Download Sequent-systems and groupoid models. I

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.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
The language GP
The system GL
Additional structural rules
The system G/~
Cut-elimination in G/J
Cut-elimination with additional structural rules
The system GC
2. Hilbert-formulations
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
The system L
L and urlogs
GL and L
The ( ~ , o) and implicational fragment of L
Extensions of L
The extensions E +
Extensions of S
The ( ~ , A, V, .1.) fragments of extensions of S +
354
K. Dogen
2.10.
Implicational fragments of extensions of L
Negation in L and its extensions
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.
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
2.9.
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.
with:
Let P be the language of the propositional calculus
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.
357
Sequent-systems...
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
,
SO ' S 1
-.,~
So
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:
(Cut)
x1 F- Y EA]
Operational rules:
(-+)
(o)
X, EA] F- Y2
Xz[X1] b- Y, EY2]
X, A t- Y[B]
X b- Y [ A - , B]
X [A, B] [- Y
X[AoB]t- Y
K. DoWn
358
(^)
X ~ YEA] X [- Y [B]
X k- YEA^B]
(v)
X [A] [-- r
X [B] ~- r
X[A v B] F Y
(T)
(_L)
X[O] [- Y
xEZ] k Y
X I- Y[O]
X }- Y[_l_]
It is easy to check that if X [- Y is a theorem of GL, t h e n 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)
(Perm)
X [ Z t ' (Z2, Zs)] 1- Y
X[(Z1, Z2), Z3] I-- Y
x [ z l , Z23 k Y
X[Z2, Z1] L- Y
(Contr)
X[Z, Z] [- g
X[Z] t- Y
(Expans)
X[Z] t- Y
x[z,z]F y
(Thin)
X[O] I- Y
X[Z] }- Y
(ThinR).
YF x [ o ]
Y}- X[A]
(IntrO/)
X[Z] }- Y
X [0, Z] [- Y;
(IntrOr)
X[Z] F Y
X[Z, 0][- Y
(ElimOl)
X[O, Z].}- e
X[Z] }- Y '
(Elim Or)
X[Z, O] }- g
X[Z] F Y
Sequent-systems...
359
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:
(Contr/)
X[Z, Z]
Z] [-[-- Y
X[O,
Y
(or (Contr r)
X [Z, Z] t- Y
X[Z, O] [- Y)"
Analogously, in the presence of (IntrO/) (or (IntrOr)) we could use:
(Expans/)
xX[O'[Z,z]Z][-]--YY
X[Z, Z] [- e
(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:
(Cane/)
X [Z 1[Z], Z 2[Z]] t- g
X [ Z I [ O ] , Z2[Z]] I-Y;
(Caner)
X [Z 1[Z], Z 2 [Z]] t- g '
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:
(Thin 0
X [Z] t- Y
"
X[ZI; Z] ~ Y'
(Thin/)
X [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 t u d i a L o g i c a 4/88
K. Dogen
360
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.
X 2[A ~ B, Xl] [- Y '
(o L)
X [A o B] [- Y;
(--, R)
X,A[-B
X[-A~B
(o R)
XI[--A X 2 [ - B
,X1, X2 t- A o B
X[A, B] [- Y
(^L)
(v L)
X[A ^ B] k Y; X[A ^ B] t- Y; (^ R)
XF-A Xk-B
X[-A^B
X[A][-- Y X[B]~- Y
;
X [A v B] t- Y
XF-A
XFB
X F A v B;X
Av B
X [A] [-- Y
X [B] F Y
( v R)
361
Sequent-systems...
(T L)
x[o] F Y
X[T] I- Y;
(T R)
(_L L)
• }- 121;
(_L R)
OI-T
xko
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
with
with
with
with
with
(--*)1" or ( ~ L ) we can prove A ~ B , A t - B ;
(o)T or (oR) we can prove A, B 1-- A o B;
(A)]" or (AL) we can prove A A B I - - A and A A B I - - B ;
(v)'~ or ( v R) we can prove A ~-A v B and B 1-A v B;
(I-)T or ( T R) we can prove 121[-- T ;
(_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
without (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
K. DoWn
362
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:
(-+L)
(Cut)
x~ l- B
X~[C] ~- A
XT[B+C,X'~]t-A
X2[A]]-- Y
X~ [C] t-- A
X]~-B
X 2[A] [-- Y
X 2[X~[C]]t- Y
x~ [x'~ [B-+ C, x~]] I- A];
X~[X~[B c, x~]] ~- r
X 2[A], B t-- C
(+R)
X~ k-A Xz[A] t- B-+C
X 2[X~] t- B ~ C;
(Cut)
X I~-A X 2[A],Bt-C
X 2 IX1], B I'- C
x2rxl] ~--B+C,
(Cut)
(~L)
(cut)
(+R)
(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)
(Cut)
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)
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 Of 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-elimination with additional 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
Sequent-systems...
