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”

3

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

Brief summary:

(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.]

Summary

[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.]

[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 perhaps only entails themselves).]

We will examine three theories of negation. What distinguishes the theories is the different sorts of “roles they allow, or assign to, contradiction” (205). The “role” here is then specified as an “inferential role”. [As we will see, the question will be what, if anything, can be derived from a contradiction. And what can be derived will be called the “logical content” of the contradictory formulation.] There are three possibilities for the inferential role that a contradiction can play. They may have {1} no inferential role, by implying nothing except possibly just themselves, {2} a total inferential role, by implying everything, or {3} a limited inferential role, by implying some things, as for example implying simply their contradictory components, but not implying other things. Along these lines, we will classify three theories of negation, doing so in terms of the relation ~A to A. [So we consider if we have A and also that we have ~A. How did we get them? I am not exactly sure how. With the Hegel motion examples that we examine later (and that you can find discussed in Graham Priest’s *In Contradiction *section 12.3), perhaps we obtain them by philosophizing and positing certain notions or claims. So we might for example say that at a moment of motion, the object both is at a certain point (call that statement “A”) and it is not at that point (~A). We here are simply making both claims, although we are doing so with certain philosophical motivations or considerations. Another way I suppose could be from some sort of an inference involving paradoxical formulations, as perhaps might happen with a liar sentence. I am not sure, but maybe in that case we can derive its negation from itself, but I am just wondering. I really do not know if this example works of if any other could work. And yet another way I suppose could be just an arbitrary formal experiment. We just say A and then ~A as pure symbolic formulations, and we see what happens given our semantics or given what we can derive from that on the basis of our derivation or tableau rules. So in the following, when we talk about A and ~A, my impression is that we are thinking of this mainly like a proof, where on one initial line we say A, and on another we say ~A, then we ask, what can be derived from that, particularly from their conjunction? Depending on which theory of negation we are using, that will determine which derivation rules will apply to the contradiction. What Routley and Routley seem to be doing here is looking to see what derivation rules one thinks can be applied to a contradiction in order to determine (or at least to characterize features of) the theory of negation that one is using.)]

Theories of negation differ, very obviously, in the rules they allow, or assign to, contradictions. Contradictions may be allowed no inferential role (they imply nothing, except perhaps themselves), a total inferential role (they imply everything), or some limited inferential role (they imply some things, such as their contradictory components, but not others). There are, correspondingly three initial ways to classify theories of negation, in terms of the relation of ~A to A.

(205)

[__Theory 1__: The Cancellation Model. Connexivism.]

[The first theory of negation is the cancellation model. It says that ~A cancels A, such that were we to conjoin them, nothing can be derived from that. 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).]

[This first view says that if you have a contradiction like A and ~A, then you can derive nothing else from that. (Suppose you have the conjunction A ∧ B. Normally by means of a derivation rule, called “conjunction elimination” in Agler’s *Symbolic Logic *section 5.3.2, you can derive either A or B (or both individually) from this. So you would think that for A ∧ ~A you would be able to derive either A or ~A. But] under this theory of negation, you cannot derive even A or ~A from A ∧ ~A. This means that the negation of A cancels, erases, deletes, neutralizes, etc., A. For this reason, when you combine A and ~A, there is no inferential content. We call this the *cancellation model *(or the *erasure model *or the *neutralization model*). (In order for this cancelation idea to work, it would seem we need to see things in the following way. From A you can derive A and from ~A you can derive ~A. But when they are conjoined or are at least found in the same line of reasoning, that means, under the cancelation view, that A cancels the content of ~A and ~A cancels the content of A, leaving no content left. If that is not what is happening, then I do not know what is being canceled. Maybe we are to think of A having other content like B, which is also canceled by ~A. I am really not sure how it works yet.) Routley and Routley note how this model is related to connexivism. They define connexivism as involving two theses. The first is that contradictions do not entail their components, like we discussed above. The second is that, a formula is not entailed by its negation. Reasoning for this second thesis (that a formula is not entailed by its negation) can be found in the first thesis (that contradictions do not entail their components), which is what defined the cancellation model. Recall that we say that something is a logical content of something else if the first can be derived from the second. So suppose we take the cancellation view of negation. What would happen were A to entail ~A? That would mean that A includes ~A as part of its logical content. But under our cancelation view, this formula that A contains, namely, ~A, cancels A such that their combination leaves nothing. This means it leaves us with neither A nor ~A. (Routley & Routley seem to be saying, as far as I can tell, that because this eliminates what you start with, that you cannot begin under the assumption that something entails its negation. And thus, a connexivist could hold their connexivist views on account of them also holding a cancellation model of negation. But I am not sure if I follow that correctly, so please consult the text to follow.

