Models of A.I. in legal reasoning


Lawyers must confess that their job is not 100% Logical: it uses, according to Chailm Perelman, only some soft argument techniques, which Perelman defined as non-formal arguments. They are mainly concerned with the adherence of an audience, so, without having a strict demonstration of propositions proper to formal logic, the lawyer-orator can only work on persuasion of the audience to adhere each successive element of an argument. I agree with Thomas that the value of precedents pertains to a non orthogonal space of discourse, where premises are uncertain, asymmetric and polarized to a certain audience, so that the language of law is, therefore, full of persuasion and of fuzziness.

At the same time, the reasonableness of law is not inferior to that of math, at the contrary in any circumstances the truth or empirical correspondence of legal propositions can be superior to that ot math (see that case of certain legal definitions applied to legal concepts).

At this regard, Perelman said that there are two ways the orator may achieve this acceptance or adherence:

The first responds with definitions to incompatible opinions through the dissociation of concepts (here law could be on the same floor of math, if formalized with any logic of fuzziness or, at the inverse, classical logic or deontic logic), and
The second involves associations according to quasi-logical arguments, appeals to reality, and arguments that establish the real (this approach could have the same empirical value of, for example, geology).
A new philosophical scenario focused on fuzzy logic, applied to legal reasoning, enables us to understand the concept of many-valued logics as fundamental in Jan Łukasiewicz works and as a solution for Aristotle’s problem of future contingents (also for modal logics).

About “paradoxical” and paraconsistent logic with an infinite number of truth values, actual computer can run software using fuzzy algorithms, so…why not thinking to new formalization of “legal reasoning”?

Many philosophers do not think that fuzzy logic can be a generalization of all MVL logics, maybe there is some paraconsistent language more general than a MVL logic (we suspect it would be something very similar to our quotidian language).

In our opinion, it is interesting the use of Peirce Arrows and Peirce graphs…they could be useful for representing chains and sets and sytems of propositions, in the same way as the analytic geometry of Cartesian plane represents algebric equations. A new scenario for legal reasoning formalization.




We could use the Peirce graphs as a planar representation of linguistic chaines of multi-value linguistic juridical rules or multi-value linguistic sets (words instead of numbers….see Computing With Words of Lofti Zadeh).

We can think that matrixes could be useful for handling such MVL systems of propositions, converting these “words” (linguistic sets are words, words are polysemic, so words are MVL sets) into numbers systems. Bbut these numbers should be “imprecise” or vague numbers, and they are like anything which just exist and has a name: they are the “fuzzy numbers”, they could represent “words”, – and their geometric representation could be fuzzy matrixes of fuzzy sets, while further we suspect that some engineer just is using fuzzy tensors for representing equations of fuzzy sets.

About the relation with binary logic of MLVs, we think these logics are each a transformation of the other (MVLs are an extension of binary logic [Zadeh], while we personally think that a self-referential application of fuzzy logic to itself could give foundation to binary logic).

About the problem of programming a binary computer, many algorithms employ rules of binary defuzzification (Sugeno inference) and fuzzification of binary data, without needing any infinite number of bytes.

Material technologic application of MVLs: Artificial intelligence of legal reasoning, robots more adequate to imprecise reality (indeterministic), vague reasoning, translations, auto-parking cars, ethics reasoning, giving sense of justice to programmes, etc.


Philosophical application: rehabilitation of induction (vague induction), of dialectic reasoning, of ethics and religious and theological reasoning (which are vague and imprecise), stating the logical and rational value of faith beliefs, etc…sweet refutations will be considered logically valid (no more simply dichotomic refutations or strict contradiction)… a logic for emotions.

Physical applications: a new MVL analytic math, which will give raise to a new view over physics of complexity (for example the non-perfectly analysed nature of black hole complex singularities or the nature of quanta via Popper analysis of propensity).