363
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:
X1FA
X~ [A] ~- Y
(*)
X 2[A]F Y
S~ [- A X~ [A] [--'Y
X~[X,]F Y
(Cut)
X2[Xl] [- Y;
(Cut)
(,)
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~ [-- A X2[A ] [- V
(Cut)
X~l-A
X~[A]F Y
(Cut)
X'g[Xl] F Y
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:
(Contr)
X[Z, Z] [-- Y
XEZ] ~- r
or
(Thin)
X[O] 1- Y
X[Z] ]- Y
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
(Contr)
Xl F- A
X 2 [Z [A]] [- Y
(Cut)
X2[Z[X1] ] [-- Y
we shall replace it by the following proof:
XI ]- A X2[Z[A], Z[A]] [-- Y
(Cut)
X 1~ A
X2[Z[X1], Z[A]] ]-- e
(CuO
X2[Z[XI], Z [ X , ] ] [- Y
(Contr)
x~ [z Ex,]] ~- r.
In the analogous case with (Thin), the proof on the left will be replaced by the
proof on the right:
x~ [0] ~- Y
xl ~- A X2 [Z [A]] F r
Xz[Z[X13] [---Y;
(Thin)
(Cut)
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
364
K. DoWn
problematic cases like-the following:
Xi [- A
X 2 [Z [A]] ~- r
X2[Z[A], Z[A]] [-- e
x2[z[xl], Z[A]]
If now we apply (Cut) to
F- Y.
(Expans)
(Cut)
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
(Cut}
X z[A]~- Y
x2Exi] F- r.
This proof can be replaced by:
Xl~-O
X1]- Y
9"
eventually (Thin R)
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'
365
Sequent-systems...
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 GJ, 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
(3)
A-~A 1 B~B 1
(Ai ~ B ) ~ ( A oB1)
(6)
A~A t B~B 1
(A o B)-->(A 1 oB1)
(9)
C~A
C~B
C ~ ( A ^ B)
(10) A-~(A v B)
(12)
A~D
B~D
(A v B ) ~ D
(11) B ~ ( A v B)
(13)
C-~(A~D) C-~(B~D)
C ~ ( ( A v B)--*D)
(4) ((A --. B) o A) ~ B
(5) A ~ ( B ~ ( A o B))
(7) (A ^ B ) ~ A .
(8) (A ^ B ) ~ B
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:
(3')
B~B 1
(Al ~ B ) ~ ( A x _ ~ e l ) ;
(3")
A~A 1
(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:
(14)
(C o A)--> B
C~(A~B) ~
367
Sequent-systems...
Next, we easily obtain that the following rules are derivable in L:
A~A1
B~Bx 9
(15) (A A B)-*(A 1 ^ B O'
(16)
A~Aa
B~B1
9
(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:
(17)
A~B
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))
( 2 2 ) ((A v B ) v C ) ~ ( A v ( B v Ci)
(19)
(23) (A v / 3 ) ~ ( B v A )
(A ^ B ) ~ ( B A 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 ) ) ~ ( ( C o A ) 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).,
A~B
(30)
(A v B ) ~ 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)
(ii)
,Oii)
(iv)
(ix)
(x)
(xi)
(anb)nc=ac~(bnc)
(v)
anb=bna
(vi)
(vii)
a=ana
(a w b) c~ a = a
(viii)
c ' ( a u b) = (e" a) w (c" b)
(a w b)" c = (a" c) w (b "c)
a <<.b \ c r a . b <<.c, where a ~< b is an
(awb) wc=aw(buc)
awb=bwa
a=awa
(anb) wa=a
abbreviation for a = a n b .
Note that in Definition 4 nothing specific is assumed about 1 and 0, save
that they are elements of d .