1.~A deletes, neutralizes, erases, cancels A (and similarly, since the relation is symmetrical, A erases ~A), s0 that ~A together with A leaves nothing, no content. The conjunction of A and ~A says nothing, so nothing more specific follows. In particular, A ∧ ~A does not entail A and does not entail -A. Accordingly, thecancellation(erasure, or neutralization)modelleads towards connexivism, a position (much discussed in RLR) distinguished by the following two theses – First, that already cited, that explicit contradictions do not entail their components, and secondly, that A does not entail ~A. The second thesis emerges naturally under the neutralization view, for instance, as follows. Entailment is inclusion of logical content. So, if A were to entail ~A, it would include as part of its content, what neutralizes it, ~A, in which event it would entail nothing, having no content. So it is not the case that A entails ~A, that is Aristotle's thesis, ~(A → ~A) holds.^{3}(205)

3. This assumes – what is not unreasonable, but strictly calls for further argument – that the inner and outer negations are the same.

(227)

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

[This cancellation/connexivist view may have been held by Aristotle but certainly by many others, including Boethius, Berkeley, Strawson, and Körner.]

[We are not certain whether or not Aristotle adhered to the connexivist thesis that bears his name. But a number of other philosophers throughout its history have taken the cancellation view, including Boethius, Strawson, Körner, and others. Berkeley explicitly explains the reasoning behind the above inference that were you to start with one proposition, then negate it, that would destroy the first supposition, and thereby destroy its consequences, including that very negation doing that destroying.]

There is reasonable, but not conclusive, evidence that Aristotle did adhere to Aristotle’s thesis. And assuming that he did certainly has great explicative advantages, for example the full theory of the syllogism translates into connexive quantificational logic without loss or qualification (as Angell, and also McCall, has pointed out); the theory of immediate inference also emerges intact (for inferential but not implicational form). Whether or not Aristotle was operating with connexive assumptions, there is a long historical line of logicians and philosophers who have assumed a cancellation picture, from Boethius in Medieval times through to Strawson, Korner, and many others in modem times (see RLR). One striking intermediate examples is Berkeley, who advances the following claims in his attack on the calculus (

The Analyst, p. 73) :Nothing is plainer than that no just conclusion can be directly drawn from two inconsistent premises. You may indeep suppose any thing possible: But afterwards you may not suppose anything that destroys what you first supposed: or, if you do, you must begin

de nove... [When] you ... destroy one supposition by another ... you may not retain the consequences, or any part of the consequences, of your first supposition so destroyed.(205)

[Hegel as Not Using the Cancellation Model.]

[The cancellation view is often misapplied to Hegel.]

[We see the cancellation view misapplied to Hegel notion of contradiction, who really takes a different view on negation (see section below).]

Cancellation views are prevalent in one place where they are particularly damaging, in so-called expositions of Hegel (but there is a basis for this ascription in Hegel himself, as will appear). Given this phenomenon it is not surprising that Hegel's logic has appeared so intractable to commentators. Here, for instance, is what the Marxist logician Havas has to say as regards Hegel's theory:

... the Aristotelian principle of non-contradiction is a general principle of metalogic, which can be said to bring out a necessary condition to be satisfied by all human thought and all of the systems of logic; namely, the condition that it is a logical contradiction, and therefore, a logical mistake to assert both something and its opposite. This is one of the elementary but necessary conditions of sound reasoning, because if one asserts something to be true and, insisting on this assertion, one also asserts that this very assertion is not true, then his assertions will neutralize each other and, in consequence of this, no knowledge will be acquired (p.7).

Apart from being unfaithful to Hegel, who (correctly) says that there is nothing unthinkable about contradictions, thereby repudiating the laws-of-thought myth, and who accepted no such simple neutralization view, the Aristotelian principle is not a metalogical principle concerning the logic of assertion.

(206)

[(Transition to Explosion Model).]

[Next we examine the explosion model used in classical and intuitionistic logics.]

We turn now to the second model for negation, which is found in classical logic and intuitionistic logics.

The second model for negation is that embodied in contemporary classical and intuitionist logics:

(206)

[__Theory 2__: The Explosion Model.]

[In the *explosion model *of negation, the conjunction of a formula yields any other arbitrary formula, no matter how irrelevant. This is found in classical logic, for example.]

