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Buddhism, Critical Thinking, Middle Way Philosophy, Philosophy

The trouble with paradox

God is both everywhere in the universe and beyond it. I am both the same and different from how I was two minutes ago. The set containing all sets that are not members of themselves is a member of itself. All of these are paradoxes – apparent contradictions that may possibly allow of solutions when considered more closely, or possibly just point to a higher mystery. So they say.

I, however, am a paradox sceptic. I think that paradoxes are the result of unnecessary dualistic thinking – by which I mean not the way we represent the world in terms of subjects and objects, but the belief that this kind of representation is capable of being true or false. Paradoxical statements reveal incoherence in the way our language represents the world: but this is only a problem if you expect that it is capable of representing the world in the first place. So, if you “solve” a paradox all you are doing is smoothing out a ruffle in your representation of the world – making it coherent again. If, on the other hand, you worship a paradox as a source of higher mystery, the paradox itself, if you are not rather careful, can easily take the place of the mystery. Paradoxes may show us a general point about the inadequacy of our understanding of things, but no specific paradox is any more significant than that. Just because a form of words is odd and contradictory doesn’t necessarily make it more profound than any other form of words.

My objection to paradox, then, is due to the special status that paradox is given, whether in maths or theology or elsewhere. If we think that this particular paradox is special, either in the sense of a problem or a profundity, we miss the point that it is no more special, no more contradictory, and no more mysterious than any other set of words. “Fred walked down the street” is just as contradictory and mysterious as “God is a person who is infinite and eternal.” For the boundaries of Fred in space and time are just as indistinct as God’s, the action of walking as echoingly archetypal and potentially infinite in meaning as God’s existence, and the street potentially an endless journey as infinite as God. How can Fred, an immortal soul and male archetype, do anything as transient as walking down a street? How can a street, that could be endlessly divided (see Zeno’s paradox), ever come to and end, and how could Fred, who traverses this infinitely divisible space, ever come to the end of it? All we need to do is play with it a little to turn any statement into a paradox – so paradoxes are not as smart and special as they think they are.

There are not certain things that we can talk about (and by implication represent fully, and be certain of), and other ‘ineffable’ things that we can’t talk about without contradicting ourselves. All things are ineffable in the sense that we will never know whether our representations are adequate or not, but not in the sense that we can be sure that they are not. It is not just talking about God that is contradictory, it is talking about anything. It is not just sets of sets that are not members of themselves that are problematic: all sets are problematic, because they have to be applied in the context of an indeterminate experience. They are sets of “things” in the complex context of our experience of what “things” mean to us, both cognitively and emotionally. Paradox as a special problem arises only from the false isolation of merely cognitive meaning.

We need to stop being irrationally selective in our scepticism, and face the scepticism of representation as the most basic form of scepticism: we can be no more certain what a word means, in the complexity of the real experience in which meaning resounds, than we can of anything else (see my earlier post on What is a sceptic?). However this scepticism need not be feared as a signal for nihilistic despair, for it is not a denial of anything and shows nothing to be false: rather it is a signal for provisionality, not just of belief but even of meaning.

One way to avoid paradoxes that at first seem to be built into our way of thinking is to incrementalise. That is, take the apparently contradictory absolutes and turn them into incremental qualities. This procedure works, for example, in the paradoxes that surround the idea of a mind. A mind seems to be a brain, which is situated in time and space and has all the attributes of other physical objects; but from the inside, it has no spatial location and only an internal view of time. It apparently has no measurements, no mass or energy, or other physical attributes. We will run into this paradox as long as we think of the mind (or the brain) as an absolute quantity. But if we enquire more carefully into what we actually experience in mind or brain, it is incremental qualities: a degree of awareness, a degree of rationality, a degree of integration, a degree of experienced temporality, spatiality, mass etc. An increment cannot contradict: it can only be compared with other increments. Incrementalisation seems like the ideal tool to defuse paradoxes where necessary. (For more on this see A Theory of Moral Objectivity 6b).

The history of Buddhism more than anything seems to trace the unfortunate effects of getting too hooked on paradoxes. Recognising that reality is mysterious because it is beyond our representation, a man called Siddhartha is believed to have discovered a profound paradoxical truth. This paradoxical truth is worshipped as the essence of enlightenment. As Buddhism evolves, new schools (such as the Mahayana, and then particularly Zen) realise the limitations of this perspective, and recognise that the paradoxes are all around us in everyday experience. The ko’ans illustrate this: “What is Zen? A dead cat.” But now it is the paradoxes of everyday experience that are reified into special contradictory truth, seen by the masters and missed by the ignorant. Zen becomes profoundly absolute and non-incremental, stressing sudden insight into everyday things and endlessly playing with paradoxes about its own claims. The effect of this play, though, disappointingly, is often just to reinforce the claims as special. We still become focused on the paradoxes themselves rather than the mystery that was there all along and did not actually depend in any way on the paradox.

