131
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journal for constitutional theory and philosophy of law
revus (2015) 26, 131–146
Giovanni Battista Ratti* & Jorge Luis Rodríguez**
On Coherence as a Formal Property  
of Normative Systems
The paper deals with the notions of consistency, completeness, and coherence within the 
normative domain. It investigates their mutual relations by singling out relative (to nega-
tion) and absolute consistency, weak, strong and trivial completeness, and three different 
functions of coherence (simplification, integration, and defeasance). The main upshot 
of the inquiry is that coherence may be regarded as a complex combination of weak 
completeness and possible absence of consistency and strong completeness of a system 
of rules regarding a non-trivially complete/non-absolutely inconsistent system of under-
lying principles.
Key words: consistency, completeness, coherence, conditionals, normative systems
1 INtrOduCtION
‘Consistency’ and ‘completeness’ are age-old concepts which are traditional-
ly related to logic and logical calculi. They have received a great elaboration and 
their mutual relations have been widely studied in the field of logic. They have 
also been extensively analyzed by legal dogmatics and analytical jurisprudence 
regarding their application to normative systems. 
By contrast, ‘coherence’ is a relatively new concept, which has raised some 
issues in the epistemological and legal philosophical debate during the last dec-
ades.1 The intensional properties and the boundaries of this concept are quite 
blurry, and its relations with consistency and completeness are consequently 
uncertain.2
* gbratti@unige.it | Professor in Legal Philosophy, Tarello Institute for Legal Philosophy, De-
partment of Law, University of Genoa.
** jorgerodriguez64@yahoo.com | Professor in Legal Theory, Faculty of Law, National Univer-
sity of Mar del Plata.
1 The formal analysis of coherence was introduced in Rescher (1973). In the domain of juris-
prudence, among the major contributions to understanding the applications of the concept 
are MacCormick (1984) and Peczenick (2008). For recent surveys, see Araszkiewicz & Šavelka 
(2013) and Amaya (2015).
2 For a formal approximation, see Amaya (2007) and Alonso (2013).
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The present paper, by elaborating on some notions of propositional calculus, 
aims at providing a basic reexamination of the notions of consistency, com-
pleteness, and coherence, when applied to the normative domain, and, on such 
a basis, elaborating on which mutual relations they have.
2 CONSISteNCy aNd COmPleteNeSS IN 
PrOPOSItIONal lOgIC
In propositional logic,3 consistency is usually and broadly defined as the 
property of a set of sentences of being free of contradictions. It is not uncom-
mon to distinguish between consistency with respect to negation and absolute 
consistency. A set S is consistent with respect to negation if and only if for every 
proposition p, if p belongs to S, then ~p does not belong to S. By contrast, a set 
S is absolutely consistent if and only if there is at least one proposition p that is 
not implied by S. 
S is consistent with respect to negation iff ∀p (p ∈ S ⇒ ~p ∉ S)
S is absolutely consistent iff ∃p (p ∉ Cn(S))4
Two corresponding notions of inconsistency may be defined as the negations 
of these properties: a set S may be called inconsistent with respect to negation if 
and only if there is at least one proposition p such that both p and ~p belong to S, 
and absolutely inconsistent if and only if every proposition p is implied by S.
S is inconsistent with respect to negation iff 
~∀p (p ∈ S ⇒ ~p ∉ S) = ∃p (p ∈ S ∧ ~p ∈ S)
S is absolutely inconsistent iff
~∃p (p ∉ Cn(S)) = ∀p (p ∈ Cn(S))
From these characterizations it is interesting to note that the properties of 
consistency and inconsistency with respect to negation are defined in terms of 
the elements that belong to a given set, while the properties of absolute consist-
ency and inconsistency are defined in terms of the logical consequences of a 
given set. 
In fact, as Žarnić and Bašić claim, the notions of logical closure and absolute 
consistency are interdefinable according to the following equivalence:5
S = Cn(S) iff ∀p (p ∉ S ⇒ (⊥ ∉ Cn(S ∪ {~p}))
3 See Hughes & Londey (1965: 137-140).
4 Strictly speaking, a specification of the logical system under consideration should be added 
to each formula (e.g. ‘S is absolutely consistent in logic L iff ∃p (p ∉ CnL(S))’). For the sake of 
simplicity we shall omit such a qualification here.
5 See Žarnić & Bašić (2014).
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133On Coherence as a Formal Property of Normative Systems
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This means that a set S is closed under logical consequence if and only if the 
negation of any element not belonging to S may be added to S preserving its 
absolute consistency. 
