BARTOSZ BROEK

LEGAL LOGIC. MYTHS AND CHALLENGES

1. Introduction

The controversy over the role of logic in legal reasoning dates back to the 19th Century. The proponents of legal positivism, endorsing the syllogistic structure of legal think-ing, put forward the claim that logic is an essential tool in the legal arsenal. This claim was strongly opposed by scholars belonging to various schools and supporting differ-ing views of the nature of law. Among them, one should mention legal realists, the rep-resentatives of the school of the free law, the Critical Legal Studies Movement, or the proponents of the topic-rhetorical conception of legal reasoning.14 I believe, however, that in most cases the debates between the proponents and the opponents of using logic in legal discourse were of no real signicance; these were rather pseudo-contro-versies, based on a number of false assumptions and misinterpretations.

Below, I would like to review the most fundamental misunderstandings and myths surrounding the application of logic in law. They are mostly of a theoretical or me-ta-theoretical character. In the concluding section I shall raise the question of whether a specically legal logic exists.

2. The myth of inadequacy

The thesis that all kinds of legal reasoning have a syllogistic (logical) structure is not a descriptive one; if it were so, the thesis would be evidently false, as it is easy to pro-vide examples of arguments utilizing norms which are logically incorrect. To put it differently, the only reasonable way of understanding the claim pertaining to the logi-cal character of legal reasoning is to treat it as normative: it says how legal reasoning should look like. It is only on this reading that the thesis may become the subject for fruitful debate.

Thus, a number of objections levelled against the thesis that logic is useful in law miss the target. For example: the proponents of various incarnations of legal realism rejected the so-called formalism for the sole reason that it misrepresented actual legal practice. Meanwhile, there are no grounds for denying the normative character of the logical models of legal reasoning. Logic is usually considered the minimal requirement of rationality. Therefore, one should not attack it on the basis that some people do not follow its precepts; they simply behave irrationally.15

It must be added, however, that there exist conceptions of rationality which seem, at least prima facie, to reject the usefulness of logic in the sphere of practical reason (including law). Two famous such conceptions are the topic-rhetorical theory of legal reasoning and legal hermeneutics.16

14 Cf. Stelmach & Broek [2006: chapter 1].15 Cf. B. Broek [2007: chapter 2]. 16 Cf. Stelmach & Broek [2006: chapters 4 and 5].

50According to the topic-rhetorical approach, any rational argumentation is efcacious

argumentation. It does not need to meet any formal requirements, or strive for fairness; the only thing that counts is to reach the pre-determined discursive goal. In such theories e.g., in Chaim Perelmans New Rhetoric logic is but one of the possible sources of topoi or common places. It must be stressed, however, that even in the topic-rhetorical concep-tions logic plays an indispensable role. It is clearly visible as soon as one considers the strategic level of reasoning. When one chooses a certain argumentation strategy, one that is the most efcacious, one follows an argument. This argument is inherently normative (practical) and must comply with the rules of logic. It is modelled after the structure of the hypothetical imperative: If a person X is willing to achieve the goal G, and M is the means of achieving G, X should do M. X is willing to achieve the goal G. M is the means of achieving G. Therefore, X should do M. This argument is a simple modus ponens:

p q r

p

q

r

More difcult is the case of hermeneutic models of legal reasoning, which ex deni-tione abandon such notions as rationality, logic or justication. They also question such distinctions as subjective-objective, or normative-descriptive. The easiest way of dealing with legal hermeneutics is to either claim that such theories are nonsensical, or admit that hermeneutics develops a novel kind of ontology, one that differs so much from the more traditional conceptions that it does not reject legal logic but makes any questions pertaining to the logic of the normative ill-stated. I believe, however, that at least some insights provided by hermeneutics are interesting and given a suf-ciently broad understanding of logic may be accounted for with the use of formal tools. I shall come back to this problem below.

3. The myth of triviality

The controversies over legal logic are usually centered around the so-called legal syl-logism. This comes as no surprise, since the legal syllogism (an argument consisting of two premises: the general legal norm and the description of the state of affairs, and one conclusion: the individual legal decision) is considered the paradigmatic example of legal argument.

