Time and norms: a formalisation in the event-calculus
RAFAÉL HERNÁNDEZ MARÍN - GIOVANNI SARTOR
Introduction
Temporal concerns are ubiquitous in the law, and have been addressed in various ways by
legal doctrine and legal theory (e.g., when dealing with periods, deadlines and conditions,
or with prescription and usucapion or with the creation and derogation of legal norms).
However, very few researches has expressly and directly addressed the link between law
and time in a formal framework (cf. for all, Bulygin 1982). This gap has recently been
addressed by AI & law research (cf. for all Mackaay et al. 1990), which has been employing
various AI formalisms for addressing the temporal aspects involved in representing and
processing legal knowledge.
Among those formalisms a special place must be recognised to the event calculus both
for its representative power (which makes it intuitively suitable for many legal contexts)
and for its simplicity (which makes it easily accessible also to an audience having a limited
formal training). The event calculus has promoted many studies in AI, where the seminal
paper by Kowalski and Sergot (1986) has been followed by a large number of contributions,
providing various axiomatisations, extensions and applications. There have been a few
event-calculus applications in the legal domain (cf. Provetti 1992, Sergot 1995), but we
think that the resources provided by the event-calculus could provide further insights into
legal issues, and possibly support some useful applications.
Our paper will not move into applications (although it includes a Prolog program) but it
will link between an analysis of certain temporal aspects of the law, and the definition of a
computable representation.
1. External and internal time of legal norms
In our analysis we will use the language of first order logic, including the usual
propositional connectives and quantifiers (this approach was advocated, among others, by
Quine 1960). We are aware that philosophical logic and artificial intelligence (cf. for all
Thomason 1984) has developed many elegant and sophisticated formalisms for dealing with
time. However, we prefer to limit ourselves to first order logic since this representational
device is sufficient for our concerns, and it allows as to express most clearly the temporal
structures in which we are interested.
In addition to the resources of first order logic, we use a special connective, , to
express normative implication (the relation between the antecedent and the consequent in
conditional norms)1. Sometimes we will write predicates as sequences of words and will
put individual terms (variable or constants) within the corresponding predicate, in order to
obtain a natural language-like formulation. We assume a discrete representation of time,
that is we represent time as an infinite succession of discrete instants ... Tj ... Tk ... We
denote as Ti+1 (Ti-1) the instant immediately following (preceding) Ti.
We only consider conditional norms, that is those having the form
Condition  Effect,
where the effect is assumed to be produced or brought about by the realisation of the
condition.
In particular, we focus on event-state norms, that is norms which establish that a certain
legal state (or property) starts to hold when a certain event takes place. To indicate that
event E happens at time T we write at T, E and similarly, to indicate that situation S holds
at T, we write at T, S. So, for example, the norm
XT ( at T, X is born in Italy 
at T, X is an Italian citizen )
establishes a new situation (qualification), the Italian citizenship of X starting at time T, as
a consequence of the event of X's birth in Italy exactly at that time (rules on Italian
citizenship are in fact more complex than this suggests, but the example suffices for present
purposes). Deontic norms, conferring duties or permissions, can also be represented as
event-state norms. For example the norm
XT (at T, X damages Y =>
at T, X has the duty to repay Y )
establishes new instances of a normative state (e.g., Philip being obliged to repay Peter) as a
consequence of an instance of an event-condition (e.g., Philip damaging Peter). We do not
address the issue of the logic of deontic modalities.
We do not consider the possibility of providing an event-calculus account of norms
with are not reducible to the event-state form, and in particular of norms having an eventevent structure. While leaving this to future research, we prefer to focus here on the eventstate structure, which is sufficient for dealing with the examples here considered.
Our analysis will be centred upon the distinction of two temporal profiles in legal
norms.
1We do not address the difficult problem of the meaning of normative implication. For our purposes it is
sufficient to assume that  behaves as a conditional connective in modus ponens inferences (from p  q
and p, we derive q).
2
One is the so-called external time) of a legal norm, i.e., the answer to the question:
during what time-interval(s) has the norm been included in the legal system? This question
concerns, in other terms, the interval during which the legal norm has been valid (by
validity of a norm we mean here its partaking to the legal system).
The other time is the so-called internal time of a legal norm, which includes all
temporal issues which are involved in the application of a valid norm. In particular it deals
with the answer to the question: what time-interval(s) was the norm intended to cover? This
question concerns the interval in which the conditions contemplated by the norm should
happen in order that they will produce the norm's effect, i.e., the subsumption interval or the
norm (cf. Hernández Marín 1996, 29-30).
Interestingly, those legal notions present a similarity (which we cannot examine here)
with certain notions which have been developed within research on databases, and in
particular with the notions of 'valid time' (the time at which something occurs) and
'transaction time' (the time at which this is recorded in the database or belief system).
