an algebraic analysis of normative systems

An Algebraic Analysis ofNormative Systems

LARS LINDAHL AND JAN ODELSTAD

Abstract. In the present paper we study how subsystems of a normative system canbe combined, and the role of such combinations for the understanding of hypo-thetical legal consequences. A combination of two subsystems is often accomplishedby a normative correlation or an intermediate concept. To obtain a detailed analysisof such phenomena we use an algebraic framework. Normative systems arerepresented as algebraic structures over sets of conditions. This representation makesit possible to study normative systems using an extension of the theory of Booleanalgebras, called the theory of Boolean quasi-orderings.

I. The Algebraic Representation of a Normative System

1. Introduction

By a normative system jurists often mean the law of a country, like Swedishlaw, or part of the law of a country, such as the Swedish law of contracts.Legal philosophers and logicians have raised the issue of how a logicalreconstruction of a normative system should be made. In this connection acentral question is what kind of entities a normative system is composed ofand how it should be represented.

In their well-known book Normative Systems, Carlos E. AlchourroÂn andEugenio Bulygin conceive of a reconstructed normative system as a set ofsentences deductively correlating pairs of sentences (AlchourroÂn andBulygin 1971, 54f.). According to them, a set a of sentences deductivelycorrelates a pair hx,yi of sentences if y is a deductive consequence of the setobtained by adding x to a, or, in symbols, if y[Cn({x}[a). For a to be anormative system the additional requirement is made that there is at leastone pair hx,yiwhere y[Cn({x}[a) such that x is a `̀ case'' and y is a `̀ solution.''(A solution is a normative sentence expressed in terms of deontic operatorsfor command, prohibition or permission.) As observed by AlchourroÂn andBulygin, the statement y[Cn({x}[a) is equivalent to (x�y)[Cn(a) where � isthe symbol for truth-functional implication.

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Ratio Juris. Vol. 13 No. 3 September 2000 (261±78)

The approach in the present paper is similar to that of AlchourroÂn andBulygin in important respects. We study normative systems essentially asdeductive mechanisms yielding outputs for inputs. Also, we are interestedin a rational reconstruction of normative systems, where the result is morelogically elaborated than the original version. Finally, a central problem isthat of finding a suitable formal framework for representing the logicallyelaborated version.

Our framework in the paper is explicitly algebraic. Fundamental com-ponents of the framework are conditions and binary relations applied tothem. Let S be a normative system and let a, b be two conditions. Thesubject-matter of our inquiry is a relation R such that aRb represents that itfollows from S that a implies b. For example, according to Swedish law, nothaving a medical degree implies not being entitled to get a license as aphysician. Saying this is tantamount to saying that a particular relation Rholds between two conditions a and b. Or, consider the statement:`̀ According to Swedish law, if x is a child of y, then x is entitled to inherity.'' If a is the binary condition `̀ to be a child of,'' and b is the binary condition`̀ to be entitled to inherit,'' we represent the original statement by aRb (orha,bi[R).1

In order to illustrate the relationship between our approach and that ofAlchourroÂn and Bulygin, let us define a relation R8 by: xR8y if and only ify[Cn({x}[a). Such a relation will have many similarities to our relation R.2

2. Conditions

One of the central notions in our analysis of normative systems is condition.Conditions can be of different arity (unary, binary, etc.). Let us use a,b,c,...,a1,b1,c1,..., a2,b2,c2,... for referring to conditions. Examples of conditions are: tobe a woman, to be a parent of, to be a citizen of.

Sometimes we represent a condition by an expression a(x1,...,xn). Forexample, the condition `̀ to be older than'' is represented by `̀ x is older thany.'') We note that, where in this way, a condition is represented by anexpression a(x1,...,xn), it is presupposed that x1,...,xn are free variables whichfunction as place-holders, and that a(x1,...,xn) is a sentence-form.

If a and b are n-ary conditions, we form compound n-ary conditions asfollows.

1 Conditions can be denoted either by expressions, using the sign of the infinitive, such as `̀ to bea woman,'' `̀ to be a child of,'' `̀ to be entitled to inherit,'' or by expressions in the ing-form, like`̀ being a woman,'' `̀ being a child of,'' `̀ being entitled to inherit.''2 In many cases, the sentence xR8y, as defined above, will express what AlchourroÂn and Bulygincall a `̀ normative proposition.'' See AlchourroÂn and Bulygin 1971, 121. The distinction betweennorms and normative propositions leads into questions treated by Carlos AlchourroÂn andEugenio Bulygin in various works, in particular AlchourroÂn 1969, 1972; and AlchourroÂn andBulygin 1984. Cf. also Lindahl 1997. In the present paper, this area of problems will not bein focus.