The solution for Buddhism, as for any other tradition, I think, is to train much more deeply in scepticism: to apply that scepticism thoroughly, not as denial, but as agnosticism and provisionality. There is both a philosophical and a psychological aspect of this provisionality: it means working with our mental states as well as thinking. And we need to let go of paradoxes. Paradoxes are no more significant than anything else – which is, of course, highly significant.

Related links on the main Middle Way Philosophy site:

Meaning (concept article)

The unhelpful assumption of meaning as representation

The Middle Way and God

The Middle Way and logic


Picture: Lightmatter tortoise by Aaron Logan (Wikimedia Commons): picture only is freely reproducible under Creative Commons licence provided it is attributed.




About Robert M Ellis

I am an independent philosopher, with a Ph.D. in Philosophy, and a distance tutor in Critical Thinking, Philosophy and Politics. I also have experience of Buddhist practice. I developed Middle Way Philosophy to apply what I see as the central insights of Buddhism in an entirely Western context.


5 thoughts on “The trouble with paradox

  1. Bravo, I love it! I’ve always thought paradox was something special about my way of thinking, it pointed to some greater mystery, etc. Now I realise I’ve been putting put my capacity for imaginative wordplay on a bit of a pedestal. Well, Oops!, it just fell off – not a great distance, but it’s not going back on a pedestal, and I’ll think of another use for it (the pedestal).

    Posted by bagmanhattan | November 3, 2012, 1:03 am
  2. Hi Robert,

    These are issues I find particularly engaging, so it’s great to hear your reflections on them. To the extent that I’m confident I’ve understood you, however, I’m confident that we disagree. I think that a great deal hangs upon what we say about the paradoxes, and that some of them may compel deep and wide revisions to – or indeed, abandonment of – cherished and foundational commitments. If anything lies within the proper remit of the philosopher, I increasingly think, it is the resolution of the paradoxes.

    Moreover, I maintain that not all paradoxes are born equal: the import of Russell’s paradox is very different to that of Zeno’s, and accordingly it is customary to speak not of the paradoxes generally (though I have followed your lead in this regard), but to speak of the semantic paradoxes, the set-theoretic paradoxes, the epistemic paradoxes, and so on. Paradoxes are also unequal in respect of solubility: the ‘paradox’ with which you begin your piece is very straightforwardly evaded. You can guess the evasive strategy I favour, but since this is likely to strike you as glib, I offer as further cases in point the problem of divine foreknowledge, which is soluble on classically theistic assumptions, and the paradoxes of fatalism. Despite being responsible for the spilling of much ink, these paradoxes just rest on fallacious reasoning (the fallacy in question is a scope equivocation concerning modal terms – substantiation on request). On the other hand, the Sorities and the Liar seem obstinate in the extreme. The intractability of the Liar, for example, partly consists in its proclivity to revenge: purported resolutions often fall prey to a new, ‘strengthened’ version of the Liar, formulated in their own terms. ‘This sentence is false’ really is – to use your terms – more special, more contradictory, and more mysterious than, say, ‘Carbon monoxide contains oxygen’. The latter is just true, I submit (knowing that this is likely to be a point of divergence), while the simplest and most natural thing to say about the Liar sentence is that it is *both* true and false – the argument for this verdict is familiar. But this is in direct contravention of that most venerable principle of classical logic: the law of non-contradiction. You maintain elsewhere that classical logic doesn’t require revision: the Liar appears to force just such revision.

    The vengeful Liar also illustrates, I think, why your own prescription for the afflcitions of paradox – incrementalisation – won’t offer unproblematic diffusion of the antinomies; or, more cautiously, it illustrates why incrementalisation alone won’t work. Maybe Middle Way Philosophy does allow you to reject premises or inferences essential to the paradoxes, but if it does it won’t be solely because your theory “take(s) the apparently contradictory absolutes and turn(s) them into incremental qualities”. Here’s why I think this. One school of responses to the Liar deviate from classical logic by admitting additional truth values: some say, for example, that sentences can be ‘indeterminate’ or ‘neuter’ as well as true or false, and the Liar is one such sentence. The incrementalist version of this approach is to conceive of truth as a continuum along which sentences lie – to admit, as it were, *infinitely* many truth values, that could be modelled by the real numbers between 0 (=completely false) and 1 (=completely true). But this incrementalist solution, like others in this school, suffers from a revenge problem: consider the sentence ‘This sentence has a truth value not equal to 1’. The reasoning familar from the original Liar now gives us the paradoxical result that this sentence is both completely true (of truth value 1) and not completely true (of truth value <1). Contradiction.

