Abstract. The past ten years have shown a great variety of approaches for formal
argumentation. An interesting question is to which extent these various formalisms
correspond to the different application domains. That is, does the appropriate argu-
mentation formalism depend on the particular domain of application, or does “one
size fits all”. In this paper, we study this question from the perspective of one rela-
tively simple design consideration: should or should there not be contrapostion of
(or modus tollens) on defeasible rules. We aim to show that the answer depends
on whether one is considering epistemical or constitutive reasoning, and that hence
different domains require fundamentally different forms of defeasible reasoning.
Keywords. epistemical reasoning, constitutive reasoning, contraposition
1. Introduction
Recent years have brought a large variety of research regarding argumentation for-malisms. These formalisms tend to differ in the way that arguments are constructed andthe defeat relation is defined [5,20,6], as well as in the abstract argumentation semantics[9,1,8], and in the various additional forms of functionality that can be provided [4,15].
Given the wide variety of formalisms and approaches, one can ask the question how these relate to each other as well as the question of how this variety is related tothe different domains in which argumentation can be applied. As for the first question,some work has already been done by for instance Baroni and Giacomin, who provide anumber of abstract properties according to which various argumentation semantics canbe compared [3,2]. As for the second question, much less work has been done. Moreover,there is little clarity on whether one should aim for a general argumentation formalismthat is applicable to a large variety of domains, or whether different domains requiredifferent and perhaps fundamentally incompatible classes of reasoning.
In this paper, it will be argued that the approach of “one size fits all” has serious disadvantages. We claim that there are two fundamentally different forms of reasoning,epistemical and constitutive, and that these require fundamentally different propertiesregarding formal entailment.
2. Default Contraposition in Epistemical Reasoning
Although the issue of applicability of default contraposition seems to be a fundamentalone, it has until now received relatively little attention. Many authors treat the validity or invalidity of contraposition as an aside when describing their respective formalisms fornon-monotonic reasoning. Our analysis will therefore begin with an overview of someof the comments by various authors in the field. Brewka, for instance, provides the fol-lowing counterexample against the validity of contraposition: “Men usually do not havebeards, but this does not mean that if someone does have a beard, it’s usually not a man.”Another example would be: “If I buy a lottery ticket then I will normally not win anyprice, but this does not mean that if I do win a price, I did not buy a ticket.” Given the last two examples, it seems that there are perfectly legitimate situations in which contraposition does not hold, and that contraposition (or modus tollens) shouldtherefore be rejected as a general principle for defeasible reasoning. The point is, how-ever, that once one starts to accept counterexamples against default contraposition, thenone should also take into consideration counterexamples against various other principlesfor defeasible reasoning: irrelevance “Tux the bird”: Birds fly and Tuxes are birds.1 Do Tuxes fly? Perhaps not,
because Tuxes may belong to a special subclass of birds that do not fly.
left conjunction “jogging in the rain” [19]: If it is hot, I tend not to go out jogging. If it
is raining I also tend not to go out jogging. Does this mean that if it is hot and it israining, I tend not to go out jogging?“Marry both of them” [17] If you marry Ann you will be happy, if you marryNancy you will be happy as well. Does this mean you will be happy if you marryboth of them? transitivity “unemployed students” [16] Students are usually adults and adults are usu-
ally employed. Does this mean that students are usually employed? The above counterexamples against irrelevance, left conjunction, contraposition and transitivity look appealing at first sight. The point of each counterexample, however, isthat it involves implicit background information. Tux does not fly because it is a pen-guin; marrying two persons generally does not make one happy (one may end up in jailinstead); women have no beards at all; and students are a special class of adults that tendto be unemployed. The view that the counterexamples against contraposition, like theones above, are flawed is shared by Ginsberg [10, p. 16], although he treats a differentexample himself (“Humans usually do not have diabetics”).2 The “all or nothing” approach to the above mentioned properties of irrelevance, left conjunction, transitivity and contraposition is confirmed when one tries to give the de-feasible rules a statistical interpretation. In ε-semantics [17], for example, none of theseprinciples are satisfied, whereas the Maximal Entropy approach [11] satisfies all of them,but as defeasible principles only (that is, their instances can be blocked if specific in-formation against their applicability is available). It appears that if one wants to make aconsistent choice that includes the (defeasible) validity of properties like irrelevance, leftconjunction and transitivity, then one should accept the (defeasible) validity of contrapo-sition as well. Yet, it is striking to see that formalisms for defeasible reasoning tend notto be based on any consistent choice on these issues. A similar observation can be made 1Tux is the well-known penguin logo of the Linux-community.
