The Semantics and Proof Theory of Linear Logic

Home , Linear logic

The Semantics and Pro of Theory of

Linear Logic

Arnon Avron

Department of Computer Science

Scho ol of Mathematical Sciences

Tel-Aviv University

Tel-Aviv

Israel

Abstract

Linear logic is a new logic which was recently develop ed by Girard

in order to provide a logical basis for the study of parallelism. It is

describ ed and investigated in [Gi]. Girard's presentation of his logic is

not so standard. In this pap er we shall provide more standard pro of

systems and semantics. We shall also extend part of Girard's results

by investigating the consequence relations asso ciated with Linear Logic

and by proving corresp onding strong completeness theorems. Finally,

we shall investigate the relation b etween Linear Logic and previously

known systems, esp ecially Relevance logics.

1 Intro duction

Linear logic is a new logic which was recently develop ed by Girard in order

to provide a logical basis for the study of parallelism. It is describ ed and

investigated in [Gi]. As we shall see, it has strong connections with Relevance

Logics. However, the terminology and notation used by Girard completely

di ers from that used in the relevance logic literature. In the present pap er

we shall use the terminology and notations of the latter. The main reason

for this choice is that this terminology has already b een in use for many

years and is well established in b o oks and pap ers. Another reason is that

the symb ols used in the relevantists work are more convenient from the p oint

of view of typing.

The following table can b e used for translations b etween these two sys- 1

tems of names and notations:

Rel ev ance l og ic Gir ar d

M ul tipl icativ e I ntensional ; Rel ev ant

Additiv e E xtensional

E xponential M odal

W ith (&) And (^)

P l us () O r (_)

E ntail ment ( ) E ntail ment (!)

P ar ( ) P l us (+)

T imes ( ) C otenabil ity ()

1; ? t; f

!; ? 2; 3

2 Pro of theory

2.1 Gentzen systems and consequence relations

The pro of-theoretical study of linear logic in [Gi] concentrates on a Gentzen-

type presentation and on the notion of a Pro of-net which is directly derivable

from it. This Gentzen-type formulation is obtained from the system for

classical logic by deleting the structural rules of contraction and weakening.

However, there are many versions in the literature of the Gentzen rules for

the conjunction and disjunction. In the presence of the structural rules all

these versions are equivalent. When one of them is omitted they are not.

Accordingly, two kinds of these connectives are available in Linear Logic (as

well as in Relevance Logic):

The intensional ones (+ and ), which can b e characterized as follows:

` ; A; B i ` ;A + B

; A; B ` i ;A B `

The extensional ones (_ and ^), which can b e characterized as follows:

` ;A ^ B i ` ;A and ` ;B

A _ B; ` i A; ` and B; `

In [Av2] we show how the standard Gentzen-type rules for these connectives

are easily derivable from this characterization. We characterize there the

rules for the intensional connectives as pure (no side-conditions) and those

for the extensional ones as impure. The same rules, essentially, were used

also by Girard. He preferred, however, to use a variant in which only one-side

sequents are employed, and in which the negation connective can directly

b e applied only to atomic formulas (the negation of other formulas b eing

de ned by De-Morgan rules, including double-negation). This is convenient

As explained in [Av2], this distinction is crucial from the implementation p oint of

view. It explains, e.g. why Girard has found the intensionals (or multiplicati ves) much

easier to handle than the extensionals (additives).

This variant is used also in [Sw] for the classical system. 2

for intro ducing the pro of-nets that he has invented as an economical to ol

for developing Gentzen-type pro ofs in which only the active formulas in an

application of a rule are displayed. For the purp oses of the present pap er it

is b etter however to use the more usual presentation.

Girard noted in [Gi] that he had given absolutely no meaning to the

concept of a \linear logical theory" (or any kind of an asso ciated conse-

quence relation). Hence the completeness theorem he gave in his pap er is

of the weak kind. It is one of our main goals here to remedy this. For this

we can employ two metho ds that are traditionally used for asso ciating a

consequence relation with a Gentzen-type formalism. In classical and intu-

itionistic logics the two metho ds de ne the same consequence relation. In

Linear Logic they give rise to two di erent ones:

LL LL

B i the ) : A ;:::;A ` The internal consequence relation (`

1 n

Kl Kl

corresp onding sequent is derivable in the linear Gentzen-type formalism.

The external consequence relation (` ) : A ;:::;A ` B i the

LL 1 n LL

sequent ) B is derivable in the Gentzen-type system which is ob-

tained from the linear one by the addition of ) A ;:::; ) A as

1 n

axioms (and taking cut as a primitive rule).

It can easily b e seen that these two consequence relations can b e char-

acterized also as follows:

A ;:::;A ` B i A ! (A ! ( (A ! B ) :::)) is a theorem

1 n 1 2 n

of Linear Logic.

A ;:::;A ` B i ` B , for some (p ossibly empty) multiset

1 n LL

of formulas each element of which is:

Intensional (multiplicati ve) fragment: identical to one of the A 's.

The full prop ositional fragment: identical to A ^ A ^ ^ A ^ t.

1 2 n

In what follows we shall use b oth consequence relations. We start by

developing a natural deduction presentation for the rst and a Hilb ert-typ e

presentation for the second.

2.2 Natural deduction for Linear Logic

Prawitz-style rules:

[A]

B B

A A

Since linear logic has the internal disjunction +, it suces to consider only single-

conclusioned consequence relations.

We use ) as the formal symb ol which separates the two sides of a sequent in a Gentzen

typ e calculus and ` to denote (abstract) consequence relations. 3

[A]

A A ! B

A ! B B

[A; B ]

A B C

A B

A B C

A t

A B A ^ B A ^ B

()

A ^ B A B

[A] [B ]

A _ B C C

B A

()

A _ B A _ B C

Most of the ab ove rules lo ok almost the same as those for classical logic.

