine, Routley, Meyer, and Maksimova) are given for our logics. These model structures are based on groupoids parallel to the sequent-systems. This paper lays the ground for a kind of correspondence theory for axioms of logics with implication weaker than Heyting's, a correspondence theory analogous to the correspondence theory for modal axioms of normal modal logics.

The first part of the paper, which follows, contains the first two sections, which deal with sequent-systems and Hilbert-formulations. The second part, due to appear in the next issue of this journal, will contain the third section, which deals with groupoid models.

Contents

Introduction

1. Sequent Systems 1.1. The language GP 1.2. The system GL 1.3. Additional structural rules 1.4. The system G/~ 1.5. Cut-elimination in G/J 1.6. Cut-elimination with additional structural rules 1.7. The system GC

2. Hilbert-formulations 2.1. The system L 2.2. L and urlogs 2.3. GL and L 2.4. The ( ~ , o) and implicational fragment of L 2.5. Extensions of L 2.6. The extensions E + 2.7. Extensions of S 2.8. The ( ~ , A, V, .1.) fragments of extensions of S +

354 K. Dogen

2.9. 2.10.

3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

3.10. 3.11. 3.12.

Implicational fragments of extensions of L Negation in L and its extensions

Groupoid models Kripke frames aad models Groupoid frames and models Heredity in groupoid models Soundness of L Completeness of L Correspondence between schemata and conditions on groupoid frames Soundness and completeness of extensions of L J groupoid models Pure groupoid models Correspondence for implicational schemata Groupoid models where negation is not purely implicational Systems with distribution

Introduction

The first impression left by models for the better known systems of relevant logic, R and E (see [8], [15], [21], [22], [23], [24]), was that these models are much less intuitive and much more irregular than Kripke models for intuitionistic or modal logic. However, this impression might be due not so much to the type of models used, but more to the systems these models were used for. The systems R and E are "composite" systems. In the negationless fragment of R, which is a subsystem of Heyting's logic, distribution of conjunction over disjunction does not arise from the ordinary assumptions about conjunction and disjunction, but is added as an extra assumption. Matters are further complicated by .having in R a nonintuitionistic negation, which is idempotent and satisfies de Morgan's Laws. The system E adds to all this considerations about modality, which make its implication a kind of strict implication.

If we do not try to model composite systems with them, models of the type used for relevant logic will prove easier to handle. They will become quite regular, and perhaps even more intuitive. Moreover, they will be suitable not only for relevant propositional logics, but for a whole family of propositional logics included in Heyting's logic. We shall present here this family of propositional logics. For these logics we shall give models called oroupoid models, which are related more to the models of [24] and [81, than to those of [21]. Groupoid models will be rather easy to handle. But their main interest should be in the connection they have with proof theory. The logics in our family have simple and clean sequent-formulations, which are parallel with the groupoid models. This contrasts with the rather problematic position of sequent-formulations for R and E.

Sequent-systems... 355

The weakest logic in our family will be called L. The logic L has the most general sequent-system of the type we shall consider, and it is sound and complete with respect to all groupoid models. It is in a position analogous to the position of the weakest normal modal logic K, which is sound and complete with respect to all Kripke modal models. All extensions of L will be obtained by moves in the sequent framework which have their exact parallel in the model-theoretic framework. This foreshadows a correspondence theory between syntax and semantics, analogous to the correspondence theory which exists for normal modal logics. In our family of extensions of L we shall find, among others, systems related to the Lambek calculus of syntactic categories (see [131, [141, [71, [41), relevant systems without distribution and with intuitionistic negation, systems related to BCK algebras (see [111, [171), Johansson's propositional calculus, and finally Heyting's propositional calculus.

In the first part of our work we shall consider the sequent-systems for L and its extensions. These sequent-systems have the peculiarity that the comma joining foraiulae in sequents is taken as a binary operation (cf. [141, [21), Next, a cannonical set of rules is given for the logical operators. By moving up from L to stronger systems, these rules are never changed. What changes are structural assumptions, involving essentially properties of the comma (cf. [6t). We shall give cut-elimination and decidability results for a number of our sequent-systems.

In the second part of our work we shall give Hilbert-style formulations of L and its extensions. In that part we shall also present some simple algebraic completeness results with respect to algebras which we shall call urlo#s.

Finally, in the third part we shall present groupoid models for L and its extensions. Groupoid models will be based on groupoids whose binary operation will mirror properties of the comma on the left of the turnstile, in our

- sequent-systems. For example, if the comma is associative, the groupoid operation will be associative. Completeness proofs with respect to classes of groupoid models will be as elementary as the soundness proofs: we never use the usual infinitary argument, of the Lindenbaum Lemma type, connected with Henkin-style completeness proofs*.

This work is written as a chapter in an intermediate course on propositional calculi. We presuppose for it a certain familiarity with nonclassical propositional calculi, which the reader may have acquired by studying, for example, parts of [11. A basic understanding of Gentzen-style proof theory, not going beyond [101, is also presupposed. Finally, we assume the reader has some knowledge of Kripke semantics for intuitionistic propositional logic: a familiarity with a few ideas of, for example, [51(Chapter 5) will suffice.

* I suppose that propositional logics recently investigated in [25] belong to the family of logics investigated here. They should correspond to logics where the comma is not associative.

356 K. DoWn

Otherwise, this work is rather self-contained. It includes a number of elementary results, many of which are probably known to experienced workers in the field, in one form or another. However, these results do not seem to have ever been collected together. We have thought it useful to have them all in one place, because we are not looking for striking single results, but surveying a "landscape". This way we hope to show that behind the intricacies of some results in the literature there is a simple and easily understandable picture.

Acknowledgements

I would like to express my debt to Dr Milan Bo~i6: his thesis [3], and the discussions we have had on relevant models, helped me enormously in understanding the topics treated here. Professor Hiroakira Ono has been very kind to send me the manuscript of [17], the subject of which overlaps with the subject of this work. I am very grateful to Professor Ono for drawing my attention to a serious error in the first draft of my manuscript. I have also profitted from contacts with Dr Wojciech Buszkowski, whose work on the Lambek calculus of syntactic categories bears some similarities with parts of this w o r k ( a s witness [4] and papers cited therein). Finally, I would like to thank Professors Johan van Benthem, Branislav Bori~i6, Robert Bull, J. Michael Dunn, Josep Font, Peter Schroeder-Heister and Andrzej Wrofiski for the encouraging comments they made on my manuscript.

1. Sequent-systems

1.1. The language GP. Let P be the language of the propositional calculus with:

denumerably many propositional variables, the binary connectives ~ , o, ^ , and v , the propositional constants -V and _L, the left and right parenthesis.

We shall use p, Pl . . . . , as schematic letters for propositional variables, and A, B, C, D, A1, ..., as schematic letters for formulae of P. In the metalanguage we shall use =~, *},/ff, &, or, V and 3 with the usual meaning they have in classical logic.

Starting from P we construct the sequent-language GP as follows. The vocabulary of GP is the vocabulary of P plus:

O, the comma, and the turnstile [-.

Next we give the following recursive definition of G-terms:

D~FrNmON 1. Every formula of P is a G-term, and 0 is a G-term. If X and Y are G-terms, then (X, Y) is a G-term.


We shall use X, Y, Z, X1, . . . , as schematic letters for G-terms. A subterm of a G-terrn is specified b y the following natural recursive definition:

DEFINITION 2. The G-term X is a subterm of itself. If(Y, Z) is a subterm of X, then Y and Z are subterms of X.