It is easy to show that for any urloo we can prove:
(xii)
(xiii)
(xiv)
a=anbcc-awb=b
~< is a partial order on d
(a <~ a l & b <<.b l ) ~ a l \ b
<<.a\b 1
= ~ a ' b <~ a l ' b 1
= ~ a n b <<.al n b 1
= , . a w b <<.a 1 u b 1.
Next, we introduce the following definition:
DEFINITION 5. Let V be the set of propositional variables of P, and let
F be the set of formulae of P. Given an urlog with domain d , and a mapping
Vo: V ~ d , we define the valuation v: F ~ d
by the following recursion:
v(p)
= Vofp)
v(A--, 8)
v(A o B)
v(A ^ B)
v(A v B)
=
=
=
=
v(A)\v(B)
v(A)" v(B)
v(A) m v(B)
v(A) u v(B)
v(T)
v(_L)
=
1
= O.
By a straightforward induction on the length of proof of A ~ B in L, we can
prove the following soundness lemma for L:
LEMMA 2. I f A ~ B
is provable in L, then for every urlo9 and every
valuation v we have v(A) <<.v(B).
Next, let IAI-- {n: A ~ B is provable in L}, and let:
Ial\lnl
IAI'IBI
IZl n Inl
IZl u Inl
=
=
=
=
Ia-~nl
Ihonl
Ih ^ nl
IA v nl.
369
Sequent-systems...
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 IT [ # 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) <. v(B).
F r o m left to right we use Lemma 2. For the other direction, suppose that for
every urlog and for every v we have v(A) ~ v(B). Then by Lemma 3, for t h e
Lindenbaum algebra of L we have IAI ~< IBI, since v(A) = [AI is a valuation
from F to La. F r o m Ial ~< IBI, by definition we obtain [A[ = IAI c~ IBI, i.e.,
IAI = IA ^ BI. Hence, A ~ (A ^ B) is provable in L, from which by (29) we
obtain that A ~ B is a theorem of L.
We can also prove the following lemma:
LEMMA 4. The Lindenbaum algebra of L is a free algebra in the class of all
urlogs, the set fq = {lPI: P e V} being the set of free generators of this algebra.
PROOF. It is clear that fr generates all elements of ~e (where ITI and Ill
are nullary operations). Let d be the domain of an urlog and let f : fr ~ d .
Then let Vo(p) = f(IPl). Starting from v0 we define the valuation v: F ~ d. Next
let h([AI) = v(A). For h to be well-defined we must have that IAI = IBI implies
v(A) = v(B); i.e. we must have that if A ~ B is provable in L, then v(A) = v(B),
and this follows from Lemma 2. It is easy to see that h: s162
~ d is a homomorphism, q.e.d.
2,3. GL and L. Let t'(Z) be the formula of P obtained from a G-term Z by
replacing every comma by o, and every O by $ , as in Section 1.3; and let ~'(Z)
be the formula of P obtained from Z by replacing every comma by v , and
every O by • as in Section 1.7. In sequent-systems with the single-conclusion
property, ff X ~ Y is provable, there are no commas occurring in Y. Since in
the sequel we shall deal only with such sequent systems, t'(Y) will be either
Y itself if Y is not O, or l if Y is O. We shall now prove the following lemma:
LEMMA 5.
I f X 1- Y is provable in GL, then f ( X ) ~ t(Y) is provable in L.
PROOF. By induction o n the length of proof of X ~ Y in GL.
I f X ] - Y i s A ~ A , then in L we have A ~ 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
<~e,\,., ~<, ITI, I-LI> be the Lindenbaum algebra of L o, where IAI ~< IBI iff
A ~ B is provable in L o. Right-residuated partially ordered groupoids are
algebras <~, \ , . , ~<, 1, 0>, where ~< is a partial order on d , and (xi) and the
first two implications of (xiv) hold (as a matter of fact, the first implication of
(xiv) follows from the second implication and (xi)).
It is not difficult to show that L o is sound and complete, in the sense of (0),
with respect to right-residuated partially ordered groupoids, and that the
Lindenbaum algebra of L o is free in the class of all such groupoids. If GL o is the
system in GP without A and v , obtained from GL by discarding ( A ) and ( v ),
then it is easy to demonstrate the analogue of (I) for GL o and L o.
