16 Feb 2018

Goldschmidt (CBSC) Le système stoïcien et l'idée de temps, collected brief summaries and contents


by Corry Shores


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[The following collects the brief summaries and contents from entries summarizing Goldschmidt’s text. Note that my numbering system is not currently optimal. Please check the page numbers to be sure.]



A directory without the summaries can be found at this link.





Collected Brief Summaries of


Victor Goldschmidt


Le système stoïcien et l'idée de temps


Première partie:

La théorie du temps et sa portée


A. La théorie du temps


III. La théorie du temps Définition


For Chrysippus the Stoic, time is the interval of movement in the sense of giving measure to the motion’s speed or slowness. It is also the movement of the world by which all things not only move but also exist. For, existence  is a matter of being actually active in the present, and this furthermore is to be true, because to exist means to be an activity currently belonging to a subject. Thus “walk” truly belongs to you and thus exists when you are actually walking right now, and it does not truly belong to you and it does not exist when you are currently sitting or lying down. This also means that the past and the future do not exist. But we still say that they “subsist”. [For, they have somethinghood as incorporeals.] The present is infinitely divisible, without that infinite divisibility ever being completable. This means that any piece of time no matter how small will always include a pastmost and futuremost extremity. Thus no part of time is precisely and completely present, but it is partially so.


Théorie d’Aristote


For Aristotle, time is the “number” of movement, which means it is the enumerable aspect of movement. It quantifies the speed or slowness of the motion.


[Aristotle’s Theory]

[Aristotle’s Time as Number of Movement]

[Aristotle and the Time of Physical Motion]


Pluralité des mouvements et mouvement cosmique



The Stoics, like Aristotle, understood time as having to do with the way that the cosmos’ regular circular motion relates to particular activities that have a finite interval during which they begin and end. For Aristotle the focus is on how the regular circular motion of the heavens sets a standard interval of duration [being perhaps the yearly circular revolution of the stars] by which other motions can be numerically quantified. The Stoics, while keeping this same structure, were less concerned with the numerical quantification involved and more in the way that the plurality of particular movements [or activities of bodies] are unified in the greater cosmic motion on account of their shared temporality.


[The Plurality of Motions and the One Cosmic Motion]

[Motion as Mathematical Number or as Dynamic-Vitalistic Interval]

[Zeno’s Pluralistic Definition of Time]

[Chrysippus’ Monistic Motion/Time as Being Like Aristotle’s Notion of a Cosmic Standard of Motion for Measuring Units of Time]

[The Non-Mathematical Sense of the Stoic Unit (or Unity) of Time and the Non-Opposition between Chrysippus’ and Zeno’s Theories of Time]

[Stoic Time Must Be Understood in Terms of the Relationship between Monism and Pluralism]

[The Stoic Plurality of Motions Ultimately Unified in the Singular Cosmic Motion]


Temps infini et temps limité


For the Stoics, time can be understood as a line stretching infinitely into the past in one direction and infinitely into the future in another. Although time is infinite, it has parts which are either infinite or finite: time is “bound” on the far extremities by the limitless limits of the infinite past and infinite future; but past and future are limited on the inside by the finite present, being a limited limit to both past and future.


[Infinite and Finite Time]

[Stoic Time as “Interval” of Motion, and the Opposition of Past and Future to the Present]

[Time in Terms of Part-Whole Structures and the Totality Encompassing the Corporeal and Incorporeal]

[Time’s Unlimited Ends and Limited Present]