Let X [ Y ] stand for a G-term in which Yoccurs as a subterm, and let X[Z] be the result of substituing Z for a single Y in X [ Y ] . It will always be clear from the context which X[Y] we have started from in order to obtain X [ Z ] . Next we have the following definition:

DEFINITION 3. If X and Y are G-terms, then X [- Y is a sequent.

In order to formulate our systems in GP we shall use the following abbreviation. If So, S~, . . . , S, are sequents, then the double-line rule:

$1 ... S~

So

will be an abbreviation for the following list of rules:

S 1 ... Sn S O S o , - . , ~

S O ' S 1 S~"

If (R) is the name of the double-line rule above, (R) ~ is the name of the first rule in the list, and (R)I' designates any of the remaining rules in the list. We shall use an analogous abbreviation with systems in P, where S o, $1, . . . , S, are formulae of P.

In general, we shall omit the outermost parentheses of G,terms.

1.2. The system GL. Now we formulate our basic sequent-system, which we shall call GL. The system GL is formulated in GP. It has the following axiom-schema:

(Id) A ]- A

and the following rules:

Structural rule:

x1 F- Y EA] X, EA] F- Y2 (Cut) Xz[X1] b- Y, EY2]

Operational rules:

X, A t- Y[B] (-+) X b- Y[A- , B]

X [A, B] [- Y (o)

X[AoB] t - Y

358 K. DoWn

X ~ YEA] X [- Y [B] (^) X k- YEA^B]

X [A] [-- r X [B] ~- r (v)

X[A v B] F Y

X[O] [- Y (T)

xEZ] k Y

X I- Y[O] (_L) X }- Y[_l_]

It is easy to check that if X [- Y is a theorem of GL, then Y is either a formula of P or O. We shall call this property of GL the single-conclusion property, and sequents X [- Y such that Y is either a formula of P or 0 will be called single-conclusion sequents. Because of the single-conclusion property, we could have given (Cut) for GL in the form:

X 1 ~ A X 2 [A] [- Y

X2[Xl] [-- Y

and analogously for (-,), ( ^ ) and (_L).

1.3. Additional structural rules. We shall consider extensions of GL with some (possibly all) of the following .structural rules:

(Assoc) X[Zt ' (Z2, Zs)] 1- Y X[(Z1, Z2), Z3] I-- Y

x [ z l , Z23 k Y (Perm)

X[Z2, Z1] L- Y

X[Z, Z] [- g (Contr)

X[Z] t- Y

X[Z] t- Y (Expans)

x [ z , z ] F y

X[O] I- Y (Thin)

X[Z] }- Y

YF x [ o ] (ThinR).

Y}- X[A]

X[Z] }- Y (IntrO/)

X [0, Z] [- Y;

X[O, Z].}- e (ElimOl)

X[Z] }- Y '

X[Z] F Y (IntrOr) X[Z, 0][- Y

X[Z, O] }- g (Elim Or)

X[Z] F Y


By (ContrO) and (ExpansO) we shall designate respectively (Contr) and (Expans) where Z is O.

It is clear that in the presence of (ElimO/) (or (ElimOr)) the effect of (Contr) could be obtained with:

X [Z, Z] t- Y X[Z, Z] [-- Y (or (Contr r) X[Z, O] [- Y)" (Contr/) X[O, Z] [-- Y

Analogously, in the presence of (IntrO/) (or (IntrOr)) we could use:

X[Z, Z] [- e (Expans/) xX[O'[Z, z]Z] [-]-- YY (or (Expans r) X[Z, O] [- Y)

to get the effect of (Expans). We could also envisage a form of Contraction where the Z omitted is not

necessarily adjacent to the other Z, as in the following rules:

X [Z 1 [Z], Z 2 [Z]] t- g X [Z 1 [Z], Z 2 [Z]] t- g ' (Cane/) X[ZI [O] , Z2[Z]] I-Y; (Caner) X[ZI[Z], Z2[O]] t-Y"

The' converse rules, corresponding to Expansion, will be called (Dupl0 and (Dupl r) respectively. In the presence of (Assoc), (Perm) and (Contr/) (or (Expans/)), the rules (Cane/) and (Caner) (or (Dupl/) and (Duplr)) are derivable.

It is also clear that with (IntrO/), (IntrOr) and (Thin) the following rules are derivable:

X [Z] t- Y X [Z] [- Y (Thin 0 " (Thin/)

X[ZI; Z] ~ Y' X[Z, Z1] F- Y"

Next, it is easy to see that with (Contr) and (Thin) the rules (ElimO/) and (ElimOr) are derivable. Also, in the presence of (Perm), only one rule in the pair (IntrO/) and (IntrOr) (or (ElimO/) and (ElimOr)) gives the effect of the whole pair. Finally, it is easy to show that with (Cut), the rule (Expans), where Z is A, is interreplaceable with the rule:

(Mingle) Xt [- A X 2 [- A X~, X 2 ~ A

(Starting from A [-A, by (Expans) we obtain A, A [-A, which with two applications of (Cut) yields (Mingle). Conversely, starting from A [-A and A [-A, by-(Mingle) we obtain A, A [-A, which with (Cut) yields (Expans) where Z is A.)

Let t'(Z) be the formula of P obtained from the G-term Z by replacing every comma by o, and every O by T. Then it is easy to see that with (o) and (T) we have the double-line rule:

(?) X[Z] ~ Y X[~(Z)] [-- Y

4 -- S tud i a Logica 4/88

360 K. Dogen

Hence, in the presence of (o) and (-[), the double-line rule (Assoc) could have been given with the following instance of (Assoc):

X[A1, (A2, A3)] t- Y

X[(A~, A2) , A3] [- Y

Analogously, in the remaining structural rules, we could replace Z, Z~ and Z 2 by A, A 1 and A 2 respectively. With (t') and (Cut) we can also show that (Assoc) could have been given with the following instance of (Assoc):

Z1, (Z2, Z3)[- Y

(Z1, Z2), Z3 F Y

To obtain (Assoc) from this double-line rule we first prove (Z 1, Z2), Za t- ~(Z1, (Z2, Z3)) and Z1, (Z2, Z3) [- t'((Z~, Z2), Za), and then we apply (t') and (Cut). We can proceed analogously with the remaining structural rules (with the exception of (Thin R), where we need (.1_) and (Cut)).

It is easy to check that none of the structural rules considered in this section can spoil the single-conclusion property.

1A. The system GE. Now we consider a system equivalent to GL, called GE, where the operational rules are not given by double-line rules, but by pairs of rules (or axioms) for introducing the connectives and constants on the left and right of the turnstille. The axioms of GE are of the form:

(Idp) P F- P;

its only structural rule is (Cut) in the form:

X 1 [-- A X 2 [A] [- Y X2[Xt] [- Y

and its operational rules and axioms are:

( ~ L) X~ F A X 2 [B] F Y. (--, R) X 2 [A ~ B, Xl] [- Y '

X[A, B] [- Y (o L) X [A o B] [- Y; (o R)

(^L) X [A] [-- Y X [B] F Y

X[A ^ B] k Y; X[A ^ B] t- Y; (^ R)

X[A][-- Y X[B]~- Y (v L) ; (v R)

X [A v B] t- Y

X , A [ - B X [ - A ~ B

XI[--A X2[-B ,X1, X2 t- A o B

XF-A Xk-B X [ - A ^ B

XF-A X F B X F A v B;X Av B


x[o] F Y (T L) X[T] I- Y; (T R)

(_L L) • }- 121; (_L R)

OI-T

x k o X [ - _L"

To establish that in GL and GE we have the same theorems, let us first note that by an easy induction on the complexity of A we can show that every instance of A t-- A is provable in GE. Next let us note that:

with (--*)1" or ( ~ L ) we can prove A ~ B , A t - B ; with (o)T or (oR) we can prove A, B 1-- A o B; with (A)]" or (AL) we can prove A A B I - - A and A A B I - - B ; with (v)'~ or (v R) we can prove A ~-A v B and B 1-A v B; with (I-)T or (T R) we can prove 121 [-- T ; with (_L) T or (_LL) we can prove _L 1-121.