2As an aside, it appears that the “counterexamples” against contraposition involve rules where the antecedent contributes negatively to the consequent. That is, the consequent holds in spite of the antecedent. See [7] for amore elaborate discussion.
regarding the to contraposition related principle of moduls tollens. Both Reiter’s defaultlogic [22] and the formalism of Prakken and Sartor [20] sanction a defeasible form ofmodus ponens, but do not sanction any form of modus tollens. A systematic analysis ofthe actual meaning of a default is often not provided. Yet, it is this analysis that shouldserve as a basis for determining which principles should or should not be sanctioned.
The current trend seems to be to sanction various principles, but not those of (defeasible)modus tollens or contraposition. It is an anomaly that is rarely questioned, and one maywonder whether this is because many researchers have become acquainted with it. Or, asGinsberg states when discussing the reasons behind the opposition against contraposition[10, p. 16]: (.) although almost all of the symbolic approaches to nonmonotonic reasoning doallow for the strengthening of the antecedents of default rules, many of them do notsanction contraposition of these rules. The intuitions of individual researchers tendto match the properties of the formal methods with which they are affiliated.
3. Default Contraposition in Constitutive Reasoning
In the current section, we again ask the question whether contraposition should be sanc-tioned, this time not from the perspective of probabilistic empirical reasoning, but fromthe perspective of constitutive reasoning. The difference between these two forms of rea-soning can perhaps best be illustrated using a mirror example, which is a small logicalformalization that can be given two informal interpretations with opposite conclusions[7, section 2.2.5].
Informal Interpretation 1 (II1): The goods have been ordered three months ago(T M A) and the associated customs declaration is still lacking in the customs infor-mation system (LIS). If the goods have been ordered three months ago, then theywill probably have arrived by now (T M A ⇒ A). If the goods have arrived, thenthere should be a customs declaration for them (A ⇒ CD). If the registration ofthe customs declaration is still lacking in the customs information system, then thereprobably is no customs declaration (LIS ⇒ ¬CD).
Informal Interpretation 2 (II2): John is a professor (P ) who is snoring in the univer-sity library (S). Snoring in public is usually a form of misbehaviour (S ⇒ M ). Peo-ple who misbehave in the university library can be removed (M ⇒ R). Professorscannot be removed (P ⇒ ¬R).
In the “arrival of goods” example (II1) it seems reasonable to apply contraposition on A ⇒ CD to construct an argument for ¬A. In the “snoring professor” example (II2),however, it would be very strange to have contraposition on M ⇒ R since this wouldallow us to construct an argument for ¬M . In fact, example II2 has been taken from[18, p. 185] where it is claimed that the justified conclusions should include M , but notR or ¬R. Hence, the above pair (II1, II2) can be considered as a mirror example inthe sense of [7, section 2.2.5]. The next question then is how this situation should bedealt with. That is, do we (1) reject at least one of the formalizations as “incorrect” or(2) acknowledge that the two examples are related to fundamentally different forms ofreasoning? In this paper, we choose for the second option. That is, we claim that there isa fundamental difference that makes contraposition applicable to II1 but not to II2.
In order to understand the nature of constitutive reasoning, it is useful to distinguishbetween statements that have a word to world direction of fit, and statements that have aworld to word direction of fit [24,25]. It should be noted that also one of the differencesbetween II1 and II2 concerns the direction of fit.
The defeasible rules of II1 are meant to describe when a certain fact holds in the object-world. This object-world has an existence that is independent of the rules thatexpress our knowledge about it. These rules, therefore, have a word to world direction offit. Their correctness depends on a validity that has an independent existence.
