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31 Jan 2018

Goldschmidt (2.1.4.1.34) Le système stoïcien et l'idée de temps, “Divination et connaissance dans le présent, 1”, summary

 

by Corry Shores

 

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[The following is summary. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, which means typos are present, especially in the quotations. So consult the original text. Also, I welcome corrections to my interpretations, because I am not good enough with French or Greek to make accurate translations of the texts.]

 

 

 

Summary of

 

Victor Goldschmidt

 

Le système stoïcien et l'idée de temps

 

Deuxième partie:

Aspects temporels de la morale stoïcienne

 

A

La Connaissance

 

Chapitre IV

L’interprétation des événements

 

I

L’interprétation a l’échelle cosmique

 

2.1.4.1.34

Divination et connaissance dans le présent, 1

 

 

 

 

Brief summary:

(2.1.4.1.34.1) The events of the world happen by causal necessity and are guided by God’s wisdom. This means that to interpret an event is to understand its place in a series of rationally ordered events that unravel in this ordered way. The practice of divination may not involve actually knowing the causes, but it can still operate by recognizing the signs of causes. (2.1.4.1.34.2) The image of the uncoiling of a rope that Cicero gives when describing an ordered time that can be divined is not about the (metrical or qualitative) homogeneity of time (like the geometrical “time-line”) but is rather about the idea that all events are from the beginning found together and that the temporal succession only deploys an initially given set.

 

 

 

 

 

Contents

 

2.1.4.1.34.1

[The Unraveling of Providential Destiny’s Coil and the Interpretation of Its Events through Divination]

 

2.1.4.1.34.2

[The Togetherness of Moments in the Coil of Time]

 

 

 

 

 

Summary

 

 

2.1.4.1.34.1

[The Unraveling of Providential Destiny’s Coil and the Interpretation of Its Events through Divination]

 

(p.79-80, “On sait que le stoïcisme identifie Destin et Providence...”)

 

[In sum: The events of the world happen by causal necessity and are guided by God’s wisdom. To interpret an event means to understand its place in a series of rationally ordered events that unravel in this ordered way. The practice of divination may not involve actually knowing the causes, but it can still operate by recognizing the signs of causes.]

 

The Stoics identified Destiny with Providence. This means that when we interpret an event, we are both assessing a causal explanation and finding a justification for it in terms of ends. This involves understanding its connectedness within the series of events that unfold as providential Destiny. Goldschmidt quotes Cicero’s On Divination Book 1, LVI, 127:

he who knows the causes of future events necessarily knows what every future event will be. But since such knowledge is possible only to a god, it is left to man to presage the future by means of certain [p. 363] signs which indicate what will follow them. Things which are to be do not suddenly spring into existence, but the evolution of time is like the unwinding of a cable: it creates nothing new and only unfolds each event in its order. This connexion between cause and effect is obvious to two classes of diviners: those who are endowed with natural divination and those who know the course of events by the observation of signs. They may not discern the causes themselves, yet they do discern the signs and tokens of those causes.

(Cicero 1923, copied from Perseus, boldface mine)

Here is the Goldschmidt text:

34. On sait que le stoïcisme identifie Destin et Providence5. En ce sens, interpréter un événement signifie indiffé- | remment l’expliquer (cause) ou le justifier (fin), en le rattachant ou, plutôt, en comprenant son rattachement à la série des événements que déroule le Destin providentiel « Qui tiendrait, en effet, les causes des événements futurs saur a nécessairement tout l’avenir. Cela, nul n’en est capable, excepté Dieu ; mais il reste à l’homme, d’après certains signes qui impliquent des conséquences, à prévoir l’avenir. L’événement futur, en effet, ne surgit pas brusquement, l’écoulement du temps d’un moment à l’autre ressemble au déroulement d’un câble qui ne produit rien de nouveau, mais qui déploie, à chaque fois, ce qui était auparavant. C’est là ce que voient aussi bien ceux qui ont reçu le don de la divination naturelle que ceux qui connaissent le cours des choses par l’observation ; s’ils n’aperçoivent pas les causes elles-mêmes, ils en aperçoivent au moins les signes et les indices »1.

5. Chalcidius, in Plat. Tim., 142 (Chrysippe). S.V.F., II, 933.

(79-80)

1. Cic., de diu, LVI, 127.

(80)

 

 

 

2.1.4.1.34.2

[The Togetherness of Moments in the Coil of Time]

 

(p.80, “La pleine connaissance des événements...”)

 

[In sum: The uncoiling of the rope is an image that is not about the (metrical or qualitative) homogeneity of time (like the geometrical “time-line”) but is rather about the idea that all events are from the beginning found together and that the temporal succession only deploys an initially given set.]

 

Only God has full knowledge of all events and their causes. We have already seen that God sees all things in the cosmic period in the mode of the present. The image of the uncoiling of a rope does not necessarily signify that all things are “perfectly uniform”; rather, it simply conveys the idea that all things are present together: the temporal succession only deploys an initially given set. [I am not entirely sure I understand the implications of that claim. Is the emphasis on ‘initially’ there to suggest that the series can change?] Bergson uses a similar image of an unfolding fan to portray the atemporalism that he was arguing very firmly against. It is primarily in divination that humans imitate, to the extent that they are able, this vision of the whole of time, by beginning with a singular event.

La pleine connaissance des événements et de leurs causes n’appartient qu’à Dieu. Nous avons déjà vu qu’elle saisit toutes choses dans le mode du présent, l’ensemble d’une période cosmique étant présent au regard de Dieu2. L’image du déroulement d’un câble ne signifie pas nécessairement que le cours des choses soit « parfaitement uniforme »3 ; elle exprime simplement l’idée que les choses sont présentes ensemble : la succession temporelle ne fait que déployer un ensemble initialement donné (« traductio temporis... primum quidque replicantis ») ; c’est par une image tout à fait comparable, que Bergson figurera l’intemporalisme qu’il combat4. – C’est principalement dans la divination, que l’homme, autant qu’il en est capable, imite cette vision d’ensemble, à partir d’un événement singulier.

(80)

2. Cf. §§ 17-18.

3. Ch. Appuhn, éd. Garnier, n. 178. – Il est vrai que, en dernier ressort, il y a équivalence entre l’homogénéité de la substance, le présent cosmique des périodes et l’identité des événements à travers les périodes (§§ 25, 2; 101; cf. p. 39, n. 6).

4. « L’éventail qu’on déploie pourrait s’ouvrir de plus en plus vite, et même instantanément ; il étalerait toujours la même broderie, préfigurée sur la soie » (La Pensée et le Mouvant, p. 11).

(80)

 

 

 

From:

 

Goldschmidt, Victor. 1953. Le système stoïcien et l'idée de temps. Paris: Vrin.

 

 

Otherwise:

Cicero. 1923. De Senectute De Amicitia De Divinatione. With An English Translation. William Armistead Falconer. Cambridge. Harvard University Press; Cambridge, Mass., London, England. 

http://www.perseus.tufts.edu/hopper/text?doc=urn:cts:latinLit:phi0474.phi053.perseus-eng1

 

 

 

 

 

 

 

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Goldschmidt (2.1.4.0.33) Le système stoïcien et l'idée de temps, “Interprétation des événements”, summary

 

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Stoicism, entry directory]

[Victor Goldschmidt, entry directory]

[Goldschmidt, Le système stoïcien, entry directory]

 

[The following is summary. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, which means typos are present, especially in the quotations. So consult the original text. Also, I welcome corrections to my interpretations, because I am not good enough with French or Greek to make accurate translations of the texts.]

 

 

 

Summary of

 

Victor Goldschmidt

 

Le système stoïcien et l'idée de temps

 

Deuxième partie:

Aspects temporels de la morale stoïcienne

 

A

La Connaissance

 

Chapitre IV

L’interprétation des événements

 

0

[Introductory Material to Ch.4]

 

2.1.4.0.33

Interprétation des événements

 

 

 

 

Brief summary:

(2.1.4.0.33.1) We must live our lives in accordance with nature, which means wanting whatever happens. To do this, we must understand events, which requires interpretation. Everything in the cosmos is well adapted to everything else. This provides the basis for semiological relations. One interpretative task is interpreting event-signs happening now in the real present as indicative of future situations.

 

 

 

 

 

Contents

 

2.1.4.0.33.1

[Interpreting Event-Signs]

 

 

 

 

 

Summary

 

 

2.1.4.o.33.1

[Interpreting Event-Signs]

 

(p.79, “Pour « vouloir les événements comme ils se produisent » ...”)

 

[In sum: We must live our lives in accordance with nature, which means wanting whatever happens. To do this, we must understand events, which requires interpretation. Everything in the cosmos is well adapted to each other. This provides the basis for semiological relations. One interpretative task is interpreting event-signs happening now in the real present as indicative of future situations.]