The di erence is due to the interpretation of what is written. For Linear

logic we have:

1. We take the assumptions as coming in multisets. Accordingly, exactly

one o ccurrence of a formula o ccurring inside [ ] is discharged in ap-

plications of I nt, ! I nt, E l im and _E l im. The consequences of

these rules may still dep end on other o ccurrences of the discharged

formula!

2. Discharging the formulas in [ ] is not optional but compulsory. More-

over: the discharged o ccurrences should actual ly be used in deriving

the corresp onding premiss. (In texts of relevance logics it is custom-

ary to use \relevance indices" to keep track of the (o ccurrences of )

formulas that are really used for deriving each item in a pro of.)

3. For ^I nt we have the side condition that A and B should dep end on

exactly the same multi-set of assumptions (condition (*)). Moreover,

the shared hyp othesis are considered as app earing once, although they

seem to o ccur twice.

4. For _E l im we have the side condition that apart from the discharged A

and B the two C`s should dep end on the same multiset of assumptions

((**)). Again, the shared hyp othesis are considered as app earing once.

5. The elimination rule for t might lo ok strange for one who is accustomed

to usual N.D. systems. One should then realize that the premiss A and

the conclusion A might di er in the multiset of assumptions on which

they dep end!

Notes: 4

1. Again we see that the rules for the extensional connectives are impure,

while those for the intensional ones are pure (no side-conditions!)

2. the rule for I nt is di erent from the classical (or intuitionistic) one,

since no o ccurrence of A on which B dep ends is discharged. In fact

we have that the dual:

[A]

B B

is derivable, but the classical version

[A] [A]

B B

is not valid!

3. It is not dicult to prove a normalization theorem for the p ositive frag-

ment of this system. As usual, this is more problematic when negation

is included. This case might b e handled by adding the ab ove derived

intro duction rule as primitive and then replacing the given elimination

rule with the two rules which are obtained from the intro duction rules

by interchanging the roles of A and A.

It is easier to see what`s going on if the N.D. system is formulated in

sequential form:

A ` A

` A ;A ` B ` B

1 2

; ` A ` A

1 2

` A ` B ` A B ; A; B ` C

1 2 1 2

; ` A B ; ` C

1 2 1 2

;A ` B ` A ` A ! B

1 2

` A ! B ; ` B

1 2

` A ^ B ` A ^ B ` A ` B

` A ^ B ` A ` B

` A ` B A; ` C B; ` C ` A _ B

` A _ B ` A _ B ; ` C

` A ` t

1 2

` t

; ` A

1 2

Again ; denote multisets of formulas. Note also that weakening is not

al lowed, i.e ` A do es not imply ; ` A.

Lemma: The N.D system has the following prop erties: 5

1. If ` A and A; ` B then ; ` B

1 2 1 2

2. If ;A ` B and ` B then ; ` A (but from ;A ` B and

1 2 1 2 1

;A ` B do es not follow that ; ` A!)

2 1 2

3. If ` A then ` A

4. If ;A ` B then ; B ` A

De nition: Let A ;:::;A ) B ;:::;B b e a sequent. An interpretation

1 n 1 m

of it is any single-conclusion sequent of one of the following forms:

A ;:::;A ; B ;:::; B ; B ;::: B ` B (1 i m)

1 n 1 i1 i+1 m i

A; : : : ; A ;A ;:::;A ; B ;:::; B ` A (1 i n)

i1 i+1 n 1 m i

Theorem: ) is provable in the Gentzen-type system i any interpre-

tation of it is provable in the N.D system. Moreover, ` A i there is a

pro of of A from in this N.D. system.

We leave the pro of of b oth the last theorem and the lemma ab ove to the

reader.

2.3 Hilb ert systems and deduction theorems for the inten-

sional fragment

There is a standard metho d for obtaining from a given pure N.D. formalism

an equivalent Hilb ert-typ e system with M :P as the only rule of inference:

One needs rst to intro duce some purely implicational axioms which suce

for proving an appropriate deduction theorem. The second step is then to

replace the various rules by axioms in the obvious way. For example, a rule

of the form:

[A ;A ] [A ]

1;1 1;2 2

B B B

1 2 3

will b e translated into the axiom:

(A ! (A ! B )) ! ((A ! B ) ! (B ! C )))

1;1 1;2 1 2 2 3

Obviously the rst part of this pro cedure is the more dicult one. This is

esp ecially true when a non-standard logic is treated. Accordingly, we start

by formulating some intuitive versions of the deduction theorem:

Classical-intuit ioni stic: There is a pro of of A ! B from the set i

there is a pro of of B from [ fAg.

RM I : There is a pro of of A ! B from the set which uses al l the formulas

in i there is a pro of of B from [ fAg which uses all formulas in

[ fAg.

See e.g. [Ho], pp. 32

The names R and RM I b elow are taken from [AB] and [Av1].

! ! 6

R (Implicational fragment of R): There is a pro of of A ! B from the

multiset in which every (o ccurrence of ) formula in is used at least

once i there is such a pro of of B from the multiset ;A.

HL (Implicational fragment of Linear logic): There is a pro of of A !

B from the multiset in which every formula of is used exactly once

i there is such a pro of of B from the multiset ;A.

As we said ab ove, these are intuitive formulations. They involved ref-

erences to \the number of times (an o ccurrence of ) a formula is used in a

given pro of ". This notion can b e made precise, but it easier (and more illu-

minating) to take the di erent versions as referring to stricter and stricter

notions of a \pro of ".

In the following assume Hilb ert-typ e systems with M.P as the only rule

of inference:

A Classical (or intuitionistic) pro of is a sequence (or directed graph)

of formulas such that each formula in it is either an axiom of the system

, or an assumption, or follows from previous ones by M.P..

A S-strict (M-strict) pro of is a classical pro of in which every (o ccur-

rence of a) formula other than the last is used at least once as a premiss

of M.P..

A linear pro of is a classical pro of in which every occurrence of formula

other than the last is used exactly once as a premiss of M.P..