Completeness, in turn, is regarded as the property of a finite set of sentences 
(axioms) – or sentential forms (axiom schemata) – of being exhaustive qua axi-
omatization of a certain field, so that every correct sentence, relevant for that 
field, is derivable from – i.e. is a theorem of – the axiomatic basis. 
It is common to distinguish between sets which are weakly complete, for they 
are capable of generating the set of all its correct theorems, and sets which are 
strongly complete, i.e. those sets which cannot be further supplemented regard-
ing their axiomatic basis without being rendered inconsistent. To these two no-
tions, one can add trivial completeness, the property of a set of demonstrating 
any sentence whatsoever. 
In formal terms, this translates into:
 S is weakly complete iff ∀p (p ∈ C ⇒ p ∈ Cn(S)) (where C is the set of correct 
or relevant sentences)
S is strongly complete iff ∀p (p ∉ S ⇒ (⊥ ∈ Cn(S ∪ {p}))
S is trivially complete iff ∀p (p ∈ Cn(S))6
From these definitions it follows that a set may be called weakly incomplete 
if and only if at least one correct sentence is not derivable from it; strongly in-
complete if and only if it can be further supplemented regarding their axiomatic 
basis without being rendered inconsistent, and trivially incomplete – though the 
label may sound a little awkward – if there is at least one sentence that it cannot 
demonstrate.7
In formal terms:
S is weakly incomplete iff
~∀p (p ∈ C ⇒ p ∈ Cn(S)) = ∃p (p ∈ C ∧ p ∉ Cn(S))
6 Along with these notions of completeness, which may be deemed semantic, several syntactic 
notions of completeness are conceivable, the most natural of which may be defined as follows: 
S is syntactically complete iff ∀p (p ∈ Cn(S) ∨ ~p ∈ Cn(S)) (see Hunter 1971: 33). This syn-
tactic concept of completeness is closely related to the notion of normative completeness that 
will be presented above.
7 Jan Woleński suggested in private communication that the notion of post-completeness 
should be introduced into our analysis for the sake of clarity. Post-completeness is usually 
defined as follows: ‘A system of logic is post-complete if every time we add to it a sentence un-
provable in it, we obtain an inconsistent system.’ We tend to reject such a suggestion, since we 
are not dealing here with completeness of the calculi (in meta-language), but with complete-
ness of the object-language. At any rate, if we reinterpret post-completeness as a first-order 
notion, and drop the ‘unprovable’ connotation, we think it collapses into what we call strong 
completeness. 
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S is strongly incomplete iff 
~∀p (p ∉ S ⇒ (⊥ ∈ Cn(S ∪ {p}))) = ∃p (p ∉ S ∧ (⊥ ∉ Cn(S ∪ {p})))
S is trivially incomplete iff
~∀p (p ∈ Cn(S)) = ∃p (p ∉ Cn(S)) 
The mutual relations between these notions of consistency and completeness 
are as follows. First, trivial completeness is equivalent to absolute inconsistency 
and, consequently, trivial incompleteness is equivalent to absolute consistency. 
Second, trivial completeness (as well as absolute inconsistency) implies weak 
completeness, because if any sentence is derivable from a certain set, then the 
partition of all sentences that are ‘correct’ theorems are derivable from it. Third, 
if one accepts the ex falso quodlibet principle, then inconsistency with respect 
to negation entails absolute inconsistency (as well as trivial and weak complete-
ness). Finally, if strong completeness is defined in the way proposed supra, then 
an absolutely inconsistent set satisfies strong completeness vacuously. 
For the sake of the argument to follow in the paper, particular stress should 
be put on the formalization of consistency (and completeness) regarding condi-
tional sentences. In formal logic, states of inconsistency stemming from condi-
tional sentences are usually represented by the following conjunction of a con-
ditional with the corresponding denied conditional:
[1] (p ⇒ q) ∧ ~(p ⇒ q)
Which, by definition of the conditional ((p ⇒ q) ⇔ ~(p ∧ ~q)), is equivalent 
to:
[2] ~(p ∧ ~q) ∧ (p ∧ ~q)
From [1], or [2], any sentence whatsoever follows under the assumption of 
the ex falso quodlibet principle. In such a case, the resulting set is weakly and 
trivially complete. By contrast, the following sentence:
[3] (p ⇒ q) ∧ (p ⇒ ~q)
brings about no contradiction, despite what may appear at first sight. The con-
junction of a conditional with its corresponding conditional denial is materially 
equivalent to the negation of the common antecedent of both conditionals.