The second model is the *explosion model *(also called the *destruction model*). [I do not know why it is also called the destruction model, because from all appearances, everything is preserved and nothing is destroyed.] It says that when you conjoin A and ~A, it yields everything. Thus A∧~A has total content. [Let me try to fashion and understanding here. Previously I offered the explanation that we understand negation as what cancels the content of the non-negation. So we might thing of the content of ~A as like the anti-content of A’s content, in the sense that ~A’s content is something like it pushes A’s content out of the field of content, but since likewise A’s content pushes ~A’s content out of the field of content, when A and ~A are combined, their contents mutually push one another equally out of the field of content, leaving nothing. Now with the explosion model, we do not thing of ~A’s content as being the anti-content of A. Rather, it is like the absolute complement content to A’s content. In other words, ~A’s content is anything and everything that is not in A’s content, but it is complementary rather than anti-thetical, because the content of ~A does not push A’s content out of the field of content; rather, aggregates to it, thereby jointly creating a field of content that contains every formula whatsoever. (Let me try an example. Suppose in the cancelation view we have “the cat is on the mat” and suppose we also have “the cat is not on the mat”. They cancel, meaning that we can no longer know what is going on with the cat and the mat. Now take the explosion view. We have “the cat is on the mat” and “all that which does not mean ‘the cat is on the mat’.” This would give us in total every possible formulation/statement of fact/sense or what-have-you, with all the contradictions and other absurdities that would result. I realize that I am misconstruing the situation with this sort of example, but I am not sure how else yet to think of the content of ~A under this view. I would just suppose that it is B, C, etc., and whatever else is not A, where in the cancelation view, ~A’s content does not extend into every other formula.)] This means that A∧~A entails any arbitrary formula B, no matter how irrelevant. [The authors then say that the explosion model is paradoxical. I am not sure why. Here I would need a definition of paradox. Quoting from the 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 since concluding things that are entirely irrelevant from our intuitions goes against our intuitions about how an inference should probably work, it might for that reason be paradoxical. I am not sure.]

2.~A explodes, or fully implodes, A (and similarly A explodes ~A) in such a way that ~A together with A yields everything, total content. The conjunction A and ~A says everything, so everything follows. A ∧ ~A entails B, for arbitrary and irrelevant B, so theexplosion(or destruction)model: is inevitably paradoxical. The paradoxical character of classical logic for example, can accordingly be obtained with very few further assumptions from the character of its negation.(206)

[Classical Negation and Explosion.]

[In the semantics for classical 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.]

[The authors define classical negation in the following way: “the classical semantical rule **I**(A, a) = 1 iff **I**(A, a) ≠ 1, i.e. A holds at world a iff A does not hold at a, for every world a.” In this formulation, it seems we use a truth-valuating function, and we designate both the formula and the world in which it has some value. In Graham Priest’s *Introduction to Non-Classical Logic* section 2.3.4, he gives a similar sort of formulation for modal logics:

ν(¬_{w}A) = 1 ifν(_{w}A) = 0, and 0 otherwise.(Priest,

Introduction to Non-Classical Logic, p.21)

It seems we will eventually arrive upon a semantics that is like First Degree Entailment (*FDE*), which allows for the sorts of value arrangements that we will find in the third type of negation. Priest explains the semantics of *FDE* in section 8.2 of *Introduction to Non-Classical Logic*. What is notable about *FDE* is that it uses a truth valuating relation rather than a function. In section 8.4, he gives a 4-valued semantics that is equivalent to *FDE*, and it thus uses a truth-valuating function rather than a relation. And in section 8.5, Priest deals specifically with Routley star semantics, which could be what we will be dealing with here.

We now have two equivalent semantics for

FDE, a relational semantics and a many-valued semantics.^{5}For reasons to do with later chapters, we should have a third. This is a two-valued possible-world semantics, which treats negation as an intensional operator; that is, as an operator whose truth conditions require reference to worlds other than the world at which truth is being evaluated.(Priest,

Introduction to Non-Classical Logic, p.151, section 8.5.1)

So what it seems that we will get to is a four-value situation with really only two values, but as there are two worlds involved, we can have in total four value situations that accord with those of *FDE *(for that equivalence of Routley Star semantics and *FDE*, see Priest’s *Introduction to Non-Classical Logic*, section 8.5.8.) I mention this all now, because it might seem odd that we are defining classical negation using possible world semantics, but as we will see, that will be compatible with the next sort of negation.] The authors add that under other conditions this model would yield intuitionistic negation. I have not learned intuitionistic logic yet, so I cannot say anything about that at the moment. But for now, I will simply quote from Priest’s *Introduction to Non-Classical Logic* what seems to be a candidate for intuitionistic negation:

v(⇁_{w}A) = 1 if for allw′ such thatwRw′,v_{w}_{′}(A) = 0; otherwise it is 0.(Priest,

Introduction to Non-Classical Logic, p.21)

The authors then write:

classical negation, i.e. negation conforming to the classical rule, yields the explosion view, since there is no world where both A and ~A hold but B does not, whence, on the semantical theory,

^{4 }A&~A→B. Thus under weak conditions the explosion view is that of classical negation.(206)

