The Legacy Of Paradox

No set of data is incoherent with itself. This includes everything from cognitive data to cosmological data. But, when one person simultaneously views a set of data completely in each of two different ways, that person gets a sense that the data conflicts with itself. This conflict is called a paradox. Illusionists, or ‘performance magicians’, often make use of the fact that a person’s brain seems able simultaneously to interpret something completely in each of two different ways.

A simple visual example of paradox is demonstrated in the image to the right. If you try to see all of it at once as representing a three-dimensional object, the image itself seems impossible. But, the image is just data. In fact, though quite difficult at first for most people, if you manage to see it as the mere two-dimensional pattern that it is, the conflict disappears.

But, paradox is not simply a phenomenon: to be taken as-is with no possibility of understanding either it or the object according to which it occurs. Rather, paradox is a legacy: the legacy of the fact that there is a host of various valid frames of reference by which a given set of data may be interpreted. The history of contentions over what to think is the ‘obvious’ facts of cosmology is a wonderful case in point. But, more about that case in a moment.

People whose interpretation of their visual experience is normalized to the environment in which they live are used to assuming that their perception of three-dimensional objects is accomplished by a simple means of viewing the object. But, actually, a person’s three-dimensional visual perception is accomplished by at least two different ways of interpreting visual data. Each of these ways normally work together for a common practical goal: to establish an accurate, or at least possible, sense of how the image their brain sees represents the three-dimensional object which they know—or, at least, expect—that it is.

But, by way of its image on our retinas, there is a lot of complexity involved in how it is that we ever correctly interpret that image as the object by which that image actually is produced. This complexity is analogous to that by which an aircraft pilot, if he is trained in instrument flying, can fly and land an aircraft without being able actually to see what’s outside his aircraft: by knowing the physical referents for, and the interactions between, the several core flight intruments. Of course, even then, he must have a correct original location. But, if he has, or else actually knows, that original location, and if he keeps constant track of his progress, then he can land his aircraft without actually seeing the ground or the landing strip.

But, just like trying to fly by only one of the several core flight intruments, three-dimensional visual perception cannot be accomplished by only one of its ways of interpreting visual data; any one of these ways cannot alone give a person a sense of three dimensions. In fact, when you manage to see the entirety of any image in only one of those ways, you recognize the actual image for what the image itself is, apart from any complex interpretation of the image: a two-dimensional pattern, just like the puzzle-piece image below.

Even the subject of ‘pure’ mathematics allows paradox. For example, when you try to fully define a multiple-element set simultaneously from each of two counting bases: in Base Ten, a three-element set, (e e e), is expressed as a single symbol, because the quantity is less than the base; whereas in Base Two (binary/1-0), that same set is expressed in a multiple-symbol unit. 

But, regardless of the subject, there may be three basic questions about the phenomenon of paradox as such. One, what makes it even possible. Two, what causes it ever to happen. Three, why would we ever want it to happen. I already suggested an answer to that third question: paradox is a legacy.

Now, the first of those three questions asks how it is possible for a single observer to get a sense that something is inherently paradoxical. We can imagine each of two persons getting a different view of the same thing. For example, they each can be standing on a different side of a third person. But, for one person simultaneously to get two complete, different, views of something, that person experiences a paradox. The image below is commonly seen in just such a way, which is what causes it to seem to shimmer and shift and, thus, to strain the eyes.

But, to answer both the first and the second of those three questions requires actually knowing what a given paradox involves. As Einstein said, ‘mathematics [including mathematical logic] can’t tell you anything about the world’. In other words, math, and mathematical logic, can’t tell you how to apply math and logic to the world: you have to know about the world in order to begin to know how to apply them to the world. For example, two musical instruments, each of which outputs four decibels, does not amount to eight decibels, but actually amounts to four decibels. You have to know what a decibel is.

Remember earlier I mentioned that I would get to the case of cosmology? I shall now lay out a little of my views on what may be The First, and Most Abiding, case of cosmological contention: the non-visual topic of cosmology called…

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…Time 

It seems to me that Einstein believed that time is not an independent dimension apart from space, but that time is a kind of measurement of change of the relative orientation of masses in space. I justify the idea that he believed this by observing that he seems to have believed that gravity is the change of the relative orientation of masses in terms of space and time.

