Content uploaded by L. Thorne McCarty
Author content
All content in this area was uploaded by L. Thorne McCarty on Nov 28, 2022
Content may be subject to copyright.
A LANGUAGE FOR LEGAL DISCOURSE IS
ALL YOU NEED
L. THORNE MCCARTY
PROFESSOR EMERITUS
COMPUTER SCIENCE AND LAW
RUTGERS UNIVERSITY
1. Introduction
My first paper on a Language for Legal Discourse (LLD) was pub-
lished at the International Conference on Artificial Intelligence and
Law in 1989 [24]. I used the language subsequently for several small
projects: [50] [28] [30] [31], and it motivated much of my theoretical
work on Knowledge Representation and Reasoning in those years.
At the time, no one was attempting anything as ambitious as the
“Rules as Code” movement, and thus I never wrote an interpreter for
the entire language or used it to encode a complete statute. But I
think this is a feasible project today. Even without a full-scale imple-
mentation, I think the design choices embodied in LLD provide useful
guidelines for anyone trying to translate legal rules into executable
computer code. I will describe these choices in this short paper.
This document was originally prepared as a position paper for two
conferences: the first Workshop on Programming Languages and the
Law (ProLaLa 2022), which was part of POPL 2022, in Philadelphia,
in January, and the conference on Computational Legal Studies 2022,
in Singapore, in March. See Section 3, infra, for links to the slides and
notes for my presentations at these conferences.
2. What is LLD?
My Language for Legal Discourse (LLD) is . . .
2.1. An Intuitionistic Logic Programming Language,
Among the several major programming paradigms — imperative,
functional, logical, and object-oriented — the best representation for
legal rules, in my opinion, is a logic programming language. This is
(obviously) the position taken by Robert Kowalski, see [51] and [13],
Date: February, 2022.
1
2L. Thorne McCarty
and it is also the position taken by Jason Morris in his recent MS Thesis
[44]. One reason for this preference is that it is easy to encode a proof
theory for various kinds of legal rules in a logic programming language,
such as PROLOG: You get proof search and unification for free in the
meta-interpreter. For an example of how this works, see [37], which
includes code for several meta-interpreters.
However, I have also taken the position for several decades that the
proper setting for logic programming is intuitionistic logic, not classical
logic. To see the difference, consider a Horn clause written as an axiom
in a sequent calculus, as follows:
Q1∧Q2∧. . . ∧Qn`CP
where Cis a context containing the free variables appearing in the
predicates. If all of our axioms are Horn clauses, then the logic is
the same whether it is interpreted in classical or intuitionistic logic.
But in intuitionistic logic, we can write universally quantified embedded
implications, as follows:
Q1∧. . . ∧ ∀y(R1∧R2∧. . . ∧Rk⇒Q)∧. . . ∧Qn`CP
An example from John McCarthy is: “xis a sterile container if every
bug yinside xis dead.” If we allowed this syntactic construction in
classical logic, we would have a full first-order logical language that
requires an unrestricted resolution theorem prover, but in intuitionis-
tic logic we have a proper subset of a first-order language that retains
both the definite-answer-substitution property and the goal-directed
proof procedure that we want for logic programs. Motivated by both
common sense reasoning and legal reasoning, I proposed such an intu-
itionistic logic programming language in [22, 23]. At about the same
time, motivated by applications in programming, generally, Dale Miller
proposed a similar language in [41]. For a full account of Dale Miller’s
work, see [42].
Missing from an intuitionistic logic program, by design, are rules
for disjunctive and existential assertions. We can add these rules sep-
arately, as in [37], or we can take the approach advocated in [38]. A
statute often includes provisions that state necessary and sufficient con-
ditions for a defined predicate, P. For the sufficient conditions, we can
use the proof rules shown above, but in some contexts we need to assert
Pand expand the necessary conditions, which means that we need to
reason with disjunctive and existential assertions. Ron van der Meyden
and I showed, in [38], that we can model this reasoning, semantically,
with John McCarthy’s circumscription axiom [19], but this sometimes
leads to inductive proofs.
