A Box of Delights
June 29, 2016
Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.
Russell's paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)
Is this a real problem? I believe several ways of looking at it help show it for what it is.
The thing which hits a person in the face straightaway is the notion of sets which are ‘not members of themselves’? In any in-common use of the word ‘set’ a person will believe members of any set, are set members per se because they have something in common with all the other members of the set. And that this in-common-ness is what enables the set to be labelled ‘a set’. Without any exceptions. Exceptions are those items which do not posses the in-commonness per se which enables the set to be a set; and logic tells us that from any such set these exceptions must be excluded.
So what are these ‘sets which are not members of themselves’? It’s easier to run through if we use the barber example.
The barber is definitely in the set of persons called ‘the group of barbers’. But the grounds for adding this group of barbers into the group of barbers ‘who shave only those men who do not shave themselves’ are not sound. Firstly, the barbers either do or else do not shave themselves. But the absolute limitations which the word ‘only’ enforces upon the barbers’ acts of either shaving or else not shaving men are problematic. This ‘only’ acts to exclude all men who shave themselves, and to include all men who do not shave themselves. But this collective item ‘all men’ includes the sole agency by which this ‘only’ is able to categorise men into one or the other of the two groups of ‘all men’. Yet this sole agency is also one of the subjects, one of the acted-upon, and these acted-upon collectively are subjects upon which the sole agency acts to divide them into two absolute and logically mutually exclusive groups.
And so the sole agency is also a subject among subjects, and as a subject, is called upon by the terms of the language to place himself into both of two absolute and logically mutually exclusive groups. But this agency; and the subjectivity associated with this agency; are not two separate things, but the shaving-oneself is a single act. Also, collaterally, it is a being-acted-upon. Thus it is as much as to say that a resolution to the problem might be found in making an analysis of the usage of grammatical arrangements which describe the problem.
Let us put this in our terms then. These terms being that for the sole agency, ‘to shave’ collaterally necessarily entails that sole agency as a subject ‘being shaven’; whereas for all other men, being all those subjects who are singularly and solely subjects, ‘being shaven’ is not a collateral necessity of ‘to shave’’; in that they are not agencies.
Thus the act to which the word ‘only’ refers; that ‘only’ which serves to divide absolutely and logically mutually exclusively men from men; and which applies all its force upon the sole agency which takes on the power endowed by that ‘only’ so to divide up the men thus; this endowed power cannot have force on the agency itself since the agency is the arbiter of the valuation (vis of either shaved or not shaved by the agent barber) and in this role of arbiter cannot thus estimate its own valuation, (i.e. into which category it should fall, by use only of its own arbitrational powers endowed by that ‘only’). It has to be adjudged in some other way and by an object beyond itself. Ask the teleological question: what is the value of money? One cannot answer in terms of dollars or pounds because something beyond money is required to stand the idea of money up against, so as for its value to be measured and weighed, and a judgement made. Ask what category should the barber go into; the shaven or the not shaven; and one cannot answer because the criteria for judging which are embedded in the given situation are applicable solely to subjects who are solely subjects, but not to those subjects who are simultaneously and identically also agents or objects – i.e. the barber who is the actor and also the acted-upon; without making these two views of the same thing a differentiation into two events.
To get to an answer to the dilemma of categorisation of the barber one has to place the barber and his unique circumstance against a larger framework which encompasses and contains and to some extent explains his predicament; which is what this ‘answer’ of mine here is attempting to do.
To move on: The dilemma of the barber is reminiscent to that of Schrödinger’s Cat. The cat is for all practical purposes considered when in the box to be both dead and alive at the same time.
a cat imagined as being enclosed in a box with a radioactive source and a poison that will be released when the source (unpredictably) emits radiation, the cat being considered (according to quantum mechanics) to be simultaneously both dead and alive until the box is opened and the cat observed.)
Can we not do something with this idea so as to help unravel the dilemma of the barber’s shave? Can he not be said to be shaven by himself yet not shaven by himself? No? This is a contradiction, which defies the laws of logic? Is Schrödinger’s Cat of the same order as the barber – or something of a different kind?
Death and life are not opposites in the way that shaven and not shaven are opposites. Death and life are biological states; albeit mutually exclusive; but only empirically perhaps and maybe not necessarily logically? Perhaps this is the point Schrödinger was making when he devised the thought experiment?
That is, that the cat logically can be looked at as both dead and alive? Certainly to do so helps physicists to get answers to their sums; and their sums provide us working gadgetry which we find so astoundingly good to use.
David Bowie sang: ‘Knowledge comes with death’s release’ and The Moody Blues sang; ‘And the only truth we know comes so easily’. For humans, maybe for animals too, there is something, at least psychologically, absolute about death which puts it on a par in our minds with absolutes like ‘Yes’ and ‘No’ and with contrasts (contradictions) like ‘x’ and ‘not x’.
Is this merely our subjective apprehensions at work; so that death’s threat of possible utter oblivion – of ‘not x’ where once was ‘x’ – represents just so starkly an utter difference of state for us that we cannot but feel and think and believe death to be other than an absolute, and therefore opposable by an opposite? The clue is no doubt in the apprehension of our loss of selfhood; of all that make us individual and unique and a moving thinking purposive agency. To imagine ourselves otherwise than alive as we are is not possible for us. The best we can visualise is the world without us. Nothing, if death is a ticket to nothingness for us, is just infeasible.
Justly I believe we believe thus. Death is unthinkable in every sense. Shakespeare calls it ‘that bourne from which no traveller returns’ – nice here the word ‘bourne’ – which may be more significant (as a pun) concerning death than most of us can bring ourselves to hope for? J M Barrie in his Peter Pan says: ‘To die will be an awfully big adventure’.
We look upon ourselves as spirits, even though we might deny the fact to ourselves consciously, and even violently gainsay my contention (that we are spirits and) that we look upon ourselves as such. You may feel I am getting into the realms of the incredible right now; but you are already there in it with me. The world, being in the world, being an autonomous agency in the world is all fantastical; some days when I think about it I can hardly grasp it as an actuality; a real and palpable thing that I am here and we are here and everything is here. To be blasé and offhand about what is around us and what is going on to ‘make it so’ (as Captain Kirk used to say); is to have lost one’s sense of hearing that hears the Christmas decorative bell ringing in Tom Hanks’ movie ‘The Polar Express’.
You can get it back. Ponder these things.