I might not get all of this right, but I will guess. The explosion view is that a formula conjoined with its negation entails any arbitrary formula. Classical negation says that a negation is true in a world only if the non-negated form is not true in that world. We are now saying that the classical view of negation leads to explosion. The reasoning to me at least seems to be related to the idea that an inference is valid if when its premises are true then its conclusion is also true. Now recall from Priest’s *Introduction to Non-Classical Logic* section 2.3.11 the definition of validity in possible worlds semantics:

An inference is valid if it is truth-preserving at all worlds of all interpretations. Thus, if Σ is a set of formulas and

Ais a formula, then semantic consequence and logical truth are defined as follows:Σ ⊨Aiff for all interpretations ⟨W,R,v⟩ and allw∈W: ifv(_{w}B) = 1 for allB∈ Σ, thenv(_{w}A) = 1.(Priest,

Introduction to Non-Classical Logic, p23)

Here we might also need the evaluation for conjunction to follow through the reasoning here. This is from Priest’s *Introduction to Non-Classical Logic* section 2.3.4:

v(¬_{w}A) = 1 ifv(_{w}A) = 0, and 0 otherwise....

v(_{w}A∧B) = 1 ifv(_{w}A) =v(_{w}B) = 1, and 0 otherwise.(Priest,

Introduction to Non-Classical Logic, p21)

This means that in classical logic, if you have something and its negation, then you have a conjunction of them valued 0. Thus in such a world, no valuation can make all the formulas true. That means it does not matter what you want to take as a semantic consequence, it will be valid. In terms of the worlds situation, that means in a world where you have A and ~A, you have made it impossible to prevent there from being any other (and in fact every other) formula to also be in that world. That might be the idea behind Routley and Routley’s claim that in classical logic “there is no world where both A and ~A hold but B does not”. But probably not, because it is just my best guess for the time being. So please check the quotation below.]

Under weak, and relatively noncontroversial, conditions on other connectives (~, ∧, ∨), the explosion model model delivers classical negation, according to which negation ~, is evaluated according to the classical semantical rule

I(A, a) = 1 iffI(A, a) ≠ 1, i.e. ~A holds at world a iff A does not hold at a, for every world a (deployed in the semantic evaluation of entailment). (Under alternative conditions the model yields intuitionistic negation.) Conversely, classical negation, i.e. negation conforming to the classical rule, yields the explosion view, since there is no world where both A and ~A hold but B does not, whence, on the semantical theory,^{4 }A&~A→B. Thus under weak conditions the explosion view is that of classical negation.(206)

4. Assumed, for the time being, at least, is a metatheory for the semantics that can be interpreted classically: cf. RLR, 3.2.

(228, note, in my pdf copy, the

Iin the first “I(A, a)” is not legible. It might actually be something else. I am just guessing at what it is here. I also added a “~” at the beginning of the sentence “i.e. ~A holds at world a iff A does not hold at a, for every world a” even though it does not show in my copy. That might also be wrong.)

[The Total Exclusion in Classical Negation.]

[Under the classical view of negation, we understand 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.]

[Classical negation’s model is wholly exclusive: “Classical negation offers a complete exclusion model of negation, more precisely, an exclusion and exhaustion view” (207). In some world, we would understand ~A to cover all formulas that are not covered by A. (The authors refer to a text by John Hospers when making a diagram for this. Let us take a look at the cited text (with some preceding parts for context:

1. The Law of Identity: A is A.

2. The Law of Non-contradiction: Nothing can be both A and not-A.

[...]

“A is A” does not tell you what the properties of A are, whether the A in question is round or heavy or soft; it just tells you that A is A, that the thing is itself. It doesn’t tell you whether A lasts a long time or, like a flash of lightning, exists for an instant and is gone forever.

(Hospers 1967 [2006] p.209, boldface and underlining mine)

Nor does the Law of Non-contradiction say anything specific about anything in the world: it says that if this is a table, it isn’t also

nota table; and that if snow is white, it isn’t alsonotwhite. The Law of Non-contradiction tells us that it can’t beboth;[...]

no matter what it is you are talking about, the thing you are talking about is itself, namely, the thing you are talking about (A is A). It is not both itself and not-itself, nor does it have a property and not have it (not both A and not-A);

(Hospers 1967 [2006] p.210, boldface and underlining mine)

“Everything you say presupposes that A is A: if you speak of a table, you are presupposing that the table is a table; if the table were

nota table, what could you even be talking about? A table or not a table? Or again: we defined analytic statements, in one sense at least, with reference to the Law of Non-contradiction. If you say that a square is a circle, you are in effect saying that the figure is both four-sided and not four-sided, which is to contradict yourself. That is, you are violating the Law of Non-contradiction, which says that nothing can be both A and not-A (both four-sided and not four-sided).