In any case, Einstein proposed the idea of spacetime as a single, inter-related, dimension. So, Einstein’s view of physics was a radical departure from the Newtonian view of physics. The Newtonian view of physics is that space and time have no dynamic relation either to each other or to gravity. It seems Einstein wanted to see if these three dimensions could be reduced to less than three independent dimensions, because his Theory of Relativity is based on the dual demands that 1) everything that’s cosmologically real must be dynamically interrelated, and 2) everything that’s cosmologically real must avoid useless and over-complicating redundancy. I think Einstein appreciated ‘Ockham’s razor’: “Don’t allow or involve more numbers and kinds of entities than is necessary to a particular explanatory task.” And, the visual paradox of the first image on this page is a visual case of such an allowance.

As to what ‘time’ itself is, I tend to think that the sense we have that time ‘flows’ is based on a combination of analogy-to-space and the fact that we exist as a continual change/rhythm within an ever-present(-ness of spacetime). It seems to me that, as analogy-to-space, our sense of change within the ever-present is what gives us the sense that time is an empty continuous ‘flow’ not only within which change occurs, but within which objects themselves ‘endure’.

But, it seems to me that there is no way to establish how the ‘emptiness’ of this ‘empty flow’ notion of ‘time’ is connected to the ‘emptiness’ of space. In fact, it seems that the ‘things’ which ‘exist’-and-occur within such a ‘flow’ implies that this ‘flow’ is an endless procession of meaninglessly redundant, ‘empty moments’. And, this, in turn, seems to imply that these moments, despite being past or future, nevertheless somehow exist presently in all their inter-relative instantiations: past -5, past -4, past -3, past -2, past -1, present 0, future +1, future +2, etc..

So, I think that our sense of this ‘flow’ of ’empty moments’ is merely a cognitive shortcut for the purpose of referring to the actual changes we observe and experience. And, notice that one thing never actually changes: the present never stops being present.

So, the problem of conceiving ‘time’ as an empty flow of ’empty moments’, and in which changes occur in the relative orientation of objects, may be compared to the complexity and redundancies implied by maintaining that the mechanical model of the relation of the heavens to the Earth is equivalent to what a ‘quick glance’ observation of their relation seems to be from the viewpoint of someone standing on Earth: that the Earth remains motionless, and the heavenly bodies all move in a very complex ‘dance of spheres’. Every practical frame of reference is objective in terms of itself, but the point is to not get caught up in thinking that any one of them is the complete picture.

Now, on to the bulk of this page, which is about the classical ‘perfect’ liar paradox. It’s presented in three sections: The ‘arrow’, the ‘aim’, and the ‘target’.

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THE PARADOX OF THE ‘PERFECT’ LIAR

 

  

THE ARROW OF THE ‘PERFECT’ LIAR

 

In order to lie, you have to lie about something: you have to make genuine reference to something in order to misrepresent it. So, all lies are inherently unstable. This is like an X29 jet airplane, the control surfaces of which constantly must keep correcting the tendency of the jet’s wing design to flip the airplane’s direction of travel in a radically contrary direction to that given to the airplane by it’s propulsion.

But, all normal statements which are false are false not by being self-referential, but by misrepresenting a subject, either within the statement or implied to be outside the statement, which is referenced by the statement. For example, for a person who is not Abraham Lincoln to assert, ‘I am Abraham Lincoln’, actually references various objects, specifically ‘Abraham Lincoln’ and himself.

But, removing the personal element from the classical ‘Liar Paradox’ does not seem to remove the fact that the statement seems to make reference to a subject: 

 

This statement is false.

 

Taken as reference to itself, the question becomes,  ‘About what is that statement actually stating? It seems to be stating that itself is false. But, what is false about it? It sounds objective. But, if it is objective, then it seems confusing. Is it objective? Or, instead, is it like the first and last images on this page? I say it is exactly like those images, and I’m going to prove it by the end of this page.

Maybe there are things about that confusingly self-referential statement which are false, and other things about it which are true. But, if it is both partly false and partly true, then it is a little bit like me saying ‘I got four hours of sleep last year’: In matter of fact, I did get four hours of sleep last year; I just didn’t mention all the other hours of sleep I got last year. But, I definitely did get those four hours of sleep last year. 

There seems to be no way normally to make sense of the statement, ‘this statement is false’. What is the thing it is asserting to be false? I think it might be reformulated as: 

 

The assertion of fallacy is a fallacy.’