Then the equivalence of GL and Gs follows by straightforward applications of (Cut).

1.5. Cut-elimination in GE. The point of considering GE is to show that (Cut) can be eliminated from it, i.e. that every theorem of GE can be proved in GE without using (Cut). This cut-elimination can be proved as follows.

Let the degree of an application of (Cut) be the number of occurrences of~ ~ , o, A, V, T and _1_ in the cut-formula A, i.e., the formula A eliminated by applying (Cut). Then we shall prove the following lemma:

LEMMh 1. For every proof of X t - Y in GE, with a single application of (Cut), there is a proof of X ~ Y in GE, with no application of(Cut), or with one or two applications of (Cut) of smaller degree.

Since no degree is negative, Lemma 1 establishes that (Cut) can be eliminated (we eliminate applications of (Cut) in proofs starting from the top).

PROOF OF LEMMh 1. (1) If the left or right premise of (Cut) is A [- A, then the conclusion of (Cut) is identical With the remaining premise, which is proved wi thout (Cut).

(2) If the left premise of (Cut) is 121 ~ T , then in the proof of the right premise X 2 [ T ] [- Y replace every T on the left of the turnstile by 12121. This way we obtain a (Cut)-free proof of X 2 [121] 1- u which was the conclusion of our (Cut).

(3) If the right premise of (Cut) is _L t- 121, then in the proof of the left premise X 1 ]-- _L replace every .L on the right of the turnstile by 121. This way we obtain a (Cut)-free proof of X1 t- t2I, which was the conclusion of our (Cut).

(4) If the left or right premise of (Cut) is obtained by an operational rule of GE, such that the connective or constant introduced is not the main connective or constant of the cut-formula A, then we must work through all possible cases

362 K. DoWn

in order to show that these operational rules need never precede (Cut) immediately in a proof. For example, the proofs on the left will be replaced by the proofs on the fight:

x~ l- B X~[C] ~- A (-+L)

XT[B+C,X'~]t-A X2[A]]-- Y (Cut)

x~ [x'~ [B-+ C, x~]] I- A];

X 2 [A], B t-- C (+R)

X~ k-A Xz[A] t- B-+C X 2 [X~] t- B ~ C; (Cut)

X~ [C] t-- A X 2 [A] [-- Y (Cut)

X]~-B X 2[X~[C]]t- Y (~L) X~[X~[B c, x~]] ~- r

X I~-A X 2[A],Bt-C (cut)

X 2 IX1], B I'- C (+R)

x2rx l ] ~-- B+C,

(5) If both premises of (Cut) are obtained by introducing the main connective of the cut-formula A, then we must work through all possible cases in order to show that in each case we need not apply the operational rules in question: instead we shall have one or two applications of (Cut) of smaller degree. For example, the proof on the left will be replaced by the proof on the fight:

(~R) X ~ , B ~ C X'2[--B X~[C]~-Y (-+L) (Cut) X'2~-B X~,B[-C X 1 ] - B ~ C X~[B~C,X'2]k- Y XI, X'2t--C X~[C]k- Y

(Cut) (Cut) X2[X 1 , Xt2] ~-- Y; X'~[X1, X'2] t- Y

and both applications of (Cut) on the fight are of smaller degree than the application of (Cut) on the left. This exhausts all possible cases, q.e.d.

A consequence of this cut-elimination is that GE, and hence also GL, are decidable systems. In order to verify whether X t- Y is provable in GE, we attempt to construct a proof O f it in GE working from the bottom up, whithout using (Cut). Every upward step eliminates an occurrence of a connective or of a constant, and there is only a finite number of ways to make this step. It follows easily that the total number of proofs that can be attempted is finite. The sequent X k- Y is provable iff one of the attempted proofs is successful.

Another consequence of cut-elimination is the subformula property for Gig, i.e. if X t- Y is provable in Gig, there is a proof of X t- Y in Gig such that all formulae of P occurring in this proof are subformulae of those formulae of P which occur in X ~ Y.

1.6. Cut-el imination with addi t iona l structural rules. Let us now consider the extensions of Gig with some (possibly all) of the structural rules (Assoc), (Perm), (Contr), (Thin), (IntrO/), (IntrOr), (ElimO/), (ElimOr)o (ContrO) and (ExpansO) (note that (Expans) and (Thin R) are not in this list).

I n order to show that in any of these extensions (Cut) is eliminable, it is enough to prove analogues of Lemma 1. To prove these analogues, we shall


now show, as in case (4) of the proof of Lemma 1, that none of our additional structural rules need ever precede (Cut) immediately in a proof; the rest is proved as in Lemma 1.

If we have the proof on the left, we shall replace it by the proof on the right:

X~ [A] ~- Y S~ [- A X~ [A] [--'Y (*) (Cut)

X 1 F A X 2[A]F Y X ~ [ X , ] F Y (Cut) (,)

X2[Xl] [- Y; X2[X1] [- V where (,) is any of the additional structural rules we have envisaged.

If we have the proof on the left, we shall replace it by the proof on the right:

X~[A][- Y X ~ l - A X~[A]F Y (,) (Cut)

X~ [-- A X2[A ] [- V X'g[Xl] F Y (Cut) (,)

X2 [X1] F IT; X2 [X1] F Y

where (,) is an application of (Assoc), (Perm), (IntrO/), (IntrOr), (ElimO/), (ElimOr), (ContrO), (ExpansO), or an application of:

X[Z, Z] [-- Y (Contr) or

XEZ] ~- r

where the cut-formula A is not a subterm of Z. If we have the proof below, where the cut-formula A is a subterm of the

Z involved in (Contr):

X 2 [Z [A], Z [A]] [- Y Xl F- A X 2 [Z [A]] [- Y

X2[Z[X1] ] [-- Y

we shall replace it by the following proof:

(Contr) (Cut)

XI ]- A X2[Z[A], Z[A]] [-- Y (Cut)

X 1 ~ A X2[Z[X1], Z[A]] ]-- e X2[Z[XI], Z[X, ] ] [- Y

X[O] 1- Y (Thin)

X[Z] ]- Y

x~ [z Ex,]] ~- r. (Contr)

In the analogous case with (Thin), the proof on the left will be replaced by the proof on the right:

x~ [0] ~- Y (Thin)

xl ~- A X2 [Z [A]] F r (Cut)

Xz[Z[X13] [--- Y;

x210] F Y x ~ [ z [ x , ] ] ~- z (Thin)

This exhausts all possible cases and shows that none of the additional structural rules we have envisaged need ever precede (Cut) immediately in a proof.

We had to exclude (Expans) from the considerations above because of

(CuO

364 K. DoWn

problematic cases like-the following:

X 2 [Z [A]] ~- r (Expans)

Xi [- A X2[Z[A], Z[A]] [-- e (Cut) x2[z[xl], Z[A]] F- Y.

If now we apply (Cut) to X1]--A and X2[Z[A] ] [--Y, we obtain x2[zEx,]] 1- r, and it seems we can pass from that to X=[Z[X~], Z[A]] 1-- Y only with the help of (IntrOr) and (Thin). But in the presence of (Intr Or) and (Thin), the rule (Expans) is derivable, and hence need not be considered as an independent rule.