[4] (p ⇒ q) ∧ (p ⇒ ~q) ⇔ ~p
This happens since a conditional and its conditional denial are subcontraries; 
they are consequently neither contraries nor contradictories. Indeed, a condi-
tional is contradictory to its negation, namely the conjunction of its antecedent 
and the negation of its consequent. This in turn is contrary to the conjunction 
of the antecedent and the consequent of the original conditional. These rela-
tions are expressed in the following square of propositional opposition:
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On the top side, there is contrariety: the conjunctions that appear there 
(which are equivalent to denied conditionals) cannot both be true, but they 
can both be false. On diagonals, there is contradiction: if one sentence is true, 
the other is false (and vice versa). On the verticals, there is subalternation (viz. 
implication): if the conjunction (i.e., the negation of the conditional denial) is 
true, then the corresponding conditional is also true. On the basis of the square, 
there is subcontrariety, which states that such conditionals can both be true, 
but they cannot both be false: indeed, when one of the two is false, the other is 
necessarily true, but when one of the two is true, it is not the case that the other 
is necessarily false.
When contradictions or contrarieties are contained by a certain set, this set 
becomes of course inconsistent, and also trivially and weakly complete. But 
when subcontrary sentences like (p ⇒ q) and (p ⇒ ~q) are contained by a cer-
tain set, such set will not necessarily be inconsistent for that reason. It will be 
strongly complete regarding those worlds in which p is true, since p cannot be 
added to the set without generating an inconsistency.
Indeed, the subcontrariety of conditionals is exhaustive of a certain world, 
in the sense that it connects a certain atomic sentence (in our example, p) to a 
partition of sentences which are mutually exclusive and jointly exhaustive (in 
our example, q and ~q). When both conditionals are simultaneously asserted 
(and hence the sentence p is connected, at the same time, to q and ~q), one can 
derive the falsity of p and, as a consequence, the truth of ~p.
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3 CONSISteNCy aNd COmPleteNeSS  
IN NOrmatIve SyStemS
In the legal philosophical domain, consistency and completeness have been 
frequently examined. In the most relevant literature, ‘normative consistency’ de-
notes the situation where the same set of operative facts is not connected to in-
compatible normative consequences, whereas ‘normative completeness’ denotes 
the situation where there is a legal consequence for any set of relevant operative 
facts.8 Nevertheless, formal logical relations between normative consistency and 
completeness have seldom been taken into account.9 In order to do so, it is nec-
essary to distinguish genuine norms from propositions about norms.10
Though the thesis that norms are proposition-like entities and, as such, are 
apt to truth-values, is highly controversial, there is no disputing the proposi-
tional character of the discourse about norms. Von Wright characterizes norm-
propositions as propositions ‘that such and such a norm exists’.11 But since 
norms usually exist as a part of complex systems of norms, the syntactical form 
of deontic statements expressing norm-propositions often hides a relevant part 
of their conceptual content.12 Thus, a statement like ‘Parking is not allowed here’, 
understood as a norm-proposition, though apparently referring to the deontic 
status of an action without qualifications, is in fact a relative statement that re-
ports the deontic status of an action according to a particular normative system. 
Norm-propositions are thus true or false depending on the set of norms taken 
as relevant. Therefore, the structure of norm-propositions may be represented 
as ‘Op ∈ S’, where Op is the norm that is being mentioned (not used) by this 
norm-proposition, and S is the normative system to which, according to the 
proposition at hand, Op is said to belong.
The standard system of deontic logic validates the principle of normative 
consistency, namely, that no action can be both prohibited and permitted, and 
the principle of normative completeness, namely, that every action is either pro-
8 See Rodríguez (2002: 62 ff).
9 The most sophisticated model of normative systems – the one developed by Alchourrón & 
Bulygin (1971: 98-101) – analyses in depth the relations between the two concepts regarding 
universes of cases characterized by different levels of specificity. 
10 See Alchourrón & Bulygin (1971). With different names, the distinction has been noticed by 
several authors, e.g. Bentham (1970), Hedenius (1941), Wedberg (1951), Hansson (1969), 
among others.
11 Von Wright (1963: 106).
12 Referring to the legal domain, Hart (1983: 329) observes ‘There is frequent occasion for lawyers 
to describe what they might call the ‘legal position’ in relation to some subject without referring to 
the particular enactments or regulations or other sources of the relevant law, though of course it 
would be always understood that ‘the legal position’ thus described is that arising under the laws 
of a particular system, and a more accurate formulation would make this explicit by including 
such words as ‘according to English law’.’