262 Lars Lindahl and Jan Odelstad

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a' is the condition defined by a'(x1,...,xn), if and only if not a(x1,...,xn),a^b is defined by (a^b)(x1,...,xn) if and only if a(x1,...,xn)-and-b(x1,...,xn),a_b is defined by (a_b)(x1,...,xn) if and only if a(x1,...,xn)-or-b(x1,...,xn).

Thus, we assume that ' is the operation for forming negations of conditions,^ is the operation for forming conjunctions, and _ the operation for formingdisjunctions.3 A condition a is simple if it is not compound.

Note that the arity of a' is always the same as the arity of a. For example,since being a woman is a unary condition, not being a woman is unary aswell. For conjunction the situation is more complicated as the followingexample shows. If a is the condition `̀ to be a woman'' and b is the condition`̀ to be a parent of,'' then a^b is the condition of being a mother of. However,the arity of a and b is not the same since a is unary and b is binary. In formingconjunctions we therefore adopt the rule that the arity of a^b equals thegreatest of the arities of a and b.4

The procedure of forming compounds can be iterated. For example, if a^bis the condition `̀ to be the mother of,'' and c is the condition `̀ to be anadoptive parent of,'' then (a^b)_c is the condition of being mother oradoptive parent of.

The n-ary empty condition is the condition F such that for no x1,...,xn,F(x1,...,xn). The n-ary universal condition is the condition T such that for allx1,...,xn, T(x1,...,xn).

Conditions have many affinities with relations, if, as is usual, relations areregarded extensionally as sets of ordered n-tuples. Obviously, theoperations of negation, conjunction and disjunction for conditions have ascounterparts the operations of complement, intersection and union forrelations. However, if R1 and R2 are relations of different arity, theirintersection R1\R2 is empty and their union is not a relation. For example,the intersection between a set of pairs and a set of triples is empty, and theunion of a set of pairs and a set of triples is not a relation. As appears fromthe foregoing, the case is different with conditions.

Perhaps it is appropriate to say that a theory of conditions in the way it isdeveloped here is a modified theory of relations. In such a modified theory,sameness of arity is not presupposed for intersection and union. Thetreatment can be made algebraic, and relations between conditions can beintroduced. In what follows, a small fragment of a modified theory ofrelations will be developed, in terms of conditions.5

3 More formally, the operation _ for disjunction is defined in terms of the operation ^ and '.Thus the expression a_b is defined by a_b=(a'^b')'.4 This means that if a is m-ary and b is n-ary and f = max{m,n}, then, for all x1,...,xf, (a^b)(x1,...,xf)if and only if a(x1,...,xm) and b(x1,...,xn).5 In choosing our approach of considering Boolean algebras of conditions, we have beeninspired by some lectures of Stig Kanger's, given in the Fall of 1977. In these lectures, Kangerstarted developing an algebraic theory of conditions, based on Boolean and cylindric algebras.

An Algebraic Analysis of Normative Systems 263

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If M={a,b,c,...} is a set of conditions, then M* is the set M closed under theoperations ^ and '. Formally, M* is defined as follows.

(1) If a [ M, then a [ M*.(2) If a [ M, then a' [ M*.(3) If a,b [ B, then a^b [ M*.(4) The only members of M* are those resulting from a finite number of

applications of (1), (2) and (3).

The identity relation = for conditions is such that hM*,^,'i is a Booleanalgebra, with = as its identity relation.6 That hM*,^,'i is a Boolean algebrameans that the following holds for any conditions a, b, c from M*:

a^b=bâ a_b=b_a Commutativitya^(b^c)=(a^b)^c a_(b_c)=(a_b)_c Associativitya^(a_b)=a a_(a^b)=a Absorptiona^(c_b)=(a^b)_(a_c) a_(b^c)=(a_b)^(a_c) Distributivitya^T=a a_F=aaâ'=F a_a'=T

Here F is the empty condition and T is the universal condition. Note thatin the Boolean algebra hM*,^,'i, F is the zero element and T is the unitelement. The arity of T and F is the maximal arity of any conditionin M.7

3. Implicational Condition Structures

As indicated above, in our representation of a normative system there is arelation of implication between conditions. As an example, consider thequestion of citizenship according to the system of the U.S. Constitution.Amendment XIV, section 1, reads as follows:

All persons born or naturalized in the United States, and subject to the jurisdictionthereof, are citizens of the United States and of the State wherein they reside. NoState shall make or enforce any law which shall abridge the privileges or immunitiesof citizens of the United States; nor shall any State deprive any person of life, liberty,or property, without due process of law; nor deny to any person within itsjurisdiction the equal protection of the laws.