    I also, unsurprisingly, disagree with your conclusions about ineffability. My own view is that reflection on the paradoxes – specifically, the semantic paradoxes – reveals that there are indeed truths, properties and relations that it is irremediably beyond the power of language to express; for the supposition that such contents *are* expressed by some actual or possible linguistic expression leads demonstrably to contradiction. Berry's paradox, Grelling's paradox, and – I suspect – the Liar itself all show this. The ineffabilia in question all concern language itself. Wittgenstein wasn't far wrong when he said that 'the inexpressible is … perhaps the background from which all I am able to express receives its meaning'. If I'm right here, this is an example of the discernibility of the paradoxes, and their discernibility from ordinary discourse: these particular paradoxes demonstrate limitations on linguistic expression where other paradoxes (e.g. Zeno's) and ordinary discourse do not.

    The special status accorded to paradox – in mathematics, logic and the philosophy of language at least – is justified by the enormous and proper infuence the antinomies have had upon these disciplines: Russell's and Cantor's paradoxes in set theory, for example, shook the very foundations of mathematics, and we do well to remember that Godel's proofs – some of the most remarkable results in logic and maths, illustrating essential limitations of the axiomatic method – are largely modelled upon one of the semantic paradoxes (Richard's paradox).

    Though we disagree about the special significance of the paradoxes in these disciplines, we are very much in agreement about the tradition of paradox in Buddhism. The mystery-mongering of Zen and the apparently willful, jubilant incoherence of texts like the Heart Sutra seem to me to muddy the waters of down-to-earth engagement with experience. It is perverse that incoherence should strike us as indicative of profundity.

    Hope this finds you well,


    Posted by Mike Price | November 4, 2012, 5:30 pm
    • Hi Mike,
      Thanks very much for your comments. The underlying point on which we disagree is that of meaning, and all the other points follow from this. You rely entirely on a cognitive understanding of meaning dependent on truth-values. For me, although this account of meaning works in its own (left-brain) terms, it only relates to half of our experience of meaning: the other half, the affective aspect of meaning, does not work in terms of truth-values and makes use of the non-representational right hemisphere of the brain. I do not recognise the right of analytic philosophers to impoverish our understanding of meaning in this way, particularly as this then has further implications for (equally impoverished) analytic epistemology and ethics.

      What I mean by saying that classical logic does not require revision is that it works in its own representational terms – as logic must. To try to stretch logic to cover non-representational, non-truth-conditional terms indeed seems impossible: and very likely just to turn out to be another self-deceptive way of appropriating the right brain to the terms of the left. It is our understanding of meaning that needs revision rather than that of logic.

      To “resolve” a paradox by detecting an equivocation in the assumptions that determine it is possibly an exercise of some value within the sphere of cognitive meaning, but when you are dealing with terms like God, which are as much (probably much more) archetypal as truth-dependent, you can only “resolve” the paradox by conveniently ignoring most of its meaning in practice for those who use it. The majority of those who worship God on a daily basis are wholly ignorant of the theological debates around the representation of the term ‘God’, so if you want to create a philosophy that actually stands a chance of being relevant to their lives, then it needs to address what God means for them (see my earlier post on ‘Jung and the meaning of God’).

      I also would not accept your version of incrementalisation, because all you have done is reduce it to truth-conditional incrementalisation and then shown how it does not conform to the truth-conditionality you assumed to start with. Incrementalisation is a strategy in the use of logic to avoid absolutisation of the two extremes, but of course we can still apply binary logic to any of the increments. So incrementalisation points to a change in psychological approach whereby we prefer to understand things in increments rather than creating a scenario that it is impossible to analyse non-incrementally. In other words, we acknowledge that the meaning of the contradictory elements of a paradox is not just a representational meaning in which we distinguish objects as present or absent, but also an affective meaning in which phenomena have more or less practical significance.

      I’m happy to concede that semantic paradoxes tell us something about the limitations of representational language. Indeed it has always seemed odd to me that analytic philosophers don’t take this point as an incentive to try a different theoretical approach. This failure goes right back to the foundations of analytic philosophy in Frege (see http://www.moralobjectivity.net/thesis4d.html) who despaired of overcoming the contradictions he encountered between logic and representation. But for me the solution seems obvious – chuck the representationalism. However, as I said in my post, any other statement also tells us just as much about the limitations of representational language when closely considered. Any statement involving assumptions about space or time raises Zeno’s paradox, and any vague term raises all the paradoxes surrounding infinity. My argument above that any statement is potentially paradoxical is not one you have responded to.