That L o is indeed the ( ~ , o, T , _1.) fragment of L can be shown with the
help of GEo, which is obtained from GL o as GE is obtained from GL. The
system L o in P with only ~ and o, gives the (--->, o) fragment of L.
Suppose A I ~ A 2 is a purely implicational formula of P, and suppose
A, ~ A 2 is provable in GL. It is not difficult to show (using the results of
Sections 1.4 and 1.5) that in that case A 1 }--A z is provable in the sequentsystem given by:
X , A ~ B C[--A (..,(B,BI).... ,Bn-O[-Bn
P~-P; X[-A~B; (..((A~B,C),B,),...,Bn-I)[-Bn
from which it easily follows that AI is the same formula as A2. So, the
implicational fragment of L is given by A ~ A, i.e. nothing but identities is
provable in this implicational logic. Note that nothing but identities follows
from (I), (2) and (3).
Sequent-systems...
371
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 +
(31) ((AloA2)oAa)-~(Alo(A2oA3))
(32) (AIo(A2oA3))-~((AxoA2)oA3)
(33) (A2 oA1)--" (At oA2)
(34) A--*(A o A)
(35) (A o a ) ~ A
(36) A ~ T
(37) J_ --*A
(38) (-[- oA)--*A
(39) (Ao -[-)--',A
(40) A--*(T oA)
(41) A-~(Ao T)
(42) Y -,(Y o T)
(43) (T o T ) - ' Y
(44) (T oA)-*(AoA)
(45) (A o T) ~ (A o A)
(46) (A o A) ~ (T o A)
(47) (A o A) ~ (A o T)
(48) (B o A)---~A
(49) (A o B) -~ A
urlog +
(al"a2)'a 3 <~al"(a2'a 3)
a~'(a2"a 3) <<.(a~'a2)'aa
a2"al <~al'a2
a<.a.a
a.a<<.a
a<~l
O<~a
1.a<~ a
a ' l <~a
a<~ l . a
a<~a.1
1~<1.1
1-1~<1
1.a <<.a.a
a.1 <~a.a
a.a <~ 1 .a
a.a<~ a.1
b.a<<, a
a.b <~a
GL (GL') +
(Assoc)~
(Assoc)$
(Perm)
(Contr)
(Expans)
(Thin)
(Thin R)
(IntrO/)
(Intr Or)
(Elim O/)
(Elim Or)
(Contr 0)
(Expans O)
(Contr/)
(Contr r)
(Expans/)
(Expans r)
(Thin/)
(Thin r)
It is easy to find the expressions in the first and second column which would
corrrespond to (Canc/), (Canc r), (Dupl/) and (Dupl r). For example, to (Canc/)
would correspond ( B I [ T ] o B 2 [ A ] ) ~ ( B I [ A ] oB2[A]) and b 1 [1].bz[a ]
<~ bl[a]'b2[a], where a schema B[A] is used as in Section 211, and b[a] is
defined analogously.
Note that in all extensions of L we have envisaged above, every theorem is
of the form A -o B.
The systems in P without ^ and v , obtained by extending L o of Section
2.4, i.e. (1)-(6), with schemata from the first column will be sound and complete,
in the sense of (0), with respect to right-residuated partially ordered groupoids
which satisfy the corresponding conditions in the second column. Also, their
Lindenbaum algebras will be free in the classes of all corresponding groupoids.
K. DoWn
372
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:
(horiz)
AI-B
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/):
zt-Y
o,z~-Y
If moreover we have (Cut), this last double-line rule amounts to (IntrO/) and
373
Sequent-systems...
(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:
O 1- A ~ B is provable in GE iff A ~ B is provable in E.
(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 r E + 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:
(neces)
O1-A
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 ~ 0 1 - 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:
01-A
01-B
01-C
Finally, if A is obtained by (50), we use (neces).
q.e.d.
K. DoWn
374
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)
and
(52)
B
adjunction:
A
B
AAB"
In systems where we have A ~ A and
(17) amounts to the following rule:
A~B
modusponens,the Rule of Replacement
C
In L, and similar systems without modusponens,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)
. / = B C K j + (3Z,.)
RAM =
~
J
'
B C K H = B C K , + ( 37 )
R A = hi + {3/,,).
= M ~- (36)
h/= S § (33)v
L + [31), (32), (38)-(/-.1)
Sequent-systems...