6 This implies that for a,b[M*, a = b expresses that a and b are extensionally equal due to logicalone. If a is m-ary and b is n-ary and f = max{m,n}, a=b implies that, for all x1,...,xf, a(x1,...,xm) ifand only if b(x1,...,xn).7 Since, for any condition a,aâ'=F and a_a'=T, if the arity of F (and of T) is n, and the arity ofcondition a is m, with m4n, then, for all x1,...,xn,

aâ'(x1,...,xm) if and only if F(x1,...,xn),a_a'(x1,...,xm) if and only if T(x1,...,xn).



According to this system, a compound of the three conditions

b: to be a person born in the U.S.n: to be a person naturalised in the U.S.s: to be a person subject to the jurisdiction of the U.S.

implies the condition

c: to be a citizen of the U.S.

That the implicative relationship holds according to the system isrepresented in the form ((b_n)^s)Rc. We observe that, according to the samenormative system, since it is a settled matter that citizens who are minorshave not the right to vote in general elections, (b_n)^s does not imply thecondition

e: to be entitled to vote in general elections.

Therefore: not ((b_n)^s)Re.We now introduce a notion that will be central in the sequel. A structurehM*,^,',Ri, where M* and R are as stated in sections I.1-I.2, will be called animplicational condition structure. We say that such a structure hM*,^,',Ri repre-sents a normative system S if, for any a,b[M*, aRb if and only if from S itfollows that a implies b.

The important function of a normative system is to express what can becalled normative correlations (AlchourroÂn and Bulygin 1971, 55). If hM*,^,',Rirepresents the normative system S, we will say that aRb describes a norma-tive correlation for S if a is a descriptive and b is a normative condition.

It should be added that an implicational condition structure can beapplied to individuals by deductions of the following kind, where i, j areindividual constants, for instance names of individuals. Let hM*,^,',Ri be animplicational condition structure representing the normative system S. Thenwe will say that b(i,j) is deducible according to S from a(i,j) if it holds thataRb.

In the next part of the paper, we introduce a number of formal propertiesof implicational condition structures. The notions of Boolean algebra andquasi-ordering will play an important role. The framework introduced,however, is of a more general character and implicational conditionstructures are merely one among several kinds of models of the theory.

4. An Algebraic Framework

We begin this section by introducing what will be called Boolean quasi-orderings with domains of conditions.

DEFINITION. A structure hB, ^, ',Ri is a Boolean quasi-ordering if hB, ^, 'i isa Boolean algebra with T as the unit element and F as the zero element and



R is quasi-ordering on B (i.e., a reflexive and transitive relation on B), whichsatisfies the following conditions for all a, b and c in B:

(1) aRb and aRc implies aR(b^c).(2) aRb implies b'Ra'.(3) (a^b)Ra.(4) not TRF.

The indifference part of R is denoted Q and is defined by: aQb if and only ifaRb and bRa. Similarly, the strict part of R is denoted S and is defined by: aSbif and only if aRb and not bRa.

Let 4 be the partial ordering determined by hB, ^, 'i where 4 is definedby a4b if and only if a^b=a. We note that from (3) it follows that a4b impliesaRb. For, by the definition, a4b implies a^b=a and, since a^bRb it follows thataRb.

There are two Boolean algebras which should be kept apart. First, there isthe Boolean algebra (B, ^,') the domain B of which is the field of the relationR. After adding the relation R we get the Boolean quasi-ordering hB, ^, ',Ri.By switching to the equivalence classes defined by the indifference part Q ofR, we get another Boolean algebra, which is determined by the Booleanquasi-ordering hB, ^, ',Ri.8

Although, by a transition to equivalence classes, from a Boolean quasi-ordering we get a new Boolean algebra, in what follows we will not makethis transition. The point is that, for two conditions a, b, although it holdsthat aQb, and therefore a, b belong to the same Q-equivalence class, we maywant to distinguish a and b because they may have different meaning.Therefore, there is a point in remaining within the framework of Booleanquasi-orderings as defined above.

II. Fragments, Connections and Couplings

1. Informal Introduction

Traditionally, a legal system is divided into a number of different sub-systems, such as private law, criminal law, constitutional law, adminis-trative law, tax law, and so on. These subsystems, in turn, are divided intosmaller subsystems. Thus, for example, private law is divided into the lawof property, the law of contract, the law of tort.

8 That a second Boolean algebra is obtained can be seen in the following way. Let us use thenotation that,

(1) aR={b[B|bQa}.(2) BR={aR|a[B}.(3) 4R is the relation on BR defined by aR4RbR if and only if aRb.

