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Graham Priest
Doubt Truth To Be a Liar
Part 1 Truth
Ch.1 Aristotle on the Law of Non-Contradiction
1.13 Some Modern Variations III: Negation as Cancellation
Brief summary:
Asserted formulas have content. And we can negate and conjoin formulas. This raises questions when we conjoin to a formula its own negation, thereby creating a contradiction. What can be said about their contents? With regard to this issue, there are three accounts of the relationship between content, negation, and contradiction. {1} The cancellation account: the content of ¬α in α∧¬α cancels the content of α, thereby leaving the whole conjunction without content. Here the contradiction entails nothing, because it has no content to carry over into an inferred conclusion. This view is seen in Ancient and Medieval logic. And, we see this sort of argument in J. Lear, who takes inspiration from Aristotle’s law of non-contradiction. He argues that you cannot at one point assert S and then later assert not-S. For, by doing so, the not-S both cancels the content of S while adding no new content, thereby rendering their conjunction meaningless. {2} The complementation account: the content of ¬α includes all of the other content not contained in α, thus α∧¬α contains total content (all content whatsoever). This is the view of classical and intuitionistic logics. {3} An intermediate account: the content of ¬α is a function of the content of α, but only in such a way that α∧¬α contains partial content, being neither null nor total; thus contradictions can entail some things but not others. This is the view held in relevant and paraconsistent logics. Now, the cancellation account does not work, for a number of reasons. {A} There are borderline situations that are aptly expressed as contradictions, like, it is both raining and it is not raining. But this is not saying nothing at all. {B} Our beliefs can be contradictory. We conclude one assertion from our beliefs at one point, and at another point we conclude the negation of that assertion. The negation is not nothing more than a cancellation of the prior one. It is a belief that we have that is inconsistent with another belief that we have. And since we can continue working with inconsistent information while trying to resolve the contradiction, that cannot mean it is contentless. We see this for example when there are inconsistent scientific theories. {C} The paradox of the preface. A person writes a book. In its body, the author makes many assertions, and thus she asserts the conjunction of all of the assertions as well. But in the preface, she acknowledges that certainly at least one assertion is mistaken and is thus false. So she both asserts the conjunction of the assertions in the body while also asserting the negation of this conjunction in the preface.
Summary
1.13.1
[J. Lear argues that you cannot first assert S and then afterward assert not-S. For, were you to do so, the not-S does not constitute a second assertion in addition to the first; rather, it merely cancels the first assertion.]
[Note, this first paragraph refers to ideas stated previously in the text, but I have not summarized them yet. Here I will quote this paragraph in full, and I will fill out the summary later if I post those prior sections.]
Let us now return to Lear. The last passage of his that I quoted (attempting to give an argument for the LNC that is independent of Aristotle’s view of substance) continues:57
One cannot assert S and then directly proceed to assert not-S: one does not succeed in making a second assertion, but only in cancelling the first assertion. This argument does not depend on any theory of substance or on any theory of the internal structure or semantics of statements. It is a completely general point about the affirmation and denial of statements.
Note that this argument is quite distinct from the one given in the first part of the paragraph. That one is about an arbitrary proposition: if the LNC fails, it rules nothing out, and so is not meaningful. The argument is specifically about contradictory propositions, and is to the effect that such a proposition has no content; a fortiori, it has no true content. The argument hinges on quite specific claims about the behaviour of negation. Let us return to it in a moment, after a few appropriate background comments.58
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57. Lear (1988), 263 f.
58. These draw heavily on Routley and Routley (1985)
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1.13.2
[There are three accounts of the relationship between content, negation, and contradiction. {1} The cancellation account: the content of ¬α in α∧¬α cancels the content of α, thereby leaving the whole conjunction without content. Here contradiction entails nothing. {2} The complementation account: the content of ¬α includes all of the other content not contained in α, thus α∧¬α contains total content (all content whatsoever). {3} An intermediate account: the content of ¬α is a function of the content α, but only in such a way that α∧¬α contains partial content, being neither null nor total; thus contradictions can entail some things but not others.]