2. The internal time of legal norms (subsumption time)
In the following paragraphs we will try to make the best use of our rather scanty formalism,
by applying it to the different aspects involved in what we called the internal time of legal
norms.
2.1. Universal temporal quantification covering both condition and effect
The temporal structure more frequent in legislation is characterised by a universal time
quantifier covering the whole norm (both the condition and the effect). Those norms are
temporally abstract: whatever instance of the condition produces a corresponding instance
of the effect. They do not include any specific limitation of their subsumption interval. This
does not mean that their subsumption interval necessarily covers all the past and the future,
since it may be limited according to general rules or principles, as we shall see in the
following.
Let us consider first a personally individualised norm, which is temporally abstract.
Thus, for example, the father of John, in a morally questionable, but probably effective,
attempt to stimulate his child's school performance, can issue the following promise:
whenever you get a good mark, you are entitled to receive £ 1. This will amount to the
formalisation
T( at T, John gets a good mark 
at T, John is entitled to receive £ 1 )
3
Temporal abstractness is the most sensible interpretation for all those cases in which no
temporal dimension is expressly specified in legislative language. So, for example, the
norm which says “he who is born from Italian parents is an Italian citizen”, would stay for
T X( at T, X is born of Italian parents  at T, X is an Italian citizen )
Such a general-abstract norm is doubly quantified, over instants (abstractness) and over
persons (generality). The two quantifiers cover both the condition and the effect.
2.2. Partial abstractness
The temporal domain of a norm may explicitly be restricted to a specified interval. Events
falling outside the specified time interval do not satisfy the norm condition, and therefore
do not trigger the norm effect. Let us consider for example a rule establishing a £ 10 fine for
anyone who parks in front of the station from 10/6/1997 to 16/6/1997 (perhaps, for
example, a sporting event is taking place over those days). This could be formalised as:
T {10/6/1997  T  16/6/1997}
X ( at T, X parks in front of the station  at T, X is liable to a £ 10 fine ).
2.3. Existential temporal quantification of the condition or the effect
Some prescriptions contemplate events which must happen within a certain interval of
time. The temporal structure of the condition of those norms can be captured by means of
an existential quantifier covering only the antecedent (which corresponds to a universal
quantifier covering the whole norm, when the quantified variable does not occur in the
consequent). Consider, for example, an insurance rule saying that if one causes any accident
during 1997, one's insurance fee is doubled from the beginning of 1998:
X [ T(1/1/1997  T  31/12/1997 
at T, X causes an accident) 
at 1/1/1998, X’s insurance fee is double what it was]
In order to satisfy the condition of such a norm, it is sufficient that a person causes just
one accident in one instant falling within the given time interval (he/she does not need to
continue causing accidents during the whole year).
It may be doubted whether norms exist having an effect which is existentially
quantified in its temporal variable. Such norms would establish that a certain legal effect is
to happen within a certain interval, without establishing the exact moment in which the
effect is to take place. Consider for example the absurdity of a norm stating that if one
marries a citizen, then one shall acquire citizenship at some moment included in the
following year (such a norm does not tell when this is going to happen, within the
considered year):
4
X Y T1 [ ( at T1, X marries Y 
at T1, Y is an Italian citizen ) 
T2 ( T1 T2  T1 + 1 year 
at T2, X is an Italian citizen ) ].
Norms prescribing that a certain performance is to be accomplished within a certain
time length do not fall within the (empty) category of the norms having a temporally
undetermined effect. On the contrary, those norms establish their effect, i.e., the beginning
of the obligation, for a determined moment, although the obligation concerns an action to
be performed in any moment within a certain interval. Therefore, in such norms the time of
the effect is determined, although the time of the performance is partially undetermined.
Consider for example the following clause, agreed by John to Tom: if John gives Tom £
100 today, then Tom is obliged to give John back the money tomorrow. In our model, such
a clause would be represented as follows (assuming that today is the 14/4/1997).
T1 {0am on 14/4/1997  T1 < 0am on 15/4/1997}
( at T1, Tom gives John £ 100 
at T1, it_is_obligatory_that
[ T2 (0am 15/4/1997  T2 < 0am 16/4/1997
 at T2, John gives Tom back £ 100)]
Note that in such a case the effect, i.e., the birth of the obligation, is determinate and
indeed immediate (it takes place at T1), though the time of performance is only fixed within
a subsequent interval. It would have been wrong to formalise the consequent of this
sentence as
T2 [ 0am on 15/4/1997  T2 < 0am on 16/4/1997 
it_is_obligatory_that (
at T2, John gives Tom back £ 100 ) ]
putting the whole effect (the deontic operator included) within the scope of the
temporal quantifier. This would mean that there exists a precise instant T2 (although the
sentence does not tell the time of T2) in which the performance should take place
(performance in any other instant would be irrelevant). On the contrary, the clause is
indifferent to the specific instant of Tom’s performance, provided that his performance
occurs within the indicated interval.