A Language for Legal Discourse is All You Need 3
Logic programming also offers a simple way to write default rules:
negation-as-failure. Sarah Lawsky argues persuasively in [16] that most
statutes are drafted and should be interpreted as rules with exceptions,
and she uses Ray Reiter’s default logic [49] to illustrate this point with
a translation of Section 163(h) of the Internal Revenue Code. I took
the following position in [29]: “If we used only stratified negation-as-
failure with metalevel references in our representation language, we
would have a powerful normal form for statutes and regulations,” but
I argued in the same paper that a well-drafted statute should never
contain unstratified default rules. Nevertheless, if unstratified rules
happen to occur in a statute, inadvertently, it may be comforting to
know that the circumscription of intuitionistic embedded implications
offers a reasonable interpretation of the possible inferences [40], with
provable relationships to both the (2-valued) stable model semantics
[10] and the (3-valued) well-founded semantics [52]. For the general
theory, see [25].
2.2. With Sorts and Subsorts,
Here is an atomic formula that appears frequently in my encodings
of legal rules:
(Own ?o (Actor ?a) (Stock ?s))
In this example, an Actor can be either a Person or a Corporation,
and Stock is a subsort of Security. The unification algorithm is re-
quired to respect these sorts, although the details will depend on how
the sort hierarchy is defined. Also, Actor is a count term, while Stock
is a mass term which can have a measure attached to it. For some
examples of legal rules that use mass terms with measures, see [30].
One feature of this syntax which is unusual in a first-order logic is
the variable ?o. Think of ?o as an instance of the ownership relation,
so that Own can be interpreted as either a predicate or an object, de-
pending on the context, and there is no syntactic distinction between
an atomic predicate and an atomic term. Thus, in some contexts,
(Own ?o)
could be an argument in a higher-order expression. One way to for-
malize this interpretation is to define a type theory and a categorical
logic for LLD. We will return to this point in Section 2.7, infra.
2.3. Events and Actions,
4L. Thorne McCarty
There are many ways to represent a world that changes over time:
Temporal Logic, in various forms [48, 46, 7]; Dynamic Logic [47, 11];
the Situation Calculus [18, 20]; the Event Calculus [14]; and many
more.
My current choice in LLD is to represent an Event as a predicate
defined over a linear temporal order, and to represent an Action as an
Event paired with a responsible Actor. Let’s assume the existence of a
set of basic events and define a set of abstract events using Horn clauses
and (optional) order relations. We can then represent concurrent and
repetitive actions, with indefinite information about their order. For
the theoretical details and several examples, see [39]. In that paper,
to reason about these actions, Ron van der Meyden and I generalized
the results in [38] and analyzed two methods for answering queries: (1)
an inductive proof procedure for linear recursive definitions, which is
sound but not complete; and (2) a decision procedure, which works for
a natural class of rules and queries.
The semantics of my action language has always been influenced by
the necessity of embedding it inside the deontic modalities, which will
be discussed in Section 2.4, infra. In the earliest version, in [21], the
basic actions were defined on partial states (called substates) and se-
quences of partial states (called subworlds) using the notion of strict
satisfaction. The intention was to construct the denotation of an ac-
tion, recursively, although the logic itself was still classical. Later, in
[27], the background logic was intuitionistic, as described in Section 2.1,
supra, and strict satisfaction was replaced with a construction based on
the principal filters of a final Kripke model. This construction worked,
technically, but it was painfully complex. I will suggest a simpler se-
mantics in Section 2.7, infra.
2.4. Modalities Over Actions,
I published my first paper on a deontic logic for legal reasoning at
IJCAI in 1983 [21], several years before my paper on LLD. Today, this
system would be described as a dyadic deontic logic with a Condition
and an Action, in which the Action is defined in a first-order dynamic
logic [11]. I revisited the subject in 1994 [27], swapping out the dy-
namic logic and replacing it with the action language discussed in [39].
The 1994 paper also incorporated into the deontic language the inter-
pretation of negation-as-failure that had been developed previously in
[40].
The deontic semantics itself remained basically the same from 1983
to 1994. There are three modalities: Ohφ|αi(obligatory), Fhφ|αi
A Language for Legal Discourse is All You Need 5
(forbidden), Phφ|αi(permitted), and each one can be negated with a
monotonic intuitionistic negation. There are no negated actions in the
language, which means that Oand Fmust be defined separately. P
is a “free choice” permission, or a strong permission. Phφ|αimeans:
“Under the condition φ,all the ways of performing the action αare
permitted.” Fhφ|αimeans: “Under the condition φ, all the ways of
performing the action αare not permitted.” Thus ¬Fhαiis a weak
permission. Ohφ|αimeans: “Under the condition φ,only the ways
of performing the action αare permitted.” Formally, the semantics of
all three modalities are determined by a single construct, the Grand
Permitted Set, P, which designates among all possible denotations of
all possible actions those that are permitted and those that are not.