Indeed, (Cut) is not eliminable from GE plus (Expans), as witness the sequent Pl, P2 ]-- P l v P2, which is provable in this system only with the help of (Cut).

With (Thin R), which was also excluded from the considerations above, we have the following problematic case:

O (Thin R)

X t } -A X z[A]~- Y (Cut}

x2Exi] F- r.

This proof can be replaced by:

Xl~-O eventually (Thin R)

X1]- Y �9 " eventually (Intr O/), (Intr Or) and (Thin)

X2[X1] 1- r

so that in the presence of (Intr el/), (Intr 0 0 and (Thin), we can also eliminate (Cut) from extensions of GE with (Thin R).

The rule (Cut) is not eliminable from GE plus (Thin R), as witness the sequent _L, pl 1- P2, which is provable in this system only with the help of (Cut) (to prove this sequent apply (Cut) to _1_ ]-pl--->p2 and Pl--*P2, P~ ~ P2)"

From the considerations above it is easy to infer in what cases we obtain cut-elimination with the remaining additional structural rules of Section 1.3.

As in Section 1.5, we can try using cut-elimination to prove decidability and the subformula property. Whereas the subformula property follows easily for all our extensions where (Cut) is eliminable, with decidability the situation is not so clear, and we would have to consider the matter case by case. Since decidability is not our main concern here, we shall just make the following remarks.

The decidability of the extension of GE with (Assoc), or of the extension of GE with (Assoc) and (Perm), is obtained by a simple adaptation of the proof of decidability for GE. Of course, this also means that the equivalent extensions of GL (which we shall consider in Section 2.5) are decidable. Let GJ' be the ex-tension of GE with (Assoc), (Perm), (Contr), (Thin) and (Intr O/), and let GH'


be the extension of GJ' with (Thin R). It is not difficult to recognize in GJ' and GH', and in the equivalent systems GJ and GH extending GL, sequent-formulations of respectively the Johansson and Heyting propositional calculus. Techniques very well known from [10] would show the decidablity of GJ' and GH'.

�9 1.7. The system GC. We have remarked at the end of Section 1.2 that GL could be formulated with a modified presentation of (Cut), (~), ( ̂ ) and (3_), which takes into account the single-conclusion property. We have not introduced GL in this" form ha order:t 0 be able to pass easily from GL to the system GC, which corresponds to the classical propositional calculus. The system GC does not have the single-conclusion property: it is obtained by extending GL with (Assoc), (Perm), (Contr), (Thin), (Intr D0, and the rules obtained from these by putting G-terms which are on the left of the turnstile to the right, and vice versa. If ( . . . ) is the rule we start from, let (. . . R) be the rule obtained by so permuting left and right G-term (as in Section 1.3 (Thin R) is obtained from (Thin) where Z is A).

It is easy to show that with (Id), (Cut), (v), (Contr R), (Thin R), (Intr O IR) and (Intr OrR) (this last rule is derivable in GC) we have the double-line rule:

x t- Y[A, B]

X ~- Y[A v B]

With this double-line rule and (_1_) we can derive:

X [-- Y[Z]

X F- Y[t'(Z)]

where t'(Z) is the formula of P obtained from the subterm Z by replacing every comma by v , and every 0 by 3_. With (t') we have that (Assoc R) and (Perm R) can be obtained from:

A 1 v ( A 2 vA3) t--(A1 vA2) v A 3

(A t V A2) v A 3 [- A 1 V (A 2 v A3)

A t v A 2 F-- A2 v A 1

which are provable in GL. It is also easy to show that with (Id), (Cut), (^) , (Contr), (Thin), (Intr O/)

and (IntrOr) we have the double-line rule:

X [A, B] [- Y

X [ A ^ B] [- Y

Hence, in G J, GH, and GC, the connectives o and ^ are equivalent.

366 K. Dogen

2. Hilbert-formulations

2.1. The system L. The system L is formulated in the language P of Section 1.1. It has the following axiom-schemata and rules:

(1) A-'*A A - * B B ~ C (2) A ~ C

A - ~ A 1 B ~ B 1 (3)

(4) ((A --. B) o A) ~ B (6)

(5) A ~ ( B ~ ( A o B))

(7) (A ^ B ) ~ A . (9)

(8) (A ^ B ) ~ B

(10) A-~(A v B) (12)

(11) B ~ ( A v B) (13)

(Ai ~ B ) ~ ( A oB1)

A ~ A t B ~ B 1

(A o B)-->(A 1 oB1)

C ~ A C ~ B

C ~ ( A ^ B)

A ~ D B ~ D

(A v B ) ~ D

C - ~ ( A ~ D ) C - ~ ( B ~ D )

C ~ ( ( A v B)--*D)

It is easy to check that every theorem of L is of the form A -~ B. Note that T and _1_ are idle in L, i.e. there are no specific assumptions involving T and ]_. So, T and 2 behave in L as arbitrary propositional variables, and L in P without T and ]_ is essentially the same system as L. If we define -~ A as A ~ L, the negation of L ~$r be of the same type as the "purely implicational negation" of the Johansson propositional calculus (see Section 2.10).

We shall now survey some theorems and derived rules of L. First, it is easy to show that (3) is replaceable in L by the rules:

B ~ B 1 A ~ A 1 (3') (Al ~ B ) ~ ( A x _ ~ e l ) ; (3") (Ax ~ B O ~ ( A _ . B i ) .

It is also easy to show that (3), (4) and (5) are replaceable in L by the double-line rule:

(C o A)--> B (14)

C ~ ( A ~ B ) ~


Next, we easily obtain that the following rules are derivable in L:

A ~ A 1 B ~ B x �9 (16) A ~ A a B ~ B 1 �9 (15) (A A B)-*(A 1 ^ B O' (A v B ) ~ ( A 1 v B1)

These two rules together with (3) and (6) guarantee that L is closed under the following Rule of Replacement:

A ~ B (17)

where A ~ B is an abbreviation for A --, B and n--, A, and C~ is obtained from C by substituting zero or more occurrences of A by B.

It is easy to check that in L we have the following theorems:

(18) ((A ^ B ) ^ C ) ~ ( A ^ ( B ^ C)) (22) ((A v B ) v C ) ~ ( A v ( B v Ci)

(19) (A ^ B ) ~ ( B A A ) (23) (A v / 3 ) ~ ( B v A )

(20) A ~ ( A ^ A ) (24) A ~ ( A v A )

(21) ((Av B) ^ A) A (25) ((A ^ B) v A) -A

(26) (Co(A v B) )~( (CoA) v (COB))

(27) ((A v B ) o C ) ~ ( ( A o C ) v ( B o C ) ) .