It is often claimed that legal reasoning is trivial yet this thesis is Janus-headed. First-ly, it is highlighted that legal syllogism is used only after all the important decisions in the process of applying law i.e. the validation decision, the interpretation decision, and so on have already been made.17 Doubtless, this is correct. However, legal syl-logism does not exhaust the logical structure of legal reasoning. Each of the partial decisions in the process of applying law may be reconstructed logically.18

17 Cf. Peszka [1996: 43ff].18 Broek [2007: chapter 2].

Bartosz Broek

51Secondly, the triviality of legal logic is claimed to result from the thesis (endorsed,

inter alia, by Chaim Perelman) that the essence of legal thinking is valuation, which has nothing to do with logic. This claim is based on a misunderstanding. Irrespective of how far one develops the logical structure of a legal argument, one ultimately needs to assume some premises. Yet this is the case both in law, and in any sphere of reection where logic is applied. It is no real discovery: it follows immediately from the formal character of logic. It is also not the case that valuations have nothing to do with for-mal logic. Like any argument, a rational valuation has a certain structure which may be captured with various formal systems. A case in point is Robert Alexys Weight For-mula (Gewichtsformel).19 Valuations do have structures, although those structures do not exhaust the entire process of valuation. To put it differently: an argument against legal logic which amounts to the claim that it is trivial as it is valuation that is the essence of legal thinking, is similar to claiming the triviality of the Aristotelian logic because in the argument All people are mortal, Philosophers are people, therefore Philosophers are mortal logic cannot help one to establish that people are mortal and that philosophers are people.

One can also formulate a positive argument against those who consider legal logic trivial. Let us observe that the logical schemata used in legal discourse, and legal syl-logism in particular, structure legal thinking. Put simply: without the syllogism as a ra-tionality standard, a lawyer would not know what partial decisions need to be made in order to arrive at a nal decision. The key role of legal syllogism has nothing to do with the algorithmisation of legal reasoning; such a view would be naive. However, legal reasoning cannot dispense with legal syllogism as it provides it with a structure.

4. The myth of paradoxicality

The next objection raised against legal logic boils down to the observation that the attempts at constructing the logic of norms lead to the occurrence of a number of para-doxes. It follows, as some argue, that the very idea of constructing a logic of norms or a legal logic may be paradoxical: the existence of paradoxes speaks to the impossibility of there being a logic in law.

This is a gross misunderstanding. Firstly, some of the paradoxes of deontic log-ic are not paradoxes at all. For example: the notorious Ross Paradox is connected to a logically valid argument in the standard deontic logic in which from the norm You ought to send the letter one deduces the norm You ought to send the letter or burn it. It is claimed, sometimes, that this argument is unintuitive. Meanwhile, a lawyer is perfectly happy with the derived conclusion. The fact that a legal system contains the norm You ought to send the letter, as well as its logical consequence You ought to send the letter or burn it, does not mean that one would obey the law by burning the letter. Such a behaviour would full the latter norm, but violate the former. It is only through sending the letter that our behaviour is in compliance with the requirements of the law.

Secondly, there exist real deontic paradoxes, and in particular the contrary-to-duty ones. The contrary-to-duty paradoxes arise in connection with the so-called contra-

19 See Alexy [2007].

Legal logic. Myths and challenges

52ry-to-duty rules. Such rules say what ought to be done in case some other rule has been violated. It means that while speaking about the contrary-to-duty paradoxes two rules have to be taken into account, e.g.:

(A) It ought to be the case that if q then p.(B) It ought to be the case that if q and not-p then r.(B) is here a contrary-to-duty norm, for it says what is obligatory (r) when some

other rule (A) has been violated (q and not-p). A very bold exposition of the contrary-to-duty paradoxes (such as Chisholms, For-

resters, etc.) may take the following form. All the sets of sentences that entail paradox-ical consequences contain, at least, the following three elements: a non-CTD rule like (A), a CTD-rule that has as a condition the violation of the non-CTD rule, a proposition stating that the violation of the non-CTD rule has occurred. This is all not to say that the CTD-paradoxes are irresolvable. An interesting way of dealing with them consists in declaring the legal rules defeasible.20

Importantly, the existence of such paradoxes does not lead to the conclusion that legal logic is impossible. When a paradox emerges, one does not need to resign from the attempts to develop a normative logic; rather, one is in a better position to develop new, more adequate formal systems.

