## Sunday, 6 November 2016

### Graham Priest's Dialetheic Logic: Negation, Consistency, and the World

Dialetheism isn't a formal logic. Thus, it that respect, this piece may well be barking up the wrong tree most of the time. Dialetheism, instead, it's argued that it's "a thesis about truth". It's this emphasis which leads dialetheists to construct a logic which deals with truth. (Or, at the least, with truth as it's thrown up primarily by various paradoxes in set theory and elsewhere.) Indeed Graham Priest himself said that "the whole point of the dialetheic solution to the semantic paradoxes is to get rid of the distinction between object language and meta-language". (Clearly, this is a Tarskian focus on the nature of truth.) Nonetheless, in what follows I question what amounts to the truth-for-dialetheism problem by asking questions about dialetheism's relation to the world (or Priest's “reality”).

There's also a problem with the Liar Paradox as it's expressed by some dialetheists. Such formulations of the Liar Paradox are used to justify dialetheism; or, at the least, to justify a dialetheic logic.

¬Boolean Negation

Graham Priest makes the claim that “[e]ven dialetheists, after all, need to show that they don't accept 1 = 0” (660). On the surface it would seem that if dialetheists don't – or can never - accept 1 = 0, then neither must they accept ¬A A. This isn't the case simply because one proposition involves numbers and the equality sign, and the other doesn't (though that may be part of it).

Priest clarifies this 1 = 0/¬A and A situation when he tackles Boolean negation.

According to Priest “if not-A is compatible with A, then asserting not-A cannot constitute a denial” (660). A classical logician will be dumbfounded by the claim that “¬A is compatible with A”. However, this is because dialetheists have their own take on this. Priest explains matters. What he says, at first sight, seems to make sense. If we “deny A one must assert something which is incompatible with it” (660). True. That must mean that ¬A is not incompatible with A – it must be, well, compatible with it. Thus, to Priest, the ¬A, in A ¬A, isn't an example of Boolean negation. Is it an example of any kind of negation? It's hard to say. The dialetheic ¬A, however, “cannot constitute a denial”.

So what does Priest mean by “compatible” and, therefore, “incompatible”?

Priest repeats his position by saying that “[t]o deny A is simply to assert its negation”. Thus a dialetheic ¬A isn't the same as a Boolean ¬A.

Negation and Dialetheic Consistency

Priest explains his position thus:

... we all, from time to time, discover that our views are, unwittingly, inconsistent.”

Priest continues:

A series of questions prompts us to assert both A and ¬A for some A.”

Prima facie, this may seem to be the case. Perhaps it's wise to assert both A and ¬A if both appear to be true; or if both have equal evidential, logical or philosophical weight. Nonetheless, at the end of the day we'll hope that either A or ¬A will prove to be true (or perhaps simply more acceptable). Our initial acceptance of A and ¬A, in other words, isn't a commitment to the “inconsistency of the world” (as Priest puts it). Our acceptance of A and ¬A tells us more about us than it does about the world or even about logic. After all, there may be equal evidence for the propositions “Jones was shot” and “Jones killed himself”. Nonetheless, it couldn't have been the case that he was murdered and he killed himself.

So let Priest continue. He asks us this question: “Is the second assertion [¬A] a denial of A?” Yes, it seems so. Priest disagrees. Priest finishes off by saying that ¬A “is conveying the information that one accepts ¬A, not that one does not accept A”. In terms of classical logic only, this seems false. However, if we return to the two propositions mentioned earlier - then, yes, my acceptance of the proposition “Jones killed himself” doesn't mean that I must also deny the proposition “Jones was shot”. In tandem with my remarks about equal evidential or logical/philosophical weight, my acceptance of “Jones killed himself” doesn't mean that I will - or that I must – also deny the proposition that “Jones was shot”.

I've cheated a little here in order to make things simpler. For a start, propositions aren't usually symbolised with capital letters like A or B. My two propositions, in the tradition, should be symbolised by the letters p and q. Thus perhaps A and its negation aren't propositions. They may be truth conditions (or simply conditions). That is, the conditions Jones being shot and Jones killing himself. (It can still be argued that those two conditions are still expressed - and even caught - by propositions and, indeed, ultimately by sentences.)

In addition, the proposition “Jones killed himself” isn't a strict (or clear-cut) negation of “Jones was shot” (i.e., not a Boolean negation). A strict and clear-cut negation of “Jones was shot” would be “It's not the case that Jones was shot”, not “Jones killed himself”. “Jones killed himself” can be seen as some sort of denial - or rejection - of the other proposition; though it's not a strict (Boolean) negation. Perhaps that explains Priest's position of rejecting Boolean negation as well as accepting dialetheic... what?

