Jusletter IT

The thing, which applies in all the possible worlds, is necessary . This definition, which is attributed to Leibniz, became popular in the last decades. This trend is connected with the rapid development of modal logic, namely the logic of principles concerning «possible and necessary entities». What is the meaning of the wordworlds ? A connoisseur of the science fiction literature will not hesitate a moment to cite a bunch of novels as examples, beginning from H.G. Wells (The War of the Worlds ) but worlds can still have some other different meanings. Hughes and Cresswell (1968), the authors of the most well-known primers to the modal logic, brought in new concept that, humans and their knowledge are worlds, while they reject any anthropological or ontological requirement emphatically. With such restraint they are qualified to be pure logicians; but in the scope of social philosophy, people may recall Heinrich Heine:Each individual person is a world, which is born with him and dead with him, and under each gravestone there is a world’s history¹ .

Hughes and Cresswell have created a game to illustrate the rules of modal logic: There are people sitting on chairs; each of them has a piece of paper on their knees, with sentences which the participants regard as true for themselves individually. The sentences can be shortened for the sake of simplicity throughp, q, r, s and so on. For the illustration of the simple, but not yet modal logic of sentence, one single person is enough. For example, what the sentence of an excluded third person concerns, people first ask him whether it is sentencep (or any other sentence). If the sentence is listed in his note, he will raise his hand up, otherwise he will not. Then people ask if it is «not p». Now he will raise the hand up, if the sentence is not listed in his note. Otherwise he will not do it (this requires the rule for negation being set). Finally people ask him whether he has lifted his hand at the firstor  at the second time (in accordance with the rule for the disjunction). The person being asked will lift his hand now: anyway, as long as the sentence of the excluded third is true.

For the modal logic we need several people, each one of them can only see one person or another person, but not both² . In the simplest case everyone can see every other: for example the participants are sitting around a round table. If people ask forp , again the players withp written on their notes will lift their hands. Right now, they can also be asked forNp , which means, whether p is necessary. Now if one has seen before, that everyone had raised their arms³ , thus he will give a positive answer to the question of whetherp applies inall worlds . If, s before, at least one person has lifted his hand (it may be the person himself), then they will show their hands up on the question aboutPp to show their agreement: p is possible because it applies in at least one world.

People can ask further questions, for example forNNp ,NNNp or NPp etc., but no further answers can be received at this constellation. Because everyone, who sees anyone else, has also noticed every previous gesture of each other and can confirm this on further questions repeatedly. Expressions of the graded modalities such asnecessarily possible ,necessarily necessary and so on, can be reduced without the loss of meaning to a one-level necessary or possible. Among logicians the system of modal logic, which arises from this visual arrangement, has the nameS5 .

Certainly, this reducibility applies only for the arrangement where everyone sees anyone else. It is, for example, conceivable if every participant is sitting in a cabin, from which he or she can see the one next to his right side; but not the one besides his right neighbor, who is in the cabin that bends at the right angles. He was not able to see his neighbor on his left hand side, because of the one-way windows; but she can perceive there is someone there!

Right now let’s say, we have such an arrangement, in which the participants can only have a one-way and short vision. If only the person at the right sees a p on his note and nobody else then the question aboutp , then only this rightmost participant will lift his hand. And related to the question aboutMp , only he and additionally his left neighbor will rise their hands up. Finally, concerning the question aboutPPp also the left neighbor of the person next to the rightmost one will give a signal. If there are no remaining participants involved into this game, we can conclude: people receive no new information by additional questions. So many grades of(possible) possibility will arise, which is the same as the numbers of participants in the game, and it is calledprincipally infinite . The system is called T. People note that here and in the systems being described, the expression ofall worlds has a relative meaning: they are only those worlds that people can perceive from a world each time. In order to modify the Leibniz’ quotation stated at the beginning: Something is possibly necessary, but only if people have an idea about some worlds and it applies to all the worlds from the viewpoint of some other worlds.

It is obvious that people can arbitrarily arrange many different seating orders with likewise different kinds of relations of perception. If one wants to limit oneself to systems with pure forms of relations, one will probably first think of symmetrical and transitive ones. Using features like Spanish walls and one-way windows can be helpful to realize it.