A factor of a formula A of P is defined recursively as follows: A is a factor of A; if n o C is a factor of A, then B and C are factors of A. Let A [B] stand for a formula of P in which B occurs as a factor, and let A [C] be the result of

_ substituting C for a single B in A ['/3] (it will always be clear from the context which A [B] we have started from in order to obtain A [C]). Then with the help of (26) and (27), it is not difficult to show by induction on the complexity of C that (12) and (13) can be replaced in L by:

(28) C[A] ~ D C[B] ~ D C[A v n]- D

Finally, it is easy to see that in L we have the following double-line rules:

(29) A ~ ( A ^B)., (30) (A v B ) ~ B A ~ B A--*B

2.2. L and urlogs. First we give the definition of algebras which we shall call uni-residuated lattice-ordered groupoids (abbreviated by urlog), and which are right-residuated lattice-ordered groupoids in the sense of 19] (pp. 189-191):

368 K. DoWn

DEFINmON 4. The algebra ( d , \ , . , n , w, 1, 0) is an urlo# iff I and 0 are elements of ~ ' (not necessarily distinct), ~r is dosed under the binary operations \ , . , n and w, and for every a, b, c ~ zr we have:

O) ( a n b ) n c = a c ~ ( b n c ) (ii) a n b = b n a

,Oii) a = a n a (iv) (a w b) c~ a = a (ix) c ' (a u b) = (e" a) w (c" b) (x) (a w b)" c = (a" c) w (b " c)

(xi)

(v) ( awb) w c = a w ( b u c ) (vi) a w b = b w a

(vii) a = a w a (viii) ( a n b ) w a = a

a


The Rule of Replacement (17) guarantees that these equalities define genuine operations on the set d = {[A]: A~F} . The Lindenbaum algebra of L is the algebra defined on ~e by these operations, i.e. the algebra ( ~ , \ , . , c~, w, IT[, [_1_[). Then we prove the following lemma:

L~MMA 3. The Lindenbaum algebra of L is an urlog.

PROOF. The equalities (i)-(x) follow immediately from (18)-(27) of Section 2.1, whereas (xi) is a consequence of (14) and (29). q.e.d.

(Note that in the Lindenbaum algebra of L we have I T [ # I-1-1, since T ~ / is not provable in L: if we identify o and ^ , all theorems of L are tautologies.)

With the help of this lemma we can now prove the following soudness and completeness result:

(0) A ~ B is provable in L iff for every urlog and every valuation v we have v(A)

370 I,;. Dogen

If X 2 [Xt ] ~- Y is obtained by (Cut) from X I ~ A and X 2 [A] f - Y, then in L from t'(X~)~ A, by using (6), we can pass to t'(X 2 [X~] )~ t'(X 2 [A]), and then with ? ( X 2 [ A ] ) ~ ? ( Y ) and (2) we obtain t '(X2[X~])~?(Y).

I f X t- Yis obtained by (~) , we use (14). The case when X ~ Y is obtained by (o) is trivial.

If X ~- Y is obtained by ( A )~, we use (9), and if it is obtained by ( A )T, we use (7) (or (8)) and (2). If X J-- Y is obtained by ( v ) ~, we use (28), and if it is obtained by (v)~', we use (10) (or (11)), (6) and (2). The cases when X J- Y is obtained by ( T ) ' a n d (J-) are trivial, q.e.d.

As an immediate corollary of Lemma 5 we have that if A ~ B is provable in GL, then A ~ B is provable in L. The converse of this implication is proved by a straightforward induction on the length of proof of A ~ B in L. (In Section 3.5 we shall give another proof of this converse implication.) Hence, we have the following: (I) A 1--B is provable in GL iff A ~ B is provable in L.

2.4. The ( - , , o) and implicational fragment of L. Let L o be the system in P without A and v , given by the axiom-schemata and rules (1)-(6). Let


2.5. Extensions of L . We shall now consider a number of extensions of L. These extensions are obtained by extending L with some (possibly all) of the axiom-schemata or axioms given in the first column of the table which we shall produce below. (An extension of L inherits from L both its axioms and rules.) In the second column of this table we indicate what we must assume about urlogs in order to obtain a soundness and completeness result like (0) for our extensions. Also the Lindenbaum algebra of a given extension will be free in the class of all corresponding urlogs. Finally, in the third column we indicate with what structural rules we extend GL (or GE) in order to obtain a sequent-system which corresponds to a given extension of L as GL corresponds to L in (I):

L + urlog +

(31) ((AloA2)oAa)-~(Alo(A2oA3)) (al"a2)'a 3

372 K. DoWn

We can assert that these systems give the (--*, o, T , A_) fragments of the corresponding extensions of L whenever cut-elimination is available for the corresponding sequent-systems of the third column.

In the remainder of this section, and in Sections 2.7 - 2.9, we shall concentrate on some extensions of L which are of interest either because similar systems have been considered in the literature, or because they correspond to natural urlogs, or natural sequent-systems.

Let LA be the system obtained by extending L with (31) and (32), and let LAP be LA extended with (33). The (--,, o) fragment of LA (i.e., (1)-(6), (31), (32)) corresponds closely to the Lambek calculus of syntactic categories of [13]. In the Lambek calculus we have an ordering relation instead of the main ~ of theorems of LA, and we have two operations, corresponding to the left and right residual respectively, instead of the single ~ of LA, which corresponds to the right residual. In the same way L corresponds to the nonassociative Lambek calculus of [14]. We could extend L and LA with an additional binary connective ~ , corresponding to the left residual; for this connective we would have the double-line rule:

(A o C)--, B

C--* (B *-- A)

Analogously, in sequent-systems we would have:

A , X F Y[B] X t-- Y[B~-A]

However, in LAP the connectives ~ and ~ would become equivalent, as the corresponding operations become equivalent in a commutative version of the Lambek calculus.

2.6. The extensions E +. Let us now consider an extension of L from Section 2.5 in which (38) and (40) are provable, and let us call this extension E. It is easy to show that in the corresponding sequent-system the following double-line rule will be derivable:

A I - B (horiz)

O[-- A-- .B

(In sequent-systems with the single-conclusion property, in the presence of (-,), (o), (T) and (_1_), , the double-line rule (horiz) is interderivable with the following special form of (IntrOl) and (Elim O/):

z t - Y

o , z ~ - Y

If moreover we have (Cut), this last double-line rule amounts to (IntrO/) and


(Elim O/), as we have shown at the end of Section 1.3. This double-line rule also amounts to (Intr O/) and (Elim O/) in the presence of (Assoc) and (Perm) only.)

If we have (horiz), the statement corresponding to (I) of Section 2.3 is equivalent to the following statement:

(E) O 1- A ~ B is provable in GE iff A ~ B is provable in E.

We can then ask for what system E? we have that:

(I +) O 1- A is provable in GE iff A is provable in E +.

The system E is contained in E +, i.e., the theorems o rE + of the form A ~ B must coincide with the theorems of E. But E + will also have theorems which are not of the form A-~B, and hence cannot be theorems of E. For example, T is a theorem of E +.

It is easy to show with the help of (T) and (horiz) that the following double-line rule is derivable in GE:

O1-A (neces)

01- T ~ A

Concerning the analogous double-line rule for systems in P:

A (50)

T ~ A we can prove the following lemma:

LEMMA 6. 0 1- A is provable in GE iff A is provable in E plus (50) iff A is provable in E plus (50)]'.

PROOF. We have:

O1- A in GE ~ 01- T ~ A in GE, by (neces)

T---' A in E, by (E).

So, if O 1- A is provable in GE, then A is provable in E plus (50)T, and hence also in E plus (50).

For the converse implications we proceed by induction on the length of proof of A in E plus (50). If A is an axiom, we use (E). If A is obtained by a rule of E of the form:

A B

C

(this rule is either (2), (3), (6), (9), (12) or (13)) with the help of (horiz), we can show that in GE we can derive:

0 1 - A 0 1 - B 0 1 - C

Finally, if A is obtained by (50), we use (neces). q.e. d.

374 K. DoWn

By this lemma, we can take either E plus (50), o r E plus (50)T, as our system E +. However, since GE is closed under (neces), and not only (neces)T, our system E + will be dosed under (50)4, even if we do not assume this rule as primitive. Because of that, we shall take in the sequel that E + is the system obtained by extending E with (50).