Thirdly, the very fact of there being logical paradoxes is nothing new or trouble-some. The most commonly accepted logic classical logic is lled with many para-doxes, e.g. the paradoxes of the material implication. For example, the following argument is valid in the propositional logic:

If John is in Paris, then he is in France; and if John is in London, then he is in England. Therefore, if John is in Paris, he is in London or if he is in France, he is in England.

5. The Jrgensen myth

In 1938 in Erkenntnis the Danish logician Jrgen Jrgensen published a paper en-titled Imperatives and Logic. In the article, he developed a dilemma pertaining to imperatives which also applies, mutatis mutandis, to legal norms and legal logic. It may be reconstructed as a set of the following four claims:

(I) Only true or false sentences can serve as premises or conclusions in logically valid arguments.

(II) Norms cannot be ascribed truth values.Hence:

(III) Norms cannot serve as premises or conclusions in logically valid arguments.But:

(IV) Intuitively, correct normative arguments do exist.Thus, the Jrgensen Dilemma boils down to the observation that there can be no

logical relations between norms (as they cannot be ascribed truth values), yet we con-struct arguments based on norms and our intuition considers them valid.

The two typical strategies of resolving the dilemma are the following. Firstly, the prem-ise (II) is sometimes questioned, which leads to accepting cognitivism, i.e. a theory accord-

20 Cf. Torre [1997]. In the recent years, the contrary-to-duty paradoxes have been the subject of intense re-search. Accepting a kind of defeasible logic is not the only solution to the paradoxes that seems plausible. See Carmo & Jones [2001] and the references given there.

Bartosz Broek

53ing to which norms do have truth values. This strategy is questionable as it rests on the acceptance of a peculiar theory of truth. Secondly, a distinction between norms and deon-tic sentences is introduced; the deontic sentences are understood as either descriptions of norms or descriptions of the duties expressed in norms. Such sentences, it is argued, may be ascribed truth or falsehood. However, we are also in this case facing a non-standard ac-count of truth (e.g., in the present context one would need to say that the truth of a sentence hangs together with the felicity of a speech act through which the norm was enacted).21

It seems that one does not need such complex strategies to deal with the Jrgensen Dilemma, as it rests on a misunderstanding. The misunderstanding pertains to the rst premise of the dilemma which holds that logic operates only with true or false sentences. In order to explain the sources of this claim we need to devote a few words on the history of the problem.

The strongest argument backing the rst thesis of the dilemma is connected to the so-called metalogical theorems, and the soundness and completeness theorems in particular. They take advantage of the fact that sentences can be ascribed truth or falsehood. In other words, a logic of propositions which cannot be true or false would have no corresponding metalogical theorems. According to some views, such a system is not a logic at all.

It is easy to understand it as soon as one realizes that the soundness and complete-ness theorems express the fact that there is a certain adequacy between the syntax and the semantics of the given formal system. Speaking loosely, it says that our lan-guage (syntax) ts the world (semantics). Moreover, the very notion of logical conse-quence is essentially connected to the notion of truth. In his 1936 paper Alfred Tarski dened the relation of logical consequence in the following way:

A sentence A follows logically from the set of premises G if and only if in every case in which the premises belonging to G are true, A is also true.