[...] suppose that someone were to deny the Law of Non-contradiction. “This is a table and also not a table,” he says, thus saying that it can be both (whereas the Law of Non-contradiction says that it can’t). What could we say to him? Could we even understand what he meant? What situation is he trying to describe?

“This is a table, and it also isn’t a table? But you already said in the first part of your statement that it

isa table – so now what do you mean in the latter part of the same statement by saying that it isnota table? Is it a table that you’re talking about, or isn’t it?”“I’m saying simply that it is, and is not, a table.”

“But in calling it a table, and then denying it in the same breath, you are contradicting yourself.”

“O.K., so I’m contradicting myself. What’s wrong with that?”

“What’s wrong with it is that what you say is

unintelligible: if it’s a table you’re talking about, then it isn’t alsonota table. If you say it’s both, whatareyou talking about?”“I’m sorry if you can’t understand it; it may be unintelligible to you, but it isn’t to me.”

“But please explain to me

what you meanby saying it’s both a table and not a table; first you say it is, then you say it isn’t, so whatisit you are talking about? A or not-A?”“Both. The thing I am talking about is A and not-A.”

What has happened here? Why has he passed beyond the pale of significant discourse?

Out of all the infinite array of things in the universe (the large circle), he picks out one (small circle), which we call A. Everything other than this is called not-A. If he claims to be talking about A, then it is something in the small circle; if he is talking about something other than A, it is something in the large circle (exclusive of A).When he says that the thing he is talking about is both A and not-A, he is in effect saying that it is in the small circle and also not in it (outside it). Thus he is contradicting himself.What more can we say? He may admit, even boast, that he is contradicting himself, but he goes on talking anyway. But about what?About A? Or about other-than-A (not-A)?There we are again – A or not-A. | We may not have stopped him from talking further, but we are sure thatif he talks about anything, A, that excludes not-A (everything other than A),and that to the extent that he himself is able to talk about anything, he too must obey the Law of Non-contradiction whether be knows it or not, whether he denies it or not.When he says A, he implicitly denies not-A; and when he thinks A, he cannot think it also as not-A.Otherwise, once again, what is it that he is thinking about?(Hospers 1967 [2006] p.211-212, boldface mine)

[...]

If you mean what

we now mean by “not,”then there is no alternative to the Law of Non-contradiction. We meanto exclude all alternatives by saying that “not-A” will cover all theterritoryother than what is covered by “A.”(Hospers 1967 [2006] p.222, boldface and underlining mine)

So here, the A can be either an object or a predicate (from p.210: “the thing you are talking about is itself, namely, the thing you are talking about (A is A). It is not both itself and not-itself, nor does it have a property and not have it (not both A and not-A)”.) But really A would seem to be a statement of identity (from p.211: “This is a table and also not a table”.) Or otherwise it is a predicating statement, like ‘snow is white’ (from p. 210: “the Law of Non-contradiction [...] says that if this is a table, it isn’t also *not* a table; and that if snow is white, it isn’t also *not* white.”). But with the terrain idea, if would seem that ~A would include for example all other identity statements (it is a chair, etc.), but not all statements whatever (snow is white). This is an ambiguity in the text. The way Routley and Routley seem to be implementing this text is by taking the general idea of ~A being being whatever is outside A, but without making any further specification whatsoever. But let me quote so you can see:)]

Classical negation offers a complete exclusion model of negation, more precisely, an exclusion and exhaustion view: for each world a and each statement A, ~A excludes A from holding in a, and ~A united with A exhausts a, one or other must hold in a. The picture is that commonly offered for the real world (as e.g. in Hospers, p.212) simply relativised to world a, namely

where the ellipse represents the whole of a, all statements of a. “‘Not A” will cover all territory (of a) other than what is covered by “A” (p.223).

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[Quine and Arguments for Classical Negation.]

[Many philosophers who seem to argue for classical negation really have only assumed that classical negation has already been settled as being “ordinary” negation. Quine exhibits this.]