Symmetry : B perceives the behavior of C and C sees the behavior of B, too. Furthermore, B also sees the behavior of A and A sees that of B, too. However A does not see – beyond B – the behavior of C, but only, how B reacts to it. Accordingly, C’s vision cannot reach up to A, either. The players only notice each other mutually in pairs. But of course everyone can be involved in several partner relations. The system has the name B. Together with the system T, B also has the small range of vision, which reaches only up to the neighbor. If – as it is assumed in the second example – p only applies in the world of the participant on the rightmost, but not in all the other worlds, then so many questions and answers are required, just as the number of the participants in the game, until the participants have passed down the possibility of a p one by one to the leftmost one, and further questions lead to no new answers. Again, that means that infinite amounts of grades for modality are possible. A difference to our example of T with the same seated order shows up in the fact that the rightmost one recognizes immediately that also a world is possible, in which there is no p: He sees it from the behavior of his partner. In the example of T the rightmost participant could only see his own world (even this is by no means taken for granted) and regarded obviously the conditions in it as necessary.  

Transitivity : A sees the behavior of B, and B sees that of C. But A sees even more – through B’s cabin – the behavior of C, too. However, because there is a one-way-window on the cabin, the view is blocked in the reversed direction. The name of this system isS4 . Here people can recognize the worlds with one look from left to right. Multi-graded modalities, which cannot be reduced, do not arise. Therefore only those with two stages and sometimes with three stages arise. They are based on the fact that the one, who can look forward, cannot look backward at the same time (in this aspect it is just the same as system T). In the distribution of seats which we have selected as example, the participant in the outer edge cannot look into another world. Thus he is not convincible regarding the conditions in his own world and therefore there exists a two-grade modality MN).⁴

Where is the relation of reflexivity – that is the self-perception here? In their introduction of 1968, Hughes and Cresswell left the reflexivity unquestioned, perhaps because allegedly they have arranged all the plausible constellations. For the new introduction in 1996 it was not the case anymore. Now also the relation of reflexivity is being questioned. However the authors remarkably omitted to present a suitable variant of the game, although that would be not so difficult at all. People imagine a group of seniors, who suffer from presbyopia but have no reading eyeglasses at hand. Although they can recognize the behavior of their neighbors, they cannot read what is written on the paper of their own. (Moreover the seeing-relation could be different: transitive or symmetrical or both or neither.) Certainly this variant seams a little absurd. In this case, the relation of self-perception is philosophically important.Recognize yourself! is carved at the entrance to the Oracle of Delphi and was one of the keywords of ancient philosophy.

That Hughes and Cresswell did not introduce such variants may have two reasons. First of all: In pure form it is not applicable. None of the participant can recognize a sentence as true or false and then answer the following question about M or N. In order to be realistic, at least one player should have normal eyesight; and he might serve as the model for others.

Secondly: systems of modal logic without reflexivity seem to offer a possibility for interpretation, which looks desirable at first sight, but is incompatible with the concept of the possible worlds and the game that illustrates it. These systems are often marked with the letter D – of «deontic». The deontic logic is a variant of the modal logic, which concerns sets of the orders and permissiveness. They are indeed obviously similar. «You must do such and such tomorrow!» will often mean: «You should … tomorrow!» and «You can do such and such!» mostly means: «You may…!» The expressions «must» and «can» often appear more naturally in the daily use of language than «should/shall/ought to» and «may». Even Kant occasionally spoke ofpractical necessity andpractical possibility . And the conclusions are as just as the sentence: «What is necessary is also possible» is generally valid in the ontic modal logic, the sentence «What is required, is also permitted» applies also plausibly in the deontic logic. Certainly the other sayings such as «What is necessary, is also real» as well as «What is real, is also possible» apply in the standard modal logic, too. The corresponding conclusion forms are however excluded from the deontic logic, out of the reason that they express the inadmissible conclusions from «being supposed to» to «be being» as well from «being» to «being allowed to». If one takes the relation of reflexivity out of the system of the modal logic, the two conclusion forms disappear; the modality of the «real one» is skipped over. From the representation of Hughes and Cresswell that can be seen easily.

In my opinion the follower of the deontic logic certainly would have to abandon the whole conception of the possible worlds and their description by using different view arrangements. Here might be the second reason, why Hughes and Cresswell present no deontic variant of their game. Because as long as one obeys its general rules, then the situations of «What all do, should be done by him⁵ too!» as well as: «What at least one does (he is again left out of consideration), is also allowed for him to do» would apply. That would be most problematic rules. They would also have no similarity with the legal rules, whose modeling people will typically think of when creating deontic logic.

Are these rules unrealistic? By no means, if the person who obeys it limit «all» to «all the things /people», which / who are important for him, e.g. particularly to his peer group and models. If I am skating and all the skater wear vans sneakers, then I must also wear vans sneakers and spend the necessary money. And apart from that, what Mick Jagger does, may also be allowed for me.