 

Or maybe as:

 

The mere idea of fallacy is an instance of lying.’

 

But, if stating merely the idea of fallacy is itself a fallacy, then a mountain is itself an avalanche. So, statements are just data. They don’t actually say anything. We interpret them as saying things. This even is how we learn to use them to express things. In other words, words don’t mean things, people mean things by words. To treat a word as if itself knows what it means is to project onto the word the fact that oneself interprets it in a certain way. But, if you try simultaneously to interpret it completely in two different ways, you get a paradox. 

So, the last two reformulations above hopefully show the average reader what I assume the average reader already knows: that the classical Liar Paradox is of absurdly little import to the current state of the world and of imperfect human nature. 

But, the remainder of this article is an attempt to flesh out the deepest implications of taking the classical Liar Paradox as seriously as some people take it. This is not to say that all of the possible kinds of fascination with the Paradox are useless or vain: Far from it. But, to the extent that this Paradox is believed to pin down the concept of the specifically contra-normally ‘perfect’ liar, the present article is an attempt to show that this Paradox only begins to identify that concept. Consider, if you can, the following statement as a stand-alone statement.

  

‘The preceding statement is false.’

  

If successfully taken as stand-alone, that statement does not actually state anything. It’s like a dog that’s so used to running to fetch a thown stick that, when the dog’s owner merely makes the motion of throwing a stick, the dog actually thinks that there must have been a stick thrown, and so the dog goes to find it over there somewhere. So, the dog compulsively goes to look for a stick that is not there. As a stand-alone ‘statement’, there is no subject to the above ‘statement’. Yet, the form of the ‘statement’ normally—and sometimes even compulsively—is taken at face value to be referring to a ‘preceding statement’. The question is, when taken as intended, namely as stand-alone, then is it even a statement? What is it stating? 

In some sense, the classical Liar Paradox statement isn’t stating anything, either. But, that Paradox statement seems more paradoxical than the above stand-alone statement, because that Paradox statement seems more clearly to be about something: a very present ‘itself’. 

But, the supposed ‘assertion of fallacy’ constituting the supposed ‘statement’ of the impersonal form of the Lair Paradox does not in any way specify as to how itself is meant as an instance of fallacy: it merely asserts a fallacy, and lets the reader determine what is ‘meant’. But, since the first subject of note in that statement is the supposed fact that the self-named statement is a statement, there seems a problem of determining what the fallacy is. Is the fallacy the statement that the statement is a statement? And, if the statement is supposed not actually to be meant by anyone, but is supposed simply to be a statement which somehow knows what itself means, then what does it mean in seeming to state something about itself?

But, I tend to think that this phenomenon of paradox in the Liar Paradox is none other than that produced by any explicitly defined contradictory compound assertion, such as ‘All currently married men are currently unmarried’. To actually succeed in accepting the two mutually contrary ideas which most normally are meant in this statement is to produce a mental conflict which has no solution within the explicit form of the statement. This is like likewise accepting the statement that ‘a paradox is not a paradox’. If such functional acceptance, despite being fully cognizant of the reality, is temporarily achieved, then a paradox even in regard that relatively non-paradoxical statement is achieved. 

An added complication to a mutually contrary statement is to ask a question as part of the assertion. For example: ‘Roosters do not lay eggs, so, when a rooster lays an egg on the peak of a roof, down which side of the roof will the egg roll?’ To require that such a question not only be answered only one time strictly within the terms of the assertion, but that the question for that one time also be directly and simply answered according to the form which the question itself normally would require, is to produce a conflict in the mind of the person who effectively is so required to answer the question.

But, the deeper question is why anyone should be required to answer in normative simplicity any question that makes unfounded or contradictory assumptions? So, even while the classical Liar Paradox poses a problem for which the form of the paradox requires a mutually contrary answer, the real question is why, in a particular case, anyone might pose the paradoxical statement in the first place. Certainly, in many cases, it is posed for the pure entertainment of observing its absurdity.

But, from certain frames of reference in which it is possible for imperfect epistemological agents to lose themselves, the classical Liar Paradox seems epistemologically profound. So, there are other, sometimes even self-delusionally aggressive, reasons for why that paradoxical ‘statement’ is posed. 