[Some philosophers think that it is a settled matter that classical negation is “our ordinary” negation, including David Lewis, Copeland, and also Quine. Quine for example, when defending classical negation, really in the end only assumes it rather than argues for it. And there are a number of other problems with how he tries to argue for classical negation and against paraconsistent negation. (See the text to follow, but the main idea seems to be that he never explains why classical negation is better but only that after adopting it, we would fine things problematic with the reasoning of those who do not adopt it, which should not be too surprising:)]

Quine and many others (e. g. D. Lewis, Copeland) think that classical negation is “our ordinary” negation and that there is no alternative to it, for any alternative would ‘change the subject’ from negation. Of course they never

arguethat it is our ordinary negation; they simplyassumethat it is. So it is in Quine’s main defence of classical negation, which occurs in a famous passage in ‘Deviant Logics’ (p.81) where he considers two parties, α and β say, who proceed as follows: α, adopting a ‘popular extravagance’, rejects the law of non-contradiction and accepts A and ~A occasionally. β objects that this ‘would vitiate all science’ and uses the paradoxes to show that everything would follow so ‘forfeiting all distinctions between true and false’. Party α tries to ‘stave’ this off by ‘compensatory adjustments’, by rigging the logic so as to isolate contradictions (in good paraconsistent fashion). In Quine’s view,neither party knows what he is talking about. They think that they are talking about negation, ‘~’, ‘not’; but surely the notation ceased to be recognisable as negation when they took to regarding some conjunctions of the form “p. ~p ‘ as true, and stopped regarding such sentences as implying all others.

Quine’s case is however vitiated by being described in a thoroughly incoherent (indeed inconsistent) fashion; for example, party β is described as objecting that ‘everything would follow’ and as adopting what appears to be the classical view, yet Quine asserts that ‘

neitherparty knows what he is talking about’ becauseneitheradopts the classical view, having just described one of his disputants as doing so. Nor are Quine’s conclusions independently warranted. The paradoxes of strict implication are not built into the ordinary notion of negation, into the particle ‘not’. The English negation determinable ‘not’ is not so determined (as distinct from the classical negation determinate). Quine has failed to observe the distinction, and has done something which entirely begs the question at issue: equated, without any trace of argument, the natural language ‘not’ with classical negation. Thus what he goes on to | claim has no secure basis:Here evidently lies the deviant logician’s predicament: when he tries to deny the doctrine he only changes the subject (p.81).

There is no predicament: the “deviant” may be trying, with more success than the classicist, to explicate the core notion ‘not’. On Quine’s viewpoint, no distinct systems can explicate the same (preanalytic) connectives – which is a reduction to absurdity of the position. Moreover, were Quine right no “deviant” could reject the classical doctrine, he would only be changing the subject. Yet elsewhere Quine admits (and has to admit on his theory of unrestricted revisability

^{4.1}) the possibility of rejecting the doctrine (e.g. on p.84, three pages later):It is hard to face up to the rejection of anything so basic [as classical negation, etc.] If anyone questions the meaningfulness of classical negation, we are tempted to say in defense that the negation of any given closed sentence is explained thus: it is true if and only if the given sentence is not true ... However our defense here begs the question ... [since] we use the sane classical ‘not’.

Noris it the meaningfullness of classicall negation that is at issue: it is its correctness, and its uniqueness. The semantical recipe given in explanation does not separate classical negation from various other negations, e.g. the relevant negation of Π′, which [can] satisfy the same recipe. Accordingly, the recipe does not explain classical negation (without further assumptions, such as a one-world assumption), nor does it show its uniqueness.(207-208. I note that the spelling “classicall” is as such in the text).

[The Explosion Model’s Consistencizing Subtractions.]

[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.]

[**We still find subtraction-features in the explosion model of negation. In a classical world, were we to add ~A, we will need to subtract A and everything it implies in order to “consistencize” the world. In relevant logics, we can add ~A to a world with A, without needing to make any subtractions. In fact, we can combine inconsistent worlds without need of subtractions**.]

Although classical negation is not, unlike connexive negation, a subtraction operation, a taking

awayof something already given, it involves certain subtraction features. By contrast relevant negation does not involve subtraction features; ~A does not imply the taking away or elimination of A, but adds a further condition (although one related to A by certain constraints); ~A does not have entirely dependent status in the way it does classically. These differences are already reflected in the structure of the complete possible worlds of classical logic, as distinct from the worlds of relevant logic. In the classical case when ~A is added to a world, quite a bit may have to be taken out of the world, e. g. A (and what implies it) if it is there, in order to consistencize the world; whereas in the relevant case ~A can simply be added without any consistencizing subtractions. More generally, worlds can be simply combined and statements added to worlds without the need to delete anything,becausewhat is being added are further conditions, not the taking away of conditions already given. This is the route to a straightforward, and relevant, theory of counterfactuals (in sharp contrast to the irrelevant classically-based theories which presently dominate the literature^{5 }).

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5. Where the classical account is not the crude (but persistent) view that the material-conditional represents ‘if ... then’, it is a classical-based modal theory in the fashion of Stalnaker and others (see, especially, Harper, Pearce and Stalnaker). Both types are criticised in RLR, where the rudiments of an alternative relevant account are also presented. The philosophical basis of the relevant account is explained in Routley (unpublished).

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[__Theory 3__: The Paraconsistent Model.]