Divisibilité du temps


Chrysippus has a seemingly self-defeating notion of time. He says that no time is completely present, and yet only the present exists. This would seem to suggest that time does not exist. Chrysippus further clarifies that no time exists in the present in the strict sense rather than in the broad sense. The strict sense of the present is not something we actually experience. At best, we can form of concept of it as a limit between the past and future. Under such a conception,  we can think of the present as admitting of no past or future. But time can be said to exist in the broad sense when we think of how we experience the specious present as having some duration. So our senses tell us that there is time in the present, but this is only one sense of the term “present”, namely, the experienceable present. However, the other sense of “present,” the strict sense, is grasped not experientially but only mentally through mathematical procedures. If this sort of present has any reality, we can never actually grasp it as a real component of time. The reason for this has to do with the Stoic ideas regarding the infinite divisibility of continua, including bodies (as spatially extending things) and time (as a temporally extending and perhaps durational thing). When bodies or time are understood mathematically, we can divide them to infinity until arriving upon an infinity of indivisibles. This is already problematic, because suppose we divide a cone into an infinity of stacking circles. We begin by assuming that the cone has a smooth surface. So each ring and its neighbor cannot be of different sizes, because then the cone’s surface would be jagged. But, if they are all the same size, we have a cylinder and not a cone. We encounter a similar problem when we divide bodies and time infinitely. Suppose a body is divided into an infinity of indivisible parts. Those parts would need to lack extension, or else they would be divisible. But parts without extension cannot be parts of extending bodies, because their additive sum would not have extension. (Also, it is not clear how division can arrive upon them, because anything with extension when divided would seem to produce parts with extension, for otherwise the thing being divided would not have extension to begin with.) Similarly for time. Suppose we could infinitely divide time into instants. On the one hand, we cannot obtain indivisibles through division of divisibles. On the other hand, were we to have indivisibles of time and space, their sum could not be said to compose larger structures, because none have any extent or duration. So Chrysippus is saying that such indivisibles produced by a mathematical procedure of infinite division are beings of reason, and if they do have any reality, we can never know them in their actual reality. For in actual practice, we can only continue our divisions endlessly, never arriving upon the limit. Thus, in one sense (the broad sense) time is “real” and in anther sense (the strict sense) time is irreal (it subsists as an incorporeal something without existing corporeally).

[Note: it could instead be that time is simply real (lacking even non-existing susbistance) and the mathematical notion of infinitely divisible time is a misconception that tells us nothing about temporality itself. I will revise this if in the next section that seems to be the case, but I had the impression the next section would propose the Aiôn time which might capture the sense of the mathematical present and also the subsistence of the past and future.]


[The Divisibility of Time]

[The Non-Presence of Time and the Existence of the Present and Subsistence (Non-Existent Somethinghood) of the Past and Future]

[The Present in the Strict and Broad Sense. The Non-Presence of Time (Past and Future)]

[The Continuous Divisibility of Corporeality]

[The Mathematically Infinite Divisibility of Bodies and Time. The Impossibility of Actual Infinite Divisibility]

[Chrysippus and the Reality of Time]


Aiôn et présent


For the Stoics, there are two modes of temporality. {1} There is the time of the actual present, and this present exists and has some limited extensive duration. And {2} there is the unlimited time reaching infinitely into the past and future, with the division between the two being the present as an infinitely divided, mathematical instant. While Chrysippus did not make a terminological distinction despite speaking of time in each of these senses, Marcus Aurelius does in fact distinguish them terminologically by calling the eternal, infinite sort of time “αἰών”. He uses this notion to make the point that in the context of eternal time, our life is but an insignificant finite interval, and so we should not be troubled too much in life, given its ultimate insignificance. Now, aiôn-time (including both the limitless past and future and the infinitely divided mathematical instant that is the “present” and the limit between the two) is incorporeal and thus irreal, while the lived, specious present that we know by our senses is the real corporeal present.


[Aiôn and the Present]

[The Irreality of Past and Future and the Reality of the Present]

[Eternal/Infinite Past and Future “Aiôn” time and the Limited Interval of Our Lifetime]

[The Incorporeality of the Aiônic Past and Future and Its Mathematical Present]


Les divisions du temps sont déterminées par l’agent


We sense that the present is durationally extending by sensing a currently ongoing activity whose duration determines the extent of its particular extended present. But strictly speaking, the present cannot extend, because then it would contain a little of the past and a little of the future. To understand the Stoic theory of time, we need to see that they understood time under two senses. There is real time as the extending corporeal present we sense, and there is the irreal time of the past and future extending infinitely in both directions. This infinite past and future, called the aiôn-time, is not real, because for the Stoics, only the present is real, as it is corporeal, and the aiôn-time is incorporeal. [So already we see that the extending real present borrows some of its composition from that of the irreal aîon-time, because otherwise it would have no durational thickness. Likewise] the aîon-time, by accompanying real corporeal actions, adds to its irreality a real counterpart, namely, the real present that expresses it through the activities happening at that time. [For example, Dion is walking presently. But walking cannot happen in an instant. So while presently walking, some part of the action is happening at the past-most part of the present, and another part of the action is happening in the future-most part. By combining these temporally distributed moments, the action expresses these irreal past and future parts of aîon-time, thereby giving the incorporeal aîon-time a corporeal embodiment and thus a component of reality that affixes to or “accompanies” its fundamental irreality.] The Stoics in fact considered periods of time, which we normally understand primarily in terms of an abstract interval that would seem to be incorporeal, as in fact being corporeal bodies. For example, a month is not simply the temporal interval of 27 or so days. Just as much as that,  a month is the corporeal reality of the moon, which is a body, moving once around the earth, in a corporeal activity of physical motion.