Also introduced in [27] is the modality DOhαi, which is veridical and
therefore results in the event αbeing true in the successor world.
There are two theorems in [27] that have implications for the deontic
proof theory in LLD. Theorem 4.7 says (roughly) that, in a language
without P, all inferences about Oand Fcan be reduced to proofs in
the action language. Theorem 4.8 says (roughly) that, in a language
without F, the inferences about Oare independent from the inferences
about P. This means that we can construct simple deontic proofs in
the style of a logic program in two restricted versions of the language.
We can use Oand F, along with negation-as-failure on F, which is
essentially a default rule in the following form: “Every action that is
not explicitly forbidden is permitted.” Or we can use Oand P, which
is essentially a default rule in the following form: “Every action that is
not explicitly permitted is forbidden.” These are two familiar principles
from real legal systems, and there are several examples in [27] that fall
into one of these two categories.
There is now an enormous literature on the “paradoxes” of deontic
logic, with no consensus. My view is that the semantics for the deontic
modalities is very simple, and the paradoxes arise from complexities in
the action language and from our problematic formalisms for default
inference. For example, see [26] for an analysis of Chisholm’s Paradox,
based on the Grand Permitted Set, P, which is itself very simple, along
with a somewhat more complicated system for default reasoning.
2.5. Epistemic Modalities,
It is a slight overstatement to say that the current version of LLD
includes the epistemic modalities, but the facilities that we need to
model knowledge and belief are a basic part of the language.
6L. Thorne McCarty
The traditional approach [8] treats knowledge as a modal operator,
KP, endowed with a Kripke semantics. But there is a more powerful
approach in the recent literature on justification logics [3, 4]. Based on
the work of Sergei Artemov on the Logic of Proofs (LP) [2], a justifica-
tion logic adds the annotation t:Pto the proposition Pand interprets
this compound term as “Pis justified by reason t.” Essentially, tis
aproof of P, and it can be extracted from a provable modal formula,
KP, by what is known as a Realization Theorem. This is currently an
active area of research, and there are justification logics that correspond
to the modal systems K,T,K4,S4,K45,S5 and others.
From my perspective, this work is interesting because it is fairly easy
to construct and manipulate proofs in a logic programming language
such as LLD. For a further note on the structure of proofs in LLD, see
Section 2.7, infra.
2.6. A Natural Language Interface,
The syntax of atomic formulas in LLD, as shown in Section 2.2,
supra, makes it a good target language for natural language processing.
In my paper at ICAIL in 2007 [31], I developed a quasi-logical form,
or QLF, to represent the semantic interpretation of a sentence in a
legal text, and a definite clause grammar, or DCG, to compute the
correct quasi-logical form from the output of a syntactic parser. The
QLF s were intended to serve as an intermediate step towards the full
representation of a legal case in LLD. There are several examples in
the paper, but for a larger sample see the QLF s for 211 sentences
from Carter v. Exxon Co., USA, 177 F.3d 197 (3d Cir. 1999), available
online at http://bit.ly/2yhnPdC. The syntactic analysis was from
Michael Collins’ statistical parser [5], which is now more than 20 years
old. There are better parsers available today.
Can these techniques be applied to statutes and regulations? In 2009,
Tim Armstrong and I attempted to compute a semantic interpretation
of Articles 2 and 3 of the Uniform Commercial Code (UCC). The results
were poor [1]. Other researchers have also reported negative results
on parsing statutes [53, 43, 45]. The explanation seems fairly clear:
The syntax of a statute is complex, contorted, and far removed from
the syntax of the sentences on which our current parsers have been
trained. One alternative is to use a human annotator and a controlled
natural language interface, such as Robert Kowalski’s Logical English
[12]. Another alternative is to experiment with a large language model,
such as BERT [6], which might be able to learn an idiosyncratic syntax
from a small number of annotated examples.
A Language for Legal Discourse is All You Need 7
2.7. And a Prototypical Perceptual Semantics.
My current work on deep learning might appear to be far removed
from a Language for Legal Discourse, but it is actually part of a broader
effort to bridge the gap between machine-learning-based AI and logic-
based AI.