2.4. Instantaneous conditions or effects
Norms that specifically indicate the unique precise time when their condition is supposed to
happen are very rare, if they exist at all. Let us thus consider a quite artificial example: a
5
contract states that if the buyer, Mr Smith, pays the price today, at 3 p.m. on 15/7/1997, he
becomes the owner of the house of the seller (Mrs Jones). Such a sentence could be
represented as:
at 3 p.m. on 15/7/1997, Mr Smith pays the price 
at 3 p.m. on 15/7/1997, Mr Smith is the owner of the house of Mrs Jones
Note that we have specified both the time of the condition and the time of the effect
(even if the latter is not explicitly indicated in the corresponding natural language
expression). Note that even instantaneous norms may include persistency assumptions (it is
very hard to express any legal content without those assumptions). The norm just says that
Mr Smith becomes the owner of the house at 3 p.m. on 15/7/1977, but we also assume that
he remains its owner subsequently. Only if an event terminating Smith’s ownership
happens, will this assumption be abandoned.
2.5. Displaced effect
Certain norms may establish their effect for a time which is earlier or later than their
condition. In this regard let us consider first norms establishing a non deontic qualification,
i.e., the rule as according to which norms become applicable (come into force) 15 days after
their publication. The effect established by this rule (the becoming applicable of another
norm) is established to happen at time which is 15 days after the time when the condition
contemplated by this rule (the publication of that norm) took place:
TN ( at T, N is published 
at T + 15 days, N is applicable)
Similarly, the effect may precede the condition, as in the norm stating that the
recognised son is such from his birth:
T1T2 { XY[ ( at T1, X is born 
at T2, Y recognises Y as his son )  at T1, X is son of Y ] }.
In deontic norms, temporal displacement may just concern the performance of the
prescribed action. Consider for example the rule that Paul must wash his hands after
dirtying them:
T1 ( at T1, Paul dirties his hands 
at T1, it_is_obligatory_that [
Te ( at T2, Paul washes his hands 
T2 > T1 ) ].
6
In this formalisation the effect (the obligation) is contemporary with the realisation of
the norm condition (at T2), although the prescribed action was to be performed before.
2.6. Analytical and synthetical representations of the internal time
Legal language may adopt two quite different approaches in the representation of internal
time, and in particular, for delimiting the subsumption interval.
1. a synthetical approach, in which all temporal and substantial elements of the norm are
represented within the same sentence;
2. an analytical approach in which one sentence represents the substantive content of the
norm, and other sentences specify its temporal features.
In particular the analytical approach is usually adopted for specifying what we have so
far called the subsumption interval of the norm. In the above example, instead of having
just one sentence:
1. During the period from 10/6/1997 to 16/6/1997, anyone who parks in front of the
station is liable to a fine,
we could have had two sentences:
1. Anyone who parks in front of the station is liable to a fine.
2. Norm 1 is applicable2 (applies, has effect ...) from 10/6/1997 to 16/6/1997.
or even three sentences:
1. Anyone who parks in front of the station is liable to a fine.
2. Norm 1 starts to be applicable at 10/6/1997.
3. Norm 1 ceases to be applicable at 16/6/1997.
In analytical representations, norms must be able to speak about other norms. This
means that we must have a naming convention. We shall simply put in front of each rule a
univocal label to be used as its name. Let us adopt for the norm above the name pfs
2Throughout the rest of the paper we will say that a norm is ‘applicable' at a moment T to mean that T
falls within its subsumption interval (i.e., that the norm is concerned with events happening at time T).
Therefore, norm 2 states that the subsumption interval of norm 1 starts at T1, by prescribing that norm 1
starts to be applicable at T1. Similarly, norm 3 states that this interval ends at T2 by prescribing that norm 1
ceases to be applicable at T2. By applicable we just mean "temporally applicable" in the sense of "capable of
subsuming". We do not address here other aspects of the "application" of legal norms, and are using the term
applicable in this sense just because we could not find any appropriate (and not too awkward) more specific
term.
7
(parking in front of the station). So, we translate the three sentences above into the
following:
pfs: XT1( at T1, X parks in front of the station 
at T1, X is liable to a fine ).
pfsA1: at 10/6/1997, pfs becomes applicable
pfsA2: at 16/6/1997, pfs ceases to be applicable
The analytical approach has a number of advantages. Firstly, it is modular and clear,
since different items of information are expressed in different sentences. This advantage is
particularly significant when using formal languages: we can avoid writing awkward
complex formulae.