 

 

In order to “wish the things which happen to be as they are” (Epictetus Enchiridion 8, Long trans., Perseus. See section 2.1.4.0.32.1; « vouloir les événements comme ils se produisent »), we must know and understand them, which means also that we must interpret them. As Epictetus writes, “God has introduced man to be a spectator of God and of His works; and not only a spectator of them, but an interpreter” (Discourses, Book 1, Ch.6., line 19. Long trans, Perseus. “L’homme, dit Epictète, est né « pour contempler Dieu et ses œuvres, et non seulement pour les contempler, mais encore pour les interpréter ».”) To interpret means to place into contact or relation. But it is of little importance at the moment whether this contact or relation is [final or simply causal]. It must, in any case, so that the relation can be established, that there be at least two terms. Thus, when the Stoics want to establish the order of providence, they show how things are well adapted to one another. And note that for them, the semiological sciences consist of relating a sign to a signified. [I assume they assign a sign now to an anticipated future event.] The two terms, when we are dealing with events, are separated in time. The “event-sign” that we must “interpret” is given to us in the present, that is to say, in the only temporal mode that is real. Although there are different ways to interpret the event-sign, there nonetheless lies at the bases the unique reality of the present.

33. Pour « vouloir les événements comme ils se produisent », il faut les connaître et les comprendre, il faut les interpréter. L’homme, dit Epictète, est né « pour contempler Dieu et ses œuvres, et non seulement pour les contempler, mais encore pour les interpréter »2. Interpréter veut dire mettre en rapport. Peu importe, pour l’instant, si ce rapport est d’ordre final ou simplement causal. Il faut, en tout cas, pour qu’il puisse s’établir, deux termes au moins3. Ainsi, quand ils veulent démontrer l’ordre de la providence, les Stoïciens font voir comment les choses sont bien adaptées (ἁρμόζειν) les unes aux autres4 ; les sciences séméiologiques consistent à rapporter un signe à un signifié. Or, les deux termes, quand il s’agit d’événements, sont séparés dans le temps ; un seul, l’événement-signe, celui qu’il faut « interpréter », nous est donné dans le présent, c’est-à-dire dans le seul des trois modes temporels qui soit réel. L’interprétation, selon le niveau où elle se pratique, pourra se faire de différentes manières, mais l’on verra que, dans tous les cas, elle suppose la réalité unique du présent.

(79)

2. Epict., Diss., I, VI, 19: Τὸν δ᾽ ἄνθρωπον θεατὴν εἰσήγαγεν αὐτοῦ τε καὶ τῶν ἔργων τῶν αὐτοῦ, καὶ οὐ μόνον θεατήν, ἀλλὰ καὶ ἐξηγητὴν αὐτῶν.

3. Cf. p. 125, n. 1.

4. P. ex. Epict., Diss., I, 6, 3-11 ; Cic., de nat. deor., II, XIV, 37 et passim.

(79)

 

 

 

From:

 

Goldschmidt, Victor. 1953. Le système stoïcien et l'idée de temps. Paris: Vrin.

 

 

 

 

 

 

 

.

30 Jan 2018

Goldschmidt (2.1.4.0.32) Le système stoïcien et l'idée de temps, “Connaissance et action”, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Stoicism, entry directory]

[Victor Goldschmidt, entry directory]

[Goldschmidt, Le système stoïcien, entry directory]

 

[The following is summary. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, which means typos are present, especially in the quotations. So consult the original text. Also, I welcome corrections to my interpretations, because I am not good enough with French or Greek to make accurate translations of the texts.]

 

 

 

Summary of

 

Victor Goldschmidt

 

Le système stoïcien et l'idée de temps

 

Deuxième partie:

Aspects temporels de la morale stoïcienne

 

A

La Connaissance

 

Chapitre IV

L’interprétation des événements

 

0

[Introductory Material to Ch.4]

 

2.1.4.0.32

Connaissance et action

 

 

 

Brief summary:

(2.1.4.0.32.1) For the Stoics, we should live our lives according to nature. But this does not mean in a Platonic sense to see nature as providing a model for particular actions that our will can either copy or not copy. Rather, for the Stoics, the distance between model and copy must be eliminated as much as possible, such that what we will is not something that may or may not accord with nature, rather, what we will should be no different than what Destiny’s laws have mandated and thus what Nature is doing.

 

 

 

 

 

Contents

 

2.1.4.0.32.1

[Living According to Nature by Willing According to Nature]

 

 

 

 

 

 

Summary

 

2.1.4.o.32

 

2.1.4.0.32.1

[Living According to Nature by Willing According to Nature]

 

(p. 77-79, “L’enveloppement de la morale par la physique s’exprime...”)

 

[In sum: For the Stoics, we should live our lives according to nature. But this does not mean in a Platonic sense to see nature as providing a model for particular actions that our will can either copy or not copy. Rather, for the Stoics, the distance between model and copy must be eliminated as much as possible, such that what we will is not something that may or may not accord with nature, rather, what we will should be no different than what Destiny’s laws have mandated and thus what Nature is doing.]

 

 

The idea of developing morality in accord with physics is captured in Zeno’s saying, “live according to nature.” This saying may have undergone slight variation as it was written from scholar to scholar. Yet, it seems to imply two different attitudes regarding moral life that nonetheless must be made to overlap. Living is an activity, but it must conform to nature, which as well acts, and moreover is the sole, true agent of activity. (I am not certain, but maybe here an idea could be that any of our own actions are actions of nature too, as we are a part of nature. And also, nature is fundamentally the unique agent of activity.) So in order for human action to conform to nature’s activity, we first need to know what nature’s activity is. In the history of philosophy, there is nothing radically new in this idea of giving priority to knowledge or contemplation over action. Consider for example how all of Platonism subordinates such activities to a prerequisite knowledge of Models/Forms. The novelty of this Stoic doctrine is that here, on the one hand, knowledge is no longer conceived as a purely theoretical attitude, while on the other hand, knowledge completely changes its object. [I probably have this next point wrong, but I am guessing: This knowledge is no longer about realities in the basis of which we derive precepts, and then secondarily our action would be free to conform or to not conform to those precepts. Rather,nature’s activity must conform to Destiny’s laws, and it does so whether or not we act with good will. (I am guessing that the idea here is the following. Nature will act the way it does in its conformity to Destiny, and we, as part of nature and Destiny, likewise act in such a way. In other words, it is not about us seeing that in nature there are certain ideals of goodness etc. that we should strive to make our actions conform to, rather, there are certain laws of nature, dictated by Destiny, and we should make our ethical life conform to these laws, which have nothing to do with good will. They rather have more to do with causality and fatality.) Knowing nature is less a matter of knowing what must be done as much as it is a matter of understanding that what nature does is what is done (maybe: to know nature (for the sake of knowing how to act) does not mean understanding what sorts of actions we need to be taking, it rather involves understanding that whatever happens (in nature) was such by Destiny (and thus we are not to develop a sense that we must try to do better to make our free will conform to certain precepts but rather that we must submit our will to that of destiny. So it is not about having a moral sort of free will. It is about willing what destiny wills.)) Thus, as Diogenes Laertius says regarding this, “living virtuously is equivalent to living in accordance with experience of the actual course of nature” (Hicks trans. at Perseus.) The experience of the ways things happen naturally is not a theorization that precedes action; between theory and action, there is no longer the same distance that we see between model and copy and between the norm that is known and the norm that is obeyed. (In other words, we do not model our behavior after nature; we simply make our behavior be natural or at least move with the flow of nature’s way and activity.) Just as nature does not wait for our free initiative in order to reduce this distance (between the way that things happen and the way we choose for things to happen), the only difference is then between nature’s will which is being achieved and our will which, having understood nature’s will, consents to it. It is the difference between “being driven” (or “being dragged along”) and “following willingly”. In the footnote here, Goldschmidt cites Seneca, and I will take more of the quotation to give context and elaboration:

I have just been following Cicero’s example:

11 Guide me, o father, lord of the lofty firmament,

wherever you decide; I hasten to obey;

I am here and ready. But if I be reluctant,

groaning, I still must go; in wretchedness must suffer,

what might have been my own act, were I virtuous.

Fate guides the man who’s willing, drags the unwilling.

12 That’s how we should live and speak, with fate finding us ready and prepared. This is the strong character that has surrendered himself to fate. In contrast we have the puny degenerate, struggling, thinking ill of the world order, and preferring to correct the gods rather than himself.

(Seneca 2015: 425, boldface mine)

Also cited are these lines from Seneca’s The Happy Life [with some context]:

Follow god. (6) On the other hand, whoever complains and laments and groans is compelled by force to follow orders, and though he is unwilling, he is nonetheless seized away to do what has been commanded. Yet what insanity it is to be dragged rather than to follow!

(Seneca 2014: 253, boldface mine)

We say that our will has as its object what should be done. But under this Stoic view, this is because our knowledge has indicated to our will that regardless, what should be done has been done (or maybe, what we should will do be done is that which has already been done or that which is already now being done): “wish the things which happen to be as they are” (Epictetus Enchiridion 8, Long trans., Perseus.)]