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hibited or permitted. Now, as Alchourrón and Bulygin have shown, it is wrong 
to assume that logical relations among norm-propositions are equivalent to 
those that hold among norms themselves.13 A simple way to explain the differ-
ences between a genuine logic of norms and a logic of norm-proposition is to 
take due notice of the different ways in which negation affects the former and 
the latter. When applied to norms, negation seems to behave in close analogy to 
descriptive discourse: the negation of a norm (e.g. Op) is also a norm (P~p);14 
for each norm there is only one norm that is its negation; a norm and its ne-
gation are reciprocal (if P~p is the negation of Op, then Op is the negation of 
P~p).15 Moreover, a norm and its negation are mutually exclusive and jointly 
exhaustive, since ~(Op ∧ P~p) and Op ∨ P~p are valid formulas in a logic of 
norms.
By contrast, the negation of norm-propositions is more problematic. ‘In nor-
mative system S act p is forbidden’ is a complex metalinguistic statement that as-
serts that a certain norm belongs to a normative system. Therefore, its negation 
can be understood in two different ways: ‘In system S act p is not forbidden’ may 
be read as expressing that S does not contain a norm prohibiting p, in which 
case the negation operates over the membership of the norm in the system (ex-
ternal negation, represented as ‘~’); or it may be read as expressing that S does 
contain a norm not-prohibiting p (i.e. a norm permitting p), in which case the 
negation affects the norm referred to in the statement (internal negation, rep-
resented as ‘¬’). While external negation of a norm-proposition is an opera-
tion that switches the value of membership of a given norm within a normative 
system, internal negation of a norm-proposition is an operation that leads to 
another norm-proposition asserting the membership in the system of the nega-
tion of the original norm. For example:
External negation: ~FSp =df. Fp ∉ S  
Internal negation: ¬FSp =df. ~Fp ∈ S =df. Pp ∈ S
Here, the external negation of the proposition according to which p is for-
bidden in normative system S is a proposition that negates the membership of a 
norm prohibiting p in system S, whilst the internal negation of such a proposi-
tion is a proposition asserting the membership in S of a norm to the effect that 
p is not prohibited, i.e. a norm permitting p. Accordingly, ‘permission’ is an 
expression that may appear in a genuine norm, or in a norm-proposition. In 
the first case, i.e. on a prescriptive interpretation, to say that an action is permit-
ted is unambiguously equivalent to saying that it is not prohibited. By contrast, 
when ‘permission’ appears in a norm-proposition it becomes ambiguous, since 
there are two alternative senses in which an action can be said to be permitted 
13 See Alchourrón (1969), Alchourrón & Bulygin (1971).
14 See Alchourrón & Bulygin (1988: 231).
15 The expression ‘negation-norm’ was coined by von Wright (1983: 130-209). 
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according to a set of norms: a negative sense − there is no norm in the set pro-
hibiting it − and a positive sense − there is a norm in the set permitting it.16 We 
may call these two notions negative permission (P−Sp) and positive permission 
(P+Sp): 
Negative permission:    P−Sp =df. Fp ∉ S
Positive permission:    P+Sp =df. ~Fp ∈ S =df. Pp ∈ S
It is easy to see that the negative permission of p is equivalent to the external 
negation of the prohibition of p, while the positive permission of p is equivalent 
to the internal negation of the prohibition of p. 
The first presentation of the differences between a genuine logic of norms 
and a logic of norm-propositions was put forward by Carlos Alchourrón, and 
later developed in work co-authored with Eugenio Bulygin.17 Their proposal 
can be set out axiomatically as follows:
System of logic of norm-propositions (LNP):
(LNP A1) OS(α ∧ β) ⇔ (OSα ∧ OSβ)
(LNP A2) OSα ⇒P+Sα
(LNP A3) P+S(α ∧ β) ⇒ P+Sα
(LNP RI1) From |-  (α ⇔ β), it follows |- (OSα ⇔ OSβ)
(LNP RI2) From |- (α ⇔ β), it follows |- (P+Sα ⇔ P+Sβ)18
If we compare this system with the standard system of deontic logic,19 we 
should notice first that norm-propositions are relative to a given normative sys-
tem, and that is reflected in the subscripts (S) in the formulas. By contrast, the 
expressions of the logic of norms do not refer to any particular normative sys-
tem, since they represent absolute concepts. Second, in the system of logic for 
norm-propositions there is nothing like the principle of Pp ⇔ ~O~p20 which is 
accepted as valid in deontic logic. This is so because, as we have seen, there are 
16 We prefer this terminology to the more common dichotomy weak vs. strong permission, since 
the latter seems to suggest that the strong/positive concept implies the weak/negative one, 
which is not necessarily the case. 