Secondly, and most importantly, temporal elements do not need to be always directly
specified, but they can be made dependant upon future and possibly not yet known facts.
So, a norm N1 may be accompanied by a norm N2 stating that N1 is going to become
applicable when a certain authority adopts a certain implementation measure or when a
certain event takes place (e.g., a situation of emergency). In particular, general rules on
temporal features can be issued, such as those stating that a period after its enactment must
elapse, for any norm to become applicable. For example, the general rule establishing that
statutory norms become applicable 15 days after their being published may be represented
as follows
publication: NT( at T, N is published 
at T+15 days, N is applicable ).
Analytical representations of time have some drawbacks. In particular, more than one
sentence needs to be considered in order to determine both the substantial content of the
norm and its subsumption interval. Moreover, inferences from analytical representations
must take into account the interaction between substantive norms and norms which regulate
subsumption intervals. However, we stick here to the analytical mode since our purpose is
to model the temporal structures usually adopted in legal language and in legal reasoning.
3. Time and validity
The notion of applicability of a norm, as defined above (let us recall that a norm is
applicable during its subsumption interval) must be distinguished by the notion of the
validity of a norm, by which we mean its inclusion in the considered legal system. A norm
may be valid, but not yet be applicable, as when a norm is intended to cover only events
happening some time after it is issued, or a norm may be applicable to events which
preceded it validity, as when it is given retroactive effect.
8
All legal norms, or at least all positive legal norms, have a temporally restricted
validity: the validity of a norm N starts at a certain time T1 and may terminate at a
subsequent time T2 (for reasons of simplicity we do not consider the possibility that a norm
lasts longer that one validity interval). By saying that <T1, T2> is the (maximal) validity
interval of N we mean that:
1. for all T such that T1  T  T2, N is valid at T,
2. N is not valid in T1-1,
3. N is not valid in T2+1.
The beginning of the validity interval of any legal norm is determined by other rules
(which may be explicit or implicit, and may be legal or conventional, as we shall see in the
following section). For example, in some legal systems, such as Italy or Spain, a general
rule establishes that the validity of legal norms begins at the time of the publication. Such a
rule can be obtained by a (very liberal) interpretation of the Constitutional provisions
ensuring the publication of norms. Alternatively, it can be considered a generally accepted
basic convention, implicit in legal practice (corresponding to the common sense convention
or moral requirement that every prescription only holds from the moment in which it is
made accessible to its addressees).
The end of the validity interval is established by derogation rules (which also may be
explicit or implicit and may be legal or conventional). Specific legislative sentences are
usually required for explicit derogation of determined norms. Among general derogation
rules we can just mention those establishing that more recent norms tacitly derogate
previously valid ones in the case of any contradiction between them.
The usual treatment of validity in legal reasoning is based upon a persistency
assumption. The fact the N is valid throughout the interval <T1, T2> cannot be derived just
from the fact that N starts to be valid at T1 and ceases to be valid at T2. Those sentences
just tell us that N is valid at T1 (and was not valid at T1-1) and that N is not valid at T2+1
(and was valid at T2). They do not tell us anything about all instants T such that T1<T<T2.
To establish the validity of N in all those instants, we need the assumption that, once N
becomes valid, it continues to be valid until it is deprived of its validity. Such an
assumption is part of “legal common sense” (including various tacit conventions which
underlie legal practice and legal science).
3. Rules attributing applicability and validity
The representation we have adopted seems to be undermined by an infinite regress in search
for both applicability and validity. As it is well known, the recursive circularity of validity
has been frequently considered by legal theorists (and especially by Hans Kelsen). A norm
N1 is valid only if it is qualified as valid by a norm N2 which is valid (and applicable); N2
9
is valid only if it is qualified as valid by a norm N3 which is valid (and applicable); N3 is
valid ... In our framework exactly the same problem emerges as far as applicability is
concerned. N1 is applicable only if it is qualified as applicable by a norm N2 which is (valid
and) applicable; N2 is applicable only if it is qualified as such by a norm N3 which is (valid
and) applicable; N3 is applicable...
One solution to both infinite regresses consists in observing that the legal system is
underpinned by social-linguistic-moral rules (conventions) which do not partake in the legal
system itself, and therefore do not require the qualifications of legal validity and legal
applicability. Those rules allow us to stop the recursion: N1 is valid (applicable) because it
is so qualified by the non legal rule N2, for which it makes no sense to consider the
requirement of legal validity and applicability. Those underpinning rules can obviously be
assimilated to Hart’s recognition rule (Hart 1961) or to Kelsen’s basic norm of (Kelsen
1960). However, we do not insist that there be just one such rules, and deny that those
norms are legal.
