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”

8

Semantical Models: Worlds on Record and Tape

Brief summary:

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

[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]

Summary

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

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

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

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

a ⊨ ~

p

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

a* ⊨

p

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

a ⊨ ~

piff a* ⊭p

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

a* ⊨

piff a ⊭ ~p

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

The debate model can be given a more semantical turn. In the

p-issue, ~pis asserted, or presented as true, on one side, a say (i.e. a ⊨ ~pin obvious notation), while the reverse, namelyp, is asserted, or presented as true, on the opposite side a* (i.e. symbolically a* ⊨p). Now one side succeeds in a | debate, or establishes its case, iff the opposite side does not; therefore a ⊨ ~piff a* ⊭p. That is, a version of the star rule naturally emerges from the debate model more semantically considered. Statement ~pis made, or presented as, true at side or situation a iffpis made, or presented as, true at its opposite a*.(218-219)

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[The Record Cabinet Model]

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

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

The debate model leads directly to the

record cabinet model. The cabinet which can represent the files of the universe, is full of records, each record is anissue, or question, withpon one side and ~pon the other side, for every atomicp(at least). From this point of view classical negation takespas one side of one record, and ~pas everything else in the cabinet (classical theory fails to duly separate issues). Relevant negation takespas one side of the record and ~pas the other side of thesamerecord, there being many many records in the cabinet. Note well that intensional functions select a program from the cabinet. Such a program may includebothsides of a record, and may include neither side of various records – in contrast to the published classical picture (the classical picture can be suped-up to avoid the latter defect but not the former).(219)

**.**

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[The Star * Function, Worlds, and Negation]

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

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

The cabinet model maybe differently oriented. Each record, or tape, represents, e.g. it may just

describe, a world, a two-sided world. Then where a is one side of a world record, or a world, the opposite side is again a*, where * is thereversal, orflip, function which gives, whichever side one is in on, the other side. Obviously a** = a, since turning the record over twice takes one back to the initial position. The semantical rule for evaluating negated statements is, as for the debate model, the star rule, ~pholds at a iffpdoes not hold at a*. By contrast, the classical rule quite erroneously identifies a side with its opposite.(219)

**.**

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[Kripke’s Sheets of Paper Metaphor]

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

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

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

(219)

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

The

structured recordmodel corresponds exactly to a natural elaboration of Kripke’s valuable sheets-of-paper model of semantic tableaux for normal modal logics. In explainingalternative setsKripke says (63, p.73): ‘Informally speaking, if the original ordered set is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case putting in addition A in the right column of tableau t and in the other case putting B; the two new sheets correspond to the two new alternative sets’. Thus a full construction which consists of a system of alternative sets corresponds to an arrangement of sheets (a sheaf of sheets).(219)

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

N1. If ~

Aappears in the left column of a tableau, putAin the right column of that tableau.

Nr. If ~Aappears in the right column of a tableau, putAin the left column of that tableau.(Kripke, p.72)

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

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

(219)

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

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

(219)

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

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

structured recordmodel corresponds exactly to a natural elaboration of Kripke’s valuable sheets-of-paper model of semantic tableaux for normal modal logics. In explainingalternative setsKripke says (63, p.73): ‘Informally speaking, if the original ordered set is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case putting in addition A in the right column of tableau t and in the other case putting B; the two new sheets correspond to the two new alternative sets’. Thus a full construction which consists of a system of alternative sets corresponds to an arrangement of sheets (a sheaf of sheets). For relevant semantic tableaux there are only two innovations. First, whereas with strict implication new related tableaux are introduced one at a time, with relevant implication new related tableaux are introduced two at a time, i.e. in pairs. This reflects the replacement of the two-place alternativeness | relation of modal logics, by the three-place alternatives relation of relevant logics. The first innovation is not particularly germane to the present issues (and quasi-relevant systems such as the I systems which require only two-place relations could be adopted for exposition). Second, and more important, then, both sides of the sheets are used. (Relevant logics are conservation-oriented in that even if rather a lot of sheets are introduced, both sides are used; the reverses are not wasted as with modal semantical tableaux). The reversal function * accordingly reverses the page, giving back for front and front for back.(219-220)

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[Relevant Negation as Neither Cancelling nor Exploding Content]

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

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

In sum, reversal and opposition have the right properties in leading respects for (the semantics of) relevant negation. Thus the opposite side of something is not the removal of the first side or, for example, everything other than the first side;

.it iswhich is relatively independent of its reverse but which is related to it in a certain wayanotherandfurtherside,Both sides can co-occur (occur simultaneously) in a framework(e.g. controversy) and one can perfectly well consider both of them.The important point, to say it yet again, is that one side does not somehow.obliterateor wipe out or entirely exclude or exhaust its oppositeNor is the reverse, or opposite, just defined negatively asthe other– it has an independent and equal role on its own behalf.(219, boldface and underlining mine, italics in the original)

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[Relevant Negation as More Natural than Classical Negation]

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

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

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

(220)

Sources cited by the authors:

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

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

Other citations made by me:

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

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