My most recent papers [34, 35] develop a theory of clustering and
coding that combines a geometric model with a probabilistic model
in a principled way. The geometric model is a Riemannian manifold
with a Riemannian metric, gij (x), which is interpreted as a measure of
dissimilarity. The dissimilarity metric is defined by an equation that
depends on the probability density of the sample input data — in an
image classification task, for example — and this leads to an optimal
lower dimensional encoding of the data, an object that is referred to as a
prototypical cluster. The theory is illustrated by a number of examples
from the MNIST dataset [17] and the CIFAR-10 dataset [15].
The main thesis of my paper at ICAIL 2015 [32] is that the theory of
differential similarity also provides a novel semantics for my Language
for Legal Discourse. Here is an excerpt from the last page of [35], which
explains how this works in the image classification tasks:
There is one operation that appears repeatedly . . . We
construct a product manifold consisting of four prototyp-
ical clusters, and then construct a submanifold which is
itself a prototypical cluster. We can now take a big step:
We can use this construction to define the semantics of
an atomic formula in a logical language, that is, to define
apredicate with four arguments. The general idea is to
replace the standard semantics of classical logic, based
on sets and their elements, with a semantics based on
manifolds and their points. . . . The natural setting for
these developments is a logical language based on cate-
gory theory, or what is known as a categorical logic.
Thus, in [32], I presented a categorical logic based on the category
of differential manifolds (Man), which is weaker than a logic based
on the category of sets (Set) or the category of topological spaces
(Top). See [32] for the technical details, or see [33] for a more informal
exposition. A comprehensive paper addressed to the computational
logic community is currently in preparation [36].
Here are some consequences for the several questions left open from
previous sections of this paper:
8L. Thorne McCarty
From Section 2.2: The hierarchy of prototypical clusters in an image
classification task is a model for the type hierarchies that we need for
predicates like Own in LLD.
From Section 2.3: How do we represent actions in a sequence of
moving images? Suppose we take differential manifolds seriously, and
represent an action by the Lie group of affine isometries in R3, also
known as rigid body motions. See [9], Chapter 18. We can then apply
the theory of differential similarity to the manifold of physical actions,
and generalize from there to a manifold of abstract actions. This gives
us a new semantics for the action language in LLD.
From Section 2.5: In a categorical logic, a sequent is a morphism, and
a proof is a composition of morphisms. Thus, in the category Man, a
proof is a smooth mapping of differential manifolds, which means that
justifications have much more structure in LLD than they do in other
languages.
3. Conclusion
I have tried to demonstrate in this short paper that the features of
my Language for Legal Discourse (LLD) are necessary for a computa-
tional representation of statutory and regulatory rules. Are they also
sufficient? Perhaps. But there are still some aspects of the language
that are subject to revision, and we still need to implement it in full
and apply it to a range of real legal examples.
In my presentations at the Workshop on Programming Languages
and the Law (ProLaLa 2022) and at the conference on Computational
Legal Studies (CLS 2022), I discussed several concrete examples of the
abstract concepts in this paper. The title of my talk at CLS 2022
was “Bridging the Gap between Machine Learning and Logical Rules
in Computational Legal Studies,” and it covered the main ideas in
Section 2.7, supra. Here is a video:
https://youtu.be/rBPadM9tyNo
The “gap” is illustrated with an example from Article 3 of the Uniform
Commercial Code, which then motivates the discussion of “A Theoret-
ical Synthesis” in the last half of the talk. Here are the slides and notes
from CLS 2022:
https://bit.ly/3x32x25
https://bit.ly/3OatUhJ
My talk at ProLaLa 2022 discussed several additional examples from
Article 3 of the Uniform Commercial Code. Here are the slides and
notes:
A Language for Legal Discourse is All You Need 9
https://bit.ly/39aHEKd
https://bit.ly/3tf8rMq
These examples demonstrate the automatic generation of QLF s and
their translation into LLD, following Section 2.6, supra, and they cover
the deontic modalities from Section 2.4 and the epistemic modalities
from Section 2.5. Taken together, my presentations at ProLaLa 2022
and CLS 2022 outline a methodology for translating statutory and
regulatory rules into executable computer code, and they suggest a
research agenda to help us achieve this goal.
References
[1] T. J. Armstrong and L. T. McCarty. Parsing the text of the Uniform Com-
mercial Code. In Workshop on Natural Language Engineering of Legal Argu-
mentation (NaLELA), at ICAIL, 2009.
[2] S. N. Artemov. Explicit provability and constructive semantics. Bulletin of
Symbolic Logic, 7(1):1–36, 2001.
[3] S. N. Artemov. The logic of justification. The Review of Symbolic Logic,
1(4):477–513, 2008.