17 See Alchourrón (1969), Alchourrón & Bulygin (1971).
18 See Alchourrón (1993: 46).
19 One possible presentation is the following:
(SDL A1)  All tautologies of PL 
(SDL A2) O(α ⇒ β) ⇒ (Oα ⇒ Oβ)   (OK)
(SDL A3) Oα ⇒ ~O~α   (OD)
(SDL RI1) If |- α and |- (α ⇒ β), then |- β  (Modus Ponens)
(SDL RI2) If |- α, then |- Oα   (O-Necessitation). 
20 In the system LNP there is also an axiom and a rule of inference (A3 and RI2) that are not 
found in the usual systems of deontic logic. This is, however, a minor difference; were it pos-
sible to add to LNP an analogous principle to the principle Pp ⇔ ~O~p, A3 and RI2 would be 
superfluous because they would be derivable from the other axioms and rules. 
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two different forms of negation of norm-propositions, which give rise to two 
descriptive concepts for each of the deontic operators. And although external 
negation satisfies all the properties that we expect from ordinary negation, that 
is not the case with internal negation. A norm-proposition and its internal ne-
gation can both be true – in which case the system will be inconsistent – as well 
as both false –in which case the system will be incomplete.21 That is why the 
equivalence expressed in Pp ⇔ ~O~p, though valid in the logic of norms (LN), 
is not valid in the logic of norm-propositions.
In spite of these differences, however, it can be proven that, under certain 
assumptions, LNP and LN are equivalent, and consequently, that positive per-
mission P+ will be equivalent to negative permission P−. And that is so since 
an analogue of the principle Pp ⇔ ~O~p is indeed valid in LNP under the 
conditions of consistency and completeness of the normative system taken into 
account: 
[5] ~(OS~p ∧ P
+
Sp)  (ConsSp)
[6] OS~p ∨ P+Sp (CompSp)
It is important to see that the conjunction of [5] and [6], i.e.
[7] ~(OS~p ∧ P+Sp) ∧ (OS~p ∨ P+Sp) 
is equivalent to
[8] P+Sp ⇔ ~OS~p22
and since by the definitions already seen, ~OS~p is equivalent to P−Sp, then:
[9] ((ConsSp) ∧ (CompSp)) ⇔ (P+Sp ⇔ P−Sp)
Moreover, if one accepts that a contradiction implies everything, normative 
inconsistency entails normative completeness,23 and normative incompleteness 
entails normative consistency, what is tantamount to saying that, at least within 
the normative domain, implication, consistency, and completeness are unavoi-
dably connected concepts. 
21 See Bulygin (1995: 135, 137).
22 Proof: by commutativity of conjunction in [5] we obtain ~(P+Sp ∧ OS~p); by definition of 
conditional in the latter expression we get P+Sp ⇒ ~OS~p, and now, by introduction of condi-
tional from [5] to this latter step we get ~(OS~p ∧ P
+
Sp) ∧ (P
+
Sp ⇒ ~OS~p). By definition of 
conditional in [6] we obtain (~OS~p ⇒ P
+
Sp), and now by introduction of conditional we get 
(OS~p ∨ P
+
Sp) ⇒ (~OS~p ⇒ P
+
Sp).
23 One could think that if a contradiction implies everything, then does normative inconsistency 
not only imply normative completeness but also its negation, i.e. normative incompleteness. 
However, this would be an incorrect conclusion derived from the identification of norms and 
norm-propositions, since from the fact that a normative system is inconsistent what follows 
is that every norm belongs to it, not that every norm-proposition (i.e., the one affirming its 
incompleteness) is true about it.
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Indeed, i) if a normative system S is inconsistent regarding a certain action 
p, it is complete regarding that same action. This is shown in the following de-
rivation:
[10] OS~p ⇔ P+Sp
[11] OS~p    by simplification in [10]
[12] OS~p ∨ P+Sp   by introduction of disjunction 
    in [11] 
[13] (OS~p ∧ P+Sp) ⇒ (OS~p ∨ P+S p)  by introduction of conditional 
    from [10] to [12].
