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by Corry Shores
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[Logic and Semantics, entry directory]
[Richard Sylvan (Francis Routley), entry directory]
[Val Plumwood (Val Routley), entry directory]
[Plumwood & Sylvan (Routley & Routley). Negation and Contradiction, entry directory]
[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”
5
Main Themes Concerning Traditional Negation, Ordinary and Natural Negation, and Their Models
Brief summary:
(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.
[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]
Summary
[Reasons to Reject the Cancellation and Explosion Models of Negation]
[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 (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 that we know to be non-trivial to instead be trivial.]
[In section 3.2, we discussed the cancellation model of negation. It says that when you have both a formula and its negation, the content of both is nullified such that you can derive nothing from their conjunction, in other words, under the cancellation model of negation, a contradiction-forming conjunction of a formula and its negation has no logical content at all. The insight here seems to be that negation is a sort of antitheses, and while you might have a formula and its negation, the content of each formula has pushed the other’s content out of the field of content, thereby giving neither formula any “terrain” in the field of content. In section 3.6, section 3.7, section 3.8, and section 3.10, we discussed the explosion model of negation. The idea with this model is that the negation of something has all the logical content that the unnegated formula does not have. So here it seems we put aside the issue whether or not contents might be antithetical. We instead think of them exhausting all formulas in the field of content, such that the conjunction of the two gives you all content. In other words, under the explosion model of negation, the contradiction-forming conjunction of something and its negation yields all possible content, that is to say, you can derive any formula you want from such a contradiction. Classical negation is such an explosive negation.] Routley and Routley say that the explosion and cancellation views are both unsatisfactory (212). The problem with the explosion view is that it is strongly paradoxical (212). [Note, I am not exactly sure what makes something paradoxical. In section 3.6, we came across this notion, and to define it, we used Andrea Cantini and Riccardo Bruni's Stanford Encyclopedia of Philosophy article entitled: “Paradoxes and Contemporary Logic:”
By “paradox” one usually means a statement claiming something which goes beyond (or even against) ‘common opinion’ (what is usually believed or held).
(Andrea Cantini and Riccardo Bruni, “Paradoxes and Contemporary Logic.”)
So maybe Routley and Routley are saying that the explosion view goes strongly against common opinion, with that common opinion holding that we should not be able to draw wildly irrelevant and silly conclusions from contradictions.] The cancellation view, while not leading to triviality, does lead to all contradiction entailing one another [which is paradoxical.] This is because under the cancellation view, all contradictions have the same logical content, namely, nothing, which means that A∧~A↔B∧~B, for an arbitrary A and B. We should reject the explosion view, because there are non-trivial inconsistent theories, and the explosion model would disallow them [or at least render them trivial when we know them to be non-trivial.] And we should reject the cancellation view of negation, because it is not the case that all contradictions “carry the same information”; rather, we would want to say that different contradictions (involving different formulas) entail different things.
Neither the explosion nor the cancellation view is satisfactory. The explosion view is strongly paradoxical,10 the cancellation view is weakly paradoxical (at least as it stands). The cancellation view does not have each contradiction entailing everything, and all inconsistent theories trivialized in the way that the explosion theory does; but it does have each contradiction entailing each other, A∧~A↔B∧~B, for arbitrary A and B. For A and ~A, and B and ~B, say exactly the same, namely, nothing.11 The explosion view is wrong because contradictions are not so destructive: there are various different non-trivial inconsistent theories. The simple cancellation view is also defective, since not all contradictions carry the same information: they differ in what they entail, some of them entailing some things, others other things.
(212)
10. That is only symptomatic of the range of things that is wrong with it, on which see RLR, 1.
11. Strictly there are different positions here depending on whether contradictions are said to imply themselves or not, i.e. whether A→A holds quite generally or not. If not, as with peripatetic logics, weak paradox can be avoided. But then many other problems arise: see RLR, 11. An alternative, sometimes attributed to Wittgenstein, is to say that contradictions lead nowhere, that all argument stops when a contradiction is encountered. As to how unsatisfactory this view is, see [22] pp. 179-80. Moreover, on Wittgenstein’s view, contradictions may stand in some language games. They are not always destructive or self-cancelling.
(228)
[Hegel’s Paraconsistent Negation]
[Hegel used a paraconsistent model of negation. 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.]
[Hegel’s notion of negation does not involve explosion or cancellation. He clearly states that he thinks contradictions can be conceived. They have a conceptual content, so they cannot involve cancellation. In fact, one of the most important and fundamental concepts for Hegel is the notion of Being being both identical and not identical with Nothingness. This means that he uses a paraconsistent model of negation.]
The negation of Hegel’s logic, like that of any paraconsistent logic, does not, and cannot, conform to the classical view 2, nor does it can form to view 1.12 For not only did Hegel reject the idea that contradictions could | not be separately thought (‘Contradiction is the very moving principle of the world: and it is ridiculous to say that contradiction is unthinkable’, Logic p. 174; cf. too Findlay, p.75 where several references are cited: ‘it is absurd to say that contradictions are unthinkable’); he also certainly held that in thinking contradictions, one was not thinking nothing, or merely a self-cancelling thought; for, quite the contrary, in thinking that Being is identical with Nothingness and is also not identical therewith, one is thinking an explicitly contradictory thought of fundamental importance. While modern paraconsistent theories are usually not as extravagant as 0 the range, type, or centrality of the contradictions asserted, the intention is much the same: accounts of negation of type 3 are required.