If hB,^, ',Ri is a Boolean quasi-ordering, then hBR,4Ri is a Boolean algebra. We note that 4R is awell-defined partial ordering and that, since R is a quasi-ordering, Q is an equivalence relation.For a proof see Odelstad and Lindahl 1998, 140.



An important function of any normative system, and therefore of any ofthe systems just exemplified, is to express normative correlations (see abovesection I.3). However, normative systems, like the legal subsystemsdescribed above, also have, as their background, implicative sentences thatdo not express normative correlations in the sense suggested. As a back-ground of a normative system S, for example, there are sentences expressinglogical or analytical relationships. If hM*,^,',Ri represents S, the set ofimplicative background sentences can be divided in parts, these parts beingrepresented by a number of implicational condition structures

hM1*,^,',Ri,hM2*,^,',Ri,...,hMn*,^,',Ri.Some of these structures will contain no normative conditions, while otherscontain only normative conditions. The normative correlations of S then willbe represented by a joining of a structure hMj*,^,',Ri of normative conditionsto a structure hMi*,^,',Ri of descriptive conditions. In the present paper wewill use the notion of fragments (defined below) for the formal study of howstructures can be joined so as to yield a representation of a normativesystem, like a subsystem of the law.

Within the formal framework we define two technical notions useful forthe combination of fragments. These are the notions of connection andcoupling.9 We begin by giving an informal explanation of these two notions.First, let us consider the following situation.10

Let the lower fragment (with a1, b1 etc.) represent descriptive conditions andthe upper fragment (with a2, b2) normative conditions. The thick lines

a2

a1

a2∨b2

a2∧b2

a1∨b1

a1∧b1

b2

b1

Fig. 1: Situation 1

9 These notions were introduced in Lindahl and Odelstad 1999.10 Note that the following diagrams illustrate merely a part of an implicational condition structure(for example, negations of conditions are not taken into account).



represent logical implicative relationships due to the Boolean algebra, forexample that a1^b1 implies a1. The thin lines represent implications that are,

(1) `̀ legal'' implications rather than logical, and(2) `̀ direct'' links from the lower to the upper fragment (in the sense that

there is nothing between the conditions linked).

The implications represented by the thin lines, i.e., the legal and direct linkscorrespond to what will be called connections. For example, the implicationexpressed by a1Ra2 is direct and the pair ha1,a2i is a connection.

If a connection consists of a descriptive and a normative condition, therewill also be a number of implicative relationships that are indirect links fromthe lower to the upper fragment. If, in the picture, we combine the thick andthe thin lines we will obtain a number of such indirect links. For example,the implication expressed by a1^b1Ra2 is such an indirect link: In the picture,a1 lies between a1^b1 and a2. Therefore, the pair ha1^b1,a2i is not a connection.

Next, let us have a look at the following picture:

In this case, let us suppose that there is only one connection (i.e., oneimplicative relationship that is a direct link) from the lower to the upperfragment, namely the one corresponding to the implication (a1_b1)R(a2^b2).Then, all the other implications are indirect in the sense that they can bederived from this particular one. For example, since the structure is

a2

a1

b2

b1

Coupling by (a1∨b1, a2∧b2)

a2∨b2

a2∧b2

a1∨b1

a1∧b1

Fig. 2: Situation 2



assumed to be a Boolean quasi-ordering, the statement a1Ra2 can be derivedfrom (a1_b1)R(a2^b2). Since the pair ha1_b1,a2^b2i is the only connection, wecall it a coupling.

Thus, from the two pictures it appears that in the first case we have fourconnections but no coupling, whereas in the second case we have only oneconnection, this connection being a coupling.

2. Formal Development

As suggested in the previous section, the subject-matter of our inquiry is thedistinction between various parts of a Boolean quasi-ordering and thedifferent ways such parts can be combined. We therefore proceed to theformal definition of a fragment hB1,^,',R1i of a Boolean quasi-orderinghB,^,',Ri.

DEFINITION. If hB,^,',Ri is a Boolean quasi-ordering, and hB1,^,'i is asubalgebra of hB,^,'i, and R1=R/B1, then the structure hB1,^,',R1i is called afragment of hB,^,',Ri.11

(The expression R/B1 denotes the restriction of the relation R to the setB1.)

In what follows, we will only consider finite Boolean quasi-orderings,with a view to applications where there is a finite number of legal conditionsbelonging to the domain of the Boolean quasi-ordering. Thus we willassume that the domain B of the Boolean quasi-ordering hB,^,',Ri is finite.

Next, the notion of a connection is defined. To simplify the notation, in thesequel we will use the convention that Boolean quasi-orderings hB,^,',Ri andhBi,^,',Rii are denoted simply by B and Bi respectively.