As an example of such rules, let us consider the following
NT ( at T, N is published 
at T, N is valid )
which makes validity dependant upon publication according to the view of Hernandez
Marín (1966, 36ff.).
4. A formalisation in the event calculus
In the following paragraphs, we shall consider the possibility of translating our temporal
analysis of legislation into a formalism which is computable, i.e., into a computer program
which can automatically perform temporal reasoning.
Basic event calculus
The axioms of the event calculus are logic programming clauses, and can be executed
by a Prolog interpreter. Those axioms are therefore rules of this form (we use the syntax
usually adopted for Prolog):
c :- p1, , pn.
where :- represent the reversed conditional, “,” is the conjunction, c is an atomic
formula and the p1  pn are literals (atomic formulae or negations of atomic formulae).
Negation is to be understood as negation by failure (which we denote as not), rather then as
classical logical negation. This means that the formula not p does not assert that p is false,
10
but rather that there is no evidence that p, i.e., that p is not derivable from the knowledge
represented in the program.
The basic concept of the event calculus it that of event. Events initiate or terminate the
periods in which certain states of affairs (properties, relations, situations, etc.) hold.
Therefore two type of domain knowledge must be provided in the event calculus:
1. specific sentences that certain event-instances have happened,
2. general rules indicating what event-types generate what states of affairs.
In our formalisation the difference between event-instances and event-types is simply
indicated by the fact that descriptions of event-types contain free variables and descriptions
of event-instances contain only constants. For example X is_born_in_Italy refers to an event
type while giovanni is_born_in_Italy refers to an event-instance (from now on we will
follow the usual Prolog convention of writing individual variable as sequences of characters
starting with upper case letters and both individual and predicate constants as sequences
starting with lowercase letters). General rules express the capability of any instance of a
certain event-type to initiate or terminate the corresponding instance of a certain state-type:
X is_born_in_Italy initiates X is_Italian.
X acquires_non_italian_citizenship terminates X is_Italian.
The event calculus has an in-built capacity to deal with persistence. The following law
of inertia is in fact assumed: states of affairs which were started by one event are assumed
to persist in the future indefinitely, unless they are interrupted by a terminating event. For
our purposes, an elementary treatment of persistency is sufficient, since for us:
1. Only persistency in the future is relevant (we are not interested in deriving past states of
affairs from present ones).
2. Each event is expressly located in a precise moment.
Let us now introduce a formalisation of the event calculus. The first rule establishes
that the state of affairs S holds at time T2 if S was initiated at a time T1, antecedent to T2
and there is no evidence (negation by failure) that the persistency of S between T1 and T2
was broken (interrupted)3.
holds_at(S, T2) :initiated_at(S, T1),
T1 =< T24,
3What happens to S before T1 or after T2 is irrelevant for its persistence between T1 and T2.
4We write =< instead as , as is usual in programming languages.
11
not broken(S, between(T1, T2)).
The second rule establishes that state S was broken between T1 and T2, if S was
terminated between T1 and T2:
broken(S, between(T1, T2)):terminated_at(S, T),
T1 =<T, T < T2.
The third rule concerns initiation. It established that a state S initiates at time T if an
event able to initiate S happens at T:
initiated_at(S, T) :E initiates S,
happens_at(E, T).
The fourth rule has a similar function as far as termination is concerned:
terminated_at(S, T) :E terminates S,
happens_at(E, T).
Limits of the event calculus in the legal domain
Let us apply our event calculus formalisation to the following premises:
happens_at(giovanni is_born_in_italy, 25/02/59).
X is_born_in_italy initiates X is_italian.
X acquires_non_italian_citizenship terminates X is_italian.
From such premises (and the above axioms of event calculus) we can derive that
Giovanni is Italian in any moment subsequent to the 25/02/59.
Let us now add the information:
happens_at(giovanni acquires_non_italian_citizenship, 15/6/1985).
Given this additional premise we are no longer able to derive holds_at(giovanni
is_italian, 15/9/1997), since the state giovanni is_italian was interrupted between
25/02/1959 and 15/9/1997 (precisely at 15/6/1985). On the other hand it is still possible to
derive that Giovanni is Italian at any moment between 25/02/1959 and 15/6/1985.
As this example makes clear, the formalisation of legal norms as initiation or
termination rules captures the assumption already mentioned when event-state norms were
introduced: from the lawyers’ perspective, legal states of affairs (for example, an
12
obligation) come into existence as a consequence of certain facts (e.g., any fact usually
listed among the sources of obligations) and cease to exist as a consequence of other facts
(e.g., any fact usually listed among the causes extinguishing obligations).
5. Some extensions of the event calculus
The representation of legal norms in the event calculus suffers from some limitations. Let
us address three major restrictions
1. Event calculus assumes that all norms are temporally universal, whereas legal rules may
have a (partially of totally) limited subsumption interval.