It is easy to see that in E + so defined we can derive modus ponens: A A~B

(51) B

and adjunction: A B

(52) AAB" In systems where we have A ~ A and modus ponens, the Rule of Replacement

(17) amounts to the following rule:

A~B C

In L, and similar systems without modus ponens, in general we do not have this rule. (However, L is dosed under this rule provided C is not A, i.e. A is a proper subformula of C.)

Finally, it is clear that E and E § have the same Lindenbaum algebra.

2.7. Extensions of S. We shall now consider the extensions of L given in the following chart, where arrows represent proper inclusion:

: C = H + ( {A . - - ,=B) - - , .AJ - -~A

H = J + (37)

. / = BCKj + (3Z,.)

R A M = ~ J '

RA = h i + {3/,,).

B C K H = BCK, + ( 37 )

= M ~- ( 36 )

h / = S § ( 3 3 ) v

L + [31), (32), (38)-(/-.1)


If E is one of the systems in this chart, E + is E plus (50), as in Section 2.6. The urlogs corresponding to S are right-residuated lattice-ordered monoids,

and those corresponding to M are residuated lattice-ordered commutative monoids. With RA these commutative monoids are square-increasing (i.e., they satisfy a ~< a. a), and with RAM they are idempotent.

It is easy to see that G-terms on the left of the turnstile in theorems of GS correspond to finite (possibly empty) sequences of formulae of P. Analogously, the G-terms on the left of the turnstile in theorems of GM correspond to finite (possibly empty) multisets, i.e. sets of occurrences, of formulae of P. (The importance of multisets in the proof theory of relevant logic is stressed in [16]. Sequent-systems where left G-terms would correspond to ordinal sets in the sense of [16], i.e. to sequences where repetitions are omitted, could be obtained with structural rules like (Canc l) and (Canc r).)

The system M § corresponds to the negationless fragment of the relevant logic R minus distribution ((A ^ (B v C))--*((A ^ B) v (A ^ C))) and minus contraction ((A ~ ( A ~ B))~(A ~ B)). In the same way RA + corresponds to the negationless fragment of R minus distribution. The system RAM + is a mingle extension of RA +; in RAM + we have the mingle principle A ~ (A ~ A). (The logic R and its neighbours are extensively treated in [1]; for RA § see [6].)

The urlogs corresponding to BCKj are residuated lattice-ordered commutative monoids where 1 is maximal, and the urlogs corresponding to BCKn are such monoids where moreover 0 is minimal. These urlogs are related to BCKalgebras (if ( d , \ , . , c~, u , 1, 0) is such an urlog, with or without 0 minimal, and if a �9 b = b\a, 0 = 1 and a ~ b ~ b

376 K. Dogen

2.8. The ( ~ , A, v , / ) fragments of extensions of S +. It is of some interest to consider the fragments of our extensions of S + which do not contain the less usual logical operations o and T .

Let P - be the language P without o and T , and let S- be the system in P - given by:

A A--,B (1) A-+A (51) B

A (53) (A --+ B) ~ ((C ~ A) --+ (C --+ B)) (54)

(A---> B)~ B

(7) (A ^ B ) ~ A (52) A B A A B

(8) (A ^ B)--, B (55) ((C + A) ^ (C-+ B))--,(C-+(A ^ B)) (10) A-+(A vB) (11) B+(A v B) (56) ( (A+C) ^ (n - , C)) - , ((A v B)--,C).

Note that (55) and (56) are already provable in L, and that (51) and (52) are derivable in all E + systems. Hence, the only new postnlates of S- are (53) and (54).

The rule:

(57) A C~(A--,B)

C--,,B

can replace (54) in S - , since we have:

A A (A --+ B) --+ (A --+ B) (57) C ~ ( A ~ B ) (A--,B)-~B (54) (A-+B)~B.

(53), (51) C ~ B ;

Let us introduce the following abbreviation:

"~ .~,--->B= ~ AI"-*(''" o(A,,--->B)...) if n i> 1 [B if n = 0 .

We shall now show that the-following rule (analogous to (Cut)) is derivable in S- :

Bn-.->A Cm--.>(A--rD ) (58) C m --+ (/~,,--~ D) , n ~> 0, m >/0.

We have:


Cm-~ (A ~D)

eventually (53) and (51)

1~,, -~ ((/~, -~A) ~ (/3, -~D))

((B.~A)~(B,~D))-~(B.~D)

eventually (53) and (51)

(54)

( e , --r ((/3. ~ A) ~ (/~. -~ D))) -~ (em ~(/~. ~ D)) : �9 ( 5 1 )

(~,, --* (/3, ~D).

The rules (51) and (57) are instances of (58), and it is not difficult to prove (53) from (1) and (58). Hence, we can replace (53), (51) and (54) in S- by (58).

It is easy to show that the rules (2), (3), (9), (12) and (13) of Section 2.1 are derivable in S - . F rom (1) and (54) we also obtain immediately:

(59)

Let the language GP- be GP without o and ]-, and let GS" be the sequent-system in GP- obtained from GS by rejecting (o) and (T) . We shall show that:

(I-) O [-- A is provable in GS- iff A is provable in S - .

With the cut-elimination results of Section 1.6 this will establish that S - is indeed the (--., ^ , v , • fragment of S + .

First, we define a translation of single-conclusion sequents of GP- into formulae of P - . Let C 1 . . . . , C,, where n >i 0, be obtained from the G-term X of GP- by deleting all occurrences of O and all parentheses which do not belong to formulae of P - . So, C 1 . . . . . C, are all the occurrences of formulae of P - which are subterms of X, arranged in the order in which they appear in X (a formula of P - may occur more than once in C 1, . . . , C,). Then we define our translation as follows:

s(X }-- A) = Cn ~ A

s (x o) = •

We shall now prove the following lemma:

LEMMA 7. I f X [- Y is provable in GS-, then s(X [- Y) is provable in S- .

PROOF. By induction on the length of proof of x t- Y in GS' . If X 1-- Y is A t-- A, then in S- we have A ~ A. If X [- Y is obtained by (Cut), we use (58). The case with ( 4 ) is trivial. For t he cases with ( ^ ) and ( v ) we use the following theorems of S - :

((D-~(C-~ A)) ^ (D-~(C ~ B)))-~(D~(C--.(A ^ B)))

((D-~(A-~C)) ^ (D~(B-~C)))-)(D--.((A v B)-~C)).

The cases with ,(• (Assoc), (IntrO/), (IntrOr), (ElimO/) and (E l imOr)are all t r i v i a l , q.e.d.

378 K. Dogen

As an immediate corollary of Lemma 7 we have (I-) from left to right. The converse is proved by a straightforward induction on the length of proof of A in S- . For example, for (53) we have the following proof in GL plus (Assoc) ~:

C--.A[-- C ~ A A~B]-- A ~ B (--,)T

C ~ A , CI--A A ~ B , A [ - - B

A ~ B , (C- .A, C)FB J

(A~B, C-~A), C~ B . , .

(Assoc) J,

(--,)T

(Cut)

A--, B F (C ~ A)~(C--. B)

which shows that (53) is tied with (Assoc)~. For (54), in GL plus (ElimOr) we can derive:

A--.B~- A ~ B OF-A A- B, B

(CuO A ~ B , O ~ B

(Elim O r) A ~ B [ - - B

which shows that (54) is tied with (ElimBr). A careful examination of our induction would show that we don't need (Asset)T, (IntrOr) and (Elim 130 to prove (I-) from right to left.