[4] S. N. Artemov and M. Fitting. Justification Logic: Reasoning with Reasons.
Cambridge University Press, 2019.
[5] M. Collins. Head-Driven Statistical Models for Natural Language Parsing. PhD
thesis, University of Pennsylvania, 1999.
[6] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. BERT: Pre-training of
deep bidirectional transformers for language understanding. In Proceedings,
2019 Conference of the NAACL: Human Language Technologies, pages 4171–
4186, 2019.
[7] E. A. Emerson and J. Y. Halpern. Decision Procedures and Expressiveness
in the Temporal Logic of Branching Time. In Proceedings of the 14th Annual
ACM Symposium on Theory of Computing, pages 169–180. ACM, 1982.
[8] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. Reasoning About Knowl-
edge. MIT Press, 1995.
[9] J. Gallier. Geometric methods and applications: For computer science and
engineering. 2nd ed. Texts in Applied Mathematics 38. Springer, 2011.
[10] M. Gelfond and V. Lifschitz. The stable model semantics for logic program-
ming. In Proceedings, Fifth International Conference and Symposium on Logic
Programming, pages 1070–1080, 1988.
[11] D. Harel. First-Order Dynamic Logic. Springer-Verlag, 1979. LNCS No. 68.
[12] R. Kowalski and A. Datoo. Logical English meets legal English
for swaps and derivatives. Artificial Intelligence and Law, 2021.
https://doi.org/10.1007/s10506-021-09295-3.
[13] R. A. Kowalski. Legislation as logic programs. In G. Comyn, N. E. Fuchs, and
M. J. Ratcliffe, editors, Logic Programming in Action, pages 203–230. Springer
Berlin Heidelberg, 1992.
[14] R. A. Kowalski and M. J. Sergot. A logic-based calculus of events. New Gen-
eration Computing, 4(1):67–95, 1986.
10 L. Thorne McCarty
[15] A. Krizhevsky. Learning multiple layers of features from tiny images. Technical
report, Department of Computer Science, University of Toronto, 2009.
[16] S. B. Lawsky. A logic for statutes. Florida Tax Review, 21:60, 2017.
[17] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning
applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324,
1998.
[18] J. McCarthy. Situations, actions, and causal laws. In M. Minsky, editor, Se-
mantic Information Processing, pages 410–417, 1968.
[19] J. McCarthy. Circumscription: A form of non-monotonic reasoning. Artificial
Intelligence, 13:27–39, 1980.
[20] J. McCarthy and P. J. Hayes. Some philosophical problems from the standpoint
of artificial intelligence. In Machine Intelligence 4, pages 463–502. Edinburgh
University Press, 1969.
[21] L. T. McCarty. Permissions and obligations. In Proceedings of the Eighth In-
ternational Joint Conference on Artificial Intelligence, pages 287–294, 1983.
[22] L. T. McCarty. Clausal intuitionistic logic. I. Fixed-point semantics. Journal
of Logic Programming, 5(1):1–31, 1988.
[23] L. T. McCarty. Clausal intuitionistic logic. II. Tableau proof procedures. Jour-
nal of Logic Programming, 5(2):93–132, 1988.
[24] L. T. McCarty. A language for legal discourse. I. Basic features. In Proceed-
ings of the Second International Conference on Artificial Intelligence and Law,
pages 180–189. ACM Press, 1989.
[25] L. T. McCarty. Circumscribing embedded implications (without stratifica-
tions). Journal of Logic Programming, 17:323–364, 1993.
[26] L. T. McCarty. Defeasible deontic reasoning. Fundamenta Informaticae, 21(1–
2):125–148, 1994.
[27] L. T. McCarty. Modalities over actions, I. Model theory. In Principles of
Knowledge Representation and Reasoning: Proceedings of the Fourth Inter-
national Conference (KR94), pages 437–448. Morgan Kaufmann, 1994.
[28] L. T. McCarty. An implementation of Eisner v. Macomber. In Proceedings of
the Fifth International Conference on Artificial Intelligence and Law, pages
276–286. ACM Press, 1995.
[29] L. T. McCarty. Some arguments about legal arguments. In Proceedings of the
Sixth International Conference on Artificial Intelligence and Law, pages 215–
224. ACM Press, 1997.
[30] L. T. McCarty. Ownership: A case study in the representation of legal concepts.
Artificial Intelligence and Law, 10(1-3):135–161, 2002.