One does not need to see that cynically. After all, we all have our models, whose opinion we regard as authoritative or their behavior we regard as worthy of being imitated. If our models’ opinions correspond with their behavior, we will regard it as «obligatory». If they deviate from each other, we will interpret this as «permitted», «justifiable», «possible» (perhaps we will also give up one or two of our models).

For a normatively interpreted modal logic (however, as I said before, not for a deontic logic) we can let the model of the possible worlds exist. But we must still introduce the concepts of «relation of acknowledgement» other than the concept of «relationship of perception». The relation of acknowledgment obviously forms a subset for the relation of perception: what people cannot perceive – cannot be recognized. Besides, people will have to differentiate between different areas of acknowledgment. What serves as the model in scientific or artistic respects does not need to be one in the respect of moral.

Do we have to give up the reflexivity that is to give up self-acknowledgement with the relation of acknowledgement, just like it is the case with the systems which lead to forms of deontic logic? If the formal system is suitable for description of the reality, the possibility of a waiver of the self-acknowledgment must be arranged; because it is often missed in the daily life. There is an everyday life situation, which by the way is close to the modal games: Students, who attend an examination, tend to consider what they see on the test paper of their desk mate or what they get from the whisper, is more correct than the answer of their own. We may say they are lack of the self-acknowledgment, but even does it really exist? How sad could that be, if there were no humans who could say it like Frank SinatraI did it my way . Here self-acknowledgment is by no means equal to individual eccentricity.

The relation of perception and acknowledgement is literally more complex than it could be represented here. For example: although the relation of acknowledgement establishes itself first on the relation of perception; a «leader» will also have interest to know who acknowledges him – is it through direct perception or indirectly through informants. He can acknowledge those as his chieftain or reject this form of acknowledgement. Relations of acknowledgement are built on relations of perception, which however, navigate relations of perception again, and these relations of perception will again lead to new forms of the acknowledgement.

Even if the modal logic is completely inappropriate to be used as the representation of sets of the orders and permissiveness in the sense of deontic logic⁶ , it is nevertheless excellently suitable for a structural description for relation of acknowledgement. Someone thinks for instance of the phenomenon that the relation is modally gradated, because he has only one subordinate whose behavior is immediately under his nose, but does not think of the highest model. «If the leader would know that!» was temporarily a popular saying.

The term of the «acknowledgement» gains an ever greater importance in the modern social philosophy. Some names: Sartre, Simmel, Hegel, Fichte are the classical philosophers, in whose works the definition of the acknowledgement plays an important role, although temporarily, the amount of works on it has decreased. Also Leibniz belonged to one of them, and he has initiated all the ways of thinking mentioned here and all of their relatives: his theory of the possible worlds, the modal logic and the deontic or at least normative logic, has already seen the importance of the acknowledgement. Anyway I quote that from his remark:Aeqvum, debitum est qvicqvid necessarium est fieri a viro bono. Unde sapienter Iti Romani Legibus indefinita revocanda ajunt ad viri bono arbitrium.⁷

Perhaps – this only serves to be a possibility. People should also bring into play the act of «distortion» of one another «world»: it is being suggested by the aggressiveness of some worlds against the comparable parallel worlds.⁸ People should also – this can be served here only as one suggestion – beyond the snapshot for relations of acknowledgement examine the dynamics of their logically developed theory. For example: someone should seriously do the things which he has acknowledged as right or wrong. With his own behavior he should publicize it or make it known to the outside world.

At the end there are still a few examples of different relations of acknowledgement⁹ : couples which are self-sufficient; families which isolate themselves from the outside world; leaders who have followers pursuing them. The followers are an unformed mass or have been hierarchically organized. Stars, which are orbited by their planets (on their part they’re isolated from each other). Groups, maybe with outsiders who are associated with them in some way, maybe one single member of the group has acknowledged the outsider to be one of them, or maybe the outsider admires the group and pursues it, without being acknowledged as one member of it. But even loners, among them they are such people, who always acknowledge themselves, or those ones, who never do it once.¹⁰ There are more concrete interpretations, which might involve gangs of youth as well as circles of the scientists.

G.E. Hughes and M.J. Cresswell (1968): An Introduction to Modal Logic, (Routledge – London, New York);

G.E. Hughes and M.J. Cresswell (1996): A New Introduction to Modal Logic (Routledge).

---

Prof. Dr. Lothar Philipps
Universität München
loth@jura.uni-muenchen.de

---