Examples:

A classical but not strict pro of of A from fA; B g:

1. B (ass.).

2. A (ass.).

S-strict pro of which is not M-strict:

1. A (ass.).

2. A (ass.).

3. A ! A (axiom)

4. A (f1; 2g; f3g, M.P.).

A M-strict pro of which is not linear:

1. A (ass.)

2. A ! B (ass.)

A ` A according to all notions of pro of. Hence A ! A should b e a theorem according

to all the versions of the deduction theorem. 7

3. A ! (B ! C ) (ass.)

4. B (1,2, M.P.)

5. B ! C (1,3, M.P.)

6. C (4,5, M.P.)

A linear pro of of C from A ! B; B ! C; A:

1. A ! B (ass.)

2. B ! C (ass.)

3. A (ass.)

4. B (1,3 M.P.)

5. C (2,4 M.P.)

A linear pro of of A from fA,Bg in classical logic:

1. A ! (B ! A) (axiom)

2. A (ass.)

3. B ! A (1,2 M.P.)

4. B (ass.)

5. A (3,4 M.P.)

De nition: We say that B is classical ly (M-strictly, S-strictly, linearly)

provable from A :::;A i there is a classical (M-strict, S-strict, linear)

1 n

pro of in which B is the last formula and A ;:::;A are (exactly) the as-

1 n

sumptions.

Alternatively, these consequence relations may b e characterized as fol-

lows:

Classical-intuit ioni stic:

1. ` A whenever A is an axiom or A 2 .

2. If ` A ! B and ` A then [ ` B (here ; ; are

1 2 1 2 1 2

sets of formulas).

S-strict:

1. fAg ` A.

2. ; ` A if A is an axiom.

3. If ` A ! B and ` A then [ ` B .

1 2 1 2

M-strict:

1. A ` A.

2. ; ` A if A is an axiom. 8

3. If ` A ! B , ` A and is a contraction of ; then

1 2 1 2

` B . (Here ; ; are multisets)

1 2

Linear:

1. A ` A.

2. ; ` A if A is an axiom.

3. If ` A ! B , ` A then ; ` B (Again ; are

1 2 1 2 1 2

multisets).

The last example ab ove indicates that with enough axioms, every clas-

sical pro of can b e converted into a linear one. What is really imp ortant

concerning each notion of a pro of is (therefore) to nd a minimal system for

which the corresp onding deduction theorem obtains (with the obvious cor-

resp ondence b etween the various notions of a pro of and the various notions

of the deduction theorem). Accordingly we de ne:

H (Intuitionistic Implicational calculus): The minimal system for which

the classical deduction theorem obtains. It corresp onds to the notion

of a classical pro of.

RM I (Dunn-McColl): The minimal system corresp onding to S-strict

pro ofs (i.e. it is the minimal system for which there is a S-strict pro of

of A ! B from i there is a S-strict pro of of B from [ fAg.)

R (Church): The minimal system corresp onding to M-strict pro ofs.

HL : The minimal system corresp onding to linear pro ofs.

The rst example shows that A ! (B ! A) should b e a theorem of

H , the second that A ! (A ! A) should b e a theorem of RM I ,

! !

the third that (A ! (B ! C )) ! ((A ! B ) ! (A ! C )) should b e

a theorem of R , the forth that (A ! B ) ! (B ! C ) ! (A ! C )

should b e a theorem of HL . It is also p ossible to show, using Gentzen-

type formulations, that 6` A ! (B ! A), 6` A ! (A ! A) and

RM I R

! !

6` A ! (B ! C ) ! (A ! B ) ! (A ! C ).

Our next step is to present formal systems for these four logics. In all of

them M.P. is the only rule of inference:

HL (linear):

I. A ! A (re exivity)

B. (B ! C ) ! ((A ! B ) ! (A ! C )) (transitivity)

C. (A ! (B ! C )) ! (B ! (A ! C )) (p ermutation)

R (M-strict): I., B., C., and either of:

S. (A ! (B ! C )) ! ((A ! B ) ! (A ! C ))

W. (A ! (A ! B )) ! (A ! B ) (contraction) 9

RM I (S-strict): Replace axiom I: of R by:

! !

Mingle. A ! (A ! A)

H (intuitionistic): Replace axiom I: of R by:

! !

K. A ! (B ! A) (weakening)

or just take K. and S. as axioms.

The pro ofs that these systems really are the minimal systems required are

all similar and not very dicult. For the case of R and RM I they es-

! !

sentially can b e found in [AB] or [Du]. The deduction-theorem parts

are provable by induction on the length of the various types of pro of. The

minimality- by providing a pro of of the needed type of B from A ;:::;A

1 n

whenever an axiom has the form A ! (A ! ::: ! (A ! B ) :::). (Some

1 2 n

of these pro ofs were given in the examples ab ove).

Note. The names I ; B ; C ; S; W and K are taken from combinatory logic. It

is well known that H corresp onds to the typed -calculus (which in turn,

can b e de ned in terms of the combinators K and S ) while R corresp onds

to the typed I -calculus. HL may b e describ ed as corresp onding to \Lin-

ear -calculus", based on the combinators I;B and C . It is not dicult also

to directly translate the notion of a \Linear pro of " into a corresp onding

notion of a \Linear -term".

Once we have the system HL at our disp osal we can pro duce a Hilb ert-

type formulation of the intensional fragment of Linear logic exactly as de-

scrib ed ab ove. All we have to do is to add to HL the axioms:

N1 (A ! B ) ! (B ! A)

N2 A ! A

1 A ! (B ! A B )

2 (A B ! C ) ! (A ! (B ! C )

t1 t

t2 t ! (A ! A)

We call the system which corresp onds to the f!; ; ; +g fragment of Linear

logic HL . It can b e axiomatized by adding N1 and N2 to HL . 1 and

m !

2 are derivable in the resulting system if we de ne A B as (A ! B ).