Now, when taken as reference to itself, the result of the impersonal form of statement, ‘this statement is false’, is no more than that of taking the word, ‘lie’ or ‘fallacy’, as reference to itself. The most obvious question is, if it can be so self-referential, then what is the word ‘lie’ lying about? The obvious answer, to my mind, is that, when taken as reference to itself, the word ‘lie’ isn’t lying, and isn’t a fallacy, but is a failure of the mind to fully identify its own projections as such. In other words, I think that a perfect negative epistemological self-reference is, by definition, impossible: there must be some reference, if not by an actual statement, then by the individual ideas in the statement, such as all those ideas in the normal interpretation of ‘Abraham Lincoln was a little green Martian’. 

But, the problem with the classical Liar Paradox is not that of self-reference as such. For example, there is no paradox in taking self-referentially either the statements, ‘this statement is true’, and, ‘I hate making statements’. The paradox is produced, rather, by incomplete self-reference in terms of an epistemological negation of the very thing referenced (or, if I’ve botched my own attempt to explicate an idea by way of that statement, then I can only hope that the reader gets the idea despite my pathetic attempt). In the classical lair paradox, the central incompleteness consists in the fact that the mind is using its mere idea of falsehood in place of the way in which something actually is false, or at least normally is falsely described as false. 

Unlike the statement in the classical Liar Paradox, the sentimentally negative ‘I hate making statements’ normally is a device for referential efficiency, not for any aim, absurd or otherwise, for referential completeness. In other words, to state that ‘I hate making statements’ normally is used in place of some less-incompletely stated set of conditions, such as, ‘I hate having to keep stating obvious things vainly for the instruction of determinedly self-deluded fools’, or ‘I hate being questioned about things that are no one’s business but mine, so, for the love of God, leave me alone.’ 

And, referential efficiency to various objects, as the normative functional nature of statements, is what the classical Liar Paradox co-opts or borrows. Imagine a world in which no one can achieve a sense of linguistic paradox, but in which everyone knows exactly two things: everything about Abraham Lincoln, and that is linguistically possible to utter either ‘this statement is false’ or ‘Abraham Lincoln was a little green Martian’. Imagine, further, that each person in that world is capable of doing only one thing: running around while shouting whichever of these statements they prefer, and only that statement, all day, on whatever day they wake up preferring it over the other. So, some people run around all day shouting ‘‘Abraham Lincoln was a Martian!’ while the others run around all day shouting ‘This statement is false!’ In such a world, the only possible interpretation of the latter statement is that it is reference to the former statement. But, even in the real world, such is the normal usage, and the normal understanding, of statements: some statements are reference to other statements, and, at some point in any precedent of statements, at least one statement is about something other than a statement. 

 

So, the dynamics of the classical Liar Paradox is that it allows not only for the fact that the act of knowingly stating a fallacy involves knowingly making some truthful reference, but for the fact that the mind is capable of conceiving some sense in which is it coherently possible for there to be, in the limited and normal linguistic sense, a ‘perfect liar’: that of a person who, for uttering only lies, is incapable of uttering that he is a liar since that would be a true reference to his until-then-perfect lying. So, either he cannot vocally so reference himself, or, when he does, he ceases, in this limited, epistemologically-linguistically normative sense, to be a ‘perfect liar’. 

But, when the two mutually contrary objects of the classical Liar Paradox are conflated by the deep purpose of constructing the paradox in the first place, a paradox results. Those two objects are: 1) the concept of the contra-normally ‘perfect’ liar; and 2) an epistemological algorithm for achieving the mental construction of such a liar. The deep purpose of constructing the Paradox is to maintain the epistemology of fallacy and, or, triviality. The paradox results because, unlike for the epistemologically-linguistically normative liar, the far more unlimited, and arguably meaningless, concept of a contra-normally ‘perfect’ liar (which the normative concept may seem at first to promise to be) implicitly cannot include epistemological agency. The following section is devoted to explaining and demonstrating this latest claim.

 

 

 

THE AIM OF THE ‘PERFECT’ LIAR

 

The classical Liar Paradox is a device serving some motive which stands outside the paradox itself. Several such motives may be: to ponder the nature of language as a tool of reference; to determine whether it is possible for statements, as such, to be completely false; to determine whether the act of reference can ever genuinely be paradoxical; or even to construct some seemingly rational precedent for rejecting a personally disfavored claim to genuine and non-trivial knowledge. 