2. Event calculus initiate and terminate rules are assumed to be always valid, whereas legal
rules are limited in their temporal validity.
3. Event calculus makes a state start as soon as the conditioning event happens, whereas a
legal effect may precede or follow the corresponding condition.
In the following paragraphs we will overcome all those limitations, by introducing a
representation formalism and an inference mechanism especially devised for legal norms.
These will integrate the basic representation and inference mechanism of event calculus.
Temporally limited applicability
According to the analytical approach, we will preserve the basic structure of event
calculus for the substantial content of legal norms, and will regulate separately their
subsumption interval (applicability). For this purpose first need to provide the syntax of
legal norms with a naming method. Let each legal rule be preceded by his name, in the
form:
N: Condition l_initiates Effect.
where N is the name of the rule and l_initiates (legally initiates) expresses the relation
between legal conditions and legal effect. Note that the only syntactical difference between
legal rules and common event calculus rules is just the fact that the legal rules are given
names.
The subsuming capacity of a legal norm N1 needs to be conferred by a (legal or non
legal) rule N2, which fixes the beginning of the subsumption interval. However, the syntax
we have so far provided for event-state norms, does not provide any intuitive way for
dealing for unconditioned norms. In fact unconditioned norms provide no event, and
therefore cannot rely on any event-time for determining the time of the effect. Let us
consider, for example, the norm stating: The exception to the Data Protection Act (named
edpa) is applicable from the 1/1/1988”. We express this norm in a state-only form (the
event is missing):
13
edpaA1:
is_applicable(edpa) l_initiated_at 1/1/1988.
Similarly, for indicating that the applicability of a norm N1 terminates at a certain time,
for example that the application of the exception to the Data protection act ends on the
10/5/1990, we introduce an additional norm stating that:
edpaA1:
is_applicable(edpa) l_terminated_at 1/1/1988.
Let us consider finally the norm concerning the fine for parking in front of the station
from 10/6/1997 to 20/6/1997, that is:
 T{10/6/1997  T  20/6/1997}
[ X ( at T, X parks in front of the station 
at T, X is liable to a fine ) ].
In our formalism, this norm (which we call pfs) is represented as the combination of
three sentences.
pfs: X parks_in_front_of_the_station l_initiates X is_liable_to_a_fine.
pfsA1: is_applicable(pfs) l_initiated_at (10/6/1997).
pfsA2: is_applicable(pfs) l_terminated_at (20/6/1997).
Now we extend our reasoning mechanism in order to take into account this extension of
our knowledge representation. Besides the initiate_at rule for general event calculus, we
introduce an additional initiated_at rule specifically intended for legal rules (we do the same
for the terminated_at rule):
initiated_at(S, T):N: E l_initiates S,
happens_at(E, T),
holds_at(is_applicable(N), T).
Let us analyse this clause. For a legal state S to start, it is not only required that there is
a legal norm N according to which event E is able to produce state S, and that an instance of
event E takes place. In addition, it is also required that N is applicable at T (i.e., that T is
included in the subsumption interval of N).
Similarly, we have to introduce a clause for unconditioned norms.
14
initiated_at(S, T):N: S l_initated_at T,
holds_at(is_applicable(N), T).
In order to avoid an infinite regress we need some statements on applicability that are
not analytically represented legal norm. For example, a statement
initiated_at(is_applicable(n0), 10/6/1900)
gives applicability (from outside the legal system) to a legal norm n0, which might
possibly legally initiate the applicability of other legal norms.
Validity in time
With validity we proceed as with subsumption capacity. Rules conferring validity are
initiation sentences:
pfsV1: valid(pfs) l_initiated_at 1/1/1997.
Similarly, derogation norms are termination sentences, specific or general:
pfsV2: valid(pfs) l_terminated_at 10/5/1998
derogation_by_conflict:
N1 conflicts_with_subsequent_norm N2
l_terminates valid(N1).
Obviously, we need some basic validity rules (at least one) which are not required to be
valid, in order to produce their effect (the qualification of some other norm as valid). For
example, we could just assume that the fundamental rule for validity is the following, which
makes validity start at publication (note that this is not a legal norm, and the production of
its effect is not conditioned on its being valid and applicable)5:
published(N) initiates valid(N).
To take into account validity in our reasoning mechanism we need a modification of the
legal version of the initiated_at (and of the corresponding terminated_at clause).
5This representation could be easily extended to enable the representation of both the event and the
concomitant conditions necessary for it to produce its effect (cf. Sergot 1995). This would allow us to
formulate validity norms according to which the event of publication produces validity only if certain
requirements are satisfied, which can be different in different legal systems and according to different legal
theories (for example, being issued by a legally empowered authority, respecting certain procedural or
substantial rules ...).