[We will be thinking in terms of the content of the formula letters, like S or α. It seems that we understand this content as being something like their meaning, perhaps given in a propositional sort of form. Priest will then consider the relationship holding between the content, negation, and contradiction. He says there are three accounts of this relationship. {1} Cancellation. When you conjoin a formula with its negation, we understand that to represent the cancellation of the first formula’s content. (Suppose we first say, “It is raining,” then secondly we say, “it is not raining”. We might think of the first sentence’s meaning being cancelled, while its negation’s content is affirmed. However, when we join them as one proposition “It is raining and it is not raining”, then we are not affirming any new content. We are understanding “It is raining and it is not raining” to have no content, even though either assertion individually would.) (I may get the next point wrong, so please consult the quotation below. We next think that an inference is valid when the content of the premises contains the content of the conclusion, but I am not exactly sure how to grasp that. I suppose all the inference rules involve the conclusion having at least one of the atomic formulas in the premises, except for ex falso quodlibet. At any rate, since a contradiction has no content, then as premises we can infer just nothing from them. Priest also says we can infer other contradictions without content. But again I do not know how that works.) {2} Complementation. Here we understand the negation of a formula as meaning all the content not found in the unnegated formula. That means α ∧ ¬α includes all content. (It also means that it entails everything. I am not sure how, but maybe it is something like the following. We assert “a certain situation holds” ((like it is raining)), then next we assert “every other situation except this one holds ((it is doing every other thing but raining)). When we conjoin them, we have asserted every possible situation, thus we can derive any one of them.) {3} Intermediate position. Here we understand the content of ¬α as a function of the content of α. But the content of α ∧ ¬α has only partial content, meaning that it entails some things but not all. (Again I will guess how to understand this. Suppose we use a sort of formulation for motion like in Priest’s In Contradiction section 11.2. So at the instant the pen lifts off the paper, it is both on and not on the paper. From this content, we can infer that the pen is at least on the paper, for example. But we cannot infer that the moon is made of green cheese, or whatever else crosses our mind.)]
One may distinguish between three accounts of the relationship between negation, contradiction, and content. (1) A cancellation account. According to this, ¬α cancels the content of α. Hence, a contradiction has no content. In particular then, supposing that an inference is valid when the content of the premises contains that of the conclusion, a contradiction entails nothing—or nothing with any content; it may entail another contradiction. (2) A complementation account. According to this, ¬α has whatever content α does not have. Hence α ∧ ¬α has total content, and entails everything. (3) An intermediate account, where the content of ¬α is a function of the content of α, but neither of the previous kinds. According to this account, α ∧ ¬α has, in general, partial content, neither null nor total. Hence, contradictions entail some things but not others.
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1.13.3
[The cancellation account is found especially in Ancient and Medieval logic. The complementation account is found in classical and intuitionistic logics. And the intermediate account is found in relevant and paraconsistent logics.]
The third account, where contradictions have partial content and entail some but not other things, “is given in relevant and paraconsistent logics” (31). The second account, where negated content is complementary and contradictions entail everything, is found in classical (orthodox modern logics) and intuitionist logic. Priest says that the first account, where negation cancels content, “appears to have been an influential account in Ancient and early Medieval logic” (31). Priest gives some examples: “Arguably, Aristotle subscribed to something like it, since he appears to have rejected the claim that α∧¬α | entails α. (See 1.4. We will have further evidence of this later.) It appears in Boethius and Abelard. It is intimately connected with principles such as ¬(α → ¬α), which are built into modern connexive logics” (31-32).
1.13.4
[Some philosophers, like Strawson, have confused the cancellation and complementation accounts.]
Priest next notes that sometimes philosophers have mistakenly taken the cancellation and complementation accounts to be the same thing. One example comes from Stawson’s Introduction to Logical Theory (1952). In one part of the book, Strawson gives the orthodox account where contradictions entail everything. But then in another part he gives the cancellation account.