Alternatively we can (conservatively) add [1] and [2] to HL and prove

that A B is equivalent to (A ! B ) in the resulting system. As for +

|it is de nable in these systems as A ! B , but it is dicult to treat

it indep endently of in the N.D. and Hilb ert-typ e contexts. (By this we

The notions of S-strict and M-strict pro ofs were not explicitly formulated there,

though, but I b elieve that they provide the b est interpretation of what is done there. 10

mean that it is dicult to characterize it by axioms and rules in which only

+ and ! o ccurre). On the other hand t is not de nable in HL , but t1

and t2 can conservatively b e added to HL to pro duce HL . (The other

intensional constant, f , is of course equivalent to t). Once we have HL

and HL , it is easy to prove that A is a theorem of either i it is derivable

in the corresp onding N :D : system. Moreover, using the characterizations

given to ` and ` in the previous section it is straightforward to prove:

Theorem:

1. Let b e a multiset of formulas in the language of HL (HL ). Then

LL t

` A i there is a linear pro of of A from in HL (HL ), (in

which is exactly the multiset of assumptions).

2. Let b e a set of formulas in the language of HL (HL ). Then

` A i there is a classical pro of of A from in HL (HL ) in

LL m

which is the set of assumptions used. (This is equivalent to saying

that ` A in the usual sense).

Note: The last theorem provides alternative characterizations for the exter-

nal and internal consequence relations which corresp ond to the intensional

(mulplicative) fragment of Linear Logic. While those which were given in 2.1

are rather general, the one given here for the internal consequence relation is

p eculiar to linear logic. As a matter of fact, one can de ne a corresp onding

notion of a linear consequence relation for every Hilb ert-typ e system. What

is remarkable here is that for the present fragment the linear and the inter-

nal consequence relations are identical. (The internal consequence relation

was de ned in 2.1 relative to gentzen-type systems. It can indep endently b e

de ned also for Hilb ert-typ e systems which have an appropriate implication

connective).

and R ) are The intensional fragments of the Relevance logic R (R

exactly as the corresp onding fragments of Linear logic are obtained from R

obtained from HL . The corresp onding (cut free) Gentzen-type formula-

tions are obtained from those for Linear logic by adding the contraction rule

(on b oth sides). Al l the facts that we have stated ab out the Linear systems

are true (and were essentially known long ago) also for these fragments of

R, provided we substitute \M-strict" for \Linear". Similarly, if we add to

. RM I the axioms N1-N2 (and if desired also 1 and 2) we get RM I

This system corresp onds to the Gentzen-type system in which also the con-

verse of contraction is allowed, so the two sides of a sequent can b e taken

as sets of formulas. However, exactly as the addition of N1 and N2 to H

is not a conservative extension of is not a conservative extension, so RM I

is not a conserva- RM I . Moreover, the addition of t1 and t2 to RM I

is signi cantly stronger than tive extension of the latter either, so RM I

. (For more details see, e.g., [Av3]). RM I

! 11

2.4 The extensional fragment

The metho d of the previous section works nicely for the intensional (multi-

plicative) fragment of Linear Logic. It cannot b e applied as it is to the other

fragments, though. The problem is well known to relevant logicians and is

b est exempli ed by the extensional (additive) conjunction. If we follow the

pro cedure of the previous section we should add to HL the following three

axioms for ^:

^ Elim1: A ^ B ! A

^ Elim2: A ^ B ! B

^ Int: A ! (B ! A ^ B )

Once we do this, however, K b ecomes provable and we get the full e ect of

weakening.

The source of this problem is of course the impurity of the intro duction

rule for ^ in the N.D. system for Linear Logic. The side condition there

is not re ected in ^ Int (and in fact, this axiom is not derivable in Linear

logic). The situation here resembles that concerning the intro duction rule

for the 2 in Prawitz system for S4 (see [Pra]). The relevantists standard

solution is really very similar to the treatment of 2 in mo dal logic: Instead

of the axiom ^I nt they rst intro duce a new rule of proof (b esides M.P.):

Adjunction (adj):

A B

A ^ B

This rule suces for simulating an application of ^I nt (in a N.D. pro of )

in which the common multiset of assumptions on which A and B dep ends

is empty. In order to simulate other cases as well it should b e p ossible to

derive a pro of in the Hilb ert system of A ! (::: ! (A ! B ^ C ) :::)

1 n

from pro ofs of A ! (::: ! (A ! B ) :::) and A ! (::: ! (A ! C ).

1 n 1 n

The last two formulas are equivalent to A ! B and A ! C resp ectively ,

where A = A A ::: A . With the help of the adjunction rule it suces

1 2 n

therefore to add the following axiom (which is a theorem of Linear logic):

^ Int': (A ! B ) ^ (A ! C ) ! (A ! B ^ C ):

Once we incorp orate ^ we can intro duce _ either as a de ned connective or

by some analogous axioms (see b elow). The extensional constants T and 0

can then easily b e intro duced as well.

The ab ove pro cedure provides a Hib ert-typ e system HL with has exactly

the same theorems as the corresp onding fragment of Linear Logic. More-

over, it is not dicult to prove also the following stronger result:

Theorem: T ` A (in the ordinary, classical sense) i T ` A.

HL LL

The various prop ositional constants are optional for this theorem. The use of t in every

particular case can b e replaced by the use of a theorem of the form: (A ! A )^:::^(A !

1 1 n

A ).

n 12

If we try to characterize also the internal consequence relation in terms

of HL we run into a new diculty: The natural extension of the notion of

a \Linear pro of " (a notion which works so nicely and was so natural in the

intensional case!) fails to apply (as it is) when the extensionals are added.

Although it is p ossible to extend it in a less intuitive way, it is easier to

directly characterize a new \Linear consequence relation". For this we just

need to add one clause to the characterization which was given ab ove for

the previous case:

If ` A and ` B then ` A ^ B .

Denote by ` the resulting consequence relation. It is easy to prove that

the deduction theorem for ! obtains relative to it and that it is in fact

. equivalent to `

An example: (A ! B ) ^ (A ! C ) ! A ! (B ^ C ) is a theorem of

linear logic. Hence by the linear deduction theorem we should have:

(A ! B ) ^ (A ! C );A ` B ^ C:

Below there is a pro of of this fact. It is imp ortant to realize that this is not

a linear pro of according to the simple-minded concept of linearity which we

use for HL and HL , but it is a \linear pro of " in the sense de ned by

! m

the ab ove \linear consequence relation" of HL.