Hence, we have proved that S- gives the (---,, A, V, 2-) fragment of S + . Similarly we can prove that the ( ~ , A, V, 2.) fragment

of M § is given by M - = S - + (60) (A~(B--,C))--.r(B~(A~C)) (alternatively, M - = S - + (61) A-.-,.((A--. B)--, B), cf. [1], p. 80; (54) is superfluous in M-); of RA § is given by RA- = M- + (62) (A ~ (A ~ B)) ~ (A --, B); of BCK~ is given b)~ BCK~ = M - + (63) A ~ ( B ~ A ) ; of J+ is given by J - = BCKj-+ (62); of BCK~ is given by BCKir B C K i + (37) _I_~A; of H § is given by H - = J - + (37)

((1) is superfluous in BCKy and its extensions). Note that since in Section 1.6 we did not prove cut-elimination for sequent systems with (Expans), we cannot assert anything here about RAM-. For systems with (Thin R) we did have cut-elimination in the presence of (IntrO/), (IntrOr) and (Thin).

2.9. Implieationai fragraents of extensions of L. We have seen in Section 2.4 that the implicational fragment of L is made of A ~ A only. On the other hand, in the implicational fragment of the system LA of Section 2.5 we already have (53), as can be seen from the proof on GL plus (Assoc)~ after the proof of Lemma 7 in the last section.


The implicational fragment of LA is given by the system LA~ with the axiom-schema (1) A ~ A and the rule (58'), obtained from (58) by replacing n >t 0 by n t> 1. The rules (2) and (3) can be obtained from (58'): the rule (2) is just the instance of (58') where n = 1 and m = 0, and for (3) we have:

(AI~B)~(A1--*B) B-~B 1 A-~A 1 (AI-*B1)~(A1--*Bt)'

(A1 ~/3) -* (A 1 --+ B1) (A 1 ~ 31) --+ (A ~ B1) (A1 ~ B ) ~ ( A ~ Bi).

The rule (58') can be replaced in LA_, by the following two rules:

B . ~ A A ~ D (51') / ~ , ~ D , n >I 1

(54') (A - , D) ~ (/~, ~ D)' n >i 1.

The rule (2) is the instance of (51') where n = 1. The rules (51') and (54') would become (51) and (54) respectively by putting n = 0.

Let GLA_, be the sequent-system given by (Id), (Cut), (Assoc.) and (-.) . To prove that LA~ is indeed the implicational fragment of LA, we first establish that A}- B is provable in GLA+ iff A--~B is provable in LA_.. From left to fight, we use a translation like the translation s of Section 2.8 to establish a lemma analogous to Lemma 7 (note that (58') is exactly what we need for the step with (Cut) in the proof Of this lemma). From fight to left, we proceed by induction on the length of proof of A ~ B in LA_.. Once we have established our equivalence, we apply the cut-elimination results of Section 1.6.

For the implicafional fragments of our extensions of S + (these fragments coincide with the implicational fragments of the corresponding extensions of S, as explained in Section 2.6), we proceed in principle as for the (-+, ^ , v , J_) fragments in Section 2.8. It is possible to show that the implicational fragment

of S + is given by S~ = (1), (53), (51), (54) (see [-24], p, 168); of M + is given by M_. = S_. + (60) (alternatively, M_. = S ~ + (61); (54) is superfluous in M_.; of. with the axioms of BCI in [19], p. 316); of RA + is given by R_, = m _ . + (62) (cf. [1], p. 88); of BCK] and BCK + is given by BCK_. = M_. + (63); of J+ and H + is given by H_. = BCK_.+ (62)

((1) is superfluous in BCK_. and H_.; cf. [19], p. 3 1 6 ) . We shall conclude this section with a brief remark on strict implication. The

system $4_., which gives the strict-implicational fragment of the modal logic $4, is obtained by extending S+ with:

(62) (A ~ ( A --* B))~(A -~ B)

(60') (A -~((B, ~ B2)~C))-~((B 1 ~ B2)~(A -~C))

380 K. Dogen

(63') (A~ + A2)+(B+(A , +A2))I

(It is a matter of routine to prove that (53), (62) and (60') can be replaced by (C ~ ( A ~B))~( (C ~ A ) ~ ( C ~B)), and that (54) is superfluous in $4_~.)

The sequent-system GS4_. corresponding to $4_~, in the sense that O [- A is provable in GS4_~ iff A is provable in $4_., would be given by (Id), (Cut), (~), (IntrO/), (IntrOr), (ElimO/) and (ElimOr), plus (Assoc) and (Contr) with Z, Z 1, Z 2 and Z 3 replaced by A, A1, A 2 and A 3, and plus the following two quasi-structural rules corresponding to (60') and (63') respectively:

(PermS4) X[A, B 1---~ B2] ~ Y X[B 1 ~ B 2, A] ~ Y; (ThinS4)

X[AI ~ A2] ~ Y X[AI ~ A2, B] F- Y"

These quasi-structural rules indicate that a logic built on $4_~ would not belong exactly to the family of logics we are investigating here.

2.10. Negation in L and its extensions. As we have already remarked in Section 2.1, we may assume that 7 A is defined in the language P as A ~ _L. We can then enquire what are the properties of this negation in L and its extensions.

Let us first consider those systems where _1_ is idle, i.e., where nothing specific being assumed about it, / behaves as an arbitrary propositional variable. Such is the system L, and such are all the systems in the chart of Section 2.7, an their + counterparts, except those extending BCK H. We shall say that these systems have a purely implicational negation. The best known system with a purely implicational negation is the Johansson propositional calculus.

Let P7 be the language which differs from P by having instead of _L the unary connective 7 as primitive. Let also 7 be defined in P as above. Then P7 is a proper sublanguage of P: to the formulae of P7 correspond only those formulae of P in which .L does not :occur except as a consequent of implications, i.e. _L occurs only in subformulae of the form A ~ .L. We shall now show what systems in P7 capture the P7 fragments of our systems in P with purely implicational negation:

LEMMA ~.1. Let E be L, or an extension of L in P, with purely implicational negation, where all theorems are of the form B ~ C. Next, let E 7 be the system in P7 obtained by extending the axioms and rules of E with:

(64) 7 ( 7 A o A) 7 B

(65) (A~B)-- , 7 A

7 A 7 B (66)

7(A vB)


A ~ B 7 B (67)

7 A

Then for every formula A of P7 we have that A is provable i n E 7 iff A is provable in E.

PROOF. Suppose A is provable in E 7. To show that A is also provable in E it is enough to establish that in L, and hence also in E, we have the following theorem and rules:

B ~ C ((A -+ C) o A)-+ C; (A -+ B) -+ (A -, C)

A ~ C B ~ C (A v B ) ~ C

A ~ B B ~ C A-~C

Next, let A1, . . . , A, be all the subformulae of a formula A of PT, and let fa = e s ( T A I o A I ) v ... v ( T A , oA.) . For every i, I

382 K. Dogen

form -1A 1 . Suppose (A 1 ---~A2) f is provable in E 7. So, A { ~ A { is provable in E 7. Since in E 7 we have A1 ~ A{ and Az ~- A{, by using (2) we have that A 1 ~ A 2 is provable in E 7. Suppose ( 7 A1) y is provable in E~. So, A { ~ f a is provable in E 7. Since in E 7 we have A1 ~ A{, by using (2) we can prove A1 ~fA in ET, and since in E 7 we also have 7fA (see the proof above, using (64) and (66)), by using (67) we can prove 7 A1 in E 7. So, i fA y is provable in E 7, then A is provable in E 7. q.e.d.

Since in all the extensions of L we have envisaged in Section 2.5 all theorems are of the form B ~ C, Lemma 8.1 covers all of these extensions which have a purely implicational negation. In the extensions E + of Section 2.6 not all theorems are of the form B--. C. However, for these extensions we can prove the following:

LEMMA 8.2. Let E be an extension of L in P, with a purely implicational negation, in which we can derive modus ponens (51). Next, let E 7 be the system in P7 obtained by extending the axioms and rules of E with (64), (65)and (66). Then for every formula A of P 7 we have that A is provable in E 7 iff A is provable in E.

