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Position Paper: LLD is All You Need



My first paper on a Language for Legal Discourse (LLD) was published at the International Conference on Artificial Intelligence and Law in 1989 [24]. I used the language subsequently for several small projects: [48] [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 implementation, 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 extended abstract.
Position Paper:
LLD is All You Need
L. Thorne McCarty
Rutgers University
My first paper on a Language for Legal Discourse (LLD)
was published at the International Conference on Artificial
Intelligence and Law in 1989 [24]. I used the language sub-
sequently for several small projects: [48] [28] [30] [31], and it
motivated much of my theoretical work on Knowledge Rep-
resentation 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 implementation, I think the design
choices embodied in LLD provide useful guidelines for any-
one trying to translate legal rules into executable computer
code. I will describe these choices in this extended abstract.
Because of space limitations, I will not be able to compare
the features of LLD to other approaches in the literature,
but this would be a good topic for discussion. Nor will I be
able to present concrete examples of statutory encodings,
although these could be included in an oral presentation.
My Language for Legal Discourse (LLD) is . . .
2.1 An Intuitionistic Logic Programming Lan-
Among the several major programming paradigms — pro-
cedural, functional, logical, and object-oriented — the best
representation for legal rules, in my opinion, is a logic pro-
gramming language. This is (obviously) the position taken
by Robert Kowalski, see [49] and [13], and it is also the posi-
tion taken by Jason Morris in his recent MS Thesis [42]. One
reason for this preference is that it is easy to encode a proof
theory for various kinds of legal rules in a logic program-
ming language, such as PROLOG: You get proof search and
unification for free in the meta-interpreter. For an example
of how this works, see [35], which includes code for several
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However, I have also taken the position for several decades
that the proper setting for logic programming is intuitionis-
tic logic, not classical logic. To see the difference, consider
a Horn clause written as an axiom in a sequent calculus, as
Q1Q2. . . 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:
. . . ∧ ∀y(QR1R2. . . Rk). . . `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 resolu-
tion theorem prover, but in intuitionistic 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 pro-
posed such an intuitionistic logic programming language in
[22, 23]. At about the same time, motivated by applications
in programming, generally, Dale Miller proposed a similar
language in [39]. For a full account of Dale Miller’s work,
see [40].
Missing from an intuitionistic logic program, by design,
are rules for disjunctive and existential assertions. We can
add these rules separately, as in [35], or we can take the
approach advocated in [36]. A statute often includes pro-
visions that state necessary and sufficient conditions 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 existen-
tial assertions. Ron van der Meyden and I showed, in [36],
that we can model this reasoning, semantically, with John
McCarthy’s circumscription axiom [19], but this sometimes
leads to inductive proofs.
Logic programming also offers a simple way to write de-
fault rules: negation-as-failure. Sarah Lawsky argues per-
suasively in [16] that most statutes are drafted and should
be interpreted as rules with exceptions, and she uses Ray Re-
iter’s default logic [47] to illustrate this point with a transla-
tion 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 repre-
sentation language, we would have a powerful normal form
for statutes and regulations,” but I argued in the same pa-
per that a well-drafted statute should never contain unstrat-
ified 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 possi-
ble inferences [38], with provable relationships to both the
(2-valued) stable model semantics [10] and the (3-valued)
well-founded semantics [50]. 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 Corpo-
ration, and Stock is a subsort of Security. The unification
algorithm is required to respect these sorts, although the de-
tails 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, depending 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
formalize 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,
There are many ways to represent a world that changes
over time: Temporal Logic, in various forms [46, 44, 7]; Dy-
namic Logic [45, 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 repre-
sent 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) or-
der relations. We can then represent concurrent and repet-
itive actions, with indefinite information about their order.
For the theoretical details and several examples, see [37].
In that paper, to reason about these actions, Ron van der
Meyden and I generalized the results in [36] 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 in-
fluenced 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 sequences of partial
states (called subworlds) using the notion of strict satisfac-
tion. The intention was to construct the denotation of an
action, recursively, although the logic itself was still classi-
cal. 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, techni-
cally, but it was painfully complex. I will suggest a simpler
semantics in Section 2.7, infra.