15
initiated_at(S, T1):N: E l_initiates S,
happens_at(E, T1),
holds_at(is_applicable(N), T1),
holds_at_or_started_after(valid(N), T1).
Let us analyse this clause. As in the formalisation introduced above, it requires that:
1. there is norm N which states that event E is able to produce state S,
2. an instance of event E takes place, and
3. N has subsumption capacity at the time when the event took place.
Moreover it also requires that:
4. N is valid at the time of the event or starts to be valid after that time.
This last requirement corresponds to the fact that invalidity behaves asymmetrically in
regard to the past and to the future. A norm may produce its effect in a time antecedent to
the beginning of its validity (as is the case for retroactive norms), while it cannot apply to
situations subsequent to the end of its validity. However, effects already produced in the
past are not eliminated by the fact that the norm ceases to be valid: abrogation usually is not
retroactive. In other words, for a norm to produce certain effects in a certain time (included
within its subsumption interval), it is sufficient that the norm has been valid in any one
interval after that time, even if that interval is later terminated.
We define similarly the notion of termination:
terminated_at(U, T) :N: E l_terminates U,
happens_at(E, T1),
holds_at(is_applicable(N), T1),
holds_at_or_started_after(valid(N), T1).
Similar rules for initiation and termination are also introduced for unconditioned
norms.
Finally, the predicate holds_at_or_started_after(U,T) is satisfied in two cases: a) if the
predicate U holds at time T, or b) if the predicate U started at a subsequent time (we use
here the symbol “;” which means “or”).
holds_at_or_started_after(U, T) :holds_at(U, T);
started_after(U, T).
16
Finally, U started after T if was initiated in a moment subsequent to T.
started_after(U, T) :initated_at(U, T1),
T < T1.
Non simultaneous norms
The formalisation so far produced still assumes that every legal condition immediately
produces its effect. Instead, as we have considered above (for example, when observing the
rule prescribing that published norms become applicable 15 days later), legal effects may be
antecedent or subsequent to their conditions. To cope with this aspect, we need to specify
when a norm's effect is displaced (when no special displacement is specified, the norm's
displacement will be 0 by default). To do this, we introduce a further infix operator, in ,
which we use for stating displacements.
So, for example let us consider general rule establishes that laws become applicable 15
days after their publication. We will represent it as follows:
vacatio: published(N) l_initiates is_applicable(N) in 15
Since vacatio is an analytically represented legal norm, in order to produce its effect, it
needs to be supplemented by a non-legal rule stating the immediate beginning of
applicability of vacatio.
published(vacatio) initiates applicable(vacatio).
Finally, the initiated_at (terminated_at) and the clause for legal effects need to be
reformulated so as to account for a possible displacement of the effect. In the following
formulation the time of the effect indeed is T0, which is obtained by adding the
displacement D to the event time T1:
initiated_at(U, T0):N : E l_initiates U in D,
happens_at(E, T1),
T0 = T1 + D,
holds_at(is_applicable(N), T1),
holds_at_or_started_after(valid(N), T1).
Let us assume that a norm, let us call it n, is published on the 1/1/1997), which we
express as the following fact:
happens_at(published(n3, 1/1/1997).
17
By applying vacatio to this fact, we obtain the result that n became applicable exactly
on 16 January 1997.
Let us finally consider a full example in order to summarise the normative structures so
far introduced:
vacatio: published(N) l_initiates is_applicable(N) in 15.
moral_damage: X causes_moral_damage_to Y
l_initiates X must_repay Y in 0.
published(N) initiates valid(N).
published(vacatio) initiates is_applicable(vacatio).
happens_at(published(vacatio),1/1/1945).
happens_at(published(moral_damage), 1/1/1980).
happens_at(
john causes_moral_damage_to mary, 1/1/1990).
This information allows us to infer that the moral_damage norm is both valid
(according to the unnamed rule on validity) and applicable from the 16/2/1980 (according
to vacatio, which turns out to be both applicable and valid). This finally allows us also to
infer that John, having caused moral damage to Mary on the 1/1/1990, has to repay her.
6. Conclusion
In this paper we have tried to develop first an analysis of the temporal structure of legal
norms, and then a corresponding computable representation. We hope to have obtained both
a satisfying conceptualisation of some significant features of legal language, and a correct
translation of this conceptualisation into the framework of the event calculus. We also hope
that our analysis provides some notions which may be useful for investigating the relation
between time and law, and in particular for tackling the problem of legal duration. This may
contribute to convince the reader having a legal background that some computational
formalism can provide an insightful approach to certain legal issues. Sometimes (hopefully
also in our case) the famous observation of Bertrand Russell that "a good notation has a
subtlety and suggestiveness which at times make it seem almost a live teacher" may prove
to be true. Finally, we hope that the reader may forgive the preliminary and provisional
nature of our work, considering that we have dealt with features of temporal reasoning to
which legal theory has so far dedicated a very limited attention. In fact, most studies of time
18
in law have so far mainly addressed validity without considering the "internal" temporal
aspects of legal norms.