[The the third sort of negation 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).]

[(The third theory of negation is what we might call the *relevant model*, but I would prefer the *paraconsistent model*. The exact name is not stated however in this paragraph.) Here ~A does not cancel A, thereby making their conjunction have no content, and it does not explode A, thereby making their conjunction have all content. Rather, ~A “constrains” A, but it does not totally control it. (I do not know exactly what constrains means, and yet it seems to be critically important here. My guess is the following. We spoke of the territories of propositions in a world. We say that we have both A and ~A. But we do not want to say they are identical. So ~A will still need to contain things not in A. So there must be some degree of exclusion between them.) This sort of negation can be taken up from different positions, including that of relevance logics. This is in fact a reasonable and intuitive notion of negation, because it gives contradictions the same inferential powers as other statements, namely, it allows for some things to be inferred but not all things.]

3.On the third part of the trichotomous classification, ~A neither cancels nor explodes A, rather ~Aconstrainsbut does not totally control A. This allows for different positions, including one which will be of especial concern in what follows, namely relevant negation. Equally as “natural” as the cancellation model, and much more natural than the explosion view of contradiction is the relevant model, according to which contradictions have exactly the same sort of inferential status as other types of propositions, that is, they imply some propositions and fail to imply other propositions and are subject to the same laws.(209)

[Relevant Negation and Opposite Worlds.]

[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.]

[In our semantic definitions for the negation operator, we have been using the idea of worlds. We have been speaking primarily of world *a *and proposition *A *that holds in world *a*. To understand relevant negation, we need to apply a function to world *a*, and that function is symbolized with an asterisk coming after the world name, so for example, *a**. (Recall this notion from Priest’s *Introduction to Non-Classical Logic*, section 8.5.3:

Formally, a Routley interpretation is a structure ⟨

W, ∗,v⟩, whereWis a set of worlds,^{∗}is a function from worlds to worlds such thatw^{∗∗}=w, andvassigns each propositional parameter either the value 1 or the value 0 at each world.vis extended to an assignment of truth values for all formulas by the conditions:

v(_{w}A∧B) = 1 ifv(_{w}A) =v(_{w}B) = 1, otherwise it is 0.

v(_{w}A∨B) = 1 ifv(_{w}A) = 1 orv(_{w}B) = 1, otherwise it is 0.

v(¬_{w}A) = 1 ifv(_{w*}A) = 0, otherwise it is 0.| Note that

v(¬_{w*}A) = 1 iffv(_{w**}A) = 0 iffv(_{w}A) = 0. In other words, given a pair of worlds,wandw*each ofAand ¬Ais 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)

) We understand this star function as yielding the “opposite” or “reverse” world of *a*. Throughout the remainder of the text, we will elaborate on this function and on the nature of the opposite worlds. We define the negation of a proposition as holding true only if its unnegated form is not true in the opposite world. This sort of negation has the following properties: {1} “A and ~A are suitably independent though nonetheless related;” {2} “A and ~A may both fail together and differently both may hold together;” and {3} A and ~A neither cancel nor implode one another.” (209) (Regarding the first point, I am very curious about their relation. It seems like the relation is mediated by the star world, but I am not sure how exactly.) (Regarding the second point, see Priest’s *Introduction to Non-Classical Logic*, section 8.5.1 and section 8.5.8.) The authors also say that the star negation rule generalizes the classical rule. I do not know what generalizes means, but I think it might mean that the rule is extended to a wider domain, in this case, between two worlds rather than one (more specifically, between the A of one world and the negation of A in another world.) In section 1.1.2 of Priest’s “Multiple Denotation, Ambiguity, and the Strange Case of the Missing Amoeba”, we quoted from Richard Grandy’s *Advanced Logic for Applications*, where he seemed to suggest that sort of an understanding of generalization:

In the last chapter we presented several systems which are essentially equivalent to first order quantification theory with identity. In this chapter we will discuss a

natural generalizationof those theories which is slightly stronger than standard quantification theory. ... The reader should be careful to note thatthe operations in this system are generalizations of those in the previous system – they are defined on a wider domain.(Grandy,

Advanced Logic for Applications,p.151, boldface mine)

Here is the Routley and Routley quotation:]

The normal semantical rule for evaluating relevant negation which again is derivable under modest conditions (see RLR, 2.9), is as follows:

I(~A,a) = 1 iffI(A,a*) ≠ 1,i.e. ~A holds in world a iff A does not hold in world a*, the opposite or reverse of a. The normal rule, which

generalisesthe classical rule, differs from the classical rule in the occurrence of function *, a function which has generated much discussion. A major objective in what follows is further explanation of the * function. It is not difficult to show that negation so evaluated has the leading properties sought, e.g. A and ~A are suitably independent though nonetheless related; A and ~A may both fail together and differently both may hold together; A and ~A neither cancel nor implode one another.(209)

[The Need for Paraconsistent Negation.]