DEFINITION. Suppose that B is a Boolean quasi-ordering and B1 and B2 arefragments of B. Then hb1,b2i is a connection from B1 to B2 in B if the followingfour requirements are satisfied:

(i) b1[B1, b2[B2 and b1Rb2.(ii) There is a1[B1\B2 and a2[B2\B1 such that a1Rb1 and b2Ra2.12

(iii) If c[B1 and b1RcRb2 then cQb1.(iv) If c[B2 and b1RcRb2 then cQb2.

(iii)±(iv) are called the proximity principles.

11 If hB, ^,'i is a Boolean algebra and A is a non-empty subset of B such that A is closed under theoperations ^ and ', then hA, Â, 'Ai is a subalgebra of hB, ^, 'i where Â and À are restrictions ofthe operations ^ and ' to A. (That A is closed under the operations ^ and ' means that if a,b[Athen aÂb [A and a'A[A.)12 The expression Bi\Bj denotes the difference of Bi and Bj, i.e., it denotes the set of elements of Bi

that do not belong to Bj.



Note that if hb1,b2i is a connection from B1 to B2 in B, then hb2',b1'i is aconnection from B2 to B1 in B. We call hb2',b1'i the converse of the connectionhb1,b2i.

The notions of fragment and connection are interrelated in an importantway. This is shown by the following observation concerning the existence ofa connection between two fragments. Suppose that B1 and B2 are fragmentsof B. Then, if there is a1[B1\B2 and a2[B2\B1 such that a1Sa2 then there existsa connection hb1,b2i from B1 to B2 such that a1Rb1 and b2Ra2.13

Next we introduce the special case of a connection called a coupling.

DEFINITION. Suppose that B is a Boolean quasi-ordering and B1 and B2 arefragments of B. Then hb1,b2i is a coupling from B1 to B2 in B if the followingthree requirements are satisfied:

(i) b1[B1, b2[B2 and b1Rb2.(ii) There is a1[B1\B2 and a2[B2\B1 such that a1Rb1 and b2Ra2.(iii) For all a1[B1 and a2[B2: If a1Ra2, then a1Rb1 and b2Ra2.

It is easy to see that every coupling from B1 to B2 in B is also a connection. Itis also possible to prove that if hb1,b2i is a coupling and hc1,c2i a connectionfrom B1 to B2 in B then b1Qc1 and b2Qc2, which shows that if there are morecouplings than one from B1 to B2, these couplings are equal with respect to R.

DEFINITION. The set {ha1,a2i,hb1,b2i} is a pair coupling from B1 to B2 in B if

(1) ha1,a2i and hb1,b2i are connections from B1 to B2 in B, and(2) not a1Qb1, and(3) for all c1[B1 and c2[B2 it holds that if c1Rc2, then either

(i) c1Ra1 and a2Rc2, or(ii) c1Rb1 and b2Rc2.

The notion of pair coupling is interesting since the following holds. If{ha1,a2i,hb1,b2i} is a pair coupling and hc1,c2i is a connection, then either c1Qa1

and c2Qa2, or c1Qb1 and c2Qb2.The present section will be concluded by an observation concerning the

importance of the notion of a connection. As is well-known, the so-calledRoss' paradox in deontic logic is based on the fact that in standard deonticlogic from Op it follows O(p_q). As an analogy, suppose that B1, B2 are twofragments of B, where a[B1 and b,c[B2. For B it holds that aRb impliesaR(b_c), and we seem to have a counterpart of Ross' paradox. However(presupposing that not (b_c)Rb), if ha,bi is a connection from B1 to B2 in B, itis excluded that ha,b_ci is a connection from B1 to B2 in B.14 Therefore, if the

13 For a proof, see Odelstad and Lindahl 1998, 146.14 That this is excluded is due to the fact that we have aRbR(b_c) but not (b_c)Rb, i.e., it isexcluded by the fact that b `̀ lies between'' a and b_c.



`̀ primary'' normative correlations according to a normative system S arerepresented by connections, from the assumption that aRb expresses aprimary correlation it does not follow that aR(b_c) expresses one.

III. Intermediaries

1. The Notion of Intermediate Concepts in Legal Philosophy

In legal philosophy, it is a well-known idea that in the formulation of legalrules a so-called `̀ intermediate term'' such as `̀ citizen'' couples a set of legalconsequences to a set of legal grounds. In 1951, Alf Ross published his well-known essay on `̀ TuÃ -TuÃ '' dealing with this matter, but a debate inScandinavian legal philosophy had already been started in 1944±45 byAnders Wedberg and Per Olof EkeloÈf (Ross 1956±57; EkeloÈf 1945; Wedberg1951).15 The paramount example of an intermediate term within this debatewas the term `̀ owner.''