And ii) If a normative system S is incomplete regarding a certain action p, 
then it is consistent regarding that same action. This is demonstrated by the 
following inference: 
[14] ~(OS~p ∨ P+Sp)
[15] ~OS~p ∧ ~P+Sp      by De Morgan in [14] 
[16] ~OS~p       by simplification in [15] 
[17] OS~p ∧ P+Sp      assumption
[18] OS~p       by simplification in [17]
[19] OS~p ∧ ~OS~p      by introduction of conjunction in [18, 16] 
[20] ~(OS~p ∧ P+Sp)      by introduction of negation from [17] to [19].24
In fact, these two demonstrations can be generalized in virtue of an applica-
tion of the principle ex falso quodlibet as follows:
iii) If a normative system S is inconsistent regarding certain action x, it is 
complete regarding any action (∃x ~(ConsSx) ⇒ ∀x (CompSx)).
iv) If a normative system S is incomplete regarding certain action x, it is con-
sistent regarding any action (∃x ~(CompSx) ⇒ ∀x (ConsSx)).25
Now, let’s assume, with Alchourrón and Bulygin, that legal rules are correc-
tly represented as conditionals of the form p ⇒ Oq.26 Denying a norm of this 
sort would prima facie result in the following sentence:
[21] ~(p ⇒ Oq)
Sentence [21] can be translated –due to the definition of conditional and the 
interdefinability of deontic modalities – into:
[22] p ∧ P~q
24 If A leads to contradiction, then ~A.
25 For an alternative system of logic for norm-propositions where normative conflicts and gaps 
are avoided ‘whenever possible’, see Beirlaen & Straßer (2013).
26 This is far from being an unproblematic assumption, but we cannot enter into such details 
now.
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However, deontic logicians and legal theorists do not regard [21] and [22] as 
the negation of the conditional sentence p ⇒ Oq. They rather reserve this title 
for the normative sentence, which is analogous to conditional denial in propo-
sitional logic, to wit:
[23] (p ⇒ ~Oq)
Which, for the inter-definability of deontic characters, equates to
[24] (p ⇒ P~q)
Hence, regarding hypothetical normative systems,27 consistency and com-
pleteness are world-sensitive, in that they only apply to the world referred to by 
the antecedent of conditional normative sentences, whenever they are actually 
instantiated. Normative inconsistency via certain facts does not trivialize the 
system globally. In other words, the presence in a normative system of the sen-
tences p ⇒ Oq and p ⇒ ~Oq renders the system inconsistent with respect to 
negation, but only regarding those worlds in which p is the case. Here, again, if 
we accept the principle ex falso quodlibet, the system will also be absolutely in-
consistent for those worlds in which p is true, and since absolute inconsistency 
is equivalent to trivial completeness, this latter property will also be predicable 
of the system regarding p-worlds.   
Of course, all these considerations are compatible with the claim that in hy-
pothetical normative systems, completeness and consistency are independent 
properties, in that a normative system correlating normative solutions to cer-
tain cases may be complete/incomplete regarding a given case and consistent/
inconsistent regarding another case, as well as consistent/inconsistent regarding 
a given case and complete/incomplete regarding another case.
4 NOrmatIve COhereNCe
The concept of coherence is a highly contested one. There are many concep-
tions of coherence, each one predicated on a web of notions which only oc-
casionally and partially overlap. However, if one were to search for the com-
mon kernel of such conceptions, one would probably affirm that coherence is a 
justificatory relation among principles and a certain set of rules to which such 
principles supposedly underlie.28 In many cases, coherence is left at this rather 
pre-theoretical stage, which in turn is predicated on the fuzzy way principles are 
defined. 
27 By ‘hypotethical normative systems’ we mean systems which are not made solely of uncondi-
tional (viz. categorical) norms.
28 Within this compass, rules are usually understood as conditional normative sentences con-
necting certain operative facts to the deontic qualification of a state of affairs (according to 
our previous symbolization: p ⇒ Oq). 
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However, we can make a preliminary attempt to distinguish principles from 
rules by pointing to (a) their fundamental character; (b) their defeasibility; (c) 
their need of being specified prior to their application.29
Principles are fundamental in that they justify several rules, but they do 
need further justification. They are regarded as defeasible for their antecedents 
are not taken to be sufficient conditions of their consequents (what entails that 
they are not liable to being formalized as material conditionals). They need to 
be specified in the sense that they do not allow modus ponens, and must con-
sequently be developed into rules, although there are many alternative ways of 
specifying a principle into an unexpressed (viz. implicit in a non-logical sense) 
rule. 
A normative system can be said to be coherent regarding a certain set P of 
underlying principles if and only if all (or most) of its rules are instances of 
P,30 in the sense that all (or most) of its rules are justified by P. According to 
this idea, coherence – unlike consistency and completeness – is not an all-or-
nothing property, but a gradual one. 