[The Divisions of Time Are Determined by the Agent]

[Sensing the Event That Fills the Extended Present]

[The Corporeal Contamination of Incorporeal Acts That Are Presently Enacted]

[Corporeal Time Periods: A Month as Moon Movement]

[The Coporealization and Realization of Time Periods]

[The Accompaniment and Contamination between Aiôn Time Intervals and Extending Active Presents]






Deuxième partie:

Aspects temporels de la morale stoïcienne



La Connaissance


Chapitre IV

L’interprétation des événements



[Introductory Material to Ch.4]

Connaissance et action


( 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.


[Living According to Nature by Willing According to Nature]

Interprétation des événements


( 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.


[Interpreting Event-Signs]




L’interprétation a l’échelle cosmique

Divination et connaissance dans le présent, 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. ( 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.


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

[The Togetherness of Moments in the Coil of Time]

Divination et connaissance dans le présent, 2


( Under the Stoic philosophy of time, all events are arranged in an ordered way by a sequence of causes. Thus the same sort of causal relation that explains how we got from the past to present also explains how we will get from the present to the future. For this reason, divination not only looks to the future but also at the present and past. ( Nonetheless, for the Stoics, only the present is real. But the present is connected causally to all other events, and thus all events are bound up together in the whole of time. God, unlike humans, can see the present event and thereby see all other events of time. Yet, despite our human limitations, our divinations strive for this grand vision as much as possible, looking at present event-signs and trying to infer past and future events that are causally bound up with the present one. We note that it is the same sort of conditional thinking that allows us to both assess the succession of these causally related events and also to formulate prophesies (both of which involving an “if... then...” structure). Thus this Stoic philosophy of temporality explains the Stoic focus on the conditional in their logic.


[Divinatation in Terms of Present, Past, and Future]

[Divining Past and Future on the Basis of the Present. Conditional Propositions.]




Science de l’individuel


( Interpreting an event-sign by means of divination means that we must place it in the greater sequence of causally related events and thereby to assign it its explanatory importance in the whole of time. ( Stoic time combines a monistic and pluralistic view of causation. [The monistic view is that time forms one whole that is rationally ordered, and thus we can see there being one unified cause corresponding to that one unified, rationally ordered time.] The pluralistic view is that any one event in the sequence has its own antecedent cause(s) and itself serves as an antecedent cause to what comes after. Divination is largely concerned with these local causes that affect individual people, thus divination is a science of the individual.


[Divination and the Present Event]

[Divination as Pluralism and Monism]



Interprétation finaliste


( We are here concerned with divination as it is applied to human life and as it is something that confers moral significance. [And as we will see, this concern will take us away from the notion of divination and more toward moral actions in the present.] ( The Stoics seem to want two incompatible things. On the one hand they want providence, which is like final causality and thus which seems to suggest strict causal determinism, but they also want there to be some moral significance in the present, which would seem to call for an indeterminate element where moral choice comes into play. ( We might also want to think of this far-fetched and anthropocentric sort of finalism according to which all past and future events can be divined as not really being absolutely central to the Stoic system. ( And given the difficulties that the Stoic’s teleological notion of providence presents, we are encouraged to understand the philosophy of time without such an emphasis on it. ( But in fact, to understand the Stoics, we must keep their notion of providence but not the radical notions of divination and finality. The more important idea here is the moral one, and not the physical one. Providence for the Stoics was a matter of all things working together for the good of the whole. Sometimes on the individual level we see how the good that happens to us serves that greater good as well. But when bad happens to us, we should see that it is still good rendered to the whole. Thus the Stoic notion of providence is more a matter of how we temper our desires and expectations in order to correspond to divine providence, as understood in this more indirect, vague,  or implicit way. ( So under this other view, we still affirm providence, but we also acknowledge that we can only know the general ends of particular events, and we cannot know the specific ends in terms of the precise role they play in attaining that greater good. However, with this broader sense of the whole in mind, we can still look to the causes of events [which is perhaps the closest equivalent to “divination” that we will find in the Stoics.]