Thanks Rand objectivism, this philosophical position on ethics seems to encourage a kind of rationality intermediate between “is” and “ought”, a way of reasoning that lawyers love (they love to breach the Hume’s Law or Hume’s Guillotine, which he theorized in his Treatise of Human Nature, book III, part I, section I).

Rand’s ethics seems to breach, as it happen in every legal debate, the difference between “is” and “ought”, this is a kind of reasoning which I would define “vague” and intermediate (rethorical and impressive) between is and ought, and very far from rationality of classical logic, discrete math and Cartesian “evidence”.

Perelman very clearly said, similarly, that legal reasoning is the field of those arguments only plausible, probable, uncertain and escaping from the certainness of the “calculus”.

Mrs. Rand, namely, said “If [man] chooses to live, a rational ethics will tell him what principles of action are required to implement his choice. If he does not choose to live, nature will take its course. Reality confronts a man with a great many ‘must’s’, but all of them are conditional: the formula of realistic necessity is: ‘you must, if –’ and the if stands for man’s choice: ‘if you want to achieve a certain goal'” (in Peikoff, Leonard. Philosophy: Who Needs It).

This is really in accordance with Perelman ideas that, in legal debates, there is a clear opposition between Cartesian rationalism and new rhetoric (see Ch. Perelman, “Juridic Logic and new rhetoric”). So, I conclude that Legal Reasoning will never win a Nobel prize, till it will be not formalized in its specific logic and math, or till the moment it could develop any appreciable theorem (even vague and rhetorical).

I guess that to win a Nobel legal reasoning shall be calculated by a complex logic, like C. S. Peirce existential semantic networks, and/or by a fuzzy or fractal math (of fuzzy-fractal sets standing for the vague qualities of linguistic concepts), like the “CW” [Computing with Words] theory of Lofti Zadeh about Soft Computing with words.

At the contrary, it is disregarded that the abstract genus of legal reasoning, the dialectic, is a kind of reasoning absolutely necessary even for logicians and mathematicians, if you consider that two Gödel theorems had the non intentional effect to show this.

Namely, the manifest undecidability of logical grounds of math made necessary thinking to a general foundation of the same math on the basis of natural and discursive language (on dialectic), revealing that the same dialectic discussion is the general genus of every science.

According to Perelman dialectic will differ from rhetoric mainly for the circumstance that

– Dialectic, as in Aristotle of “Topics”, supposes an universal audience, and/or has the form of general discussion about systemic principles (so called Aristotelian “endoxa”); incidentally, is dialectic the thought of solitary thinker (who thinks and talks to himself), because his thought is anything like a discourse of single man to a universal audience,

– while i.e. Judicial Rhetoric supposes the limited audience of a court (lawyers, parties, judge, jury, etc.) and the restricted discussions of legal norms, relevant for certain facts.


The main statement is that the definitions of law, through the dissociation of concepts, could be on the same floor of math, if formalized with any logic of fuzziness or, at the inverse, classical logic or deontic logic.

Here the “premises” of definitory activity are the hole, which premises the lawyer or, especially, the legislator, can arbitrarily choose, and from which he can start giving a definition of concepts or a definitory difference between two or more logical concepts.

But it is like the holes in 5 postulates (especially the fifth) of Euclidean geometry: once you made a choice about chain rules and postulates, the consequences logically follow.

We can add that a now less disregarded hole is present in every logical concept (even in discrete math): every concept “comes from quotidian language” (as Godelian thought could teach), so it gets the original sin of fuzziness…but someone could not think to it as a hole (someone could guess that is another kind of logical precision).

Another relevant statement is that “fact situation is a concise description of all the occurrences and circumstances of a particular case, without any of the consequences under the law”.

The concise description is the hole (because, actually, any concision is a vague and abstract outline of any, not all, circumstances). So the fact pattern is always a theory (K. R. Popper would say), not the brute and very complex, and richer of individual circumstances, fact; this theory supposes a similarity, namely a huge range of aspects in common, inside a class of different/similar facts. This explains why “litigator may use similar case in its fact pattern and adjudication thereof as precedent”, he is extending per analogy a theory “of this fact” to other precedent cases (which suppose a similar class of facts).