In II2, on the other hand, the very nature of the rules is different. The rules do not merely describe the reality, but to some extent also construct it, especially if we assumethese rules to be taken from, say, the library regulations. The rule S ⇒ M , for instance,contributes to the definition of misbehavior in the context of the library regulations. Therule essentially makes it the case that snoring is considered to be misbehavior, as far asthe library is concerned. The defeasible rules of II2, therefore, have a world to worddirection of fit. Their application results in the creation of new (legal) facts.
Epistemic versus constitutive reasoning Based on the direction of fit, one can distinguish two kinds of reasoning: epistemic andconstitutive3. The nature of this distinction can be described as follows [12, p. 60]: “Epis-temic reasons are reasons for believing in facts that obtain independent of the reasonsthat plead for or against believing them. Constitutive reasons, on the contrary, influencethe very existence of their conclusions”.
In order to understand the differences between epistemic and constitutive rea- soning, we provide the following abstract example4 (AE): Premisses = {A; D},Defeasible rules = {A ⇒ B; B ⇒ C; D ⇒ ¬C} conflict: A; A ⇒ B; B ⇒ Cversus D; D ⇒ ¬C Now, take the following two constitutive interpretations of this example.
deontic The following example is somewhat similar to that of the Christian Soldier. An
artillery soldier is given the order to destroy an enemy military installation, andorders should generally be obeyed (order ⇒ O(shoot)). When the soldier looksthrough his binoculars, he observes some movements that probably mean thatsome people are really close to the target (movements ⇒ people), thus makingit from an ethical point of view imperative not to shoot (people ⇒ O(¬shoot)).
Thus, we have: Premisses : {order , movements} and Defeasible rules ={movements ⇒ people; people ⇒ O(¬shoot); order ⇒ O(shoot)}. Con-flict: movements; movements ⇒ people; people ⇒ O(¬shoot) versusorder; order ⇒ O(shoot)Here, the conflict is between the obligation to shoot and the obligation not to do so.
In some logics, like Standard Deontic Logic, such a conflict would lead to an in-consistency. If we would allow for contraposition, the effect would be that people 3The term “constitutive rules” was originally introduced by Searle [23]. In this essay, however, we use the 4The reader will notice that the structure of this example is similar to II1 and II2.
is no longer justified. This is, of course, absurd; the belief in empirical statementsshould not depend on the presence or absence of deontic conflicts.
legal An example of a legal interpretation is II2. Here, the reasoning concerns whether
or not certain legal facts obtain. Even though the conflict could be described indeontic terms (is the library personnel permitted to remove the person in questionor not), the conflict (P ermitted(remove) v.s. ¬P ermitted(remove)) is essen-tially not of a deontic nature, like in the previous example (Obliged(shoot) v.s.
Obliged(¬shoot)). The question is whether it is legally permitted to remove theperson or not, and this question does not rely on the specifics of deontic reasoning.
The fact that this conflict exists, however, is no reason to reject the intermediateconclusion of M . To make this point more clear, suppose that the library regula-tions contain an additional rule saying that those who misbehave have to pay a fineof ten euro (M ⇒ F ) and that no rule is available that provides professors withexemption for this fine. Then, the fact that the intermediate conclusion M can leadto R (which conflicts with ¬R) is no reason to disallow the entailment of F .
The point is that constitutive reasoning obeys different principles than epistemic reason-ing. Under epistemic reasoning it is perfectly reasonable to sanction contraposition, aswas argued in section 2. Under constitutive reasoning, on the other hand, contrapositionis not valid by default, as was discussed above. In legal reasoning, for instance, the lead-ing paradigm is that the law should be interpreted as consistently as possible. Hence, inthe snoring professor example the potential conflict between R and ¬R is not a reason tocreate a conflict between M and ¬M and hence reject M or F . The idea is to keep theeffects of possible conflicts as local as possible [12, p. 109]. A great deal of research hasbeen dedicated at stating and formalizing meta-principles (such as lex posterior, lex spe-cialis or lex superior) for determining which of the conflicting rules should be applied,and which should not. But even in the case that no determining meta-principle is avail-able, the application of both rules is blocked and the conflict does not have consequencesfor conclusions that do not depend on it. Our snoring professor, even though he may notbe removed, still has to pay his 10 euro fine.