[Divination and the Present Event]

[Providence, Final Causality, and the Morality of the Present]

[The Non-Centrality of Providence in Stoic Philosophy]

[Non-Teleological Emphasis in the Stoic Philosophy of Time]

[Providence, Morality, and the Good of the Whole]

[Providence without Divination. Knowledge of Particular Causes without Knowledge of Particular Ends]



Interprétation par les causes


( When we understand why things happen, namely, for the greater good, then even troubling events become bearable. And with regard to how sometimes bad people obtain good things, we should realize that they paid some moral price for it that we should be glad we are not paying. ( We can generalize this notion that whatever happens to us happens for the greater good of the whole to now further say that the same goes for animals and even all inanimate things. For the Stoics, the moral problem is how to actualize this general, natural law, and doing so is what is meant by the precept: “co-operate with Destiny.” ( We will next turn to another way that providence refers us back to causality.


[Providence and Theodicy]

[Cooperating with Destiny by Actualizing the Greater Good of the Whole as the Stoic Moral Problem]

[Transition to the Next Section on Causality]



Providentialisme et causalité


( Since we cannot know how events and our actions are leading finally to the greater good of the whole, we are left to wonder, what then is the good of consulting diviners to tell us about the future and about how we should behave now with respect to that foreseeable future? For, whatever it is that they tell us to do, it will still only result in the foreseen event. But that is precisely what their value is. They can guide us to do our part in bringing about that future. But why would we help a bad outcome for us to transpire? That is because there really are no bad outcomes from the perspective of divine providence. For, anything that seems bad for us is really good for the whole in the end, even though we cannot understand how or why. Thus diviners are agents of Destiny, and they help us become agents of Destiny too.


[Diviners and Divinees as Agents of Destiny]








Finalité et causalité




La cause efficiente


( In the Stoic system, the events of the world are organized in accordance with God’s rational, providential wisdom, and this means that there is both final purpose but also causal destiny. [In other words, on account of the overarching rational order of the world and also because of God’s good and wise intentions, things happen both because they must happen in order to arrive upon a pre-established purpose or end, and also things happen on account of everything up until now causing that outcome.] But, given the limitations of our human understanding, we are more equipped to understand causality than finality. ( Unlike Plato and Aristotle, who favored final cause or Forms as the primary causality, for the Stoics, efficient causality is more primary. ( The Stoics have a more straightforward notion of cause as being the acting party in causality, and thus God is understood as the primary acting cause. ( The Stoics primary cause is active and efficient. It thus can be seen as a motor cause. And while in its actualization it expresses final and formal causality, these are secondary byproducts of its more primary efficient causal nature.


[Providence, Destiny, Causality, and Finality]

[The Stoics and Efficient Causality over Formal or Final Causality]

[Causal Activity and God]

[Stoic Motor Causality]








La perfection instantanée


( The Stoics favor a sort of cause that is involved in the construction of copies. They do not place the models of the copies as having a primary causal role however. The role of the models is to inspire the creation of techniques that will produce copies. ( This “artisanal” cause is a corporeal that causes the incorporeal predicates. The incorporeals in Plato and Aristotle would serve as primary causes, being the forms that draw forward the motion of change. But for the Stoics, the incorporeals serve a secondary causal role in that it is the nature of the corporeal motion that determines what the incorporeal ends will be, and it is not the incorporeal ends that determine what the corporeal movements will be. ( Primary Causality in Plato, Aristotle, and the Stoics involves the establishment of a temporal distance that is also overcome by that very same primary causality that established the distance in the first place. For Plato, the Forms that serve as models for guiding how the demiurge will try to fashion the world are also the ends on the causal basis of which the demiurge sets upon fashioning the world in the first place. Aristotle’s Unmoved Mover, in order to cause movement, needs there to be things before it that move toward it, but since it causes every moment of the movement, it closes that temporal gap through its final causality. For the Stoics, however, it is not that the final end causes the movement toward it, even though the end is expressed in each moment of movement, because the efficient causality of the movement is really what drives it in the direction of that end. This means that the incorporeal predicate that is expressed every present moment temporally coincides in the present with that future end state, even though the end state has not been actualized yet. ( Motion for the Stoics is not determined by an end to which it must arrive at and thus the pre-existing temporal/spatial interval that it must come to occupy before doing so. Rather, the whole of all of its time and movement is built into or nested into the causal and other properties of its present nature.