1. (A ! B ) ^ (A ! C ) (ass.)

2. A (ass.)

3. (A ! B ) ^ (A ! C ) ! (A ! B ) (axiom)

4. A ! B (1,3 M.p.)

5. (A ! B ) ^ (A ! C ) ! (A ! C ) (axiom)

6. A ! C (1,5 M.P.)

7. B (2,4 M.P.)

8. C (2,6 M.P.)

9. B ^ C (7,8 Adj.)

For the reader's convenience we display now:

The full system HL.

Axioms:

1. A ! A 13

2. (A ! B ) ! ((B ! C ) ! (A ! C ))

3. (A ! (B ! C )) ! (B ! (A ! C ))

4. A ! A

5. (A ! B ) ! (B ! A)

6. A ! (B ! A B )

7. A ! (B ! C ) ! (A B ! C )

8. t

9. t ! (A ! A)

10. A ! ( A ! f )

11. f

12. (A + B ) ! ( A ! B )

13. ( A ! B ) ! (A + B )

14. A ^ B ! A

15. A ^ B ! B

16. (A ! B ) ^ (A ! C ) ! (A ! B ^ C )

17. A ! A _ B

18. B ! A _ C

19. (A ! C ) ^ (B ! C ) ! (A _ B ! C )

20. A ! T

! A 21. 0

Rules:

M.P.:

A A ! B

Adj:

A B

A ^ B

Notes: 14

1. HL was constructed here by imitating the way the principal relevance

logic R is presented in the relevantists work (see, e.g., [Du]). The

relation b etween these two systems can b e summerized as follow:

Linear logic + contraction = R without distribution.

2. Exactly as in rst-order R, the Hilb ert-typ e presentation of rst-

order linear logic is obtained from the prop ositional system by adding

Kleene's standard two axioms and two rules for the quanti ers (See

[Kl]).

3 Semantics

We start by reviewing some basic notions concerning algebraic semantics of

prop ositional logics:

for a prop ositional logic L consists De nition: An Algebraic Structure D

of:

1. A set D of values.

2. A subset T of D of the \designated values".

3. For each connective of L a corresp onding op eration on D .

De nition: Let AS b e a set of algebraic structures for a logic L. De ne:

Weak completeness(of L relative to AS): This means that ` A i v (A) 2 T

L D

for every valuation v in any D 2AS.

(Finite) Strong Completeness: (of L relative to AS): This means that for

every ( nite) theory T and every sentence A, T ` A i v (A) 2 T for

L D

every D2AS and every valuation v such that fv (B )jB 2 T g T .

Internal weak completeness: Supp ose that instead of (or in addition to) T

each D 2AS is equipp ed with a partial order on D . Then L is internally

weakly complete (relative to this family of structures) if:

. A ` B i v (A) v (B ) for every v and D

L D

Notes:

11 l

do es) then If ` has all the needed internal connectives (as `

every sequent ` is equivalent to one of the form A ` B .

It is imp ortant to note that the various notions of completeness dep end

on ` , the consequence relation which we take as corresp onding to L.

The most known systems of relevance logic include as an axiom the distributi on of

^ over _. As a result they are undecidabl e (see [Ur]) and lack cut-free Gentzen-type

formulation.

See [Av2] for the meaning of this. 15

We next intro duce the basic algebraic structures which corresp ond to

Linear Logic:

=< D ; De nition: Basic relevant disjunction structures are structures D

; ; + > s.t.:

1. < D ; > is a p oset.

2. is an involution on < D ; >.

3. + is an asso ciative, commutative and order-preserving op eration on

< D ; >.

De nition: Basic relevant disjunction structures with truth subset are

;T > s.t.: structures < D

1. D is a basic relevant disjunction structure.

2. T D .

3. a 2 T ; a b ) b 2 T .

D D

4. a b i a + b 2 T .

Let HL b e the pure intentional (or \multiplicative") fragment of HL

(; !; +; ). The corresp ondence b etween HL and the ab ove structures

is given by the following:

Strong completeness theorem: HL is strongly complete relative to ba-

sic relevant disjunction structures with truth subset (where the designated

values are the elements of the truth subset).

This is true for the external (pure) C.R.. For the internal one we have

the following easy corollary:

Weak internal completeness theorem: A ` B i v (A) v (B ) for

every valuation v in the ab ove structures. Hence the linear consequence

relation is internally complete relative to the ab ove structures.

Outline of the pro of of strong completeness: For the less easy part,

de ne the Lindenbaum algebra of a HL -theory T as follows: Let A B i

T ` A ! B and T ` B ! A. This is a congruence relation. De-

HL HL

m m

note by [A] the equivalence class of A. Let D b e the set of equivalence classes.

De ne: [A] [B ] i T ` A ! B , [A] = [ A]; [A] + [B ] = [A + B ].

This means that for all a; b: a = a; a b ) b a.

These structures were rst introduce in [Av3].

If we consider only the implicati onal fragment then HL is a minimal logic for which

this is the case! 16

These are all well de ned. The resulting structure is a basic relevant dis-

junction structure with a truth subset: T = f[A]jT ` Ag. By de ning

D HL

v (A) = [A] we get a valuation for which exactly the theorems of T get des-

ignated values.

The following prop osition provides an alternative characterization of the

ab ove structures:

Prop osition: Let < D; ; ; + > b e a basic relevance structure . The

following are necessary and sucient conditions for the existence of a truth-

subset of D :

D.S. a b + c ) b a + c

R.A. a+ ( b + b) a

Moreover, if a truth-subset exists it is uniquely de ned by:

T = faj a + a ag:

(R.A) is not a convenient condition from an algebraic p oint of view. For-

tunately we have:

Prop osition: Each of the following two conditions implies (R.A) in ev-

ery basic relevant disjunction structure in which D.S. is satis ed:

1. Idemp otency of + : 8a a + a = a.

2. Existence of an identity element f for + (8a a + f = a).

In the second case we have: T = faja tg where t = f .