2.4 Modalities Over Actions,
I published my first paper on a deontic logic for legal rea-
soning 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 re-
visited the subject in 1994 [27], swapping out the dynamic
logic and replacing it with the action language discussed in
[37]. The 1994 paper also incorporated into the deontic lan-
guage the interpretation of negation-as-failure that had been
developed previously in [38].
The deontic semantics itself remained basically the same
from 1983 to 1994. There are three modalities: Ohφ|αi
(obligatory), Fhφ|αi(forbidden), Phφ|αi(permitted),
and each one can be negated with a monotonic intuitionis-
tic negation. There are no negated actions in the language,
which means that Oand Fmust be defined separately. Pis
a “free choice” permission, or a strong permission. Phφ|αi
means: “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|αi
means: “Under the condition φ,only the ways of perform-
ing 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 F
can be reduced to proofs in the action language. Theorem
4.8 says (roughly) that, in a language without F, the infer-
ences 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 fol-
lowing 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 famil-
iar 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 se-
mantics 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.
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 jus-
tification logics [3, 4]. Based on the work of Sergei Artemov
on the Logic of Proofs (LP) [2], a justification logic adds the
annotation t:Pto the proposition Pand interprets this com-
pound term as “Pis justified by reason t.” Essentially, tis a
proof 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 Sec-
tion 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
adefinite clause grammar, or DCG, to compute the cor-
rect 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 sam-
ple see the QLF s for 211 sentences from Carter v. Exxon
Co., USA, 177 F.3d 197 (3d Cir. 1999), available online at 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 regula-
tions? 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 [51, 41, 43]. The explanation seems fairly clear:
The syntax of a statute is complex, contorted, and far re-
moved from the syntax of the sentences on which our current
parsers have been trained. One alternative is to use a hu-
man annotator and a controlled natural language interface,
such as Robert Kowalski’s Logical English [12]. Another al-
ternative 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.
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 paper [34] develops a theory of clustering
and coding which combines a geometric model with a prob-
abilistic 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 dis-
similarity metric is defined by an equation that depends on
the probability density of the sample input data — in an im-
age 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 il-
lustrated 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 se-
mantics for my Language for Legal Discourse. Here is an
excerpt from the last page of [34], which explains how this
works in the image classification tasks:
There is one operation that appears repeat-
edly . . . We construct a product manifold consist-
ing of four prototypical clusters, and then con-
struct 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 a predicate with four arguments. The gen-
eral 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 devel-
opments is a logical language based on category
theory, or what is known as a categorical logic.
Thus, in [32], I presented a categorical logic based on the cat-
egory 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.
Here are some consequences for the several questions left
open from previous sections of this extended abstract:
From Section 2.2: The hierarchy of prototypical clusters
in an image classification task is a model for the type hier-
archies that we need for predicates like Own in LLD.
From Section 2.3: How do we represent actions in a se-
quence of moving images? Suppose we take differential man-
ifolds 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 dif-
ferential 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 differ-
ential manifolds, which means that justifications have much
more structure in LLD than they do in other languages.
I have tried to demonstrate in this extended abstract that
the features of my Language for Legal Discourse (LLD) are
necessary for a computational 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.
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Conference Paper
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In a pair of papers from 1995 and 1997, I developed a computational theory of legal argument, but left open a question about the key concept of a "prototype." Contemporary trends in machine learning have now shed new light on the subject. In this paper, I will describe my recent work on "manifold learning," as well as some work in progress on "deep learning." Taken together, this work leads to a logical language grounded in a prototypical perceptual semantics, with implications for legal theory. The main technical contribution of the paper is 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). The paper also shows how this logic can be extended to a full Language for Legal Discourse (LLD), and suggests a solution to the elusive problem of "coherence" in legal argument.
Humans and intelligent computer programs must often jump to the conclusion that the objects they can determine to have certain properties or relations are the only objects that do. Circumscription formalizes such conjectural reasoning.
Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.