by Corry Shores
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Collected Brief Summaries for
Val Plumwood
(at that time as: Val Routley)
and
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.
(201)
[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.
(204)
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.
[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.]
5
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.
[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.
(214)
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.
[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.
[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.
[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.
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