Implicitly, Girard has chosen the second possibility. So did b efore him

most of the relevantists. Accordingly we de ne:

De nition: Relevant disjunction monoids are relevant disjunction struc-

tures which satisfy (D.S.) and in which + has an identity element.

It is easy now to formulate and prove completeness theorems as ab ove

for the full intensional fragment of Linear Logic (including the prop ositional

constants) relative to relevant disjunction monoids. Since this fragment is

a strongly conservative extension of that treated ab ove, these completeness

results will hold also for the more restricted fragment. It is worth noting

also that in relevant disjunction monoids condition (D.S.) is equivalent to:

a b if f t a + b

This prop osition as well as the next one are again taken from [Av3].

Compare Dunn`s work on the algebraic semantics of R and other relevance systems.

For more information and references|see [Du] or [AB]. 17

In order to get a similar characterization for the full prop ositional frag-

ment of Linear logic we have to deal with Lattices rather than just p osets.

The op erations of g.l.b and l.u.b provide then an obvious interpretation for

the extensional (\additive") connectives _ and ^. All other de nitions and

conditions remain the same. (This is a standard pro cedure in a semanti-

cal research on relevance logics). Completeness theorems analogous to those

presented ab ove can then b e formulated and similarly proved. (If we wish to

incorp orate also Girard's > and 0 then the lattices should include maximal

and minimal elements.)

Again, the standard way of characterizing linear predicate calculus is to

work with complete rather than ordinary lattices. We can then de ne :

v (8x'(x)) = inf fv ('(a))ja 2 g

v (9x'(x)) = sup fv ('(a))ja 2 g

(Where is the domain of quanti cation).

From now on it will b e more convenient to take instead of + as primitive

and to reformulate the varios de nitions accordingly. (The two op erations

are de nable from one another by DeMorgan's connections). Our last obser-

vation leads us accordingly to consider the following structures (which will

b e shown to b e equivalent to Girard's \phase spaces"):

Girard structures: These are structures < D ; ; ; > s.t:

1. < D ; > is a complete lattice.

2. is an involution on < D ; >.

3. is a commutative, asso ciative, order-preserving op eration on D with

an identity element t.

4. a b i a b f (f = t).

Note: If we demand: a a a we get Dunn`s algebraic semantic for QR.

=< D; ; ; ; t > b e a basic relevant The embedding theorem: Let D

disjunction structure with identity t such that faja tg is a truth-subset.

(The last condition is equivalent here to (D.S.) or to (4.) ab ove). Then D

can b e embedded in a Girard structure so that existing in ma and suprema

of subsets of D are preserved.

Corollaries:

1. The various prop ositional fragments of HL are strongly complete for

Girard structures.

2. The various fragments of Linear Predicate calculus are strongly con-

servative extensions of each other.(e,g: in case A and all sentences of

T are in the language of HL then T ` A i T ` A).

m HL HL

m 18

3. Linear predicate calculus is strongly complete relative to Girard struc-

tures. (The embedding theorem is essential in this case since the direct

construction of the Lindenbaum Algebra contains all the necessary inf

and sup, but is not yet complete!)

The pro of of the embedding theorem, as well as the pro of of the equiv-

alence of the notions of \Girard structures" and the \phase semantics" of

Girard, dep end on some general principles from the theory of complete lat-

tices (see the relevant chapter in Birkho `s b o ok: \Lattice theory"):

De nition: Let I b e a set. C : P (I ) ! P (I ) is a closure operation on

P (I ) if:

1. X C (X )

2. C (C (X )) C (X )

3. X Y ) C (X ) C (Y )

X is called closed if C (X ) = X . (Obviously, X is closed i X = C (Y ) for

some Y).

Theorem: For I;C as ab ove the set of closed subsets of I is a complete

lattice under the order. Moreover, we have:

inf fA g = A

j j

[

supfA g = C ( A )

j j

Standard embeddings: Let C b e a closure op eration on I such that

C (fxg 6= C (fy g) whenever x 6= y . Then x ! C (fxg) is the standard em-

b edding of I in the resulting complete lattice of closed subsets.

In case I is a structure with, say, an op eration we usually extend it to

this complete Lattice by de ning

X Y = C (XY )

where XY = fx y jx 2 X ; y 2 Y g

The most usual way of obtaining closure op erations is given in the following:

De nition: (\Galois connections"): Let R b e a binary relation on I , X I .

De ne:

X = fy 2 I j8x 2 X xRy g

X = fx 2 I j8y 2 X xRy g

Lemma: 19

+ +

1. X Y ) Y X ;Y X

+ + + ++ +

2. X X ;Y Y ;X = X ;Y = Y

3. + and + are closure op erations on P (I ).

An example: Take R to b e , where < I ; > is a p oset, and take C to b e

+. We get (for X I; x 2 I ):

X =set of upp er b ounds of X

X =set of lower b ounds of X

C (fxg) = fxg = fy jy xg.

It is easy to check then that x ! fxg is an embedding of (I; ) in the

complete lattice of closed subsets of I . This embedding preserves all existing

suprema and in ma of subsets of I . (I is dense, in fact, in this lattice.)

Moreover, if is an involution on < I ; > then by de ning:

X = f y j y 2 X g

we get an involution on this complete lattice which is an extension of the

original involution.

Pro of of the embedding theorem: It is straightforward to check that

the combination of the constructions describ ed in the last example with the

standard way to extend which was describ ed ab ove (applied to relevant

disjunction structures with identity) suces for the embedding theorem.

Another use of the metho d of Galois connections is the following:

Girard`s construction (The \phase semantics"): We start with a

triple < P ; ; ?>, where P is a set (the set of \phases",) ? P and is an

asso ciative, commutative op eration on P with an identity element. De ne:

xRy x y 2?

then

+ ?