PROOF. The proof is analogous to the proof of Lemma 8.1, Save that now we do not need (67) to show that if A f is provable in ET, then A is provable in

. E 7. This implication follows easily by using A ~ A ~ and modus ponens, which is derivable in E 7 too. q.e.d.

Note that the rule (67) is derivable in the E of Lemma 8.2, with the help of (2), 7 B 7 -~ (B--,fa_.~), (A ~fa-.B) ~ 7 A and modus ponens.

Let P-] be obtained from the language P - of Section 2.8 (i.e. the language P without o and T) by replacing I by 7 , as P7 was obtained from P. For systems in Pq we cannot have (64), and hence Lemma 8.2 is not forthcoming. However, for the system M - a n d its extensions in P - , we can prove the following 1emma, analogous to Lemma 8.2:

LEMMA 9. Let E- be M - , or an extension of M - in P - , with purely implicational negation, and let EY n be the system in PY~ obtained by extending the axioms and rules of E - with:

(68) (a ---, 7 B) --* (B --* 7 A).

Then for every formula A of P-~ we have that A is provable in E71 iff A is provable in E - .

Before proving this 1emma let us first note that in every extension of S - , and hence also in every extension of M ' , (68) can be replaced by the following two schemata:

(69) A ~ 7 7 A

(70) (A ~ B) -~ ( 7 B --* 7 A)


since we have:

(-7 A ~ 7 A) --, (A ~ 7 7 A) (68) (1), (51)

A ~ 7 7 A ;

B ~ 7 7 B (69) (3)

( A ~ B ) ~ ( A ~ 7 7 B ) ( A ~ 7 7 B ) ~ ( T B - - , TA) (68) (2)

(A--, B)-- , (7 B--, 7 a) ;

(a--, 7 B ) - - , ( 7 7 B ~ 7 a) (70) (53), (2)

(A -~ 7 n)-~ ((n-~ 7 7 B)-~ (B-~ 7 A)) (69), (57)

(A ~ 7 ~) ~ (B- , 7 A).

It is a matter of routine to show that in every extension of M - , (69) and (70) can be replaced by (70) and

(71) 7 7 (A ~ A ) .

Now we are going to prove Lemma 9:

PROOF OF LEMMA 9. Suppose A is provable in E-~. To show that A is provable in E - it is enough to establish tha t in M - , and hence also in E - , we have (6O) (A--,(I~--,C))--,(B--,(A-.C)).

Next, let A 1, . . . , A. be all the subformulae of a formula A Of P q , and let v a = a I ( A I ~ A ~ ) ^ .. . A (A .~A. ) . For every i, 1 ~

384 K. Dogen

the theorems in P_~ of these systems, we add to their postulates:

'TA (72) 7 A ~ (A --, 7 (B --, B)) (74)

B)--, -7 A (73) 7 (B--, B)--, A

The schema (72) can replace (68) in extensions of M where (A ~ A) ~ (B ---, B) is provable (already BCKj is such an extension). Since in these extensions we have -l- ~ (B ~ B), the double-line rule (74) can replace (50'). Of course, (74) is superfluous for the extensions of BCK~. The schema (73) can be replaced above by -7 B--,(B--,A). We proceed analogously for the corresponding systems in P-~.

References

[1] A. R. ANDERSON and N. D. BELNAP, Jr., Entailement: The Logic of Relevance and Necessity, vol. I, Princeton University Press, Princeton, 1975.

[2] N.D. BELNAP, Jr, Display Logic, Journal of Philosophical Logic 11 (1982), pp. 375-417. [3] M. Bo~I~, A Contribution to the Semantics of Relevant Logics (in Serbo-Croatian), Doctoral

Dissertation, University of Belgrade, Belgrade, 1983. [4] W. BUSZKOWSrd, Completeness results for Lambek syntactic calculus, Zeltsehrifl ffir

Mathematlsche Loglk und Grundlagen der Mathematlk 32 (1986), pp. 13-28. [5] D. VAN DALEN, Logic and Structure, 2nd ed., Springer, Berlin, 1983. [6] K. DO~EN, Sequent-systems for modal logic, The Journal of Symbolic Logic (50) (1985),

pp. 149-168. [7] K. DO~EN, A completeness theorem for the Lambek calculus of syntactic categories, Zeitschrift

fiir Mathematische Loglk and Grundlagen der Mathematlk 31 (1985), pp. 235-241. [8] K. FINE, Models for entailment, Journal of Philosophical Logic 3 (1974), pp. 347-372. [9] L. Fucns, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963.

1-10] G. Gentzen, Untersuchun#en fiber das lo#ische Schliessen, Mathematlsche Zeitschrifl 39 (1934-35), pp. 176-210, 405-431 [English translation in: The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, 1969].

[11] K. ISEYd and S. TANAKA, An introduction to the theory of BCK-al#ebras, Mathematica Japonica 23 (1978), pp. 1-26.

[12] K. ISEKI, BCK-al#ebras with condition (S), Mathematica Japonica 24 (1979), pp. 107-119. [13] J. LAMBEK, The mathematics of sentence structure, The American Mathematical Monthly

65 (1958), pp. 154-170. [14] J. LAMB[K, On the calculus of syntactic types, in: R. Jacobson, ed., Structure of Language and

its Mathematical Aspects, Proc. 12th Symp. Appl. Math., AMS, Providence, R.I., 1961, pp. 166-178, 264-265.

[15] L. L. MAKStMOVA, A semantics for the calculus E of entailment, Polish Academy of Sclences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic 2 (1973), pp. 18-21.

[16] R. K. MEYER and M. A. McRoBBm, Multisets and relevant implication, I and II, Australasian Journal of Philosophy 60 (1982), pp. 107-139, 265-281.

[17] H. O14o and Y. KOMO~, Logics without the contraction rule, The Journal of SymboUc Logic 50 (1985), pp. 169-201.

[18] M. PALAS~SKI, On BCK-algebras with the operation (S), Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logle 13 (1984), pp. 13-20.

[19] A.N. PRIOR, Formal Logic, 2nd ed., Oxford University Press, Oxford, 1962.


[20] H. RASIOWA and R. StKORSrd, The Mathematics of Metamathematics, Pafistwowe Wy- dawnictwo Naukowe, Warsaw, 1963.

[21] R. ROUTLEY and R. K. MEYER, The semantics of entailment I, in: H. Leblanc, ed., Truth, Syntax, Modality, North-Holland, Amsterdam, 1973, pp. 199-243.

[22] R. RotrrLEY and R. K. MEYER, The semantics of entailment II, Journal of Philosophical Logic 1 (1972), pp. 53-73.

[23] R. ROUTLEY and R. K. MEYER, The semantics of entailment II1, Journal of Philosophical Logic 1 (1972), pp. 192-208.

[24] A. URQUHART, Semantics for relevant looics, The Journal of Symbolic Logic 37 (1972), pp. 159-169.

[25] Dj. VUKOMANOVI~, Implication and Lattices (in Serbo-Croatian), Doctoral Dissertation, University of Belgrade, Belgrade, 1985.

MATEMATI~KI INSTITUT KI~Z M1HAILOVA 35 11000 BELGRADE YUGOSLAVIA

Received October 24, 1987

Studia Looica XLVII, 4

sequent-systems and groupoid models. ikosta/dosen radovi/[p][21... · groupoid models will be based...

Documents