[Stoic Artisanal Causality]

[Corporeal and Incorporeal Causality]

[Overcoming Incorporealized Temporality]

[Eternity in the Present]







L’usage des représentations




Coopération avec la causalité du Destin


( Divination of the entirety of God’s providence by interpreting present event-signs is impossible. We turn now to a different kind of interpretation of event-signs that will involve matters of wisdom and conformity with nature. ( Our human limitations make us unable to understand the whole of the cosmos and thereby to act exactly like God does. However, by accepting the present moment in a certain way, we can at least act and understand the world in the general way that God does. ( Because causality is always active and rational, it is thereby always perfect, in that what happens always does so by necessity. We ourselves should cooperate with God in this divine causality. However, we are such a small part of that causality that we cannot cooperate with God on the cosmic scale. Nonetheless, insofar as we act in accordance with the necessarily causality acting upon us, we can play our small role in the divine causality. That is to say, our causality can partake in the perfection of the divine causality, because perfection here means being necessary. ( God knows everything, so God knows how the causal nature of any situation will ultimately result. With that knowledge, God does not need time to deliberate while calculating outcomes of actions that God can decide to take. God knows it all already. There is thus not a delay between God first having an intention for how things happen and then secondly taking on the action that will lead to that end. Even though there is logical order, they transpire in the same instant, namely, the first instant of time. Humans, however, have limited knowledge, and we find ourselves in the middle of a chain of determinate efficient causal relations. This means that we cannot change what happens, and we cannot have acted otherwise, even though it felt like we were choosing one option or another (because in fact the whole decision-making process we underwent followed a course determined by antecedent causality), and we also cannot discern where current events are ultimately leading. As God did not consult us at the beginning of time, the course of events are not in any sense decided by us. That means that although we are the necessary efficient cause for the consequences of our actions, whatever happens does not really depend on us in the sense that it was our choice or intention that really played any role in how things transpire. This would seem to place us into a passive role where it does not matter how we live our lives. For, we have no free will anyway, and whatever happens does not depend on our choices. However, we do have one particular freedom that is not determined by fate, even though it also will have no influence on the course of events. While for God both the intention which comes first logically and the action that comes secondly and that will lead to fulfilling that intention in fact occur simultaneously for him (they are logically ordered but temporally simultaneous), for us, our action or our undergoing of action comes first and our intention, understood as what we want, hope, or intend to happen, comes second, both logically and temporally. Our action comes first logically, because it was determined long before we were even alive, and when it happens, it was really something provided by the antecedent causality. So first we act or undergo action, and only secondly can we decide whether or not we want it, that is, decide that it is something that we have been hoping for, wanting, intending, deem just, and so on. Yet despite this necessary interval, the Stoic sage strives to make their intentions be as simultaneous as possible with their actions and undergoings of actions. They try to keep themselves in a constant state of always wanting what is in fact happening and anticipating that they will want what must necessarily happen next. So, our only freedom is to choose what we want, that is, to choose our intentions. Now, since we cannot change the course of events anyway, and since we cannot know how any particular event must happen necessarily for the sake of the cosmic good, we are best to choose to want whatever is happening. For, it is both necessary and good, regardless of whether or not it is agreeable to our preferences. In that way, we can participate in the divine causality, namely, by placing our value-assigning activities into accord with the fundamental nature of reality, even though what transpires does not actually depend on our affirmational choices.