Much work remains to be accomplished in order to complete the framework here
sketched, in the spirit of event-calculus:
1. an analysis of the temporal dimension of different types of norms, which are not
reducible to the event-state model here considered;
2. a more refined treatment of events, which also considers their length;
3. a treatment of the temporal aspects involved in action, especially those included within
deontic operators;
4. a computable formalisation of those features of our formalisation that are not amenable
to the extension of the event calculus here provided.
Bibliography
Bulygin E. 1982. Time and Validity. In Deontic Logic, Computational Linguistics and
Legal Information Systems. Ed. A.A. Martino, 65-81. Amsterdam: North Holland.
Hart, H.L.A. 1961. The Concept of Law. London: Oxford University Press.
Hernandez Marín, R. 1996. Dos lecciones de filosofía del derecho. Murcia: DM.
Kelsen, H. 1960. Reine Rechtslehre. Wien: Franz Deuticke.
Kowalski, R.A., & M.J. Sergot. 1986. A Logic-Based Calculus of Events. New Generation
Computing 4: 67-95.
Mackaay, E., D. Poulin, J. Frémont, C. Deniger, & P. Bratley. 1990. The Logic of Time in
Law and Legal Expert Systems. Ratio juris 3: 254-271.
Quine, W, van O. 1960. World and Object. Cambridge (Mass): MIT.
Provetti, A. 1992. The Law of Contracts in the Event Calculus. In GULP'92 - Proceedings
of the Ninth Italian Conference on Logic Programming. Milano: Clup.
Sergot, M.J. 1995. Unpublished slides.
Thomason R.H. 1984. Combinations of Tense and Modality. In Handbook of Philosophical
Logic. Volume II. Extensions of Classical Logic. Ed. D.M. Gabbay, & F. Guenthner,
135-165. Dordrecht: Reidel.
Appendix. The Prolog program
Let us now summarise the results of our work in a Prolog program, which we build by
combining the clauses previously introduced. Such a program should run on any Prolog
interpreter, after executing the following operator definitions (or similar ones):
operators:
19
op(100,xfx, initiates),
op(100,xfx, terminates),
op(100,xfx, l_initiates),
op(100,xfx, l_initiated_at),
op(100,xfx, l_terminates),
op(100,xfx, l_terminated_at),
op(90, xfx, in),
op(110,xfx, ':').
To keep the program simple, we avoid any further "syntactic sugar". Therefore, instead of
writing dates in the usual format 22/10/1990, we write them 19901022. Similarly, we stick
to the functional notation for predicates expressing legal knowledge. So instead of X
causes_moral_damage_to Y we write causes_moral_damage_to(X, Y).
/* PROGRAM NORMS AND TIME */
/* General Event Calculus */
holds_at(S, T2) :initiated_at(S, T1),
T1 =< T2, !,
not broken(S, between(T1, T2)).
broken(S, between(T1, T2)):terminated_at(S, T),
T1 =<T, T < T2.
initiated_at(S, T):E initiates S,
happens_at(E,T).
terminated_at(S, T) :E terminates S,
happens_at(E,T).
/* Legal Event Calculus */
initiated_at(S, T):N : E l_initiates S in D,
happens_at(E, T1),
T = T1 + D,
20
holds_at(is_applicable(N), T1),
holds_at_or_started_after(valid(N), T1).
initiated_at(S, T):N : S l_initiated_at T,
holds_at(is_applicable(N), T),
holds_at_or_started_after(valid(N), T).
terminated_at(S, T) :N : E l_terminates S in D,
happens_at(E, T1),
T = T1 + D,
holds_at(is_applicable(N), T),
holds_at_or_started_after(valid(N), T).
terminated_at(S, T) :N : S l_terminated_at T,
holds_at(is_applicable(N), T),
holds_at_or_started_after(valid(N), T).
holds_at_or_started_after(U, T) :holds_at(U, T);
started_after(U, T).
started_after(U, T) :initiated_at(U, T1),
T < T1.
/*Example */
published(N) initiates valid(N).
published(vacatio) initiates is_applicable(vacatio).
vacatio : published(N) l_initiates is_applicable(N) in 15.
moral_damage : causes_moral_damage_to(X, Y) l_initiates must_compensate(X,Y) in 0.
happens_at(published(vacatio),19450101).
happens_at(published(moral_damage), 19800101).
happens_at(causes_moral_damage_to(john, mary),19900101).
21