[31] L. T. McCarty. Deep semantic interpretations of legal texts. In Proceedings
of the Eleventh International Conference on Artificial Intelligence and Law,
pages 217–224. ACM Press, 2007.
[32] L. T. McCarty. How to ground a language for legal discourse in a prototypical
perceptual semantics. In Proceedings of the Fifteenth International Conference
on Artificial Intelligence and Law, pages 89–98. ACM Press, 2015.
[33] L. T. McCarty. How to ground a language for legal discourse in a prototypical
perceptual semantics. Michigan State Law Review, 2016:511–538, 2016.
[34] L. T. McCarty. Clustering, coding, and the concept of similarity (Version 2.0).
Preprint, arXiv:1401.2411v2 [cs.LG], 2018.
A Language for Legal Discourse is All You Need 11
[35] L. T. McCarty. Differential similarity in higher dimensional spaces: Theory
and applications (Version 3.0). Preprint, arXiv:1902.03667v3 [cs.LG, stat.ML],
2021.
[36] L. T. McCarty. Manifold logic and the theory of differential similarity. Forth-
coming, 2023.
[37] L. T. McCarty and L. A. Shklar. A PROLOG interpreter for first-order intu-
itionistic logic (abstract). In Proceedings, 1994 International Logic Program-
ming Symposium, page 685. MIT Press, 1994.
[38] L. T. McCarty and R. van der Meyden. Indefinite reasoning with definite
rules. In Proceedings of the Twelfth International Joint Conference on Artificial
Intelligence, pages 890–896, 1991.
[39] L. T. McCarty and R. van der Meyden. Reasoning about indefinite actions.
In Principles of Knowledge Representation and Reasoning: Proceedings of
the Third International Conference (KR92), pages 59–70. Morgan Kaufmann,
1992.
[40] L. T. McCarty and R. van der Meyden. An intuitionistic interpretation of finite
and infinite failure. In L. M. Pereira and A. Nerode, editors, Logic Program-
ming and NonMonotonic Reasoning: Proceedings of the Second International
Workshop, pages 417–436. MIT Press, 1993.
[41] D. Miller. A logical analysis of modules in logic programming. Journal of Logic
Programming, 6(1):79–108, 1989.
[42] D. Miller and G. Nadathur. Programming with Higher-Order Logic. Cambridge
University Press, 2012.
[43] L. Morgenstern. Toward automated international law compliance monitoring
(TAILCM). Technical Report AFRL-RI-RS-TR-2014-206, Leidos, Inc., 2014.
[44] J. P. Morris. Spreadsheets for legal reasoning: The continued promise of declar-
ative logic programming in law. Master’s thesis, University of Alberta, 2020.
[45] M. A. Pertierra, S. Lawsky, E. Hemberg, and U. O’Reilly. Towards formalizing
statute law as default logic through automatic semantic parsing. In 2nd Work-
shop on Automated Semantic Analysis of Information in Legal Texts (ASAIL),
at ICAIL, 2017.
[46] A. Pnueli. The temporal logic of programs. In Proceedings of the 18th Annual
Symposium on Foundations of Computer Science, pages 46–57. IEEE Com-
puter Society, 1977.
[47] V. Pratt. Semantical considerations on Floyd-Hoare logic. In Proceedings, 17th
IEEE Symposium on Foundations of Computer Science, pages 109–121, 1976.
[48] A. N. Prior. Past, present and future. Oxford University Press, London, 1967.
[49] R. Reiter. A logic for default reasoning. Artificial Intelligence, 13(1-2):81–132,
1980.
[50] D. A. Schlobohm and L. T. McCarty. EPS-II: Estate planning with prototypes.
In Proceedings of the Second International Conference on Artificial Intel ligence
and Law, pages 1–10. ACM Press, 1989.
[51] M. J. Sergot, F. Sadri, R. A. Kowalski, F. Kriwaczek, P. Hammond, and H. T.
Cory. The British Nationality Act as a logic program. Communications of the
ACM, 29:370–386, 1986.
[52] A. Van Gelder, K. A. Ross, and J. S. Schlipf. Unfounded sets and well-founded
semantics for general logic programs. In Proceedings, Seventh ACM Symposium
on the Principles of Database Systems, pages 221–230, 1988.
12 L. Thorne McCarty
[53] A. Wyner and G. Governatori. A study on translating regulatory rules from
natural language to defeasible logic. In Proceedings of RuleML 2013. CEUR,
2013.