      I think we’re correct to give attention to the antinomies, if only so as to note how much we are often slaves to them. The antinomies are a side-effect of the development of our tool-using left brain, where the representational language centre is close to the tool-using centre. We need to distinguish one thing from another in a non-contradictory way so as to be able to focus on our various tasks in the world and accomplish them effectively. But even when we are busy using our tools we also needed to keep an eye open for external (or maybe internal) threats, requiring a sensitivity to stimuli that lie beyond our represented world. This openness is just as much an aspect of meaning for us. If we try to engage with the world beyond our representation, the paradoxes dissolve in the immediacy of incremental experience. Ignoring the other aspect of meaning strikes me as a bit like continuing to focus on the knapping of your flint as the tiger approaches.

      Posted by Robert M Ellis | November 4, 2012, 8:01 pm
  3. Hi Robert,

    Thanks a lot for this. You may well be right that the source of our disagreement over these issues is disagreement about the nature of meaning. I’m afraid I’m not confident that I understand your account of meaning, so I’m hestitant to affirm this diagnosis. I’m also not qualified to comment on your claims about the neurological bases of our apprehension of (kinds of) meaning. I would, though, like to come back on a couple of your remarks and to press you on a couple of points.

    You maintain that I ‘rely entirely on a cognitive understanding of meaning dependent on truth-values’. From your allegation that analytic philosophers *impoverish* meaning, I take it that you think that I (and my fellow fellow errant analysts) conceive of meaning as exhausted by the cognitive. I make no such assumption. I’m quite happy to accept that the meaning of linguistic expressions outstrips the cognitive and includes affective components. Moreover, I don’t think anything I said anything in my response that commits me to denying this. Accepting classical logic, as we both do, means (amongst other things) holding that every proposition has exactly one (classical) truth value; it doesn’t mean holding that the meaning of a proposition consists solely in its truth value or the conditions under which it has the truth value True. The classical logician can happily maintain that meaning is a much richer, emotional affair. But providing we are classical logicians we are in serious trouble with, for example, the Liar.

    So – given that the discussion has been so far quite general – could I press you to give your specific verdict on the sentence ‘this sentence is false’? I ask because just don’t see how enriching one’s account of meaning by admiting affective elements will make this sentence any less toublesome. Could you use this case to illustrate the dissolution of paradox your alternative approach affords?

    Could you also use this case to illustrate how incrementalisation diffuses paradox? I still maintain that “tak(ing) the apparently contradictory absolutes and turn(ing) them into incremental qualities” won’t suffice: I’ve given an example of an approach satisfying this description that offers no diffusion, and which shows that it’s not true that ‘an increment cannot contradict’. Perhaps you intend this description as merely schematic, and my example doesn’t count as incrementalist on the fleshed-out characterisation. But then, perhaps the flesh should be exposed as a matter of course, given that the schematic version has counterexamples.

    You might reply that you’ve already met the demand for specific application: you describe an application to issues concerning the mind-body problem, and I know you’ve written extensively on the application of non-dualism to problems elsewhere in philosophy. But I don’t think we’ve yet heard how incrementalisation diffuses the paradoxes more strictly and narrowly conceived: the ostensibly conflicting convictions we have about the mind and brain are paradoxical in a much broader and looser sense than that in which the Liar sentence and the Sorities reasoning are paradoxical.

    You weren’t satisfied that I’d responded to your argument that any statement is potentially paradoxical –
    that “if we think that this particular paradox is special, either in the sense of a problem or a profundity, we miss the point that it is no more special, no more contradictory, and no more mysterious than any other set of words”. Well, given the sheer pervasiveness of vagueness in natural language, it’s certainly true that a great many statements will be attended by their own Sorities paradox. But you’re making a very strong universal claim here. It’s not *obvious*, for example, that vagueness is utterly ubiquitous. Consider the predicate ‘… is a molecule containing oxygen’, as uttered in the chemistry classroom. Does this admit of borderline cases? Do either of us know enough analytical chemistry to make a confident judgement here? I can well believe that there just might not be borderline cases concerning the application of this predicate, and thus no attendant Sorities. But I think the more important point is one that I made in my reply – that some statements are paradoxical *in ways that other statements aren’t*. The Liar sentence is a clear example: again, I claim that it really is more special, contradictory and mysterious than your run of the mill declarative sentence.