[Since we want inconsistent and incomplete worlds, that means we should use the star negation rule.]

[The next part requires more logic knowledge than I have, but I will go through it. I am going to guess that the main idea here is that it is by determining why certain paradoxes or counter-intuitive inferences should not be rules that we arrive upon a semantics for negation that we should avoid in order to avoid these paradoxes and counter-intuitive inferences. These inferences that we want to avoid are: {1} A∧~A⇒B; and {2} C⇒D∨~D. (Priest says in section 8.6.1 of his *Introduction to Non-Classical Logic* that:

As we have seen (8.4.8 and 8.4.11), both of the following are false in

FDE:p⊢q∨ ¬q,p∧ ¬p⊢q. This is essentially because there are truth-value gaps (for the former) and truth-value gluts (for the latter). In particular, then,FDEdoes not suffer from the problem of explosion (4.8).

(Priest,

Introduction to Non-Classical Logic, p.154)

In our bracketed commentary for Priest’s *Introduction to Non-Classical Logic*, section 8.5.8, we took great pains to see how these are also not valid in Routley Star semantics.) For the first case (A∧~A⇒B), we need a world where A and not A hold, but B does not. And to avoid the second kind of inference (C⇒D∨~D), we would need a sort of negation that would allow for worlds in which there are in fact members (like C) but they are not complete, because in them neither D nor the negation of D holds. (The last line is:

With only very weak (De Morgan) conditions on negation, e.g. ~(A∧B)↔~A∨~B, the normal (star) rule is inevitable.

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I cannot figure this out. Is it saying that the explosion view or the relevant view follows from such conditions? I would think he means the relevant view. But I cannot figure out why yet. I could only guess that these conditions would lead to A∧~A⇒B or C⇒D∨~D, which we do not want and which we would prevent with the star rule. I will note a passage in Priest’s *Introduction to Non-Classical Logic* section 8.3.4 that could help us here, but I am not sure:

The other rules are also easy to remember, since ¬(A ∧ B) and ¬A∨¬B have the same truth values in

FDE, as do ¬(A ∨ B) and ¬A∧¬B, and ¬¬A and A. (De Morgan’s laws and the law of double negation, respectively.)(Priest,

Introduction to Non-Classical Logic, p.145)

]

It is also not too difficult to indicate how requisite allowance for incomplete and inconsistent worlds, both sorts of which are called for in the semantical evaluation of inference, leads to the normal rule for negation. Such was the historical route: given that the paradoxes of strict implication, (1) A∧~A⇒B; and (2) C⇒D∨~D, are indeed paradoxes and false of entailment, and that entailment (at the first degree) amounts to truth (or holding) preservation over worlds, then their semantical evaluation must allow for worlds where A and ~A (strictly A∧~A) hold but B does not, i.e. for non-trivial inconsistent worlds, and for worlds where C holds but neither D nor ~D do, i.e. for nonnull incomplete worlds.

^{6 }The classical rule has to be rejected. With only very weak (De Morgan) conditions on negation, e.g. ~(A∧B)↔~A∨~B, the normal (star) rule is inevitable.(209)

6. There are independent arguments, presented in RLR, 2, that ∧ and ∨ semantical rules do not change from orthodoxy.

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[Summary and Preview.]

[We have seen the importance of the star rule, and we will now move to other related topics.]

[So we see that the star negation rule is needed, that it is a generalization of the classical negation rule, and we will see other things later.]

To both sum-up and anticipate: the star rule may be variously seen as a generalization of the classical negation rule, as a generalization that is inevitable if inconsistent and incomplete worlds are to be symmetrically allowed for, as deriving from a general analysis of negation as a certain type of one-place connective, as a way of reducing a 4-valued picture to a two-valued one (the American plan to an Australian one), as a natural reversal operation in semantic tableaux and in worlds modellings (all these explications are featured in RLR).

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From:

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

Sources cited by the authors:

J. Hospers, *An Introduction to Philosophical Analysis*, Second edition, Routledge and Kegan Paul, London, 1967.

“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.

W.V. Quine, *Philosophy of Logic*, Prentice-Hall, Englewood Cliffs, N.J., 1970 (see chapter 6, ‘Deviant Logics’).

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

Grandy, Richard. 1979 [first published 1977]. *Advanced Logic for Applications*. Dordrecht: Reidel.

Hospers, John. 1967 [Sixteenth Indian Reprint 2006]. *An Introduction to Philosophical Analysis*, Second edition. New Delhi: Allied.

Pages available for preview at:

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

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