Liar Paradox

In the Epimenides Paradox post,  I looked at the easily refuted "This statement is false," where obviously enough it is seen that there is no statement, and so the line is intellectual gibberish. If the line is changed to "This sentence is false," the same refutation basically applies as there is no claim here which justifies this sentence being described as true or false, and so it is a false construct and despite appearances not a valid sentence.

Not that I remember but when I did a number of posts on alleged paradoxes way back, maybe I hadn't initially even formulated to myself the inner motivation behind the refutations, and on a surface level was perhaps simply struck by the faulty logic of the individual cases, but after a while it becoming clear that a general truth was involved which was that language used properly cannot result in meaningless conclusions, while the paradox acolytes like Borges were claiming otherwise - that language and even by extension life were inconsistent, faulty systems. But all that has happened with the 'paradoxes' is, without noticing it, people have intellectually gone wrong.

So now onto the Liar Paradox, which is closely related it seems to Epimenides paradox but far more clever or tricky to unravel. To add, I'm not really going to look into the history of these 'paradoxes' and whether the Epimenides Paradox truly is "This statement is false" isn't the point or just considered an equivalent of it. It is the inner logic of the given sentence that I am dealing with, and even if just for the sake of demarcation we'll call that the Epimenides and the following the Liar.

So lifting from elsewhere:

The Liar Paradox is among the simplest of paradoxes. It can be traced back at least as far as Eubulides of Miletus, a fourth-century B.C. Greek philosopher.
Eubulides’ version of the paradox is this: A man says that he is lying; is what he says true or false?
However we answer this question, difficulties arise.
If we suggest that what the man says is true, then we end in contradiction: if the man’s claim that he is lying is true, then he is lying, in which case what he says is false.
If we suggest that what the man says is false, then we are no better off: if the man’s claim that he is lying is false, then he is not lying, in which case what he says is true.

So, the relevant line here is "I am lying." Again unfortunately like "This statement is false," this is a meaningless construct. About what is he lying? There is no statement which as above can be true or false. So the intended gist of this must be put a little differently to make at least more apparent sense; and so a better variation is the line "All Cretans are liars," said by, it is said, a Cretan. Here we do have an apparent statement. Also it could equally be put, "Everything I say is a lie." Again, rather than just saying "I am lying", this seems at least to be a definite statement about an aspect of reality - that everything I say is a lie. The resulting train of thought then goes or is alleged to go that if everything I say is a lie then this is itself a lie, and everything I say is hence not a lie. Or if the statement is true and so is itself a lie, this this implies everything I say is not false, which contradicts itself. And the wider philosophical implication being that language is capable of being used correctly and yet leading to untenable conclusions; this showing that language is ultimately a flawed system, and so Truth is an illusion, reality is intellectually corrupt, etc.

So with truth and coherence at stake, onwards. It turns out however that this has already been covered in the previous post, with the point that logic follows from true statements, not false ones. An intellectual process is real when every step is true, every line is in accordance with the inner logic of language and accurate representation of external reality. A mathematical or linguistic line, theorem, etc is real on such a basis, and if error occurs the overall construct which follows is unreal, does not in truth exist.  And so to begin with 'Everything I say is a lie', means that the logical process has been immediately flouted at source, and so there is no further justifiable process. It is a degradation of the logical process to go any further, since the line is not real, not in accordance with the truth to which logic must conform. Logic applies to the logical implications of true statements, not false ones.  The line explicitly describes itself an untrue and so unreal, and so reality cannot proceed from unreality.  It is not real but for someone who wrongly follows a supposed train of logic along this illogical pathway, an intellectual hallucination or something which appears to be real ensues, such as a supposed paradox resulting from imagining the false conclusion has been arrived at legitimately. The false conclusion though should be the giveaway that it is the intellectual process that has been degraded or simply gone wrong.

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To some degree I imagine the above denying of the validity of the process that supposedly leads to the paradox could be felt to be a bit unfair, and I will now look at the matter from a slightly different angle.

Is it intellectually possible - i.e. can it make sense - for me to say that everything I say is a lie? - which is the essence of the Liar Paradox.  The whole supposed point of a paradox being of such import is that it is allegedly logically coherent and yet leads to illogical conclusions. And we should try being as existential as we can and first realise the 'logical deductions' which follows the line "Everything I say is a lie" are as yet nowhere to be seen; it is that line itself that we should be looking directly at.

Quoting from an earlier piece - "To say anything in a true intellectual sense is to involve oneself necessarily in an acceptance that the language one is using is real and imbued with meaning; that the words one is using- if used correctly, i.e. meaningfully- are meaningful." This is the necessary ground and of course is the ground from which logic proceeds. So to say that everything I say is a lie is immediately a total violation of the ground upon which argument and logic sits. And so it is self-contradictory as a piece of logic. Truth doesn't dwell within falseness or lies, and so, as more or less said earlier, it makes no sense for such a line to be considered to dwell within the world of logic or logical statements. So there is no great truth revealed by this 'paradox' as the line can't be said to exist intellectually.

That it is worthy of any intellectual interest is that it is a truth claim, but here it is clearly self-contradictory for a truth-claim to deny itself as precisely that, to claim itself to be a lie, and so not to fall within the sphere of logic. Even following the supposed logic though doesn't lead to the paradox but simply to the falseness of the line's claim. So, having laboured the point, going back to looking directly at the statement, "Everything I say is a lie," and seeing if it can be legitimately said. The idea of the paradox is that the relevant line can be said but the conclusion of this valid line contradicts truth, thus revealing language's flawed nature. Generally the supposed sophistication of these paradoxes comes out as quite a childish and stupid misunderstanding however - such as with the inane "This statement is false", and its deflating riposte"What statement?"