X = X = fy j8x 2 X x y 2?g = X :

It follows that X ! X is a closure op eration. The closed subsets (X =

X ) are called by Girard \Facts". By the general result cited ab ove they

form a complete lattice. De ne on it:

X = X

X Y = (XY )

what Girard does in ch.1 of [Gi] is, essentially, to prove the following:

Characterization theorem: The construction ab ove provides a Girard`s

structure. Conversely, any Girard structure is isomorphic to a structure 20

constructed as ab ove (since by starting with a given Girard`s structure, and

taking ? to b e fxjx f g, the ab ove construction returns an isomorphic

Girard`s structure).

Corollary: HL is strongly complete relative to the \phase semantics" of

Girard.

Notes:

1. In the original pap er Girard is proving only weak completeness of

Linear logic relative to this \phase semantics".

2. It can b e checked that Girard's construction works even if originally we

do not have an identity element, provided we use a subset ? with the

prop erty ? = ? (the existence of an identity element guarantees

this for every p otential ?).

Summary: Points in Girard structures and \facts" in \phase spaces" are

two equivalent notions.

4 The mo dal op erators

4.1 Pro of theory

In this section we examine how the mo dal op erators of Girard can b e treated

in the ab ove framework. It is enough to consider only the 2, since the

other op erator is de nable in terms of 2 and . Starting with the pro of-

theoretic part, we list in the following the rules and axioms that should b e

added to the various formal systems. It is not dicult then to extend the

former pro of-theoretic equivalence results to the resulting systems. (Again

the Gentzen-typ e rules were given already in [Gi] in a di erent form).

Gentzen-type rules:

` 2A; 2A; `

2A; ` 2A; `

2A ;:::; 2A ` B A; `

1 n

2A; ` 2A ;:::; 2A ` 2B

1 n

Here the second pair of rules is exactly as in the standard Gentzen-type

presentation of S4, while the rst pair allows the left-hand structural

rules to b e applied to b oxed formulas.

N.D. rules: Essentially, we need to add here Prawitz' two rules for S4:

A 2A

()

A 2A

(where, as in [Pra], for the 2-intro duction rule we have the side con-

dition that all formulas on which A dep ends are b oxed). In addition, 21

the rules for the other connectives should b e classically re-interpreted

(or formulated) as far as b oxed formulas are concerned. Thus, e.g,

the multisets of assumptions on which A and B dep ends in the ^-

intro duction rule should b e the same only up to boxed formulas, while

for negation and implication we have:

[2A] [2A] [2A]

B B B

2A ! B 2A

Where the interpretation of these rules are in this case exactly as in

classical logic. (It suces, in fact, to add, b esides the basic 2-rules,

only the b oxed version of ! I nt to make all other additions derivable).

HL |the Hilb ert-type system: we add to HL:

K 2: B ! (2A ! B )

W 2: (2A ! (2A ! B )) ! (2A ! B )

2K: 2(A ! B ) ! (2A ! 2B )

2T 2A ! A

24: 2A ! 22A

N ec:

Again we see that the added axioms and rules for the Hilb ert-typ e system

naturally divided into two: the rst two axioms are instances of the schemes

that one needs to add to the implicational fragment of Linear Logic in order

to get the corresp onding fragment of the intuitionistic calculus. The other

axioms and rules are exactly what one adds to classical prop ositional calcu-

lus in order to get the mo dal S4. The main prop erty of the resulting system

is given in the following:

Mo dal deduction theorem: For every theory T and formulas A; B we

have:

2 2

T;A ` B i T ` 2A ! B

HL HL

The same is true for all the systems which are obtained from the various

fragments studied ab ove by the addition of the ab ove axioms and rule for

the 2.

The pro of of this theorem is by a standard induction. It is also easily seen

that except for W 2 the provability of the other axioms and the derivability

of 2A from A are all consequences of the mo dal deduction theorem.

Notes:

1. It is imp ortant again to emphasize that the last theorem is true for

the external consequence relation. 22

2. The deduction theorem for HL is identical to the deduction theorem

for the pure consequence relation de ned by S4 (in which N ec is taken

as a rule of derivation, not only as a rule of pro of ). S4 may b e

characterized as the minimal modal system for which this deduction

theorem obtains.

4.2 Semantics

From an algebraic p oint of view the most natural way to extend a Girard

structure (and also the other structures which were considered in the pre-

vious section) in order to get a semantics for the mo dal op erators is to add

to the structure an op eration B , corresp onding to the 2, with the needed

prop erties. Accordingly we de ne:

De nition: A modal Girard structure is a Girard structure equipp ed with

an op eration B having the following prop erties:

1. B (t) = t

2. B (x) x

3. B (B (x)) = B (x)

4. B (x) B (y ) = B (x ^ y )

Lemma: In every mo dal Girard structure we have:

1. B (a) t

2. B (a) b ! b

3. B (a) b b

4. B (a) B (a) = B (a)

5. a b ) B (a) B (b)

6. a t ) B (a) = t

7. If B (a) b then B (a) B (b)

8. If a a ::: a b then B (a ) B (a ) ::: B (a ) B (b).

1 2 n 1 2 n

Pro of

1. B (a) = B (a) t = B (a) B (t) = B (a ^ t) a ^ t t

2. Immediate from 1.

This consequence relation corresp onds to validity in Kripke mo dels. See [Av2].

This mo dal deduction theorem was indep endently used by the author as the main to ol

for implementing S4 in the Edinburgh LF|see [AM]. 23

3. Equivalent to 2.

4. B (a) B (a) = B (a ^ a) = B (a).

5. a b ) a = a ^ b and so:

a b ) B (a) = B (a) B (b) B (b) (by 3.)

6. Immediate from 5. and 1.

7. B (a) b ) B (a) = B (B (a)) B (b) (by 5.)

8. since B (a) a, we have that B (a ) B (a ) ::: B (a ) b whenever

1 2 n

a a ::: a b. But B (a ) B (a ) :::B (a ) = B (a ^ ::: ^ a ).