But, when used as an attempt to demonstrate certain favored philosophical problems which supposedly are implied in the theory of the ‘perfect liar’, the classical Liar Paradox fails. In other words, the classical Liar Paradox is inconsistent with the ‘standard’ upon which the paradox itself is imperfectly constructed: the theory that the concept of the ‘perfect liar’ logically may involve a referential, that is, epistemological, agent. This theory is a paradox, because it purports, and fails, to be a demonstration even of the normatively linguistic concept of the ‘perfect liar’. As a puzzle of cognition, the paradox may be solved. But, as a demonstration say, of that to which all statements rationally are reduced, it is a conflation of imperfect cognition with a sense of cognitive or rational power. 

The simple fact is that the classical Liar Paradox allows the hypothetical perfect liar to utter any actually false statement. And, when this is shown to fail the dubious philosophical aim, the hypothetic stator is replaced by an impersonal form of the statement as somehow standing as ontological equivalent to fallacy. The simpler fact is that all actually false statements necessarily make some actual, or truthful, reference, because all actually false statements state something which actually is false to which the statement actually makes reference. But, the statement, ‘this statement is false’ seems to make reference to itself as a statement, while also seeming to be a statement about nothing.

So, consider the idea that the four-word sequence, ‘this statement is false’, is not actually a statement, but merely is the idea of a statement. In other words, its ‘statement’-ness can be thought to be reducible to the word ‘statement’, which clearly normally is no more a statement about anything than is the word ‘dog’ an actual dog. Of course, the word ‘word’ is a word, but is ‘word’ a statement? That depends on how it is meant. So, it might seem that the word ‘statement’ can likewise be a statement, and indeed, it can be. But, when such a usage of ‘statement’ is part of ‘this statement is false’, what is false about it? That it is asserting that it is false, when, in fact, there is nothing false about it? Then it is a paradox. But, the paradox is no more in the form of the statement than is the word ‘statement’ necessarily a statement. It depends on its usage, and, in that case, the entire meaning is in the mind. So, linguistic paradox is semantic, not a ‘surface structure’ of language.

So, to allow even the least reference to be made by the supposedly ‘perfect’ liar, or the ‘perfectly’ false statement’, is to negate the assertion that this liar, or this statement, is a perfect instance of a failure to make any actual, truthful, reference. Even the semantically normative, but summarily false, statement ‘Abraham Lincoln was a little green Martian’ makes truthful reference to various objects. And, even if it’s word ‘green’ is actually meant for the color normally called ‘purple’, the fact that ‘green’ is correctly used as reference to color means that that statement makes some genuine, or truthful, reference.

The only way to remove the conflict in the nominally ‘perfect liar’, and thus in the classical Liar Paradox, is to preclude the very object which the classical Liar Paradox labels ‘liar’: an epistemological agent. Even the ‘impersonal’ formulation of the classical Liar Paradox fails to preclude such an agent, because only by the instantiation of such an agent as interpreter-of-the-statement is it even possible for there to be any sense whatever of the sequence of physical forms by which the sense of the paradox is expressed: There must be a functional receiver of actual knowledge at the other end of that statement—even if the only receiver is the same as the sender—-in order for that statement to be considered to mean anything. 

So, the classical Liar Paradox does not obtain as a sequence of forms; it obtains, rather, on the part of its expressing/interpreting epistemological agent, as an often unwitting action of projection-of-meaning onto those forms. An example of an opposite projection-of-meaning is that of a person who, like the proverbially ignorant ‘backwoods hick’, believes that since he cannot make any sense of the Chinese language of his unwanted Chinese-language-only tourists, that that Chinese language actually is gibberish invented-on-the-spot by these Chinese tourists for the purpose of making him think that they do not understand ‘plain English’; that Chinese is not a real language, and, that any Chinese person who ‘is willing’ to speak English is, even then, too stupid to know how to ‘speak it right’ without an ‘accent’. 

And, since two minds do not necessarily have the same purpose, much less the same accustomed forms for a given purpose and vice versa, it is true to a certain interpretation that the classical Liar Paradox obtains, at best, as one mind knowingly acting in two mutually incompatible epistemological purposes at the same time: a failure to define the contra-normally perfect fallacy, by way of a pretense of a perfect means, or algorithm, for constructing that definition.