375
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 R A M 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 t h e 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 R A M + is a mingle
extension of RA +; in R A M + 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 <<.a, then ( d , . , 0, 4 ) is
a BCK algebra in the sense of [l 1], and ( d , , , . , 0, ~ ) i s a BCK algebra with
condition (S), in the sense of [12] and [18]). The system B C K f corresponds to
the positive fragment of a logic associated with BCK algebras (see [17]), and
BCK~ corresponds to this logic with a Heyting-type negation. This logic is
essentially intuitionistic logic without contraction.
In the urlogs corresponding to J we have a. b = a n b . (Since b ~< 1, we
have a- b ~< a- 1, and hence, a- b ~< a. Similarly we obtain a- b ~< b, which gives
a'b<<.anb. For the converse, from a n b ~ < a and a c ~ b < ~ b , we have
( a n b ) ' ( a c a b ) ~ < a ' b . Then using a n b ~ < ( a n b ) . ( a n b )
we obtain a n b
~< a" b.) So, according to (xi), the operation \ in these urlogs is the relative
pseudo-complement (see [20], p. 54). Since every relatively pseudo-complemented lattice has a unit element, and since every such lattice is distributive,
we can identify the urlogs corresponding to J with relatively pseudo-complemented lattices. The urlogs corresponding to H are relatively pseudo-complemented lattices with a zero element, i.e. Heyting algebras.
The system J+ is an axiomatization of Johansson's propositional logic,
and H + is an axiomatization of Heyting's propositional logic. In J and
H, and hence also in J+ and H +, we can prove ( A o B ) ~ ( A ^ B ) and
(A ^ (B v C))~((A ^ B) v (A A C)). The system C + is the classical propositional calculus. The systems C and C + are connected to the sequent-system
GC of Section 1.7 by statements corresponding respectively to (I) and (I+).
5 -- S t u d i a L o g i c a 4/88
K. Dogen
376
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:
(1) A-+A
(51)
(53) (A --+B) ~ ((C ~ A) --+(C --+B))
(7) (A ^ B ) ~ A
(54)
(52)
A
A--,B
B
A
(A--->B)~ B
A
B
AAB
(8) (A ^ B)--, B
(55) ((C + A) ^ (C-+ B))--,(C-+(A ^ B))
A-+(A vB)
B+(A v B)
(10)
(11)
(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
C~(A~B)
A
(A--,B)-~B (54)
(53), (51)
C~B;
(A --+B) --+(A --+B) (57)
(A-+B)~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
(58)
We have:
Cm--.>(A--rD )
C m --+ (/~,,--~D)
, n ~> 0, m >/0.
Sequent-systems...
377
(54)
Cm-~ (A ~D)
((B.~A)~(B,~D))-~(B.~D)
eventually (53) and (51)
eventually (53) and (51)
1~,,-~ ((/~, -~A) ~ (/3, -~D))
( e , --r ((/3. ~ A) ~ (/~. -~ D))) -~ (em ~(/~. ~ D))
:
9 (51)
(~,, --*(/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 r o m (1) and (54) we also obtain immediately:
(59)
Let the language G P - 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.
If 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 h e 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 ,(•
trivial,
q.e.d.
(Assoc), (IntrO/), (IntrOr), (ElimO/) and ( E l i m O r ) a r e all
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) ~:
(--,)T
C--.A[-- C ~ A
A~B]-- A ~ B
C ~ A , CI--A
A~B,A[--B
A ~ B , (C-.A, C)FB
J
(--,)T
(Cut)
(Assoc) J,
(A~B, C-~A), C~ B
. , .
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
A~B,O~ B
(Elim O r)
A~B[--B
(CuO
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 R A M - . 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.
Sequent-systems...
379
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:
(51')
B.~A
A~D
/~,~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 B C K ] 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))
K. Dogen
380
(63')
(A~ + A 2 ) + ( 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;
X [ A I ~ A2] ~ Y
X [ A I ~ A2, B] F- Y"
(ThinS4)
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)
(65)
(66)
7B
(A~B)--, 7 A
7A
7B
7(A vB)
381
Sequent-systems...