L’enveloppement de la morale par la physique s’exprime dans le célèbre adage de Zénon : « Vivre conformément à la nature »1. Cette formule, malgré les légères variations qu’elle a pu subir d’un scolarque à l’autre2, implique, semble-t-il, deux attitudes dans la vie morale, encore qu’elle exige de les amener à se recouvrir. Vivre, c’est bien une activité, mais qui doit se conformer à la nature, laquelle, elle aussi, agit et même, est l’unique agent véritable3. C’est [77 | 78] donc l’action de la nature qu’il faut connaître, avant de pouvoir y conformer l’action humaine. Cette priorité donnée à la connaissance ou, si l’on préfère, à la contemplation sur l’action, n’aurait, en soi, rien d’original ; tout le platonisme, par exemple, subordonne les conduites d’action ou de création à la connaissance préalable des Formes-Modèles. La nouveauté de cette doctrine stoïcienne1 vient de ce que la connaissance n’est plus conçue comme une attitude purement théorique et que, d’autre part, elle change complètement d’objet. Elle ne porte plus sur des réalités dont il s’agirait de faire dériver des préceptes2, auxquelles, ensuite, l’action serait libre de se conformer ou non. Elle porte sur la nature qui, dans le Destin, conforme elle-même son action à ses lois3, sans rien demander à notre bonne volonté. Connaître la nature, c’est bien moins comprendre ce qu’il faut faire, que comprendre ce qu’elle fait, ce qui se fait : « Vivre selon la vertu équivaut à vivre conformément à l’expérience des choses qui arrivent naturellement4. » Cette expérience n’est pas une theoria précédant l’action ; il n’y a plus, entre l’une et l’autre, la distance du modèle à la copie, de la norme connue à la norme obéie. Comme la nature n’attend pas notre libre initiative, pour réduire elle-même cette distance, le seul écart est entre sa volonté qui s’accomplit et la nôtre qui, ayant compris cette volonté, y consent. C’est la distance, indiscernable en apparence, entre « être entraîné » et « suivre de bon gré »5. Si l’on peut dire que la volonté prend pour objet ce qu’il faut faire, c’est parce que la connais- [78 | 79] sance lui indique que, de toutes manières, cela se fait : « Vouloir les événements comme ils se produisent »1.

(77-79)

1. Selon Pohlenz (Stoa, II, 67) après M. Schafer (cf. note suiv.), il conviendrait d’accepter le témoignage de Stobée (II, 75, II = S.V.F., I, 179), selon lequel la formule de Zénon était seulement : ὁμολογουμένως ζῆν. Nous préférons, avec E. Bréhier (Chrysippe, p. 220, n. 2), attribuer déjà à Zénon l’adjonction : τῇ φύσει. 1) Il ne suffit pas, pour invalider la témoignage de Diogène Laërce (VII, 87), de le déclarer « faux », avec Pohlenz, qui, d’autre part, ne commente ni ne réfute le texte concordant de Cicéron (de fin., IV, VI, 14) ; 2) nous ne tenons pas pour résolu le problème des antécédents académiciens (Polémon) et péripatéticiens (Théophraste) de la morale stoïcienne, eu général, et de la formule en question, en particulier (cf. p. 56, n. 2et l’article de von Arnim, cité à la p. 56, n. 3). Pohlenz lui-même doit concéder que « Zénon a dû sans doute en avoir reçu plusieurs suggestions » (Stoa, I, 118.) Le problème (comme, d’ailleurs celui des antécédents pythagoriciens du platonisme) est parfois obscurci par l’ardeur des auteurs à sauvegarder la pleine originalité des Stoïciens (ou de Platon). Mais cette originalité subsiste dans tous les cas (parce qu’un système n’est pas une mosaïque de doctrines ; cf. p. 202, n. 2), et sans qu’il soit besoin, pour la soutenir, de nier de parti pris les emprunts de certaines formules. On risque, autrement, d’imiter, en sens inverse, l’attitude de Cicéron (de fin., IV) qui ressortit à la polémique plutôt qu’à la science historique.

2. Cf., p. ex. Cic., de fin., IV, VI, 14-15; citons l’historique donné par Pohlenz, Stoa, 116-118, et par Max. Schäfer, Ein frühstoisches System d. Ethik bei Cicero, Munich, 1934 (qui propose, aux pp. 10-11 et 127, n. 2, une interprétation, reprise par Pohlenz, de la formule zénorienne : ζῆν ὁμολογουμένως, sans datif complémentaire). – Notons enfin qu’un des derniers commentateurs de Zénon rétablit, dans cette formule, l’adjonction : τῇ φύσει (A. Jagu, Zénon de Cittium, Paris, 1946, p. 18).

3. Sénèque, Ep., 65, 12 (cf. p. 35) ; de benef., IV, VII, I : « quid... aliud est natura quam deus et diuina ratio toti mundo et partibus eius inserta ? » ; le Destin conçu comme αἰτία unique (S.V.F., II, 934- 938).

(77)

1 Cela sans préjudice des « deux idées neuves » qu’y signale, à un autre point de vue, E. Bréhier, Crysippe, p. 221.

2. Que ces réalités soient transcendantes (Formes platoniciennes) ou à la portée de tous (« nature », au sens que ce terme pouvait avoir chez Théophraste ou Polémon) ; les impératifs auxquels elles donnent lieu étant toujours, dans une certaine mesure, dépendants de l’initiative humaine, pour être obéis.

3. Sén., de prou,. V, 8: « Irreuocabilis humana pariter ac diuina cursus uehit. Ille ipse omnium conditor et rector scripsit quidem fata, sed sequitur. Semper parit, semel iussit. »

4. Diog. Laërt., VII, 87: Ἴσον ἐστὶ τὸ κατ᾽ ἀρετὴν ζῆν τῷ κατ᾽ ἐμπειρίαν τῶν φύσει συμβαινόντων ζῆν, ὥς φησι Χρύσιππος. Cf. Cie., de fin.., IV, VI, 14: (secuudum naturam uiuere summum esse bonum) « His uerbis tria significari Stoici dicunt, unum eius modi, uiuere adhibentem scientiam earum rerum quae natura euenirent ; hune Zenonis aiunt esse finem. »

5. Voir entre autres, Sén., Ep., 107, II: « Ducunt uolentem fata, nolentem trahunt » (citation de Cléanthe, cf. Epict., Man., 53) ; de uita beata, XV, 5 : « a Deum sequere ! … quae autem dementia est potius trahi quam sequi. »

(78)

1. Epict., Man., 8: θέλε τὰ γινόμενα ὡς γίνεται καὶ εὐροήσεις.

(79)

 

 

 

 

 

From:

 

Goldschmidt, Victor. 1953. Le système stoïcien et l'idée de temps. Paris: Vrin.

 

 

 

Also cited:

 

Diogenes Laertius: Lives of eminent philosophers. Translated by Robert D. Hicks. 2 vols. London: William Heinemann / Cambridge, Mass.: Harvard University Press 1950.

The Perseus Greek page for the Diogenes’ passages:

[1-160]

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0257%3Abook%3D7%3Achapter%3D1

[179-202]

http://www.perseus.tufts.edu/hopper/text?doc=D.+L.+7.7&fromdoc=Perseus%3Atext%3A1999.01.0257

The Perseus English page for the Diogenes’ passages:

[1-160]

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D7%3Achapter%3D1

[179-202]

http://www.perseus.tufts.edu/hopper/text?doc=D.+L.+7.7&fromdoc=Perseus%3Atext%3A1999.01.0258

 

Seneca. Epistulae morales ad Lucilium.

Available at:

https://la.wikisource.org/wiki/Epistulae_morales_ad_Lucilium

 

Seneca. 2014. On the Happy Life. In Hardship and Happiness, pp. 235-273. Translated by James Ker [other texts in this edition are translated by others]. Chicago: University of Chicago.

 

Seneca. 2015. Letters on Ethics. To Lucilius. Translated by Margaret Graver and A.A. Long. Chicago and London: University of Chicago.

 

 

 

.

28 Jan 2018

Plumwood & Sylvan [Routley & Routley]. (8) ‘Negation and Contradiction,’ sect.8 “Semantical Models: Worlds on Record and Tape”, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

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[The following is summary. Boldface and bracketed comments are mine. This text is not proofread, so I apologize for its typos and other mistakes.]