Coherence has been said to serve three main functions:31 a simplifying one, 
an integrating one, and a defeating one. Simplification is given by the circum-
stance that a certain normative system can be traced back to a limited array of 
underlying principles, liable to be further developed into new rules. Integration 
consists in the possibility of filling up gaps in the normative system by reaching 
out to its underlying principles, so that the norm introduced into the system to 
fill the gap increases the overall coherence of the normative system. Defeasance 
points to the fact that rules which bring about incoherent results can be revised 
accordingly.
Nevertheless, it is important to note that there are to different ways of un-
derstanding the relation of justification between rules and principles, and ac-
cordingly two different ways of understanding the very notion of principles, 
that have considerable impact over these three functions of coherence. A first 
interpretation of the idea that principles ‘justify’ rules, and of conceiving the 
former, is to understand ‘principles’ as general norms that are obtained through 
a process of abstraction or complete induction from the rules of a given nor-
mative system. The expression ‘general principles’ of law or of a concrete legal 
institution is usually employed to refer to this notion. On this interpretation it 
seems perfectly sensible to maintain that principles function as a simplification 
of a system of rules, and that they cannot conflict with those rules. Principles 
29 Here we are indebted to Guastini (2011: 173 ff).
30 Here we are fundamentally following MacCormick’s notion of coherence in law (see MacCor-
mick (1984: 235-244)). We shall not discuss here other possible paths, like the ones developed 
by Jaap Hage (e.g., 1997, 2005).
31 On this point, see Alonso (2010: 155 ff).
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so understood are not able to display either the integration or the defeasibility 
function, since in this sense principles do not allow deriving solutions different 
from those that follow from the rules, as they would be but mere reformula-
tions of the rules, even though some authors have correctly pointed out that so 
inferred principles are normally used by jurists to create “new” rules by means 
of strengthening the antecedent.32
As an alternative, it could be claimed that though rules ‘exemplify’ princi-
ples, the latter are not mere reformulations of the former, but express values 
or ends presupposed by the rules. On this reading, principles may eventually 
allow deriving solutions for cases not regulated by the rules, so that arguments 
based on principles will have certain ‘ampliative’ character, and they may even 
justify solutions in conflict with the rules of the system, and thus are capable of 
displaying the functions of integration and defeasibility. By contrast, it is not 
clear whether principles, on this understanding, can work as a simplification of 
a system of rules, since the idea of simplification loses its proper sense if we are 
not dealing with axiomatic bases with identical normative consequences where 
one of them is more economical than the other. But here, insofar as the con-
sidered principles allow deriving normative solutions for cases which are not 
regulated by the rules of the system, and even normative solutions incompatible 
with those that follow from the rules, it is clear that the system of rules and the 
system of principles may have different consequences. And it is only here, on 
this second understanding of principles, that the idea of coherence of a system 
of rules with a set of underlying principles is significant and not merely trivial, 
as in the previous understanding, where rules cannot but cohere with the prin-
ciples abstracted from such rules.
It is interesting to inquire into the relations between coherence (understood 
as a relation between a system of rules and a system of principles) and the pre-
viously examined properties of completeness and consistency of both systems. 
First, the simplifying function of coherence seems to be connected to weak com-
pleteness, in that it allows one to build an axiomatic basis, liable to be developed 
into all its relevant consequences. However, it should be kept in mind that this 
function of coherence only holds in a proper sense for the first understanding of 
principles, and that on this reading any system of rules is trivially coherent with 
the principles abstracted from them. 
Now, on the second understanding of principles, and when it comes to the 
system of rules, coherence is not necessarily connected to strong completeness 
or consistency (at least with respect to negation) of this latter system. In fact, 
coherence sometimes seems to require the introduction of a new (unexpressed) 
32 See Guastini (2011: 188-189).
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rule in the system which is in conflict with an older (expressed) one: the famous 
case Riggs v. Palmer33 is a clear instance of this phenomenon. 
As everybody knows, the legal question in Riggs v. Palmer was whether 
Elmer Palmer, who had poisoned his grandfather, was entitled to inherit his 
grandfather’s assets under the Statute of Wills which was in force, at that time, 
in the state of New York. The Statute of Wills was completely silent about un-
worthy beneficiaries, and only provided that one was entitled to inherit if one 
was so vested by a valid will. One could maintain – as Gray J. actually did in his 
dissenting opinion – that Elmer was entitled to inherit under both the Statute 
of Wills and his grandfather’s will. However, the majority opinion, drafted by 
Earl J., offered a different solution: the U.S. legal system could not admit, from 
an axiological point of view, the extremely unjust result of letting the inherit-
ance go to someone who killed his own testator. The rejection of such a solu-
tion was founded on the presumed existence, within the U.S. legal system, of 
the principle according to which ‘No man should profit from his own wrong’. 