A graphic expression of the idea is found in the well-known picture usedby Alf Ross:

Ross' scheme is aimed at representing a set of legal rules concerningownership (for example, the rules on ownership in Swedish law). Thus, wemay think of O as saying that x is the owner of object y, of each Fi asexpressing a possible legal ground for x's being the owner of y, and of eachCj as expressing one of the consequences of x's being the owner of y. Each ofthe lines to the left represents entailment and the same holds for the arrowto the right. Ross himself explains the scheme in the following way:

`̀ O'' (ownership) merely stands for the systematic connection that F1 as well as F2,F3...Fp entail the totality of legal consequences C1, C2, C3...Cn. As a technique ofpresentation this is expressed then by stating in one series of rules the facts that`̀ create ownership'' and in another series the legal consequences that `̀ ownership''entails. (Ross 1956±57, 820)

F1

F2

F3

Fp

O

C1

C2

C3

Cn

Fig. 3

15 Wedberg (1951, 246) refers to a lecture called `̀ On the Fundamental Notions ofJurisprudence,'' given by himself in the Law Club of Uppsala, as early as 1944. As mentionedby Wedberg, this lecture was based on a larger manuscript from that same year, on the logicalanalysis of legal science.



The role of O can be illustrated by its occurrence in two arrays of rules, asfollows.

(I):For any x1,...,xi: If F1(x1,...,xi), then O(x1,...,xi)For any x1,...,xi: If F2(x1,...,xi), then O(x1,...,xi).

.

.

For any x1,...,xi: If Fp(x1,...,xi), then O(x1,...,xi)

(II):For any x1,...,xi: If O(x1,...,xi), then C1(x1,...,xi)For any x1,...,xi: If O(x1,...,xi), then C2(x1,...,xi).

.

.

For any x1,...,xi: If O(x1,...,xi), then Cn(x1,...,xi).

The arrays can be reformulated in two rules:

(1) For any x1,...,xi: If F1(x1,...,xi), or...or Fp(x1,...,xi), then O(x1,...,xi).(2) For any x1,...,xi: If O(x1,...,xi), then C1(x1,...,xi), and...and Cn(x1,...,xi).

If the middle term O is eliminated, we get the single rule:

(3) For any x1,...,xi: If F1(x1,...,xi) or...or Fp(x1,...,xi), then C1(x1,...,xi) and...and Cn(x1,..., xi).

The most economical way to express the two sequences (I) and (II) of rulesabove would seem to be by the single rule (3). This way out, however,would often be less convenient from the point of view of legislativetechnique, since there can be good reasons for placing different grounds anddifferent consequences in different codes or statutes (private law, criminallaw, taxation law, etc.). From this perspective, a formulation in terms of thetwo rules (1) and (2) would be undesirable as well. As remarked byWedberg and Ross, however, if a formulation in terms of O and the twosequences of rules (I) and (II) is chosen, some economy of expression will beachieved. For p grounds and n consequences, this formulation will yield anumber of p plus n rules. If, in contrast, for each Fj and each Ck, a rule

(4) For any x1,...,xi: If Fj(x1,...,xi), then Ck(x1,...,xi)

were enacted, we would have p times n rules instead (Wedberg 1951, 273;Ross 1956±57, 820).

What is possibly a disadvantage connected with the use of intermediateterms in law and ethics is their uncertain status as regards the distinctionbetween descriptive and normative.



As is well-known, within the tradition from Hume and Bentham it is heldto be of great importance to distinguish sharply between empiricalstatements on one hand, and statements that are normative or deontic onthe other. As noted by Hume, when an author develops an argument, thereis often an imperceptible shift from descriptive propositions in terms of Isand Is not to normative ones in terms of Ought and Ought not:

I cannot forbear adding to these reasonings an observation, which may, perhaps, befound of some importance. In every system of morality, which I have hitherto metwith, I have always remarked, that the author proceeds for some time in the ordinaryway of reasoning, and establishes the being of a God, or makes observationsconcerning human affairs; when of a sudden I am surprised to find, that instead ofthe usual copulations of propositions, is and is not, I meet with no proposition that isnot connected with an ought or ought not. This change is imperceptible; but is,however, of the last consequence. For as this ought, or ought not, expresses some newrelation or affirmation, it is necessary that it should be observed and explained; andat the same time that a reason should be given, for what seems altogetherinconceivable, how this new relation can be a deduction from others, which areentirely different from it. (Hume 1949, 177±78)

In many cases, the imperceptible shift observed by Hume might result fromthe use of intermediate terms. For example, it might plausibly be held that`̀ to be in the public interest'' is an intermediate term within ethics, in a wayanalogous to the way in which `̀ owner'' is an intermediate term in the law. Itis not clear that the sentence `̀ the action A is in the public interest'' is whollydescriptive, neither that it is wholly normative. Rather `̀ to be in the publicinterest'' seems to be descriptive in part and normative in part.