 

 

 

 

Val Plumwood

(at that time as: Val Routley)

 

and

 

Richard Sylvan

(at that time as: Richard Routley)

 

 

“Negation and Contradiction”

 

8

Semantical Models: Worlds on Record and Tape

 

 

 

Brief summary:

(8.1) Winning a debate involves giving premises whose semantic consequence must be relevant (in an issue-restricted way) to the other party’s argument. That means we can use the star rule of negation rather than classical negation to designate one side as being the negation of the other. For, we need an issue-restricted other side in the debate model, and the star rule for relevant negation gives us such a restricted other. (8.2) Another metaphor for understanding the difference between classical negation and relevant negation is the record cabinet model. We think of a cabinet full of record albums. One side of any album we call p. Classical negation would say that ~p is everything else in the cabinet. Relevant negation would say that ~p is simply the other side of some particular record p. (8.3) We can also think of the sides as “worlds,” and we use the * function – which takes us from one world to its reverse or flip world – to define negation: “~p holds at a iff p does not hold at a*”. (8.4) We see a structure similar to that of star relevance logic in Kripke’s validity testing tableau procedure involving something like the copying of diagrams on separate sheets of paper, making certain modifications in the copies. In relevance logic, however, we use the front and back side of the paper, so to speak. (8.5) Now, relevant negation, by giving the flipside, does not remove the first side, like the cancellation model of negation supposes. Rather, it gives us an external other to the first side. But moreover, relevant negation does not give us an unrestricted or absolute other to the first side. So it does not explode the content out to the full extent of the domain, like the explosion model of negation supposes (which includes classical negation). Rather, relevant negation gives us an opposite other that is limited by its relevance to the first side. (8.6) And so, relevant negation is a far better candidate for natural negation than classical negation is. For, relevant negation captures the issue-controlled complementation of debate argumentation, and it is more able to account for intensional functions in natural language.

 

 

 

 

Contents

 

8.1

[The Star Rule of Negation as Arising Naturally from the Debate Model]

 

8.2

[The Record Cabinet Model]

 

8.3

[The Star * Function, Worlds, and Negation]

 

8.4

[Kripke’s  Sheets of Paper Metaphor]

 

8.5

[Relevant Negation as Neither Cancelling nor Exploding Content]

 

8.6

[Relevant Negation as More Natural than Classical Negation]

 

 

 

 

 

 

Summary

.

.

8.1

[The Star Rule of Negation as Arising Naturally from the Debate Model]

 

[Because winning a debate involves giving premises whose semantic consequence must be relevant (in an issue-restricted way) to the other party’s argument, that means we need the star rule of negation rather than classical negation to designate one side as being the negation of the other.]

.

[Recall the debate model of relevant negation from section 7.7. We have one party arguing p and the other side arguing the issue-restricted (i.e. relevant) ~p. I am not exactly sure I understand what we do now in this paragraph, but it might be the following. We will construe the debate situation using the concepts and notations of semantic entailment. We think of one party of the debate as putting forth its “side” (its claims that supposedly lead us to infer either p or ~p). Now also recall the star rule from section 6.6.

Routley-Routley-Negation-215b4_thumb

(See also Priest’s Introduction to Non-Classical Logic section 8.5.) Suppose we are the party arguing ~p. Our side of the argumentation (all our argumentation and thus our all premises), we call a. So we think that our side a leads us to infer ~p, or:

a ⊨ ~p

Next, take the other side, which is arguing p. Now, since we are using relevant negation, we are using the * symbol to mean the reverse world or situation, or in this case, side of the debate. The other side of the argument is not putting forth any propositions that entail any arbitrary position or conclusion other than the one their debate partner is arguing (that is, other side is not arguing for some irrelevant conclusion) but rather they put forth only issue-restricted propositions (that is, they are arguing for the opposite conclusion, which is issue-relevant). So it is not enough to say that for the other side, a ⊨ p. Rather (and I may be mistaken) we must say:

a* ⊨ p

because a* would mean the issue-restricted other side of the argument. Routley and Routley’s next point has to do with the settling of the debate. One side will win the debate if they establish their case, that is if a ⊨ ~p or if a* ⊨ p. And, one side establishes their case only if the other side does not (for otherwise there is a stalemate or draw or whatever). So

a ⊨ ~p iff a* ⊭ p

(I am not sure, but I wonder if we can also say that

a* ⊨ p iff a ⊭ ~p

Maybe that is built into the first formulation, because the * operator works both ways, as it is involutary. Again, see section 6.6.) Routley and Routley’s point is that from this semantical understanding of the situation, we can see how the star rule for negation naturally emerges. (Maybe the idea here is the following, but I am not sure. If we use classical negation and classical negation rules, one side would win the debate simply by showing that certain premises lead to some conclusion that is not the other side’s conclusion. But that is not how we understand debates to work. Rather, one side wins only when it provides sufficient argumentation to lead to an issue-restricted conclusion, which would be an opposing conclusion that is issue-relevant to the other party’s conclusion.)]

The debate model can be given a more semantical turn. In the p-issue, ~p is asserted, or presented as true, on one side, a say (i.e. a ⊨ ~p in obvious notation), while the reverse, namely p, is asserted, or presented as true, on the opposite side a* (i.e. symbolically a* ⊨ p). Now one side succeeds in a | debate, or establishes its case, iff the opposite side does not; therefore a ⊨ ~p iff a* ⊭ p. That is, a version of the star rule naturally emerges from the debate model more semantically considered. Statement ~p is made, or presented as, true at side or situation a iff p is made, or presented as, true at its opposite a*.

(218-219)

[contents]

 

 

.

.

8.2

[The Record Cabinet Model]

 

[Another metaphor for understanding the difference between classical negation and relevant negation is the record cabinet model. We think of a cabinet full of record albums. One side of any album we call p. Classical negation would say that ~p is everything else in the cabinet. Relevant negation would say that ~p is simply the other side of some particular record p.]

 

[Routley and Routley then pose the record cabinet model. Recall from section 7.6 the metaphor of the record album. The relevant negation of one side of the record is the other side, and not every other thing in the world, including all other recorded music (hence their footnoted joke that otherwise we would only need one record company and one record album for all music, as the flip side would contain all other music). The authors elaborate on that insight with their record cabinet model. We think of a cabinet full of records. We can think of each record as an issue, like in the debate model, or a question. On one side of each record (or issue, question, etc.) is p, and ~p is on the other (“for every atomic p”). Classical negation would regard p as one side of one particular record, and ~p “as everything else in the cabinet.” But relevant negation would simply regard one side of a particular record as p and ~p would be just the other side of that same record. I do not following the next sentences very well. They speak of an intensional function that selects programs from the cabinet. I do not know what the programs are. But they say the program can include both or neither sides of the record. The authors also say that this cannot be done with the classical system, although it can be suped up to allow for both sides to be selected. (I really do not know, but maybe it is something like multiple denotation or no denotation; I am very wildly guessing.  And I cannot even guess what the suping-up of the classical system would be. Please read to see:)]

The debate model leads directly to the record cabinet model. The cabinet which can represent the files of the universe, is full of records, each record is an issue, or question, with p on one side and ~p on the other side, for every atomic p (at least). From this point of view classical negation takes p as one side of one record, and ~p as everything else in the cabinet (classical theory fails to duly separate issues). Relevant negation takes p as one side of the record and ~p as the other side of the same record, there being many many records in the cabinet. Note well that intensional functions select a program from the cabinet. Such a program may include both sides of a record, and may include neither side of various records – in contrast to the published classical picture (the classical picture can be suped-up to avoid the latter defect but not the former).

(219)

[contents]

 

 

.

.

8.3

[The Star * Function, Worlds, and Negation]

 

[We can think of the sides as “worlds,” and we use the * function, which takes us from one world to its reverse or flip world, to define negation: “~p holds at a iff p does not hold at a*”.]

 

[We may also think of each record as a world. (See Routley Star semantics in Priest’s Introduction to Non-Classical Logic section 8.5.) The star * function gives gives us the reverse or flip side of whatever side or world we are on, and so we can call it the reversal function or flip function. And of course, a** gives us a. (In section 8.5.3 of Priest’s Introduction to Non-Classical Logic, he writes:  “ is a function from worlds to worlds such that w∗∗ = w” (p.151),  so it seems this property is in fact what defines the function.) To evaluate negative statements, we use the star rule: “~p holds at a iff p does not hold at a*.” The authors then write, “By contrast, the classical rule quite erroneously identifies a side with its opposite.” I am not exactly sure what that means. But it might be that classical negation, when evaluating ~p, identifies a with a*, because it is defined in terms of p in the same world, meaning perhaps that it identifies p in a with p in a*. Note that Priest in Introduction to Non-Classical Logic section 8.5.2 writes: “If w = w (which may happen), then these conditions just collapse into the classical conditions for negation” (p.151).)]

The cabinet model maybe differently oriented. Each record, or tape, represents, e.g. it may just describe, a world, a two-sided world. Then where a is one side of a world record, or a world, the opposite side is again a*, where * is the reversal, or flip, function which gives, whichever side one is in on, the other side. Obviously a** = a, since turning the record over twice takes one back to the initial position. The semantical rule for evaluating negated statements is, as for the debate model, the star rule, ~p holds at a iff p does not hold at a*. By contrast, the classical rule quite erroneously identifies a side with its opposite.

(219)

[contents]

 

 

.

.

8.4

[Kripke’s  Sheets of Paper Metaphor]

 

[We see a structure similar to that of star relevance logic in Kripke’s validity testing tableau procedure involving something like the copying of diagrams on separate sheets of paper, making certain modifications in the copies. In relevance logic, however, we use the front and back side of the paper, so to speak.]

 

[This paragraph involves a number of technical matters that I am unfamiliar with and that I cannot figure out readily at the moment. But I will still try to cover all the points, even though I cannot offer much in explanation. Let us go little by little.