Suppose a man sets out to walk to a certain place; but when he gets half way there, he turns round and comes back again. This may not be pointless. But, from the point of view of change of position, it is as if he had never set out. And so a man who contradicts himself may have succeeded in exercising his vocal chords. But from the point of view of imparting information, or communicating facts (or falsehoods) it is as if he had never opened his mouth . . . The point is that the standard function of speech, the intention to communicate something, is frustrated by self-contradiction. Contradiction is like writing something down and erasing it, or putting a line through it. A contradiction cancels itself and leaves nothing.
(32, citing Strawson 2 f.)
1.13.5
[In Lear’s argument, an assertion α normally conveys information. And also normally, any new assertion will convey additional information. However, when we add the assertion ¬α to the stock of information of α, then not only have we added no new information, we have also removed the information of α.]
We return to Lear’s argument, which is based on the cancellation account. Priest articulates it as:
Speaker’s assertions (and here, ‘assertion’ does seem the appropriate word) normally convey information. Normally, when they make a new assertion this adds to the stock of information conveyed. But when the stock contains the information α, an assertion of ¬α adds nothing, but merely removes α.
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1.13.6
[It is not obvious in Lear’s cancellation account why the negation cannot add information rather than subtract given information.]
Priest will explain why this account is “sketchy and unsatisfactory” (32). [I probably do not follow this well, so please consult the quotation below. (Suppose we obtain the following bits of information. One person tells us: “The package contains something tiny”; and another person tells us: “the package is very heavy”. These are not really negations of one another, but we also become aware that: It cannot both be that the package contains something tiny and the package is very heavy. So when we learn that the pieces of information cannot both be true, then we should be able to delete either one. But we do not know which to delete. And suppose we cannot delete both. So all three sentences we must still be assertable. We simply regard the negated conjunction formulation as adding to the information already provided by each content. Using similar reasoning, we would think that if we already have the information of α, then by adding ¬α, we are adding new information rather than subtracting it. For, suppose that we did not first assert α, but rather first asserted ¬α. That would presumably have some content on its own. Let me quote so you can see.]
Even filled out like this, the account is obviously sketchy and unsatisfactory. What does an assertion of ¬α do if α is not in the information store? It must do something; negative statements do, after all, have content. So presumably that content is merely added to the store. So why doesn’t it do this if α is already there? (Inconsistent data bases are not news.) And what happens if α and β are in the information store, and ¬(α∧β) is asserted? A natural suggestion is that we delete either α or β; but we have, in general, no way of knowing which. So presumably we can only add ¬(α ∧ β) to the store. But if we can have α, β and ¬(α ∧ β) in the store, why not α and ¬α?
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1.13.7
[The cancellation account does not work. There are borderline situations that are aptly expressed as contradictions, like, it is both raining and it is not raining. But this is not saying nothing at all. And our beliefs can be contradictory. We conclude one assertion from our beliefs at one point, and at another point we conclude the negation of that assertion. The negation is not nothing more than a cancellation of the prior one. It is a belief that we have that is inconsistent with another belief that we have.]
Priest then explains some of the greater problems with the cancellation account. {1} Borderline situations. There are certain real situations, like certain weather conditions we have experienced, that are adequately expressed as contradictions, for example, “it both is and isn’t raining” (32). [Perhaps the idea is that there are certain sorts of precipitation that are substantial enough to qualify as at least not being not raining, so we would say it is raining, but they are also insubstantial enough that we would not feel the need to affirm that it is in fact raining, so we would also say it is not raining.] Priest says that such a borderline case “shows clearly that negation does not have to function as a cancellation operator” (32). Another example are inconsistent beliefs. Suppose we examine our beliefs, and that leads us at one point to assert α, and we keep reflecting on the beliefs, and later we come to assert ¬α. This second assertion is added information about our beliefs; it does not simply do no more than cancel the prior ones.