In other words, if the statement/condition A ¬A is seen as including only the autonym A (a self-referential symbol without content), then clearly that statement can't be accepted. Only they aren't autonyms in Priest's book. They are the dialetheic acceptance of “contradictories” (with content?). Does that help?

Consistency and the World

Bryson Brown (in his paper 'On Paraconsistency') also states the importance of consistency (actually, of inconsistency) for dialetheism. (Informatively, he also says that dialetheists are “radical paraconsistentists”.)1 He writes:

[Dialetheists] hold that the world is inconsistent, and aim at a general logic that goes beyond all the consistency constraints of classical logic.” (628)

Deriving the notion of an inconsistent world from psychologistic and/or epistemological limitations (perhaps also from accepted notions in the philosophy of science and mathematics), is, I think, problematic. In other words, the epistemological conclusion of an inconsistent world can't also be applied to the logical and/or ontological world itself. This statement, however, may well be deemed to be a crudely or naively realist position. Then again, if one self-consciously “goes beyond all [ ] consistency”, then that surely implies that, in their heart of hearts, dialetheists know that the (logical/ontological) world is still consistent. (Though what of quantum mechanics?)

Another way to put this is in terms of set-theoretic paradoxes, as also mentioned by Bryson Brown. Brown says that “the dialetheists take paradoxes such as the liar and the paradoxes of naïve set theory at face value” (632).2 That is, it may be the case that dialetheists choose - for logical and/or philosophical reasons - to accept paradoxes even though they also believe that, ultimately, they aren't true of the world. Then again, Brown continues by saying that dialetheists “view these paradoxes as proofs that certain inconsistencies are true”. True of what? True only of the paradoxes (in themselves) or true of the world?

Again, this stress on the world may betray a naïve, crude and, perhaps, an old-fashioned view of logic. Nonetheless, Priest himself does mention the world (or “reality”) when he talks of consistency and inconsistency. When discussing the virtue of simplicity, for example, he asks the following question:

If there is some reason for supposing that reality is, quite generally, very consistent – say some sort of transcendental argument – then inconsistency is clearly a negative criterion. If not, then perhaps not.”  (662)

Perhaps here I have to admit to what may be a blatant misunderstanding of Priest's position. This again concerns the world; or, as Priest himself puts it, it concerns “reality”. As it is, I can't see how the world can be either inconsistent or consistent. My position on this is similar – or parallel – to Baruch Spinoza's philosophical point that the world can only, well, be. (Isn't Graham Priest a Buddhist?) Thus:

I would warn you that I do not attribute to nature either beauty or deformity, order or confusion. Only in relation to our imagination can things be called beautiful or ugly, well-ordered or confused.”

Here, yet again, I may be betraying my realism. That is, what we say about the world (whether in science, philosophy, mathematics or logic) may well be consistent or inconsistent (we may also say, with Spinoza, that the world is “beautiful” or “ugly”). However, the world itself can neither be consistent nor inconsistent. Yet the strange thing is that I've a lot of sympathy for certain brands of anti-realism in metaphysics and science. Despite that, here within Priest's logical and dialetheic context, claims of the world's consistency or inconsistency don't seem to make sense. Thus, it seems to follow, that inconsistency is neither a “negative criterion” nor a positive criterion. The only way out of this, as far as I can see, is if Priest squares his dialetheic logic with findings and/or positions in metaphysics and science; which, in a sense, he does do from time to time.

I mentioned science a moment ago, and perhaps it is science (well, physics) that's coming to Priest's rescue here.

There's a lot of talk of simplicity and consistency (along with other positive criteria) when it comes to scientific theories. Priest too, without actually mentioning science, also says that “simplicity and consistency may well pull in opposite directions” (663). More specifically, “a high degree of simplicity may outweigh a low degree of inconsistency”. Thus, if this is applied to scientific theories (or, more simply, to theories), then it must also apply to the logic/s of such theories. Priest must therefore believe that dialetheic logic does a good job of capturing these movements in “opposite directions”. Nonetheless, stressing simplicity (at the expense of consistency), or consistency (at the expense of simplicity), doesn't seem to entail - or even imply - an inconsistent reality or even any specifically dialetheic manoeuvres. Or, alternatively, there's nothing hidden in these positions which hints at the fact that it may be “rational to accept a contradiction” (663). Nonetheless, Priest does goes on to say that “there is nothing to stop the person accepting both their original view and the objection put to it, which is inconsistent with it” (663). As pointed out elsewhere, talk of acceptance and non-acceptance seems psychologistic or epistemological in character, not logical and/or ontological. In other words, the predicaments of our epistemological and psychological positions shouldn't be read into the world.