The records may be ordered or arranged in a way that reflects the relational structure of (two sided) worlds.

(219)

I will guess this means that we can arrange the record sides so that those on the * sides (the back sides) form a world whose parts relate in a way that is isomorphic to how the parts in the non-* sides (the front sides) relate. Or maybe here we have one world with internal flip sides. But I am guessing. And I do not know if this involves making front-back assignments, or if, assuming they have been made, it involves arranging the records in a particular linear order.

The structured record model corresponds exactly to a natural elaboration of Kripke’s valuable sheets-of-paper model of semantic tableaux for normal modal logics. In explaining alternative sets Kripke says (63, p.73): ‘Informally speaking, if the original ordered set is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case putting in addition A in the right column of tableau t and in the other case putting B; the two new sheets correspond to the two new alternative sets’. Thus a full construction which consists of a system of alternative  sets corresponds to an arrangement of sheets (a sheaf of sheets).

(219)

I checked the Kripke text, and the part in question is about assessing validity by finding countermodels by means of tableau formation. Please see the original text, as I cannot say much about it. I would need to do a lot of work before being able to understand it. But let us note some superficial things that might still be helpful. The sort of formulas whose validity we are testing seems to be conditionals, where there is a series of terms A, joined by conjunction and making up the antecedent, and a series of terms B, joined by disjunction and making up the consequent. The countermodel would make the antecedent true (so I would assume that every A needs to be true) and the consequent false (so I would assume all the B’s need to be false). To do this, there is a method which I cannot picture well or comprehend, but it involves making parallel columns of a tableau, set beside one another such that the A’s are on the left and the B’s are on the right. I do not know how these tableaux are constructed exactly, but they will be done so such that the A’s are made true and the B’s false. As I said, I cannot begin to grasp how this works, but let us note something that seems to be at least superficially relevant here, namely, the following two rules for constructing the tableau.

N1. If ~A appears in the left column of a tableau, put A in the right column of that tableau.
Nr. If ~A appears in the right column of a tableau, put A in the left column of that tableau.

(Kripke, p.72)

Regardless of what is really going on here, we can at least see a structure that is like the star rule. However, I am not sure if these rules have anything to do with the sheets of paper metaphor. Then we have:

For relevant semantic tableaux there are only two innovations. First, whereas with strict implication new related tableaux are introduced one at a time, with relevant implication new related tableaux are introduced two at a time, i.e. in pairs. This reflects the replacement of the two-place alternativeness | relation of modal logics, by the three-place alternatives relation of relevant logics. The first innovation is not particularly germane to the present issues (and quasi-relevant systems such as the I systems which require only two-place relations could be adopted for exposition).

(219)

As I do not understand how these tableaux work, I cannot comment here. The idea might be that we need pairings of parallel tableaux, one for each world. I do not know what the two-place and three-place alternatives relations are, and there is nothing I can say about the rest of those passages either.

Second, and more important, then, both sides of the sheets are used. (Relevant logics are conservation-oriented in that even if rather a lot of sheets are introduced, both sides are used; the reverses are not wasted as with modal semantical tableaux). The reversal function * accordingly reverses the page, giving back for front and front for back.

(219)

Here the idea might have something to do with the notion that every formula in one world has a partner in the star world, or at least that each world is the flipside of the other. I really do not know. Here is the quotation in full:]

The records may be ordered or arranged in a way that reflects the relational structure of (two sided) worlds. The structured record model corresponds exactly to a natural elaboration of Kripke’s valuable sheets-of-paper model of semantic tableaux for normal modal logics. In explaining alternative sets Kripke says (63, p.73): ‘Informally speaking, if the original ordered set is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case putting in addition A in the right column of tableau t and in the other case putting B; the two new sheets correspond to the two new alternative sets’. Thus a full construction which consists of a system of alternative  sets corresponds to an arrangement of sheets (a sheaf of sheets). For relevant semantic tableaux there are only two innovations. First, whereas with strict implication new related tableaux are introduced one at a time, with relevant implication new related tableaux are introduced two at a time, i.e. in pairs. This reflects the replacement of the two-place alternativeness | relation of modal logics, by the three-place alternatives relation of relevant logics. The first innovation is not particularly germane to the present issues (and quasi-relevant systems such as the I systems which require only two-place relations could be adopted for exposition). Second, and more important, then, both sides of the sheets are used. (Relevant logics are conservation-oriented in that even if rather a lot of sheets are introduced, both sides are used; the reverses are not wasted as with modal semantical tableaux). The reversal function * accordingly reverses the page, giving back for front and front for back.

(219-220)

[contents]

 

 

.

.

8.5

[Relevant Negation as Neither Cancelling nor Exploding Content]

 

[Relevant negation gives the flipside. By giving the flipside, we do not remove the first side, like the cancellation model of negation supposes. Rather, it gives us an external other to the first side. But moreover, it does not give us an unrestricted or absolute other to the first side. So it does not explode the content to the full domain, like the explosion model of negation supposes. Rather, relevant negation gives us an opposite other that is limited by its relevance to the first side.]

 

[Recall the three models of negation from section 3. The cancelation model says that negation cancels content such that the conjunction of a formula and its negation results in no content whatsoever. The explosion model says that negation gives the absolute, unrestricted other to the unnegated formula, and so the conjunction of a formula and its negation gives all content in the domain. Finally, the third model, which we might call the relevant model, considers negation as being non-cancelling and so as having the otherness of the explosion model, only the otherness is limited to what is relevant to the unnegated formula. We also have discussed the notions of reversal and flipside, which gives us the sorts of results we want for a relevant negation. “Thus the opposite side of something is not the removal of the first side or, for example, everything other than the first side; it is another and further side, which is relatively independent of its reverse but which is related to it in a certain way” (219). The relation seems to be a sort of relevant opposition or relevant otherthanness.]

In sum, reversal and opposition have the right properties in leading respects for (the semantics of) relevant negation. Thus the opposite side of something is not the removal of the first side or, for example, everything other than the first side; it is another and further side, which is relatively independent of its reverse but which is related to it in a certain way. Both sides can co-occur (occur simultaneously) in a framework (e.g. controversy) and one can perfectly well consider both of them. The important point, to say it yet again, is that one side does not somehow obliterate or wipe out or entirely exclude or exhaust its opposite. Nor is the reverse, or opposite, just defined negatively as the other – it has an independent and equal role on its own behalf.

(219, boldface and underlining mine, italics in the original)

[contents]

 

 

.

.

8.6

[Relevant Negation as More Natural than Classical Negation]

 

[Relevant negation is a far better candidate for natural negation than classical negation is. For, relevant negation far better captures the issue-controlled complementation of debate argumentation, and it is more able to account for intensional functions in natural language.]

 

[Recall from section 5.4 that many classical logicians seem to assume that classical negation is the one that best captures natural negation as with negation in natural languages. But as we can see now, relevant negation has a much better claim to capturing natural negation. For, classical negation, as we saw in section 7.9, is an unrealistic limit case of the more natural sort of relevant negation that we find in natural language and experience.]

There is no mystery then about relevant negation. It is an otherthanness notion; it has natural and easy reversal models. There is some mystery however about classical negation, except as an extrapolation, and much mystery as to why some logicians are tempted to apply it everywhere, especially where, as so often, it mucks things up. Indeed, given the naturalness of relevant negation as issue-controlled complementation, versus the unnaturalness of classical; the naturalness of the reversal notion; and the improved ability of relevant negation to account for actual intensional functions in natural languages, relevant negation has a far better claim to be considered the core negation relation of natural language than classical. So much for the classical claim to have the only real natural negation and that relevant negation is queer.

(220)

[contents]

 

 

 

 

 

Sources cited by the authors:

 

[17] S.A. Kripke, ‘Semantical analysis of modal logic I. Normal propositional calculi’ Zeitschrift fur Mathematische Logik and [sic]Grundlagen der Mathematik 9 (1963), 67-96.

 

“RLR”, the abbreviation for: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady, Relevant Logics and Their Rivals, Ridgeview Publishing Company, Atascadero, California, 1983.

 

 

 

 

Other citations made by me:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

.

26 Jan 2018

Plumwood & Sylvan [Routley & Routley]. (7) ‘Negation and Contradiction,’ sect.7 “Transposing the Hegelian Picture: Restricted Otherthanness, Reversal and Opposites”, summary


by Corry Shores

[Search Blog Here. Index tabs are found at the bottom of the left column.]

[Central Entry Directory]
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[The following is summary. Boldface and bracketed comments are mine. This text is not proofread, so I apologize for its typos and other mistakes.]