All metalogical theorems are based on this observation. However, Tarskis analysis, even if commonly accepted, is no dogma. To analyse this problem, let me repeat an analogy developed by John Etchemendy in his 1990 book on logical consequence.22 In the metamathematics several different formal systems characterizing the class of com-putable functions have been developed. It turned out that the results provided by the systems are coextensive. It was the basis for the claim (known as Churchs Thesis) that the class of intuitively computable functions is coextensive with the class of computable functions as dened by the systems. The problem with Churchs Thesis is that it hasnt been proven (and possibly cannot be proved) mathematically.23 Now, Etchemendys analogy is that a similar problem was faced by the early 20th century logicians whose various proof-theoretic or, in other words, syntactic systems were designed to capture the intuitive notion of logical consequence. Etchemendy calls the claim that those sys-tems captured the intuitive notion of consequence Hilberts Thesis. Hilberts Thesis has had a different fate to Churchs Thesis. As Etchemendy puts it, it has been replaced by soundness and completeness theorems and the idea of those theorems is ultimately based on Tarskis analysis of logical consequence. But what does the proof of Hil-berts Thesis consists of? This is an exercise in the formalist paradigm of doing logic and mathematics. Any soundness and completeness theorems establish certain rela-tions between two mathematical structures; it so happens that some elements of one of

21 Cf. Woleski [1980].22 See Etchemendy [1990: 5-6].23 Olszewski [2009].

Legal logic. Myths and challenges

54these structures are interpreted informally as designating truth values. This does not mean, however, that one cannot account for some interesting relations between propo-sitions which have nothing to do with their truth values. This is the case in a number of recently developed formal systems, such as some kinds of nonmonotonic logic.24

6. The myth of hard cases

It is sometimes argued that logic may be applicable in the law, but only in easy, algorithmic cases. The so-called hard cases the argument runs escape any logi-cal analysis. This claim may be treated as a rened version of the triviality objection. However, as it does not reject the role of logic in the law altogether, but only limits its application, I treat it separately.

At the outset it should be noted that it is difcult to establish a sharp criterion for distin-guishing between easy and hard cases. The best candidate for such hard or non-algorith-mic cases are those in which there is a conict between a legal rule and a legal principles (in the Dworkinian sense).25 The possibility of such conicts makes room for a potentially innite number of exceptions to any legal rule: such rules cannot be formalized in the classical monotonic logic. Thus, hard cases may be prima facie identied with situations in which there is a conict between a rule and a principle; they are hard, as they cannot be accounted for with the use of the classical logic. This does not mean, however, that there are no identiable formal aspects of such conicts. It is argued that they are well captured in defeasible systems. An example of a defeasible logic is a system which operates on two levels. On the rst level, from a given set of premises arguments are built; on the second level the arguments are compared in order to decide which of them prevails. The conclu-sion of the best argument becomes the conclusion of the given set of premises.26

This leads us to a more general observation. I believe that any reasoning even in the hardest of cases has a certain structure. If so, it is formalizable: one can develop a logical system that captures the rationality standard for the given type of arguments. This applies also to such non-algorithmic views of legal cognition as the ones pro-posed in legal hermeneutics. The hermeneutic act of cognition has a certain struc-ture, although it is vaguely dened. However, a logician who considers the famous hermeneutic circle may suggest a number of its formal interpretations: with the use of the logic of abduction, formal theory of belief revision, logical theory of coherence, defeasible logic, or a combination of these techniques.27

7. Is there a special legal logic?

There is no doubt that logic may serve to analyse and set standards for legal reasoning. The question emerges, however, whether there exists a special or dedicated legal logic. Jaap Hage answers this question in the positive.28

24 Cf. Hage [1997] and Broek [2004].25 Cf. Dworkin [1978].26 Cf. Prakken [1997].27 I explore these possibilities in some detail in Broek [in press].28 Hage [2001].

Bartosz Broek

55Hage begins by noting that it is sometimes claimed that there exists (informal) legal

logic, one in which such arguments as:

(ARGUMENT 1)

A did not commit a crime forbidden by the written law.