(212-213)
12. The theme that Hegel’s logical theory is a paraconsistent one will be argued elsewhere, as will the theme that contradictions in Hegel’s theory are genuine contradictions.
(228)
[Traditions of Negation]
[Historically speaking, the dominant view on negation has for the most part been a nonclassical view.]
Historically speaking, there is no one tradition for negation in logic. There instead have been competing traditions, which are seen for example in the competing positions going back to Ancient times regarding implication. But after the scholastic period, a dominant view has been the cancellation view. But there is still more historical work to be done on the matter. Nonetheless, “the mainstream or dominant negation of traditional logic is distinctly nonclassical” (213).
Before considering in more detail what such negations are like, it is worth inquiring, and important to inquire – since classical logic is wont to claim that history (as well as God and Truth and Language) is on its side – what traditional negation, the negation of traditional logic (if there was such a single creature) was like. What was the tradition, especially as regards negation? There wasn’t a single unified tradition, there were various competing traditions in particular as to negation and implication. These competing positions are especially evident in the debate as to implication in ancient Alexandria, and in the controversies of scholastic writings. Despite the competition, there seems, at least from post-scholastic times, to have been a dominant view, namely the cancellation view.13 It should be stressed that this is very much a working hypothesis. There is a great deal of difficult assemblage of historical evidence still to be accomplished (both for modern and for earlier periods).14 A weaker theme, on somewhat firmer ground, is that the mainstream or dominant negation of traditional logic is distinctly nonclassical. Some of the evidence supporting this first working hypothesis will emerge below.
(213)
13. Where does Hegel fit in? Hegel seems to have realized that he was doing something different from traditional logic, that he was in a sense outside of (and extending) the tradition.
14. We should be grateful to anyone who supplies historical leads to pursue. The situation is much complicated in the case of scholastic logicians by the selection of work that has so far been made available – which typically tries to see these people as anticipating modern established doctrine, the conventional classical wisdom, rather than as investigators of various alternative logic options. The bias of history impedes research of altematives, so to say. Fortunately that situation may be beginning to change especially with new research into the obligationes-literature.
(228)
[The Non-Centrality of Classical Negation]
[We will see that classical negation, despite its pretensions, is the exceptional view and not the norm.]
Many logicians currently treat classical negation as if it has always been the one true view, and the others are aberrations in relation to it. Instead, we should develop a more accurate view and see that in fact classical negation is the exceptional view.
It is also important to inquire what natural negation, negation of natural language, is like, because part at least of the logical enterprise concerning negation is to reflect key features of that negation. Again it has been assumed, with precious little evidence, that classical negation fulfils this role. Many considerations tell against this assumption (see RLR 2). It is important to see through classical negation’s pretensions to be the ordinary normal intuitive notion of natural language and logical thought – compared with which alternative negations such as relevant negation must be seen as ‘deviant’, ‘peculiar’, ‘queer’, abnormal, contrived, or purely formalistic. For seeing through its pretensions is an essential part of seeing through classical (implication) theory and seeing why relevant (implication) theory should replace it.
(313)
[Relevant Negation as Natural]
[Relevant negation is a natural negation, because it models negation in natural language, and it does so much better than classical negation.]
In fact, not only is classical negation not really the primary theory, but relevant negation more properly is so, because it is much better for modeling negation in natural language, and thus is is a “natural negation”.
In fact the situation is pretty much the reverse of the conventional pic- | ture. Relevant negation has a better claim to be the (primary) negation determinate of natural language than classical (if indeed there is a unique natural language negation, which is to be doubted). A second working hypothesis is, then, that relevant negation is a natural negation. (A, because the negation determinable is probably the most commonly occurring natural language negation; see further RLR, 2.9).
(213-214)
From:
Routley, Richard. and Val Routley. 1985. “Negation and Contradiction.” Revista Colombiana de Matematicas, 19: 201-231.
Sources cited by the authors:
[5] J.N. Findlay, Hegel. A Reexamination, Allen & Unwin; London, 1958.
[12] G.F.W. Hegel, Science of Logic (translated by W.H. Johnston and L.G. Struthers), Volumes One and Two, Allen and Unwin, London, 1929.
“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.
[22] R. Routley, Exploring Meinong’s Jungle and Beyond (Especially: “Ultralogic as Universal?” RSSS, Australian National University, 1979.
Other citations made by me:
Cantini, Andrea and Bruni, Riccardo. 2017 [2007]. “Paradoxes and Contemporary Logic.” Stanford Encyclopedia of Philosophy. Web. [First published Tue Oct 16, 2007; substantive revision Wed Feb 22, 2017.] Accessed 2018.01.17.
https://plato.stanford.edu/entries/paradoxes-contemporary-logic
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