Val Plumwood

(at that time as: Val Routley)


and


Richard Sylvan

(at that time as: Richard Routley)



“Negation and Contradiction”


7

Transposing the Hegelian Picture: Restricted Otherthanness, Reversal and Opposites



Brief summary:

(7.1) We will examine relevant (or restricted) negation. The sort of semantics we will use regards the interpretation j function as assigning propositions to our wffs. Although our semantics has only two truth values, it is not simplistically bivalent, because it allows for situations where something and its negation are both true or are both false (in the same world). (7.2) Our relevant (restricted) negation involves two worlds with parallel formulas that may or may not have the same truth evaluations, despite being paired off and being mutually determinative of each other’s values. Thus, classical negation is more limited in comparison: for it, negation is simply everything that is not the unnegated formula in that same world as the unnegated formula. (7.3) And, classical negation is structured in such a way that all contradictions entail the same thing, namely the whole domain. This means that from any contradiction we can derive any other arbitrary contradiction, thus there is a problem of relevance with the classical model of negation. (7.4) Moreover, classical negation involves an alienation of ~A to A, which is a problematic structure noted for example by Simone de Beauvoir in her commentary on the alienation of women arising from “woman” being defined as “other than man.” (7.5) But although relevant negation also gives us otherness like classical negation does (and this so far is something that we want at least in part), unlike classical negation, it gives us an otherness without it being an unrestricted otherness (which is something more specifically that we want). [We thus might say that with relevant negation we get otherthaness without alienation.] (7.6) We can picture restricted otherthaness as being like the flipside of a record album. So relevant negation gives us what is other than something without giving us everything other to that thing. We can thus think of restricted negation as being like an opposite or reversal. In contrast, the classical negation of the record side would not just give us its flipside, it would additionally give us everything else in the world too. (7.7) We see restricted relevant negation illustrated also by the debate or dialectical model, where one side argues for p; and the other side, by arguing for ~p, is not arguing every other argument but p but rather argues only the issue-restricted opposite of p. (7.8) In this debate model, it is clear that built into the structure of classical negation is irrelevance, because any irrelevant support that is not p would confirm ~p. (7.9) And in fact, classical negation is not even the sort of natural negation we encounter in experience. It is a limit case of the natural (restricted) negation. In other words, if we loosen the restriction of restricted relevant negation as far as we can go, we would get classical negation, which is like an unrealistic ideal of negation.








Contents


7.1

[The Propositional Reading of j, with De Morgan Lattice Logic]


7.2

[Classical Negation as More Limited than Relevant Negation]


7.3

[Classical Negation Is Irrelevant]


7.4

[Classical Negation and Alienation]


7.5

[Limited Otherness in Relevant Negation]


7.6

[Restricted Otherthanness of Relevant Negation as like the Flipside of a Record Album]


7.7

[The Debate or Dialectical Model of Restricted Relevant Negation]


7.8

[Classical Negation as Structurally Irrelevant]


7.9

[Classical Negation as Inexistent Limit-Case of Restricted Relevant Negation]








Summary




7.1

[The Propositional Reading of j, with De Morgan Lattice Logic]


[We will now regard the interpretation j function as assigning propositions to our wffs (like A, B, etc.), and we will use De Morgan Lattice Logic (which is perhaps a four truth-value situation structure for value interpretations) to assess the values of formulas built up using connectives.]


[I do not follow this part so well. It seems that we are to have considered the work we have done in the prior section 6 to have been about terms. So maybe when we spoke of A and used geometrical diagrams, we were thinking of A (also) as a term, as with the second reading of the interpretation function j (see section 6.3). But I am not sure. What we might be doing now is regarding A, B, etc. as names for propositions (which was the third reading, called the “propositional reading;” see section 6.4). What we will devise involves functions that are “extended not according to Boolean but according to De Morgan lattice logic”. I am not sure what this means yet. Overall it seems that we will not be using the Boolean semantic rules for evaluation the connectives, like we saw in section 6.1. Instead, we will use De Morgan lattice logic. But I have not learned it yet. Having looked at the cited source texts, I would think that a good place to begin would be Anderson and Belnap’s Entailment: The Logic of Relevance and Necessity, especially chapter 3, section 18.1, pp.190ff. But for the time being, I will make a guess. Recall Priest’s Introduction to Non-Classical Logic section 8.4.3. There we examined the diamond lattice for evaluating the connectives in First Degree Entailment:

1

↗               ↖

b                                 n

↖                   ↗

0

(After Priest, Introduction to Non-Classical Logic p.147)

In a footnote Priest also writes:

this structure is more than a mnemonic. The lattice is one of the most fundamental of a group of structures called ‘De Morgan lattices’, which can be used to give a different semantics for FDE.

(Priest, Introduction to Non-Classical Logic p.147)

For now, let us consider the possibility that what Routley and Routley are saying here is that the diamond lattice above which works for four-valued semantics is somehow related to what they mean by De Morgan lattice logic. (Maybe the idea is that a De Morgan lattice logic uses just true and false but still somehow gives us the four truth-value situation. I am wildly guessing.) See Priest’s Introduction to Non-Classical Logic section 8.2 for details, and also see section 8.5 of that text on Routley Star semantics. One final point is that negation here will be relevant negation and not classical negation. Later in section 7.3 below and following we get a better sense of what the “relevant” element is. But for now let me mention some other ideas regarding relevance for context. Recall from section 3.13 that there are a couple inferences we want to avoid: {1} A∧~A⇒B; and {2} C⇒D∨~D. The first case at least has the problem of drawing an irrelevant conclusion from the premises (the second case seems to have that problem too, but I am not sure yet if we can count it). The idea is that classical negation leads to A∧~A⇒B. As I understand it, this is because validity is understood as truth preservation, so if the premises are true then the conclusion must be true, for the inference to be valid. Classical negation toggles the value, which can be assigned either 1 or 0 in the first place. So here, we can still say it is valid, because the validity conditions are still met. If the premises are true then the conclusion must be true. But the premises are not true, so the condition is still being met. Now let us look at how relevance and negation are explained by Nolt in his Logics. In that text, section 16.3.19, he gives something like a First Degree Entailment situation, but in Nolt’s account, we are using an assignment function v to assign to one of four value situations, but only two values, where either it assigns just T, just F, both T and F, and neither T and F. Here are then the connective rules:

1.
T ∈ v(~Φ) iff F ∈ v(Φ).
F ∈ v(~Φ) iff T ∈ v(Φ).
2.
T ∈ v(Φ & Ψ) iff T ∈ v(Φ) and T ∈ v(Ψ).
F ∈ v(Φ & Ψ) iff F ∈ v(Φ) or F ∈ v(Ψ), or both.
3.
T ∈ v(Φ ∨ Ψ) iff T ∈ v(Φ) or T ∈ v(Ψ), or both.
F ∈ v(Φ ∨ Ψ) iff F ∈ v(Φ) and F ∈ v(Ψ).
(Nolt, Logics, p.443)
And here are truth-table arrangements:

Negation:

16.3.c


Conjunction

16.3.d


Disjunction

16.3.e

As we can see from this evaluation,

16.3.a

we can infer P on the one hand or ~P on the other hand from their conjunction, just like how we can derive A on the one hand and B on the other hand from their conjunction. For, if the evaluation makes the premises at least true, then the conclusion is at least true. But from the conjunction of a formula and its negation, we cannot infer any thing we please:

16.3.b

For, there is a counter-example, marked in red. But while we have here given a sort of relevance logic with a rule for negation, we will find that we probably cannot call this “relevant negation”. For, negation is still defined as toggling the 1 or 0 value. What is different here in the Nolt situation is that the formula can be originally assigned both values. In order to get such a structure from the concept of negation alone, it seems we will need to define the negation in terms of the star world, which will present the possibility of the four value situations. Now, recall Priest’s Introduction to Non-Classical Logic section 8.5.3 where he gives the evaluation rules for Routley Star semantics:

Formally, a Routley interpretation is a structure ⟨W, ∗, v⟩, where W is a set of worlds, is a function from worlds to worlds such that w∗∗ = w, and v assigns each propositional parameter either the value 1 or the value 0 at each world. v is extended to an assignment of truth values for all formulas by the conditions:

vw(AB) = 1 if vw(A) = vw (B) = 1, otherwise it is 0. .

vw(AB) = 1 if vw(A) = 1 or vw (B) = 1, otherwise it is 0.

vwA) = 1 if vw*(A) = 0, otherwise it is 0.

| Note that vw*A) = 1 iff vw**(A) = 0 iff vw(A) = 0. In other words, given a pair of worlds, w and w* each of A and ¬A is true exactly once. Validity is defined in terms of truth preservation over all worlds of all interpretations.

(Priest, Introduction to Non-Classical Logic p.151-152)

What we see is that these are the same rules for evaluating truth in possible worlds semantics (see that text section 2.3.4), only the negation rule has the * modification. In other words, I was originally thinking that perhaps one reason we might call it “relevant negation” is because built into it is this Routley Star possible worlds semantics that allows for the four-value situation, without changing the rules for conjunction and disjunction; and thus the relevance conditions are built into the negation and not into the other connectives. So we call it, “relevant negation.” But later we learn from Routley and Routley what they mean by “relevant negation”. (I kept the above notes just for context and reference.)]

The next task is to transpose the whole business (as preclassical thinkers like Joseph also tried to do) from the term to the statement level. The Hegelian picture goes over intact, and what results interpretationally are functions extended not according to Boolean but according to De Morgan lattice logic (for details see Anderson and Belnap, or RLR). The negation is no longer classical, but relevant.