    That’ll probably do for now.

    Left-hemispherically yours,

    Posted by Mike Price | November 9, 2012, 7:40 pm
    • Hi Mike,
      Thanks; I appreciate being kept on my toes here! I’m glad that it turns out that you agree that meaning is not purely cognitive. However, I wonder if you have taken into account the large possible implications of that concession!

      Perhaps I should clarify what usefulness I think incrementalisation has. It’s not that I think we can ‘solve’ all paradoxes by incrementalising them, because the challenges created by meaning being bigger than logic will remain. What incrementalisation can do is to help us apply our logic better in many cases, so as not to run into unnecessary paradoxes that create a barrier for understanding conditions. The mind/body issue seems to illustrate this much better than the liar paradox.

      However, that doesn’t mean that the recognition of affective meaning can’t help us to put he liar paradox into a bigger perspective. “The sentence is false” seems like the ultimate in closed, left-hemispheric, self-referentiality, but like any possible sentence uttered in practice its meaning is not limited to the representations given to it by the left hemisphere. For both of us, for example, it also has an affective meaning in this discussion: exactly what that meaning is will differ in the experience of either of us and also of anyone else who reads this discussion. For me it has a certain association with you as a person, for example, because of your special interest in it. It also has an association with the various emotional experiences associated with intellectual challenge and effort.

      The concrete context of meaning does not deprive “this sentence is false” of its merely logical paradoxicality, but it does mean that it has no further paradoxicality in terms of meaning. In the concrete context it is just an irrelevant piece of metaphysics, because it makes absolute claims in a context in which only incremental ones are admissible. In the concrete context, no claim is either wholly true or wholly false: truth and falsity can only work as regulative ideas “on the edge” rather than being applicable absolutely to any statement. One of those regulative uses, I would agree, is in making sense of logic, where the ideas of truth-value and false-value have to be used in the abstract to be able to make logical analyses. However, such logical anlyses do not give us a complete account of the meaning of any statement.

      The Sorites Paradox, on the other hand, is one that could create barriers that can be removed by incrementalisation, and incrementalisation here would just be another way of talking about recognising affective and physically-based meaning. Affective and physically based meaning is vague and involves incremental changes from one object or quality to another: so the lack of precision in terms like “heap” is only a problem in terms of cogntive meaning taken in isolation. If you give this a practical application in an issue like abortion, the vagueness of when a foetus becomes a person can be accepted in general and in the abstract rather than requiring an abstract resolution. It is only when a decision has to be made about a specific abortion that it becomes necessary to specify the boundary lines precisely and put the tool-using left-hemisphere fully in charge. At that point logic can be practically applied in a way that is maximally adequate to the actual conditions of that particular case, rather than on the basis of an abstract nd general resolution of that vagueness that we never had any non-dogmatic grounds to accept anyway. Logic started off as a practical tool – lets keep it to that sphere where it can be useful, rather than turning it into a totalising cognitive account of meaning.

      Let me also take up your contrasting example of what you claim is a non-paradoxical statement: since I’d rather deal with a whole statement than merely a predicate, I’ll go back to your earlier example of “Carbon monoxide contains oxygen”. I don’t think it’s necessary to be an expert in chemistry to see the same issues here as in the paradoxes above. If the cognitive sense of this statement does not exhaust its meaning, we must take into account its affective meaning for a chemist, or indeed anyone else. Its chief affective meaning for me is associated with its use as an example in this context, but perhaps there is also a sense of mild surprise that an element I am aware is essential to life also forms part of a molecule that kills. I guess your reason for choosing this example is that it is analytic – i.e. that the fact that it contains oxygen could be inferred from the very construction of the idea of carbon monoxide. However, this analyticity, as you already seem to have conceded, does not exhaust its meaning. Since paradox is created by the interplay between logic and broader meaning, this statement assumes that same kind of paradoxical features as other kinds of statement when given this broader meaning: not just the basis of surprise that I mentioned but also issues of vagueness and scope. Is it being claimed that everything we experience as carbon monoxide contain what we experience as oxygen, everywhere in the universe? Is it being claimed that there no borderline cases in practice, where either carbon monoxide is impure or difficult to identify, or mildly modified by interaction with other substances? These kinds of difficulties might be best dealt with through incrementalisation and provisionalisation: the statement needs to be interpreted in practice as a provisional claim about the large extent to which different kinds of observations in chemistry correlate on our experience so far.

      Posted by Robert M Ellis | November 10, 2012, 11:24 am

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