## 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

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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)

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.

[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]

Summary

[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:

Conjunction

Disjunction

As we can see from this evaluation,

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:

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)

[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:

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)

[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:

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.

[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)

[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)

[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.

[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)

[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)

[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.

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

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