1 2 n 1 2 n 1 n

Hence 8. Follows from 7.

Theorem: HL is sound and strongly complete relative to mo dal Girard

structures.

Pro of: The soundness follows immediately from the previous lemma. The

pro of of completeness is similar to the previous ones. The only extra step

needed is to show that we can extend the op eration B (de ned from the 2)

of the Lindenbaum Algebra LA to the completion of this algebra. This is

done by de ning:

0 0 0

B (x ) = supfB (x)jx x jx 2 LAg

Using the fact that in Girard structures distributes over sup, it is not

dicult to show that B is an op eration as required. (The fact is needed for

0 0 0

establishing that B (a) B (b) = B (a ^ b). It was used also in [Gi] for quite

similar purp oses).

Corollary: HL is a strongly conservative extension of HL.

Pro of: Let A b e a sentence in the Language of HL, and let T b e a the-

ory in this Language. Supp ose T 6` A. We show that T 6` A. The

HL HL

completeness theorem for HL provides us with a Girard structures and a val-

uation in it for which all the theorems of T are true and A is false. To show

A it is enough therefore to turn this Girard structure into a that T 6`

mo dal one. That this can always b e done is the content of the next theorem.

Theorem: In every Girard structure we can de ne at least one op erator B

which makes it mo dal.

Pro of: De ne: B (a) = supfx a ^ tjx x = xg. The theorem is a

consequence of the following facts:

1. B (a) a

2. B (a) t 24

3. B (t) = t

4. B (a) B (b) B (a)

5. B (a) B (b) a ^ b ^ t

6. B (a) B (b) B (a ^ b)

7. B (a) B (a) = B (a)

8. B (B (a)) = B (a)

9. B (a) B (b) B (a ^ b)

10. B (a) B (b) = B (a ^ b)

(1)-(3) are immediate from the de nition of B . It follows that B (a) B (b)

B (a) t = B (a). Hence (4). (5) follows then from (1) and (2). (7) is

immediate from (6) and (4), while (8) follows from (7),(2) and the de nition

of B (B (a)). (10) is just a combination of (6) an (9). It remains therefore to

prove (6) and (9).

Proof of (6): supp ose that z (a ^ b) ^ t and z z = z . Then z B (a)

and z B (b) by de nition of B (a);B (b). Hence z = z z B (a) B (b).

This is true for every such z and so B (a ^ b) B (a) B (b) by de nition of

B (a ^ b).

Proof of (9): By (7) (B (a) B (b)) (B (a) B (b)) = B (a) B (b): This,

(5) and the de nition of B (a ^ b) imply (9).

The last construction is a sp ecial case of a more general construction.

This general construction is strong enough for obtaining any p ossible mo dal

op eration, and is, in fact, what Girard is using in [Gi] for providing a seman-

tics for the mo dal op erators. The construction is describ ed in the following

theorem, the pro of of which we leave to the reader:

Theorem: Supp ose G is a Girard structure and supp ose that F is a subset

of G which has the following prop erties:

1. F is closed under arbitrary sup.

2. F is closed under .

3. x x = x for every x 2 F .

4. t is the maximal element of F .

Then B (a) = supfx 2 F jx ag is a mo dal op eration on G.

D f

Conversely, if B is a mo dal op eration on G then the set F = fx 2 Gjx =

B (x)g has the ab ove prop erties and for every a: B (a) = supfx 2 F jx ag.

In [Gi] Girard (essentially) de nes top olinear spaces to b e Girard struc-

ture together with a subset F having the ab ove prop erties. Only he has 25

formulated this in terms of the \phase" semantics. His de nition of the

interior of a fact is an exact equivalent of the de nition of B which was

given in the last theorem. The subset F corresp onds (in fact, is identical)

to the collection of op en facts of his top olinear spaces. (Note, nally, that

in every Girard structure the subset fx 2 Gjx = x x and x tg has the

4 prop erties describ ed ab ove. By using it we get the construction describ ed

in the previous theorem).

5 References

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University Press, Princeton,N.J., 1975.

[Av84] Avron A., Relevant Entailment - Semantics and formal systems,

Journal of Symb olic Logic, vol. 49 (1984), pp. 334-342.

[Av90a] Avron A., Relevance and Paraconsistency - A New Approach.,

Journal of Symb olic Logic, vol. 55 (1990), pp. 707-732.

[Av91a] Avron A., Simple Consequence relations, Information and Com-

putation, vol 92 (1991), pp. 105-139.

[AHMP] A. Avron., FA. Honsell, I.A. Mason and R. Pollack

Using Typed Lambda Calculus to Implement Formal Systems on a Ma-

chine Journal of Automated Deduction, vol 9 (1992) pp. 309-354.

[Bi48] Birkho G. Lattice theory, Providence (A.M.S) (1948)

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sophical Logic,Vol I I I, ed. by D. Gabbay and F. Guenthner, Reidel:

Dordrecht, Holland; Boston: U.S.A. (1986).

[Ge69] Gentzen G. Investigations into logical deduction, in: The collected

work of Gerhard Gentzen, edited by M.E. Szab o, North-Holland,

Amsterdam, (1969).

[Gi87] Girard J.Y., Linear Logic, Theoretical Computer Science, vol. 50

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[Ho83] Ho dges W. Elementary predicate calculus, in: Handb o ok of Philo-

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Dordrecht, Holland; Boston: U.S.A. (1983).

[Kl52] Kleene S.C. Intro duction to metamathematics, New York (D.

van Nostrad Co.), (1952).

[Pr65] Prawitz D. Natural Deduction, Almqvist&Wiksell,Sto ckholm (1965).

[Sw77] Schwichtenberg H. Proof-theory: some applications of cut-elimination,

in: Handb o ok of Mathematical Logic, ed. by J. Barwise, North-

Holland (1977). 26

[Ur84b] Urquhart A. The Undecidability of entailment and relevant impli-

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