2. Intermediaries in an Implicational Condition Structure

Within our framework of implicational condition structures there emergesan analogous subject-matter of what will be called `̀ intermediaries.'' Let usreturn to Amendment XIV, section 3.1, of the U.S. Constitution and let c bethe condition `̀ to be a citizen.'' Furthermore, let g1,...,gm be conditionsindicating the grounds for c according to the U.S. Constitutional system, andq1,...,qn be conditions indicating the legal consequences.16 Then the role of caccording to the system can be illustrated by its occurrence in two sequencesof sentences, as follows.

(I) g1Rc,g2Rc,...,gmRc,(II) cRq1,cRq2,...,cRqn.

The sequences (I) and (II) can be reformulated in two single sentences:

(1) (g1_g2_..._gm)Rc,(2) cR(q1^q2^...^qn).

16 As will be clarified in the following section, q1,...,qn can be thought of as denoting so-called`̀ hypothetical'' legal consequences.



If the middle condition c is eliminated, we get one single sentence:

(3) (g1_g2_..._gm)R(q1^q2^...^qn).

Our formal definition of `̀ intermediary'' is as follows. Let us say that m isan intermediary from B1 to B2 in B, if there exists a connection hb1,b2i from B1

to B2 in B such that

b1RmRb2

expresses the meaning of m.17

Let us apply this definition to the example. It clearly holds that,

(1) (g1_g2_..._gm)RcR(q1^q2^...^qn).

For c to be an intermediary from B1 to B2 two requirements must be satisfied:

(i) h(g1_g2_..._gm),(q1^q2^...^qn)i is a connection from B1 to B2, and,(ii) (1) expresses the meaning of c.

Whether requirement (i) is satisfied presumably depends on how thefragment B1 is chosen. It seems very probable that B1 can be chosen in such away that (i) is fulfilled. Indeed, in our specific example it is not obvious thatthere is at all a choice of B1 such that (i) does not hold. Imagine, however,that there were in B a condition h, `̀ to be an honorary citizen,'' such that thegrounds h1,h2,...,hk for h in B differed from the grounds for c while theconsequences q1,q2,...,qn were the same for h as for c. Then, if h1,h2,...,hk wereamong the elements of B1, requirement (i) would not be satisfied and cwould not be an intermediary from B1 to B2.

In the particular example of citizenship, it is reasonable to assume that

(2) (g1_g2_..._gm)QcQ(q1^q2^...^qn).

We note that (2) is equivalent to

(3) (g1'^g2'^...^gm')Qc'Q(q1'_q2'_..._qn').

It seems plausible that we can choose B1 and B2 in such a way that the pair

{hg1_g2_..._gm,q1^q2^...^qni,hg1'^g2'^...^gm',q1'_q2'_..._qn'i}is a pair coupling in the sense defined earlier (see section II.2). If this holds,c is an intermediary of a special kind and can be called a pair couplingintermediary.

3. Hypothetical Legal Consequences

Another important aspect of legal concepts is that the normative con-sequences of their applicability are oftenÐas some jurists sayÐhypothetical.

17 This does not exclude that m[B1 or m[B2. With a view to some applications, it may beappropriate to exclude this by further assumptions about B1 and B2.



Thus, for example,

`̀ x has protection of possession with respect to y''

means:

`̀ If someone takes away y from x, then y shall be returned to x.''

Indeed, as regards the central notion of ownership a large amount of con-sequences are hypothetical in this sense, regarding what is obligatory orpermissible if certain events occur. The same applies to a great number ofother legal concepts. It is our contention that legal concept formation cannotbe properly understood if hypothetical consequences are not taken intoaccount.18

Let us return to the earlier example of citizenship and let

c: to be a citizena: to be adult,e: to be entitled to vote in general elections.

Simplifying matters, suppose that,

(1) (c^a)Re.

According to the axioms for Boolean quasi-orderings, (1) holds if and only if

(2) cR(a'_e).

Going from (1) to (2) can be called exportation, and going from (2) to (1)importation.

We observe that a'_e occuring to the right in (2) can be understood as sayingthat if a then e. The sentence (2) expresses that, according to the normativesystem, the consequence `̀ to be entitled to vote in general elections'' ofcitizenship is hypothetical, since it is conditional on being adult.

4. Isolating the Intermediary

Up to this point we have looked at the joining of two fragments. As we willnow see, if hypothetical legal consequences are taken into account, threefragments, and their combinations, must be taken into account.