In our opinion, if lawyers and judge were trained since university to calculate the similarity of cases and theories about facts this would be a general improvement of the jurisprudential method. The training would be in logics.

There is logical and mathematical concept, in which lawyers/judge could be trained, the fuzzy entropy of the fuzzy sets and fuzzy rules implied in a fact pattern (and in related precedents), which tool could be used for ascertaining when any precedents could not strictly match with any facts (and be excluded).

We explain: Lofti Zadeh (approximate name of the article: “Soft Computing, Computing with Words”), Bart Kosko and many others elaborated the logical and mathematical measure of how a vague description of a certain fact (done by a certain number of fuzzy sets or fuzzy rules = a certain connection of fuzzy sets) is vague. This measure of vagueness is at the same time a measure of how a certain fact (namely, its fuzzy sets translation) is “similar” to other facts (more or less similar).

Here an example partially coming from analogous speeches of C. S Peirce: the fact pattern “chair” is near and similar to “recliner” more than the fact pattern “seat”…this is due to the fact that the fuzzy set “seat” got more fuzzy entropy (in the relation with recliner) than “chair”. If the precedent about chair is more closely related to that about “recliner”, we should exclude more precisely those about a simple “seat”…

…This is the second way, proposed by Perelman, the orator may achieve this acceptance or adherence (the empirical), because it would involves above mentioned “analogical” associations, according to quasi-logical arguments (about similarity), appeals to reality (of fact pattern), and arguments that establish the “similarity” of real facts (this approach could have the same empirical value of, for example, geology, where any stratum or layer of ground is “compared” to another).


  • It computes with words not numbers,
  • Is a soft computing,
  • Employs metaphors,
  • Support human decisions,
  • Enable authority to be aware of its logical limits,
  • Is a new university legal matter.



If law is a system of metaphorical holes in the texture of laws, than it is a consequence that legal reasoning focuses on legal “problems” (as lack of solution, namely …the hole). This is the principle and motor of legal reasoning.

Lawyers does not notoriously write equations for solving these problems, neither for theorizing the profound structures of legal reasoning or the organization of legal systems: this would be a fascinating future, not the actual panorama.

Holes of the legal system would be the “problems”.

We mean that a legal problem is the principle and motor of organization of legal systems operators, in the sense that economic, moral, social problems become legal after these “holes” outside the legal systems put at risk the rhetorical expectations of legal operators about governance of civil society…like the big “hole” which brought USA all down in 2008.

“Can fact pattern alone give a person the right to seek judicial redress or relief against another?” Absolutely not: the right comes from the legal “problem”, which compels judge and lawyers to award an analogy to prior adjudication called precedents (and relative matching between theories about facts and conjectures about rules). And this legal problem is, before, a translation into the legal systems of “holes” of civil society.

The legal problem is “only” legal reasoning, it has a rhetorical, or “metaphorical”, structure and got fuzzy mathematical nature: this is our conjecture.

We explain our reference to rhetoric, namely to epistemic rhetoric (not ornamental one).

We mention application of logic helping ensure that arguments are at least internally coherent and consistent.

It cannot be untrue that any logic is also a general topics (as production of topical arguments) directed to a universal audience, given that its logical arguments want to apply universal rules, but only to the aim of persuading a universal public about contradictions or inconsistencies (see Perelman theory of general rhetoric).

Without such universal audience, logics without interlocutors would be surely strange. Logic is, therefore, a part of epistemic rhetoric, that rhetoric which is addressed to a universal audience.

While more generic epistemic rhetoric can be addressed to both universal and to specialized audiences (i.e. the argumentation directed to the special interlocutors who are medicine doctors or lawyers/judges is a specialized rhetoric, different from logics, due to a reduced class of interlocutors).