(Im)perfect procedures versus pure procedures The difference between epistemic and constitutive reasoning is comparable to the differ-ence between (im)perfect procedures and pure procedures, as distinguished by Rawls. Toillustrate the concept of a perfect procedure, Rawls provides the example of cake-cutting[21, p. 74]: A number of men are to divide a cake: assuming that the fair division is an equal one,which procedure, if any, will give this outcome? Technicalities aside, the obvioussolution is to have one man divide the cake and get the last piece, the others beingallowed their pick before him. He will divide the cake equally, since in this way heassures for him the largest share possible. This example illustrates the two character-istic features of perfect procedural justice. First, there is an independent criterion forwhat is a fair division, a criterion defined separately from and prior to the procedurewhich is to be followed. And second, it is possible to devise a procedure that is sureto give the desired outcome.
One of the assumptions of the above cake-cutting example is that the person cutting thecake can do so with great accuracy. As long as deviations in cutting are ignored, theresult will be an equal distribution. If we assume that the deviations in cutting cannotbe ignored, cake-cutting becomes an imperfect procedure. The characteristic mark ofan imperfect procedure is that while there is an independent criterion for the correctoutcome, there is no feasible procedure which is sure to lead to it [21, p. 75].
A pure procedure, on the contrary, is the case when there is no independent criterion for the right result: instead there is a correct or fair procedure such that the outcome islikewise correct or fair, whatever it is, provided that the procedure has been properlyfollowed. An example of a pure procedure is a free election. The outcome of electionscannot be evaluated as “right” or “wrong” according to an outside objective standard.
The idea is that any resulting outcome should be accepted, as long as the election processitself was carried out in a correct way. In general, one can only fight the outcome of apure procedure by arguing that the procedure itself was not applied properly [14].
Since the process of reasoning can to some extent be seen as a procedure, it is inter- esting to evaluate how the kind of reasoning as performed in II1 and II2 can be seen interms of (im)perfect and pure procedures.
II1 is basically an instance of empirical (epistemic) reasoning. One uses potentially incomplete information and rules of thumb, with the idea that the reasoning process islikely to generate a correct result. Even though an outside criterion exists to evaluatecorrectness (the goods have either arrived or not), there is no guarantee that the reasoningprocess indeed obtains this result. Hence, the reasoning process as performed in II1 canbe seen as an imperfect procedure.
II2 is an instance of constitutive reasoning. The idea of the library regulations is that applying them defines which (legal) consequences hold in a particular situation. Thereis no outside criterion, other than the library regulations themselves, that allows us toevaluate the legal implications as far as the library is concerned. Hence, the reasoningprocess can be seen as a pure procedure.
The difference between epistemical and constitutive reasoning has implications for what principles do or do not hold in the reasoning process. Let us ask the question ofwhether some principle (like contraposition) holds in constitutive reasoning. The answer,of course, is that it depends on how the particular form of constitutive reasoning is de-fined. This definition needs not to be explicit. It may very well be that a certain typeof informal reasoning has become common in a certain community, and that it is theresearcher’s task to provide a formal model of this reasoning; this is essentially whathappens in, for instance, AI & Law.
In epistemical reasoning, an outside criterion is available for determining whether the results are considered correct or not. The task of the reasoner is to perform its rea-soning in such a way that the outcome approximates the objective criterion as closely aspossible. In a certain sense, the presence of an objective criterion forces the reasoningprocess to become of a certain shape, in which certain properties (like contraposition)hold and other properties do not hold.
In constitutive reasoning, such an objective criterion is absent. For the community of reasoners, there is nothing that forces their reasoning process to become of a certainshape. In essence, the reasoners rely only on their own opinions and intuitions regardingwhat such a reasoning process should look like and which properties it should adhereto. Wason’s card experiment, however, makes clear that a large group of people has difficulties with the principle of contraposition; it should therefore not come as a surprisethat, when no outside constraint or criterion is present that forces its validity, the type ofunreflective reasoning that a group of people comes up with does not necessarily sanctioncontraposition.
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