(216)

[contents]






7.2

[Classical Negation as More Limited than Relevant Negation]


[Relevant negation involves two worlds with parallel formulas that may or may not have the same truth evaluations, despite being paired off and being mutually determinative of each other’s values. Classical negation is more limited in comparison. For it, negation is simply everything that is not the unnegated formula in that same world as the unnegated formula.]


[In classical negation, we have one universe of propositions, and the negation of A is every other proposition that is not A. Routley and Routley says that classical negation, when seen in terms of relevant negation, is a “depauperite one-dimensional notion”. I am not sure how that terminology works in a technical sense. If they have a non-technical sense, I would think they simply mean the following. Relevant negation involves more than the one universe of propositions. So it is a richer notion (thus making the more limited classical negation notion be depauperite in comparison), and it has more than one dimension (because we are to think of the other world as another dimension. So we do not just have A. We also have another A in another universe of propositions). But I am wildly guessing there.]

In terms of relevant negation we can see classical negation as a depauperate one-dimensional notion, which forces us to consider otherness with respect to a single universe consisting of everything. In classical logic negation, ~A, | is interpreted as the universe without |A|, everything in the universe other than what A covers, as reflected in the Venn diagram:

Routley Routley Negation 217a

The universe can be interpreted as the sum of propositions. Thus where atomic wff p is interpreted, naturally enough, as the proposition it expresses, ~p amounts to every proposition in the universe other than the proposition that p.

(216-217)

[contents]






7.3

[Classical Negation Is Irrelevant]


[Classical negation is structured in such a way that all contradictions entail the same thing, namely the whole domain. This means that from any contradiction we can derive any other arbitrary contradiction, thus there is a problem of relevance with the classical model of negation.]


[The reason that we get irrelevance from classical negation is that all contradictions have the same meaning, which is V. (I need to stop here, because I cannot quite come to this conclusion yet. In section 6.1, we defined negation and conjunction in the Boolean-Venn system as:

j(~A) = V-j(A);

j(A & B) = j(A) ∩  j(b) i.e. the common part

As I would think the diagram shows:

Routley Routley Negation 217a

A and ~A have no common part, thus their conjunction would be the empty set and not V. The best way I have right now for understanding how this works is the following. The delimited spaces represent derivational content. For example, from A you can derive A, and from not A you can derive any B, C, D etc. that does not equal A. Then, conjunction here is understood as the inferential content of the conjunction. So we put aside for the moment the above ideas regarding the union and intersection of sets. These ideas do not work for inferential content, at least not in the simplistic way I mentioned before where we use the Boolean rule for conjunction to determine the logical content of conjoined terms. Suppose we have A and B. From A we derive A, so A is in the inferential content of A. And From B we derive B, so B is in the inferential content of B. Now we conjoin A and B. We say that we can still derive A and we can also derive B. But the intersections of their contents is the empty set. Therefore, we understand the conjunction of inferential contents to be more like their union. Now, what is at issue is how the negation operation is defined. So we have A, from which you can derive A. The classical situation says that from their conjunction you can derive everything. That means we must have defined ~A as everything other to A (supposing that the conjunction of logical contents is their sum total). If instead we say that from the conjunction of A and ~A we can only derive A and ~A, then we are defining negation not as the absolute remainder of V once you remove A. Rather, it seems more like a partial remainder of V once you remove A. But as you can see, I am not grasping these matters well enough. At any rate, the point it seems is not simply that negation here is structured such that a contradiction gives you all the contents including A and everything else. Routley and Routley say the relevance issue is that every contradiction has the same content, namely the entirety of V, and thus they entail one another, and this gives us paradoxes. It is not clear to me yet how this is a matter of relevance, but the idea seems to be that if we derive B&~B from A&~A, then we have derived something completely irrelevant. (See section 5.1 on explosion and A∧~A↔B∧~B.)

Relevance problems come straight out of this; for irrelevance is written in at the bottom. All contradictions have the same interpretation, namely V: hence each entails all others and indeed everything. Paradoxes are inevitable.

[contents]






7.4

[Classical Negation and Alienation]


[Classical negation involves an alienation of ~A to A, which is a problematic structure noted for example by Simone de Beauvoir in her commentary on the alienation of women arising from “woman” being defined as “other than man.”]


[I might get the next idea wrong. It might be the following. (Suppose we are using classical negation.) We have ~p. But what is ~p? It can only be understood in terms of p. It cannot be “independently identified”. Routley and Routley then see this as being involved in certain structures of alienation, as for example Simone de Beauvoir’s commentary on the alienation of women arising from “woman” being defined as “other than man.” I am not familiar with this material, so I cannot say anything about it. Routley and Routley say that the woman is alien to the primary notion, man. So it would seem the idea is that because women are not men, they become alien to men: “‘woman’ is identified as ‘other than man’; and is not positively identified, only introduced as alien to the primary notion, ‘man’. The negation ~A of A is (so to say) alien to A” (217).  (I would have thought there would be an idea of woman becoming alien not just to men, because why is that a concern? I guess it is a concern if men are considered the standard, and by being alien from men, they are then non-standard and are alienated from what is most “real”, “good”, “normal” etc. I would have thought that given these logical structures, there would be a problem of women being alienated from themselves, because they would be identified with may other non-women things, as both women and also other non-man/non-women things (like tables, etc.) are also non-man. This I thought was the problem of unrestricted negation, but I am no longer sure I follow. Also, I would think that even with relevant negation, you could have women being otherthan to men and thus alien to men in the same way as Routley and Routley seem to be talking about. So I do not quite understand the point about alienation yet, and I need to check the de Beauvoir source.]

It is corollary that ~p cannot be independently identified, it is entirely dependent on p. This relates, more than coincidentally, to alienation (compare what Simone de Beauvoir has to say to alienation of women where ‘woman’ is identified as ‘other than man’; and is not positively identified, only introduced as alien to the primary notion, ‘man’). The negation ~A of A is (so to say) alien to A.

(217)

[contents]






7.5

[Limited Otherness in Relevant Negation]


[Relevant negation gives us otherness, like classical negation does, and this is something that we want, but unlike classical negation, it also gives us otherness without it being an unrestricted otherness, which is also what we want.]


Traditional negation uses a concept of otherness. But it is too unrestricted, as we have seen. It is an otherness with respect to the entire universe. Relevant negation is also an otherness, but “with respect to a much more restricted state”. [That state is not given definition in this paragraph, so we move on for now. Also, I have the impression that we might be able to make a terminological and conceptual distinction between otherness and otherthanness. Both terms are used by Routley and Routley, and although as far as I can tell there is not an obvious distinction, maybe we can make the following one for our own purposes. Otherness is the unrestricted sort. So dog is an other of blue. But a dog is not other than blue (or not an otherthan of blue), only the other colors are. So red has otherthanness with regard to blue but not otherness (or we might say, red is other than blue, but dog is other to blue; or, red is an otherthan of blue, but dog is an otherto of blue. To put things in yet another way, we would say that red is {another color} other than blue, but we would not say that red is another color other to blue. But we would say that a dog is other to blue, and here we do not have an implied {another of this sort}.) But perhaps when Routley and Routley interchange between otherness and otherthanness, they mean no conceptual distinction, or maybe they mean one that is not like what I offered.]

Relevant negation can, however, preserve much of the otherness notion of traditional negation (without the counterproductive alienation features). But relevant and classical negation differ firstly as regards what the otherness is considered in relation to. In the case of classical negation it is otherness with respect to the universe. In the case of relevant negation it is otherness with respect to a much more restricted state, such that p and its negation do not (interpretationally) exhaust the universe between them.

(217)

[contents]






7.6

[Restricted Otherthanness of Relevant Negation as like the Flipside of a Record Album]


[Restricted otherthaness is like the flipside of a record album. So relevant negation gives us otherthanness without giving us everything other to. The classical negation of the record side would not just give us the other side, it would as well give us everything else in the world.]