[Wisdom and Event-Sign Interpretation]

[Wisdom and Event-Sign Interpretation]

[Divine Necessary (Perfect) Causality and Our Necessary (Perfect) Causality Cooperating with It]

[Divine Necessary (Perfect) Causality and Our Necessary (Perfect) Causality Cooperating with It]









Les causes parfaites et les événements, 2


( The impulses that we receive are given to us by perfect and primary, that is, by sufficient and necessary causality, meaning that our actual reception was something determinately caused by the factors acting on us; and, our reaction will be caused by the impressions we have received; however, our response results from auxiliary and proximate causality, meaning that the prior factors (the impressions) were only enough to present us with the choice to assent or not to them, and whatever the effects of our chosen reaction is will themselves be a primary and perfect causality upon something else (which likewise takes our forward-giving perfect and primary cause as being for itself an auxiliary and proximate cause to its own reaction to it). ( The way the body works is just like how the cosmos and God work. God’s own actions determine the series of events by means of the perfect corporeal causality. Likewise, the leading part of our soul is a perfect cause of how it transfers its pneumatic impulses and thereby affects other corporeals, and thereby, it causes incorporeal effects, like walking, sitting, or being angry, etc.


[The Non-Necessity of Our Caused Reactions (Perfect and Primary Causes versus Auxiliary and Proximate Causes)]

[The Leading Part of the Soul as like God]








L’accord rétabli ; usage des représentations et causes antécédentes


( God as the world soul is the source of all causality. Our souls are a part of God’s soul, so whatever our souls cause, God is thereby causing. ( Destiny dictates that we undergo certain events, meaning that we are determinately caused to have certain impressions. Impressions will be the antecedent causes of our reactions. But our reactions are not determined by a necessity of the impressional antecedent cause, which is really only an auxiliary and proximate cause. Our rational faculty thinks independently of the sensory impressions, and it is free to generate its own rational impressions. It can therefore generate impressions to counteract the causal power of the sensory impressions so to bring about a more rationally-guided, chosen reaction. Here we can observe the following picture. Something acted on us. The way it affected us, and thus the impressions we receive, are caused by a necessary (primary and perfect) causation. This is fate in the sense that whatever happens (to us) does so by necessity. But these causes in us, from our internal perspective, are received not with their perfect and primary causality pointing toward us, but rather we from our internal perspective get the “flip side” of the causal “coin”, and we experience the impressions as auxiliary and proximate causes. In a sense, causality like this is like the perfect cause ending where auxiliary cause picks up, and the perfect cause we then create likewise ends where the next auxiliary cause picks up. In that way, we have a siblinghood of freedom and fate.


[God as Cause of Causes]

[The Usage of Impressions, Given Fatefully, Used/Resisted Freely]










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




11 Feb 2018

Plumwood & Sylvan [Routley & Routley] (CBSC) ‘Negation and Contradiction,’ collected brief summaries and contents


by Corry Shores


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


[Central Entry Directory]

[Logic and Semantics, entry directory]

[Richard Sylvan (Francis Routley), entry directory]

[Val Plumwood (Val Routley), entry directory]


Entry directory without the summaries:

[Plumwood & Sylvan (Routley & Routley). Negation and Contradiction, entry directory]





Collected Brief Summaries for


Val Plumwood

(at that time as: Val Routley)




Richard Sylvan

(at that time as: Richard Routley)



“Negation and Contradiction”






[quoting from the Routley and Routley text, abstract:]

The problems of the meaning and function of negation are disentangled from ontological issues with which they have been long entangled. The question of the function of negation is the crucial issue separating relevant and paraconsistent logics from classical theories. The function is illuminated by considering the inferential role of contradictions, contradiction being parasitic on negation. Three basic modellings emerge: a cancellation model, which leads towards connexivism, an explosion model, appropriate to classical and intuitionistic theories, and a constraint model, which includes relevant theories. These three modellings have been seriously confused in the modern literature: untangling them helps motivate the main themes advanced concerning traditional negation and natural negation. Firstly, the dominant traditional view, except around scholastic times when the explosion view was in ascendency, has been the cancellation view, so that the mainstream negation of much of traditional logic is distinctively nonclassical. Secondly, the primary negation determinable of natural negation is relevant negation. In order to picture relevant negation the traditional idea of negation as otherthanness is progressively refined, to nonexclusive restricted otherthanness. Several pictures result, a reversal picture, a debate model, a record cabinet (or files of the universe) model which help explain relevant negation. Two appendices are attached, one on negation in Hegel and the Marxist tradition, the other on Wittgenstein’s treatment of negation and contradiction.