In any case, and for quite general reasons, the cancellation account of negation doesn’t stand up to inspection. For a start, in a borderline situation of, e.g. rain, one might say that it both is and isn’t raining. This is, perhaps, something of a special case; but it shows clearly that negation does not have to function as a cancellation operator. | Or consider another sort of situation. One can, in considering one’s beliefs, come to assert contradictory statements, and in so doing discover that they are inconsistent. The assertion of¬α in this context does not “cancel out” the assertion of α—whatever this might be supposed to mean. The assertion of ¬α is providing more information about what it is one believes, not less. And it is precisely the inconsistent nature of this information that gives one pause; if what one said had no content, it would have no unsatisfactory content, so it is difficult to see why one should bother to revise one’s beliefs at all.
(32-33)
1.13.8
[There are other ways to see how the cancellation account fails. When we have inconsistent beliefs or information, we often can still make use of them in our reasoning while we try to solve the contradiction. That would be impossible if their informational contents were cancelled. We see this for example when there are inconsistent scientific theories. Another example is the paradox of the preface. In the book’s body, the author makes many assertions and thus asserts the conjunction of them all as well. But in the preface, she acknowledges that at least one assertion is mistaken and thus false. So she also asserts the negation of the conjunction of the assertions in the preface. ]
Also, often when we have inconsistent information, we continues using parts of it while still in the process of resolving the contradiction. We see this with scientific principles that are known to be inconsistent. Since we can still work with the inconsistent information, we would not say that it is cancelled out. [The next example I may get wrong, so please consult the quotation below. Suppose you write a short book, and in it you make hundreds of claims. Since you assert each of them, you assert their total conjunction, which we can maybe think of being like: (α1∧α2∧...∧αn). But we know that most likely at least one of those assertions will be false, as it is close to impossible to get so many right. To warn the reader and to protect ourselves a little from harsh, unforgiving criticism, we write a preface where we state that there are most likely some mistakes in the book, certainly one at least. But if we acknowledge that at least one is false, then we are saying that their entire conjunction is false, and thus we would be asserting (α1∧α2∧...∧αn) in the body of the text, but asserting ¬(α1∧α2∧...∧αn) in the preface. These examples show that “The account of negation as cancellation therefore fails, as does the second part of Lear’s argument” (33).]
Moreover, even if one does try to resolve the contradiction in such a situation, until one succeeds, one may well continue to use parts of the inconsistent information. (Think of scientific theories that are known to be inconsistent.) It is certainly not “cancelled out”. Consider an extreme case, the paradox of the preface. A person writes a book and thereby asserts the conjoined truth of all of the claims in it. Being aware of the overwhelming inductive evidence, they also assert that there are mistakes in the book, i.e. the denial of that conjunction. This does not cancel out the claims in the book. Indeed, in this case, it might even be argued that believing the inconsistent totality of information is the rational thing to do, something that would make no sense if bits of it cancelled out other bits.60 The account of negation as cancellation therefore fails, as does the second part of Lear’s argument.
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60. See Priest (1993a) and sect. 6.2.
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1.13.9
[We now see from the last three sections that those who took inspiration from Aristotle’s first refutation were just as unsuccessful as he was.]
Priest writes in the last paragraph:
In the last three sections of this chapter we have looked at a number of philosophers who have been inspired by Aristotle’s first refutation. In the end, none are any more successful than was Aristotle himself. Let us now return to Aristotle’s text, and to his other refutations.
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Graham Priest. 2006. Doubt Truth To Be a Liar. Oxford: Oxford University, 2006.
Also cited:
Gabbay, D. and Wansing, H. (eds.) (1999), What is Negation?,Dordrecht: Kluwer Academic Publishers.
Lear, J. (1988), ‘The Most Certain Principle of Being’, in Aristotle; the Desire to Understand, Cambridge: Cambridge University Press, sect. 6.4.
Priest, G. (1999a), ‘What not? A Defence of a Dialetheic Account of Negation’, in Gabbay and Wansing, 101–20.
Routley, R. and V. (1985), ‘Negation and Contradiction’, Rivista Colombiana de Matemáticas, 19: 201–31.
Strawson, P. (1952), Introduction to Logical Theory, London: Methuen.
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