To conceptualize the restricted otherness in relevant negation, which is a notion of a “restricted other than,” we should think of negation as a reversal, giving the other side of something, and not as giving us everything other to the one side (an unrestricted otherness). They give the example of a record album. We look at one side of the record. What is the relevant negation of that side? It is the the other side of the record. The classical negation of the top side is everything else whatsoever that is not the front side. They joke that were the “other side” everything else whatsoever, that would include all other music, and so with just one record you would have all recorded music (along with every other thing in the world), thereby making it such that “there would be room for only one record company, and only one record from it.” [The next idea involves the notion of a “relevantly restricted universe”. I am not sure, but the idea might be that the other side of the album would be restricted to only those other things that are relevant to the front side.] [I note something for now, but I will probably return to this idea. In a Routley sort of semantics, the (unrestricted) otherness of negation where the values toggle is not an otherness that occurs in the same world. So ~A will be true in our world if A is false in the star world. But suppose also that A is true in our world (so in our world, A and ~A are true, and in the star world A is false). What is the relation of A and ~A within our world? Perhaps we might say that the otherness between ~A in our world and A in the star world is now found also in our world as otherthanness between the ~A and A of our world, because both A and its negation are affirmed in our world. What I mean is the following. Suppose we are working with a Routley Star world semantics (see Priest’s Introduction to Non-Classical Logic section 8.5.). We define negation in terms of the toggling relation of ~A in our world with the truth value of A in the other world. It is other to, because the values toggle. (The rule as Priest gives it is: “vwA) = 1 if vw*(A) = 0, otherwise it is 0”, p.151 of Priest’s Introduction to Non-Classical Logic, see its section 8.5.3.) So in light of what Routley and Routley are saying here, the otherness of the negation is between ~A of our world and A of the other world (the star world), with that being a sort of unrestricted otherness, because the toggle is absolute: “vwA) = 1 if vw*(A) = 0, otherwise it is 0”. But, when this leads to both A and ~A being true in our world, then it is like importing that otherness between worlds into a restricted otherthanness of our world. Let me explain. Suppose in our world ~A is true and A is false. That means A is not in our world. So there is an absolute otherness, an alienation as we might say (see section 7.4 above.) To be clear, in this case there is an otherness and alienation between ~A and A in our world, because A is not even in our world, and there is also an otherness and alienation between our worlds, because again the A is in another world and also their values are absolutely other to one another. Instead imagine that A is true in our world and A is false in the other world. That means in our world both A and ~A are true. That furthermore means that A and ~A are both in our world. Still ~A in our world bears an absolute otherness to A in the otherworld (because their values are toggled). But since ~A is true in our world along with A being true as well in our world, then it is like we imported that transworldly otherness (between ~A of our world and A of the other world) into an intraworldly otherthanness (between ~A of our world and A of our world). So suppose we reserve the term “otherness” (or maybe even, “otherwiseness”) for the absolute sort of negation, which logically manifests as a value toggle, and “otherthanness” for a restricted sort, which logically manifests not as a toggle within our world but only as one with the star world. As such, we could rename the star world as the “other-world” (or the “otherwise-world”) and our own world as the “otherthan-world”. Now let us see how this can play out. Consider these ideas in terms of selfhood. Suppose we are in an act of “becoming”. And think of it in terms of the “instant of change” that we saw in Priest’s In Contradiction section 11.2, where at the moment of transition, the pen is both on and off the paper. Likewise, at the moment of our own change in selfhood, we both are and are not our own selves at that moment. Whatever we will later become is normally other to us, and not in this world (for it is in the world of the future). But when our own (restricted) self-negation is affirmed in this world, in the present (world), it brings our own otherthaness into our own immediate proximity (or we might say, it brings our transworldly otherness or otherwiseness into our intraworldly otherthanness). We come into intimate contact with our otherthan selfhood. It affirms both what we are and what we are then becoming, even though they are reversals or logical other-thans of each other.)]

Such a restricted otherness notion is provided by reversal, which gives the other side of something. The lead side and the other, or opposite, side do no yield everything, the universe, by any means,16 any more than p and ~p yield everything with relevant negation. Reversal is in fact a restricted other than notion – on the other side is not all territory other than p, representing everything other than p. With reversal otherthanness operates in a relevantly-restricted universe. The reverse direction (or sense) is not any direction other than the forward or given one.

(217)

16 Otherwise there would be room for only one record company, and only one record from it.

[contents]






7.7

[The Debate or Dialectical Model of Restricted Relevant Negation]


[We see restricted relevant negation illustrated by the debate or dialectical model, where one side argues for p; and the other side, by arguing for ~p, is not arguing every other argument but p, but rather the issue-restricted opposite of p.]


Another illustration or metaphor is the debate or dialectical model. We would say that on one side of a debate a person is arguing p and on the other side someone else is arguing ~p. The ~p side is the opposite or reverse of the p side of the argument. But the opposite side is not every other argument but p. Rather, it is “issue-restricted.”

The reversal picture can be filled out in several apposite (and of course connected) ways, both more superficially syntactically, since in one sense the reverse of p is ~p, and less superficially semantically. Consider first the debate, or dialectical,17 model which reveals the type of restricted situa | tion with respect to which otherness (the rest of the situation) is assessed. A debate can be represented as the p-issue, or the p-question, when the issue is as to whether p or ~p. One side asserts, argues, or defends p, the other side ~p, Or, as we say, p and ~p are each sides of the issue as to whether p, one side being the opposite (X or reverse) of the other. The sides are clearly issue-restricted, and so accordingly is the complementation. To present the case for one side, e.g. the positive or affirmative, and to present the case for the other side, the negative, is not to present the case for everything, to exhaust what can be said, etc.

(217-218)

17. In one of the historical senses of ‘dialectical’. A debate can als0 be ‘dialectical’ in the other historical sense; for one side may defend both q and ~q. A related model is the  evidence model, where one side is the evidence for p, the other the evidence for ~p.

(229)

[contents]






7.8

[Classical Negation as Structurally Irrelevant]


[In the debate model, it is clear that built into the structure of classical negation is irrelevance, because any irrelevant support that is not p would confirm ~p.]


In the debate model, as we said in section 7.7 above, we have one side arguing p and the other side arguing ~p. Now suppose we take the classical notion of negation. That mean those arguing against p only need to confirm anything which is not p, no matter how irrelevant. Thus we see from the debate model that “classical negation itself carries the seeds of irrelevance” (218). And classical negation distorts the notions of “aboutness, of case, issue, relevance, confirmation and evidence” (218). [The reasoning for this is the following: “The systematic distortion is a result of the restriction to (complete) possible (consistently describable) worlds, a restriction forced by retention of classical negation.” I did not follow that. I suppose that if we do not use classical negation but rather relevant negation, we are using worlds that are not “(complete) possible (consistently describable) worlds”, so even in a debate when we are arguing relevantly for ~p, we are using such other worlds. That is not something too obvious to me, but I guess the idea might be that the only way to make the negation issue-restricted is to designate a star world. But exactly how that works is not apparent to me yet. The next idea I also do not follow, because I am unfamiliar with the concepts. The idea might be that if there are two conflicting but equally moral choices we can make, or two conflicting beliefs that are equally justified, then we have a problem in dealing with this if we just have classical negation. But please read the quotation to see.]

The debate model indicates that classical negation itself carries the seeds of irrelevance. Thus if one is debating an issue, whether p or ~p, classical negation would allow anything at all that wasn’t p as relevant to truth of one half. Thus in debating say, uranium mining one could introduce say, child care centres as relevant to one side of case. The notion of relevance is similarly destroyed, since anything confirming anything which is not p is relevant to the debate. Notions of aboutness, of case, issue, relevance, confirmation and evidence, are all seriously distorted, in a systematic way, by classical negation (as independently shown in much detail in RLR and [22]). The systematic distortion is a result of the restriction to (complete) possible (consistently describable) worlds, a restriction forced by retention of classical negation. There is a similar, and similarly forced, distortion of other intensional functors, e.g. of deontic functors such as obligation (with respect to moral conflicts), of psychological functors such as belief (with respect to inconsistent beliefs), etc. etc.

(218)

[contents]






7.9

[Classical Negation as Inexistent Limit-Case of Restricted Relevant Negation]


[Classical negation is not the sort of natural negation we encounter in experience. It is a limit case of the natural (restricted) negation. In other words, if we loosen the restriction of restricted relevant negation as far as we can go, we would get classical negation, which is like an unrealistic ideal of negation.]


So as we can see, classical negation cannot be seen as sufficient to capture everything that naturally and intuitively belongs to negation. It only captures one dimension of negation [namely, otherness] but it misses other dimensions, like restrictedness. Classical negation is a limiting case [meaning perhaps if you loosened restricted negation as far as you can go, at the limit you would get classical negation], and like a “frictionless surface or perfectly elastic body”, it is not something that actually occurs in experience.

Classical negation is a depauperate one-dimensional concept which distorts the functions of natural language and limits the usefulness of the logic it yields. Classical negation may seem natural, firstly because we (or rather some, the brainwashed among us) have become accustomed to it and perhaps impressed by its computer applications and arithmetical analogues, and secondly because (like material implication ion itself) it captures one dimension of negation, but it has rejected the other dimensions (e. g. restrictedness). Classical negation gives a simple account which is a limiting case, but one which, like that of frictionless surface or perfectly elastic body, does not occur in experience.

(218)









From:

Routley, Richard. and Val Routley. 1985. “Negation and Contradiction.” Revista Colombiana de Matematicas, 19: 201-231.




Sources cited by the authors:


[1] A.R. Anderson and N.D. Belnap, Jnr., Entailment, Princeton University Press, Princeton, 1975.

[22] R. Routley, Exploring Meinong’s Jungle and Beyond (Especially: “Ultralogic as Universal?” RSSS, Australian National University, 1979.


“RLR”, the abbreviation for: R. Routley, R.K. Meyer, V. Plumwood and R.T. Brady, Relevant Logics and Their Rivals, Ridgeview Publishing Company, Atascadero, California, 1983.





Other citations made by me:


Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.



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