[Contradiction and Incompatibility]

[quoting the Routley and Routley text, with my own boldface:]

Parasitic on negation is contradiction. A contradictory situation is one where both B and ~B (it is not the case that B) hold for some B. An explicit contradiction is a statement of the form B and ~B. A statement C is contradictory, it is often said, if it entails both B and also ~B for some B, etc. Contradiction is always characterized in terms of negation and the logical behaviour of contradictions is dependent on that of negation. Different accounts of negation result not merely in different conceptions of contradiction and of incompatibility, they likewise correspond to different accounts of what constitutes a describable world, what constitutes a logically assessible world. Classical negation restricts such worlds to possible worlds, excluding contradictory and incomplete worlds.





Basic Modellings of Negation in Terms of Different Relations of ~A to A.


(3.1) We can distinguish three theories of negation by looking for the role negation is thought to play in the inferences that can be drawn from contradictions. We call whatever can be inferred from something its “logical content.” Each of the three theories thinks that a contradiction-forming negation has a different sort of content, either: {1} no content (it entails nothing), {2} full content (it entails everything), or {3} partial content (it entails some things but not others). (3.2) The first kind is called the cancellation model of negation. It says that ~A cancels (erases, deletes, neutralizes, etc.) A, such that were we to conjoin them, nothing can be derived from that contradictory conjunction. This sort of thinking seems to be built into the connexivist view that something cannot entail its negation (because were it to do so, it would cancel what you started with). (3.3) This cancellation/connexivist view may or may not have been held by Aristotle but certainly by many others, including Boethius, Berkeley, Strawson, and Körner. (3.4) But note that this cancelation view does not apply to Hegel, even though some have mistakenly done so. (3.5) Next there is (3.6) the explosion model of negation used in classical and intuitionistic logics. Here, the conjunction of a formula yields any other arbitrary formula, no matter how irrelevant. (3.7) In the semantics for classical negation, which is an explosive negation, a negated formula is true only if in that same world its unnegated form is not true. This means that there is no world where both a formula and its negation are true, thus from such a conjunction, we can derive from that whatever we please. (3.8) We can picture this classical, explosive view of negation in terms of there being a certain total “terrain” of statements in a world. A covers a certain terrain of those statements, while ~A covers everything else, with nothing left out and with no overlap, as we see in these diagrams, one from Routley and Routley and another from a source of their diagram, by Hospers:



(3.9) Many philosophers who seem to argue for classical negation really have only assumed that classical negation has already been settled as being “ordinary” negation. We see this for example in Quine’s argumentation. (3.10) Although it leads to an expansion of valid formulas, the explosion model involves subtractions, because when adding ~A to a world with A, we need to consistencize it by removing A and all that it implies. In other words, when we recognize the contradiction-forming negation’s explosive potential, we need to make subtractions in order for that negation to be included. (3.11) Then there is a third sort of negation [a relevant or paraconsistent model] which is such that when we have a contradiction, we can infer some things but not all things. One sort of this negation is relevant negation. Here, ~A, rather than cancelling or exploding A, instead constrains A [meaning perhaps that it limits its “terrain” of propositions but not completely]. (3.12) Relevant negation defines a negated formula as true in a world only if its unnegated form is not true in an “opposite” or “reverse” world. (3.13) Since we want inconsistent and incomplete worlds, that means we should use this star negation rule, as it will allow for contradictions without explosion and also for excluded middle not to hold [so that something can neither hold nor not hold for some world.] (3.14) We next turn to some elaborations.

3 Contents


[Entailment as Determining Logical Content. Theories of Negation Divided into Three on the Basis of the Logical Content of Contradictory Negations: None, All, and Some.]


[Theory 1: The Cancellation Model. Connexivism.]


[The Cancellation Tradition in the History of Philosophy (Aristotle, Boethius, Berkeley, Strawson, and Körner).]


[Hegel as Not Using the Cancellation Model.]


[(Transition to Explosion Model).]


[Theory 2: The Explosion Model.]


[Classical Negation and Explosion.]


[The Total Exclusion in Classical Negation.]


[Quine and Arguments for Classical Negation]


[The Explosion Model’s Consistencizing Subtractions]


[Theory 3: The Paraconsistent Model.]


[Relevant Negation and Opposite Worlds]


[The Need for Paraconsistent Negation.]


[Summary and Preview.]