On the basis of this principle, an unexpressed rule was introduced according 
to which ‘The unworthy beneficiary, though granted the inheritance by a valid 
will, is not entitled to inherit’.34 Manifestly, such a rule conflicts with another 
rule (regarded as in force until the Riggs case) according to which a valid will is 
sufficient in order to inherit the testator’s assets. This case exemplifies a situa-
tion where the exigencies of axiological coherence brought the legal applicators 
to introduce a rule into the system which made it normatively inconsistent.
If this is correct, coherence sometimes runs counter to the preservation of 
strong completeness and consistency of the system of rules, for it can encour-
age legal applicators to introduce rules which bring about new inconsistencies, 
which were not present before. Recall that strong completeness has been de-
fined as the situation where no sentence can be added to the axiomatic basis of 
a certain set without rendering it inconsistent. It is easy to see that coherence of 
the system of rules with a set of underlying principles is an exigency which can 
(and do) often conflict with this ideal, for it requires the continuous introduc-
tion of new (‘apocryphal’) rules on the basis of axiological judgments regarding 
the inadequacy of the existing rules.
This last conclusion has to be balanced with another usual role assigned to 
principles: sometimes we appeal to a set of underlying principles to give priority 
to one normative solution within a set of inconsistent solutions derivable from 
the system of rules. In such case, principles would be used to restore consistency 
in the system of rules. Moreover, though this idea deserves a thorough justifica-
tion that we shall not try to offer here, it may be suggested that coherence with 
33 115 N.Y. 506 (1889).
34 Indeed, such a rule is a defeasible and frequently defeated one, as is shown in Schauer (2009: 
34).
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a set of underlying principles could even stop whatever consequences to follow 
from a normative contradiction within the system of rules, for they would not 
match the requirements set by those underlying principles. If this is correct, co-
herence would thus work as a warrant for absolute normative consistency.
When it comes to the system of underlying principles – again, on their sec-
ond understanding – a meaningful predication of coherence of a set of rules 
regarding such system seems to assume that the latter is not trivially complete 
or absolutely inconsistent, since an absolutely inconsistent/trivially complete 
system of principles justifies any rule whatsoever. 
However, consistency is a problematic property when it is predicated of a 
system of principles, if principles are regarded as defeasible norms. Indeed, if 
– as we mentioned before – the antecedent of a principle is not taken to be a 
sufficient condition (or to provide sufficient conditions) of its consequent, their 
proper formal representation should be something like:
[25] (p > Oq)
Here the corner (‘>’) is used to symbolize a defeasible conditional, one which 
neither allows modus ponens nor strengthening the antecedent.35 For this rea-
son, a system of principles can contain [25] together with
[26] (p > P~q)
with no inconsistency derivable from them, even for p-worlds, since p does 
not constitute a sufficient condition for the provided consequence to follow.
It would seem, then, that the system of principles cannot be inconsistent 
by definition, since it is composed of defeasible normative standards. If it is so, 
however, it is not clear, from a logical point of view, how the relation of coher-
ence between the system of principles and the system of rules should be framed. 
An option consists in regarding the principle as a “tentative confirmation” of the 
normative consequence provided by a certain set of rules. The more confirmed 
a certain normative solution provided by a rule is, the stronger appears the re-
lation of coherence between underlying principles and such a rule. However, 
what is important to stress here is that the idea of competing confirmations 
presuppose inconsistencies of solutions within the system of rules. Coherence 
relations would thus be, or might be used, as tools liable to solve antinomies af-
fecting sets of rules.36
As a conclusion, coherence, as a theorized logical item, may be said to be a 
rather ambiguous combination of weak completeness, and possible absence of 
consistency and strong completeness of a system of rules regarding a non-trivi-
ally complete/non-absolutely inconsistent system of underlying principles. 
35 For a recent survey on defeasible norms, see Ferrer Beltrán & Ratti (2012).
36 This is an idea expressly maintained by Dworkin (1978: 27).
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—Acknowledgments.— We would like to thank Andrej Kristan, Riccardo Guastini, Jan 
Woleński, as well as two anonymous reviewers, for very helpful comments on prelimi-
nary drafts of this paper.  
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