 

 

 

 

THE TARGET OF THE ‘PERFECT’ LIAR

 

It normally may be thought that the hypothetical ‘perfect liar’ sometimes aimed at by way the classical Liar Paradox is the most perfect definition possible of a ‘perfect liar’: that of a liar who is precluded, by definition, absolutely from all senses in which it is possible correctly to represent anything. But, to assert that a ‘perfect liar’ must, by definition, be ‘precluded absolutely from all senses in which it is possible correctly to represent anything’ is to preclude even all implicit epistemological qualifications of what it means to ‘misrepresent something’. So, to construct this ‘contra-normal’, or ‘anti-qualified’, perfect liar, one must preclude from consideration any sense in which the ‘perfect liar’ can be said epistemologically to reference anything. And, since, even all false statements necessarily make some reference, such as by the name of an actual person, the genuinely perfect liar is precluded even from making any false statement. In other words, the anti-qualified, contra-normal liar is precluded from making any reference to any object so as to be capable of mis-representing that object. At this stage, the separation forms, like, perhaps, what some paranoid schizophrenics might be thought to perceive when they say they hear voices. Thus, we have statement A:  

 

A: ‘Abraham Lincoln was a little green Martian.’

  

At face value, self-referentially, that statement is true, since it does not also state that what it states is false. That’s what a contra-normative ‘face value’ begins to look like. 

But, in so far as statements are devices by one epistemological agent to communicate referented objects to another epistemological agent, statements themselves fail to pare down the concept of the anti-qualified, or contra-normal, perfect liar to its conceptual root. And, the reason that the preclusion of making statements fails to get to the root of the genuine concept of such a perfect liar is because anti-qualification, or contra-normality, is a suspension of all epistemologically normal usages/limitations of the active term. In the case of the liar paradox, the active term is ‘lie’, ‘liar’, ‘fallacy’, etc.. And, since forms themselves do not constitute reference, all forms are allowed—even ‘Abraham Lincoln was uncommonly tall’—, while no form may be produced as reference. And, if no form is produced as reference, then it doesn’t matter whether any form produced is taken by a non-producer as reference: the production is consistent with the standard, and so no paradox is produced. At this stage, the true separation begins: statement B: 

 

B: ‘Statement A is false, and statement B also is false’. 

 

So, the anti-qualified liar is precluded even from having any object in mind. This includes even any irrational object, since to have even an irrational object in mind is an act of reference to that object. The problem, however, with irrational objects, is as to whether they ever are necessary compounds of two genuinely rational objects, or, instead, are constructs merely of two devices of referential efficiency one or both of which is mistaken as epistemologically complete. So, the contra-normally ‘perfect liar’ is precluded from any self-knowledge. And, since self-knowledge is what makes reference— including self-reference—possible, such a perfect liar cannot make any statement which refers to this liar as a liar, much less as a self. The same, then, goes for any self-nominal, impersonal ‘statement’ produced as false. And, if self-knowledge is precluded, then the anti-qualified liar is precluded from being a cognizing, that is, epistemological, object in any case (the least conscious, sentient, aware, feeling, or etc.). The separation is complete: Statement C: 

 

C: ‘Statement B is false.’ 

 

But, if an anti-qualified perfect liar cannot exist as a cognizing object, then can such a liar exist even as a non-cognizing object such as a rock? Some self-admittedly imperfect epistemological agents believe that the classical Liar Paradox demonstrates that two fully rational objects may be combined into one irrational object. But, if such agents, while believing this, fail, for lack of sufficiently ever-present self-knowledge, to be consistent with the concept of the perfect liar, then such agents, whether realizing it or not, face the following question: is the connection between imperfect epistemological agents and the objects of such agents necessarily the kind of connection which these agents may ever lead themselves into concluding? Specifically, what is the connection between knower and known in the case in which the known is known by its knower to be a fallacy—much more to be an impossibility? In so far as the least contra-normative idea of a ‘perfect’ liar is allowed to be a cognizing object, that is, an epistemological agent, that ‘perfect’ liar is an irrational object, and so is any seeming statement which purports to know what itself means as being a ’false statement’. The clanking of so much machinery does not, in itself, reference anything, falsely or otherwise: it just clanks. Like statement D: 

 

D: ‘Statement C is true, and false, and meaningless’. 

 

Hence, statement D is meaningless, and you may now return your mind to its natural position: ‘Abraham Lincoln’. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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