Once more, let us consider the example of citizenship. In section III.2 whatholds about the constitutional system concerning citizenship was illustratedby two sequences of sentences, as follows:

(1) g1Rc, g2Rc,...,gmRc,(2) cRq1, cRq2,...,cRqn.

18 The fact that many legal notions, such as the notion of being an owner, cannot be properlyunderstood without taking hypothetical legal consequences into account, was observed andemphasised in particular by the Swedish jurist Per Olof EkeloÈf. See EkeloÈf 1950, 161.



However, within the framework now available, we can elucidate that, intwo important respects, sequence (2) is not an adequate representation. First,q1,q2,...,qn are hypothetical legal consequences. Therefore, to obtain a moreappropriate representation, each qi should be replaced by a condition of theform di'_ri where di is descriptive and ri is normative. This way, instead of (2)we get

(3) cR(d1'_r1), cR(d2'_r2),..., cR(dn'_rn).

Secondly, the law is often formulated by conjunctions rather than by hypo-thetical consequences. When this is the case, rather than starting with (3) asa representation we have to start with the sequence,

(4) (c^d1)Rr1, (c^d2)Rr2,..., (c^dn)Rrn.

By exportation we can go from the sequence (4) to (3) where c has beenisolated and where the legal consequences are hypothetical.

From (1) and (3) respectively we can deduce the two sentences:

(5) (g1_g2_..._gm)Rc,(6) [cR(d1'_r1)^(d2'_r2)^...^(dn'_rn)].

The question now emerges whether c is an intermediary from one suitablychosen fragment to another. Due to the phenomenon of hypothetical legalconsequences, in order to make the question more exact, we have to takeinto account combinations of three fragments of the Boolean quasi-orderingB. Let us assume that B1, B2 and B3 are fragments of B, and that

g1,g2,...,gm are elements of B1,r1,r2,...,rn are elements of B2, andd1,d2,...,dn are elements of B3.

Let C be the fragment generated by B2 and B3, in the sense that C is thesmallest fragment of B containing B2 and B3.19 Then our previous questioncan be expressed in a more exact way: Is c an intermediary from B1 to C?

As we know, one prerequisite of c being an intermediary is that the pair

P=h(g1_g2_..._gm),(d1'_r1)^(d2'_r2)^...^(dn'_rn)iis a connection from B1 to C. The other prerequisite is that (5) and (6)together express the meaning of c.

When considering the question whether P is a connection, it is of interestto compare P (where the second component is a conjunction of hypotheticallegal consequences) with the set of pairs where the elements di of B3 occur inthe first component. Thus let g=g1_g2_..._gm, and let

K={hg^d1,r1i,hg^d2,r2i,...,hg^dn,rni}.19 More formally, the fragment C1 of B generated by B1 and B3 is the intersection of all fragmentsof B whose domain contains B1[B3.



Let D be the fragment generated by B1 and B2. Then, the interesting questionemerges in which kinds of cases there is a correlation between P being aconnection from B1 to C and K being a set of connections from D to B2. Sincethe `̀ primary'' representation of the law is often given by sentences likethose in the sequence (4) rather than by those in (3), it is desirable to havecriteria for c being an intermediary (and P a connection) in terms of K.

If P is a connection from B1 to C in B, and K is a set of connections from Dto B2 in B, the `̀ primary'' normative correlations that hold according to thenormative system can be described by either of these two choices of frag-ments and connections. For some purposes, therefore, a description in termsof K is as good as a description in terms of P. If, however, the purpose is toshow the role of c as an intermediary, the description should be made interms of P.

IV. Conclusion

In this paper we have studied combinations of subsystems and of fragments,intermediate terms and hypothetical legal consequences. It might beapparent from our discussion that using the formal framework consistingof Boolean quasi-orderings, fragments, connections, couplings and inter-mediaries, we are able to distinguish between many different cases whichotherwise might be overlooked. To give a systematic classification of thesedifferent cases is an interesting task for the future.

(For Lars Lindahl)Faculty of Law

Lund UniversityBox 207

221 00 LundSweden

E-mail: [email protected]

(For Jan Odelstad)Department of Philosophy

Uppsala UniversityDrottninggat 4753 10 Uppsala

SwedenE-mail: [email protected]

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Odelstad, J., and L. Lindahl. 1998. Conceptual Structures as Boolean Orderings. InNot without Cause. Philosophical Essays Dedicated to Paul Needham on the Occasion ofHis Fiftieth Birthday. Ed. L. Lindahl et al. Uppsala: Department of Philosophy.

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an algebraic analysis of normative systems

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