So stands the point that without rhetoric, the soundness or cogency of argumentation (also philosophical) cannot be established.


If we accept this Perelmann concept of Epistemic Rhetoric, the generic rhetoric will be only “secondarily” an ornamentation, because at first:

– An epistemic rhetoric will try to show to an audience the deficiencies or deceptions of any kind of argument,

– Either if those deficient or deceptive arguments are directed to the universal public of Logics (which is surely a substantial background of every reasoning) or to specialized audiences.

About Masters:

Or we accept any authority established on the basis of tradition or we have to trust in an unlimited sceptical “substance” of whatever new-age’s theory, which tries to self-accredit in view of its simple claim of being “the progress”.

– We need Masters.

– At the same time, we need to criticize such Masters, as expression of mere witness, and not as Gold.

This argument about masters is both expressed in ancient times by Aristotle in his treatise on Rhetoric and in his Topics, and more recently by Popper about the necessary contribution of Tradition to formulation of any kind of scientific conjectures.

Without being careful in trying to describe in too much detail any methodology, one remain without philosophical point of view. This is the death of any argumentation and free discussion.

Justly, Perelman in his “The New Rhetoric: A Treatise on Argumentation” (Part. I, frames of argumentation, par. 2, the end of this paragraph) said that “It is not sufficient talking or writing, you need also to be heard and read. It is not without importance to have an auditor, to have a large audience, to be admitted to do a speech in certain circumstances, in any assemblies, in any environments. We ought not forget that hearing anyone means to be disposable to eventually admit his point of view”.

You can easily infer from this citation (Perelman meant) that:

– A dialogue to himself or to only one auditor or to a large audience is the necessary methodological basis of every argumentation, strictly rational as for example geometry or mathematics or physics, or merely reasonable as legal argumentation or philosophy.

– Every (only) reasonable argumentation (as for example philosophy) tends to a contact between minds, which contact Perelman conceives as a primary frame of argumentation, especially for whatever reasonable (non necessary strictly rational) argumentation, as different from Cartesian demonstration (logical or empirical);

– Philosophy is the dialogue to himself (as universal audience) or to one auditor (idem) or to a non quantifiable universality of men (and women), which dialogue finds, select, criticize and show topics, eventually universal (this is philosophy) to that audience, using a methodology of persuading (this is rhetoric) a universal audience to accept those universal topics;

Our comments:

– At this level of detail, trying to define a general methodology of philosophy, you are simply put at risk of showing a point of view to an audience!

– Showing a point of view (also on methodology of philosophy) is necessary as basis of discussion, in order to be heard and read, to have an audience and to “not forget that hearing anyone means to be disposable to eventually admit his point of view” (Perelman).


No math is taught in University law schools, neither ten seconds…I am talking of any logics and math outcomes of experimental Ph.D. studies in philosophy of law and of academic speeches.

“How fuzzy is a fuzzy set compared to a normal set?”

Good question (we mean that fuzzy entropy of a normal set is 0, while a fuzzy set has entropy varying between 0% and 100%, so fuzzy logic in an extension of classical…it is how to say that Perelman New Rhetoric is an extension of Cartesian evidence).

“How much of the set matching is judgmental how much is scientific?” Nobody, to our knowledge, can reply to this question: our legal traditions has not, nowadays, begun to use “vague” computation…juridical method is till now a wild field of semi-computations: the set matching, applied to rhetoric, is a prospective future, a perspective of future…could it be scientific? Who can know it?

Basically it will be all about adherence and elimination of non or anti-adherence. All about

– “too much” non similar set and

– “sufficiently” similar set

It can be translated in modal logics (logics of fuzzy modalities of facts and precedents, modalities like “similar”, “possible”, “obligatory”, “adherent”): you are absolutely right.

Facebook Twitter Vimeo Share to Stumble Upon More...