A~B
(67)
7B
7A
Then for every formula A of P7 we have that A is provable in 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, a n d 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 <<.i <~n, ifi-E 7 we can
prove 7 A i ~ (Ai ~fa), since we have:
(1), (2), (10), (11)
(TAi~
7 A i-* (A, ~fA);
7(TAloA1)
(14)~
... 7 ( T A , oA,,) (64)
"7fa
(66)
(65)
(A, '->fa)" 7 A,.
Let A~ be a subformula of A, a n d let Af be obtained from Aj by replacing
every subformula of A t of the form 7 A i by Ai"+fA. In E 7 we can prove
+ A1 ~ Af. To show that, we use 7 A+ ~ (A i ~ f a ) and the Rule of Replacement
(17), which holds in E 7 in virtue of the rules (3), (6), (15), (16) and
A~B
7B~TA"
This last rule is derivable in E 7 with the help of (3"), (2), 7 B ~ (B-~fa-+n) and
(A -~fA-~n) -> 7 A.
Suppose n o w that A is provable in E. Since l behaves i n E as an arbitrary
propositional variable, in E we can prove the schema A B obtained from, A by
substituting uniformly the schematic letter B for 2_. But then A n is provable in
E 7, and hence A s is provable in E 7. We shall now show that it follows that A is
provable in E 7.
Since A is a-formula of P7 provable in E, and since all theorems of E in
P are of the form B-~ C, we have that A is either of the form A 1 ~ A 2 or of the
K. Dogen
382
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, ifA 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, L e m m a 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 L e m m a 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 L e m m a 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 P q we cannot have (64), and hence L e m m a 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 L e m m a 8.2:
LEMMA 9. Let E - be M - , or an extension of M - in P - , with purely
implicational negation, and let EYn 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)
383
Sequent-systems...
since we have:
(-7 A ~ 7 A) --, (A ~ 7 7 A)
(1), (51)
B~ 7 7 B
(3)
(A~B)~(A~
(2)
(68)
A~77A;
(69)
77B)
(A~ 77B)~(TB--,
TA)
(68)
(A--, B)--,(7 B--, 7 a);
(53), (2)
(69), (57)
(a--, 7 B ) - - , ( 7 7 B ~ 7 a)
(70)
(A -~ 7 n)-~ ((n-~ 7 7 B)-~ (B-~ 7 A))
(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 ) .
N o w we are going to prove L e m m a 9:
PROOF OF LEMMA 9. Suppose A is provable in E-~. To show that A is
provable in E - it is enough to establish t h a 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
va = a I ( A I ~ A ~ ) ^ ... A ( A . ~ A . ) . For every i, 1 ~<i~< n, in E~ we can
prove 7 A i ~ (A i ~ 7 VA), since we have:
(1), (52)
(1), (2), (7), (8)
(70),
(2)
(60), (51)
(68), (51)
v a --, ( A i --' A i )
Va ~ ( T A i ~
(57)
7Ai)
va
(A i ~ 7 Va) --~ (V,t ~ 7 a t )
(68)
(Ai --7-?-7 VA) ~ 7 A i.
7 Ai --', (VA--" 7 A3
7 A i ~ ( A i ~ 7 VA);
Next we let A f be obtained from A by replacing every subformula of A of
the form 7 Ai by Ai --+ 7 va, and we proceed as for the proof of L e m m a 8.2 (the
closure of E-~ under the Rule of Replacement (17) is guaranteed by (70)). q.e.d.
If we replace L in Lemma 8.1 and 8.2 by M, we can replace (64)-(67) in these
lemmata by (68) and:
(50')
7A
T--, qA"
This double-line rule is an instance of (50) of Section 2,6; so, in the extensions of
M + it is superfluous.
The systems o f Section 2.7 which extend BCK n do not have a purely
implicational negation: in these systems (37) .1_~ A is assumed. To axiomatize
384
K. Dogen
the theorems in P_~ of these systems, we add to their postulates:
(72)
7 A ~ (A --, 7 (B --, B))
(73)
7 (B--, B)--, A
(74)
'TA
B)--, -7 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-~.
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[25] Dj. VUKOMANOVI~,Implication and Lattices (in Serbo-Croatian), Doctoral Dissertation,
University of Belgrade, Belgrade, 1985.
MATEMATI~KI INSTITUT
KI~Z M1HAILOVA35
11000 BELGRADE
YUGOSLAVIA
Received October 24, 1987
Studia Looica XLVII, 4