Therefore: A is not liable to be punished.

are valid. The obvious observation is that this argument is invalid in classical propo-sitional logic (an arbitrary p does not follow logically from an arbitrary q). This fact forced Arend Soeteman to question the possibility of a logic exclusively dedicated to law.29 Soeteman observes that such arguments as the one presented above are either enthymemes (i.e., the missing premise, the proposition p q, holds), or they are logi-cally invalid. In other words, Soeteman launches an attack against (informal) legal logic which makes use of domain knowledge, [by claiming that] either the domain knowledge can be made into an acceptable additional premise which makes the argu-ment formally valid [], or it is not possible, and the verdict of formal logic that the argument is invalid turns out to be the correct one.30

Hage disagrees with this conclusion indicating that Soeteman presupposes a sharp distinction between the form and the content of arguments.31 However, such a distinc-tion is always relative to the accepted formalism. Even when we consider proposi-tional logic vis a vis predicate logic Hages argument runs the distinction between form and content becomes obscure. Let us consider the following argument:

(ARGUMENT 2)

All thieves are punishable.

John is a thief.

Therefore: John is punishable.

Reconstructed in propositional logic, this argument is invalid, while it becomes valid formalized with the resources offered by the predicate logic. Of course, one can follow Soetemans strategy and imply that there is a hidden premise to the argument, namely If all thieves are punishable and John is a thief, then John is punishable. However, the possibility of adducing such hidden premises has never been considered a reason to denounce the predicate logic as informal or no real logic. The same line of argument can be applied, Hage claims, to dedicated legal logic.

Hage reinforces his view by discussing the problem of the nature of logic. He in-vokes a Quinean holistic view of knowledge and claims that logic is but a knot in our web of beliefs. Of course, in the face of inconsistencies, we are more prone to revisit our empirical beliefs or scientic theories; however, neither logic nor mathematics is adjustment-proof. In this way, logic is not opposed to domain knowledge, as it is on the traditional view [of logic]. There is a continuum in our presumed knowledge, ranging from accidental beliefs which we are willing to revise on the slightest evidence that they are false, through rm beliefs which we are only prepared to give up on the basis of strong counter evidence, corroborated laws which we use to derive beliefs

29 Cf. Soeteman [1989].30 Hage [2001: 354].31 Cf. also Sher [1991] and Read [2002].

Legal logic. Myths and challenges

56and which we only give up if we can nd better ones, to logical laws, of which we cannot even imagine circumstances under which we are prepared to give them up.32 This leads Hage to a much more liberal view of logic than traditionally assumed: logic deals with all connections between propositions which we hold to be neces-sary because we are not prepared to change them in the case of incompatible beliefs. Such necessary connections may be based on the meanings of logical operators, and therefore logic in the traditional sense is part of the holistic logic []. But other neces-sary connections than those based on the meanings of logical operators fall under the scope of logic too. [] This more liberal picture of logic [] leaves room for a special legal logic. The task of such a logic would be to explore (semi)-necessary relations that belong specically to the domain of law.33

In order to evaluate both Soetemans and Hages stances, let us pose the question as to what may be the relationship of logical consequence between propositions analysed (captured) by various logical systems. According to the traditional view, logic is a conse-quence relation understood in terms of Tarskis analysis: A sentence A follows logically from the set of premises G if and only if in every case in which the premises belonging to G are true, A is also true. This truth-preservation view of logical consequence may be re-placed with a weaker notion pertaining to justication: logic identies such relationships between the sets of premises and their conclusions in which the conclusions are justied relative to the premises. (It is easy to observe that such a relation of logical consequence is not necessarily monotonic and thus it ts the defeasible logic referred to above.). Finally, one may adopt the broadest understanding of logic, one that allows logicians to analyse any relations between propositions. This tripartite division corresponds to three different denitions of logic: as a formal system that encodes the Tarskian relation of logical consequence, as the justication-grounding formal system, and as any formal system that encapsulates some formal relations between propositions.

In all these cases, the relations in question are formal. The question is, however, how to dene form, or, in other words, when a system counts as formal? Is there any suit-able criterion that would differentiate between form and content? When the broadest denition of logic is assumed, there is no such criterion. This seems to be Hages view: such arguments as (ARGUMENT 1) can be considered logically valid. In other words, there are no a priori limits of incorporating domain-specic knowledge into the formal apparatus. It follows that there can be a special legal logic.