Finally, Priest's point about a “transcendental argument” is apt because it can be said that in order for the world to be fully understood (in, say, a Kantian manner) it would also need to be consistent. However, Priest could either say that reality isn't fully understood or that such a transcendental argument isn't needed. That may be the case because we can happily accept that the world is inconsistent without engendering too many problems (i.e., by utilising dialetheic logic!).

The Dialetheic Use of Possible Worlds

It's difficult to tell how much Priest depends on the notion – or existence! - of possible worlds to justify dialetheism. (As he may also depend on quantum mechanics – see my 'Is Graham Priest's Dialetheism a Logic of Quantum Mechanics?'.) Take this example of possible-worlds talk:

It might be thought that the fact that ¬(A ¬A) holds at a world entails that one or other of A and ¬A fails; but this does not necessarily follow.”  (654)

Is Priest saying that ¬(A ¬A) holds at the actual world (i.e., our world); though not “necessarily” at all possible worlds? Indeed considering the aforementioned views on negation and inconsistency, it may hold at any – or every - possible world! In other possible worlds, Priest believes that A ¬A can hold because of his views on both negation and consistency. This means that the dialetheic A¬A is not necessarily false.

To go back to possible worlds, Priest offers us the following symbolisation of his position:

¬A is true at w iff is A false at w.
¬A is false at w iff A is true at w.

According to Priest, “it is possible for A to be both true and false at a world” (654). That, of course, is the dialetheic position. Does it require possible worlds when Priest, elsewhere, has already said that it's also applicable at our world – the actual world?

Not surprisingly, Priest concedes that “it is natural to ask whether there really are possible worlds at which something may be both true and false” (654). Priest thinks that this is a “fair question”. Nonetheless, he also thinks that “the best reasons for thinking this to be possible are also reasons for thinking it to be actual”. That seems to follow from possible-worlds logic. That is, since possible worlds are infinite in number, we can argue the following:

If it's possible for A¬A to be true at at least one possible world,
then, with an infinite amount of worlds,
it's also likely to be true at an actual world.

If we move away from from scientific and set-theoretical reasons for embracing dialetheism, the acceptance of ¬A A can also be justified by the claim - which some (or all) dialetheists have made - that logic can't prove anything outside itself (as it were). That's because anything is possible at a possible world. (Yes; though what about the actual world?) In terms of the epistemology of dialetheism (which was mentioned earlier), dialetheists also subscribe to the traditional (Cartesian) view that nothing is certain except our own experiences and mental states. (This is an extremely controversial claim.) The question is, then:

Is the acceptance of contradictories an acceptable (or justifiable) consequence of all these dialetheic assumptions – even if they're true?

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Notes

1 Graham Priest does make an (or even the) important distinction between paraconsistent logic and dialetheic logic. In terms of paraconsistent logic, Priest states that contradictories

may [be] set [ ] together in possible worlds, to provide paraconsistent logics, logics which allow for the sensible handling of inconsistent information and theories” (663).

In terms of dialetheism, logics

may set contradictories together in the actual world, to allow for things such as a simple and natural theory of truth”.

2 Bryson Brown puts the position of dialetheism in terms of the Liar Paradox. This paradox can be formulated in this manner:

L: This sentence is false.

Through a convoluted logical procedure (which is utterly convincing if we take L as it stands), Bryson comes to the conclusion that “L is true if and only if L is false” (632). From there he also concludes that “it follows that L must be both”. Of course accepting the nature of the Lair Paradox (in any of its forms) isn't the same as accepting the dialetheic solution. But let's leave that there.

My own problem with with L itself is that it has no content. And if it has no content, then that may be partly - or even wholly - responsible for the paradox itself.

Nonetheless, this seems like a digression. In any case, it can happily be admitted that most logicians and philosophers disagree with my fixation on L's semantic content; though some do agree with it. The other thing is that other formulations of the Liar Paradox - such as “Everything the Liar says is false” and "Everything I say is false” - seem less problematic from the point of view of semantic content.

(See my '(A) The sentence A is not true' for more on this.)

References

Brown, Bryson. (2007) 'On Paraconsistency'.
Spinoza, Baruch. (1677) Letters.
Priest, Graham. (2002) 'Logicians setting together contradictories: A perspective on relevance, paraconsistency, and dialetheism'.