Main Themes Concerning Traditional Negation, Ordinary and Natural Negation, and Their Models


(5.1) [There are three models of negation: the cancellation model, the explosion model (which includes classical negation), and the paraconsistent/relevant model. See section 3]. The cancellation model is wrong because it says that all contradictions have the same logical content, namely, nothing, when in fact they have different content. For, we should be able to derive something different from A∧~A than we do from B∧~B. The explosion model is wrong, because it would cause certain inconsistent theories to be rendered trivial when in fact we know them instead to be non-trivial. (5.2) There are major philosophers who used the third model of negation. One of them is Hegel. He certainly did not use a cancellation model, because for him there are philosophically fundamental notions that are contradictory and yet are not void of content, as for example Being and Nothing being both identical and not identical. (5.3) Over the course of philosophy’s history, the three models have been in competition, but for the most part, the dominant view on negation has been a nonclassical view. (5.4) And in fact, classical negation, despite its pretensions, is the exceptional view and not the norm. (5.5) If our criteria is, what is the best view on negation for modelling the sort of negation that we find in natural language?, we would regard relevant negation, and not classical negation, as being a natural negation.

5 Contents


[Reasons to Reject the Cancellation and Explosion Models of Negation]


[Hegel’s Paraconsistent Negation]


[Traditions of Negation]


[The Non-Centrality of Classical Negation]


[Relevant Negation as Natural]




Negation as Otherthanness, and Progressive Modification of the Traditional Picture


(6.1) We will come to a notion of negation as restricted otherthaness, beginning in the logical philosophy of the 19th and early 20th centuries. Thus we will construct a Boole-Venn sort of semantics, where we assign values (taken very broadly as we will see) to atomic formulas, and the connectives further operate on those values. Quoting the authors:

Such an interpretation j is a mapping from (initial) wff of S to V which consists of a composite with (at least two) components, e.g. a geometrical area, a set, a mereological class, such that the following conditions are met:

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

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

j(A ∨ B) = j(A) ∪ j(B) i.e. the union (of areas)

A wff C of S is said to be BV-valid iff, for every mapping j, j(C) = V, i.e. the interpretation is always the whole of V.


There are three pertinent readings of the j interpretation function. (6.2) The first is the “geometrical reading.” It sees the j function as mapping to A some geometrical area or “territory”. (6.3) The second reading of j (the “set-theoretic reading”) sees it as mapping to A some set or class of objects, like “animal,” “plants,” and “horses.” The Boolean definitions of and and or are not problematic under this reading, but problems with not lead many who take this reading to use a non-Boolean interpretation of not. (6.4) The third reading of j is the “propositional reading.” It regards j as mapping to each wff some proposition or propositions. (6.5) The prevailing view of negation in the late 19th century is that it is restricted otherthanness, meaning that ~A is other than A, but not everything other than A. (6.6) To picture restricted otherthanness [under the geometrical reading], we need to think of ~A as being some other part of the terrain which is other than A’s part, but also not the entire remainder of that terrain. So it would be the situation in the right diagram and not the left:


We use the * operator on j to get this limited negation. Thus we have:


(6.7) The picture we consider now is one where A and ~A are not mutually exclusive and also not exhaustive of the terrain.


6 Contents


[Boole-Venn Semantics]


[The Geometrical (“Territorial”) Reading of j]


[The Set-Theoretical Reading of j]


[The Propositional Reading of j]


[Late 19th Century Negation as Restricted Otherthanness]


[The * Operator on j for Restricted Negation]


[A and ~A as Non-Exclusive and Non-Exhaustive]




Transposing the Hegelian Picture: Restricted Otherthanness, Reversal and Opposites


(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.

7 Contents


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


[Classical Negation as More Limited than Relevant Negation]


[Classical Negation Is Irrelevant]


[Classical Negation and Alienation]


[Limited Otherness in Relevant Negation]


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


[The Debate or Dialectical Model of Restricted Relevant Negation]


[Classical Negation as Structurally Irrelevant]


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





Semantical Models: Worlds on Record and Tape


(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.

8 Contents


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


[The Record Cabinet Model]


[The Star * Function, Worlds, and Negation]


[Kripke’s  Sheets of Paper Metaphor]


[Relevant Negation as Neither Cancelling nor Exploding Content]


[Relevant Negation as More Natural than Classical Negation]








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