When I nd Soetemans position too restrictive, I also believe that Hages stance is too liberal (at least as it is stated). Of course, there is no a priori limit on which concepts of our knowledge (represented as predicates in the rst order logic) may be turned into logical operators. However, let us consider the following argument:

(ARGUMENT 3)

x has entered into a service agreement that requires of him to see to it that p.

Therefore: x has an obligation to see to it that p.

It is possible to design a logical system in which this argument would be valid. It would sufce to introduce two logical operators:

32 Hage [2001: 358-359].33 Hage [2001: 359-360].

Bartosz Broek

57E(x, p) for has entered into a service agreement that requires of him to see to it that; and

O(x, p) for has an obligation to see to it that;

where x ranges over persons and p over actions,

and a rule of inference (RI/E-O):

E(x, p)

O(x, p)

However, I believe that the introduction of new operators must meet at least the fol-lowing two conditions:

(Condition 1) The concepts transformed into logical operators must be general enough to be applicable in

a sufciently broad class of arguments.

(Condition 2) The introduced operators must be denable in the semantic structure of the given logical

system.

I am fully aware of the fact that both conditions are, to a certain extent, arbitrary and vague. However, I would argue that they hint at a real problem. Let us consider the operator E(x, p) I introduced above: it does not meet (Condition 1), as it is not general enough; at the same time, the operator O(x, p) seems to comply with (Condition 1). It is clearly visible as soon as one considers the need to introduce the rule of inference (RF/E-O) in order for the operator E to work. There are a number of ways in which our actions may generate obligations: one of them is entering a contract. But there are numerous kinds of contracts, the service agreement being only one of them. It follows, therefore, that the introduction of the operator E(x, p) would require the introduction of a number of other operators corresponding to all kinds of contracts (lease agree-ment, loan agreement, brokerage agreement, and so on; notice also that there exist the so-called innominate contracts, i.e. contacts whose essentialia negotii are not specied in any legal act, but which can be entered into on the basis of the freedom of contract; thus, there exists a potentially endless list of the types of contracts). Add to it that for every such new operator one would need to introduce corresponding rules of infer-ence, and then the resulting formal system would be useless.

One way out of this problem is to stay at the suitable level of generality. For exam-ple, one may argue that instead of introducing the operator E(x, p) and an indenite number of other operators standing for various types of contracts, one need only to introduce the operator C(x, p) standing for x has entered into a contract that requires of him to see to it that p. However, such a solution helps us to deal with (Condition 1) only (Condition 2) remains unanswered. The motivation behind the latter require-ment is straightforward. When one considers such logical operators as obligatory (in the standard deontic logic), necessary (in modal logic), see-to-it-that (in branching-time deontic logic), they are given intuitively correct semantic denitions (e.g., a proposi-tion p is necessary iff p is true in every possible world accessible from the actual world; a proposition p is obligatory iff p is true in every possible world belonging to the class of deontically perfect worlds, etc.). As long as there is no similar denition for the spe-cically legal logical operators (like E(x, p) or C(x, p)), there is no proof that a special

Legal logic. Myths and challenges

58legal logic is possible. I know of no such logical systems (and the examples that Hage provides e.g. the logical tools for dealing with the distinction between rules and prin-ciples are not law-specic, as they apply equally easily to ethical discourse or even to some common sense descriptive reasoning). This is not to say that it is impossible to develop such full-blooded legal logical operators. However, this manoeuvre needs to be justied and the best way to do so is to introduce a semantic structure which is typically legal and has an obvious advantage over representing legal knowledge within some domain-independent semantics.

My doubts concerning the existence of a specic legal logic have no bearing on the claim that there exist interesting and fruitful elds of studying law with the use of logic. The application of logical tools in the legal domain have led not only to the de-velopment of new formal techniques (e.g., some incarnations of defeasible logic), but also to profound questions pertaining to the very nature of logic. There is little doubt that this rich area of study will bring us more fruit.

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Legal logic. Myths and challenges