                                              Everything I say is false.

 Is it possible for everything I say to be without exception false, that is in practical terms, to be with every utterance relentlessly lying? And as this extends philosophically, can Falseness be a consistent cohesive system? And so, the reach of all this has extended with more at stake than what one might have thought.

So the key now is whether it is possible to relentlessly lie. As already shown, to use language is to accept its truth and meaningfulness, and the very fact that we are discussing whether this Liar Paradox is a viable or true language statement shows how impossible it is to escape this. But now we are seeing whether Falseness can exist relativistically, to dwell cohesively at some more minor level within the necessarily admitted greater truth of Truth. Someone can of course within a trivial range lie with everything he says, from what he had for breakfast to who won a football match, what age he is, etc. Such a life of relentless lying could happen. But that's not the issue. Is this unbroken lying possible as an entire system when pushed to its intellectual limits? In a theological sense, could a Satan console himself with the truth of his defiance of Truth, even while reluctantly accepting God's existence and primacy?

Well and perhaps surprisingly, the line itself "Everything I say is false" reveals the impossibility of the coherence of this edifice of falseness. Lying reaches its own limits, trips itself up, as if I am lying with this very statement, then this implies that not everything I say can be false. So falseness as a system collapses in on itself, and has no consolation.

Epimenides Paradox

On the continued theme of the paradox, I saw mentioned last night in a television programme on mathematics the alleged paradox "This statement cannot be proved." And this apparently a tangent of what is called the Epimenides Paradox, which seems to go in something like its pure form:

This statement is false.

So do we have here what Borges described as a crack in the architecture of reason where we see that the world is false? The supposed logic going, "Well, if it is true that it is false, then it is true. Which means it is false. Which means it is true." And so on. A unsolvable paradox. Logic has been breached. We are free!

The vital essence of language in the form of reason, rather than say poetry, is that at every level what one says is reasonable - makes sense. And here, as should be immediately obvious, is where this alleged paradox gets in trouble. "This statement is false." What statement? There is no statement here, simply the referring to one which does not appear. The sentence, without the accompanying statement, is linguistically meaningless. And so it is of course meaningless to describe a non-existent statement as true or false.

"My dog is black" is a statement which may be true or false, depending on the colour and existence of my dog, and in this correct understanding of what a statement is, no paradoxes occur. The statement exists- refers to something. "This statement" is not however a statement, and so the supposed paradox is simply resting on a foundation of gibberish. And so, alas, no paradox.
Identically - "This statement can't be proved." What statement?

Even if, and here I am giving far more respect and time to this nonsense than it merits, "This statement is false" wasn't meaningless and actually made some kind of sense, the moment it is false it makes no more sense to apply logic to it. It is illogical to treat illogical statements as logical- you don't build and form deductions from a foundation of "Two plus two equals five." Logic doesn't apply. You don't apply rational conclusions to irrational statements. That is simply, and literally, insanity: take meaningless nonsense and run with it. Logic applies to the logical implications of true statements, not false ones.

So summing up, "This statement is false" is linguistic nonsense as it isn't a statement, but in any case a logical train should not follow illogical statements.

As written in a separate post, "Just because words combined may make what appears a proper sentence doesn't mean the structure is a legitimate one, i.e. language isn't simply a matter of structure but of course meaning also, and here the meaning is absent."

A follow-up on the Liar Paradox here.

Kant's First Antinomy: The World Has a Beginning in Time

The world has a beginning in time.

I said I'd have a reluctant look at the particular antinomies and so here we are. This is part of Kant’s first antinomy which according to my intermediary of Bertrand Russell is a position which “can apparently be proved,” along with the counter-claim that the world has no beginning in time which can also apparently be proved; and so language inferentially falls apart as a truth tool.

Each language statement can be treated existentially, that is to stand alone as a sensible language statement. So for example if someone produces fifty pages of argument that end in the ‘proven conclusion’ that 2+2=5, then one can and should with full justification ignore or not bother to read the arguments since 2+2=4, and 2+2=5 being simply wrong, then the arguments are blatantly useless. Even if the whole world but oneself declared the arguments to be sound, one still knows they are imaginary and cannot possibly prove a false language statement. The ‘proofs’ can only be as erroneous as the conclusion. Anyone claiming that the ‘proofs’ have disturbing implications regarding language is merely wallowing in an intellectual hallucination; instead of realizing the argument must be false, jumping to the unsustainable notion of language itself being false. To even attempt to state this position of language being false requires the starting point of language being true so as to state it as a meaningful proposition.

And so back to Kant’s first antinomy regarding the world having a beginning in time. Kant also brings space into this affair but it is enough here to merely deal with time. Firstly it is no surprise the intellectual issue is a very abstract one, where unlike for example the 2+2 case, there is no possible observable data to compare with one’s logic; instead it is a matter where one could very easily go wrong with one’s abstract, perhaps convoluted, perhaps very convoluted, thought; and immersed in this language, lose all sight of whether one’s language is sensible or wholly gone astray. The language of mathematics is by contrast a far clearer and simpler logical tool, with one’s errors presumably blatant to any competent colleague in the discipline. This ‘ordinary’ language of words though is a far more elusive affair. So once again:

The world has a beginning in time.

So as said earlier this statement must be able to stand existentially as a sensible language statement. Any ‘arguments’ or ‘proofs’ are beside the point, and ultimately superfluous.
Taking the world to be the universe and the universe is all that is, then all that is includes time, and  it is meaningless to try and talk of ‘the world’ and ‘time’ as though they were separate phenomena. To do so is akin to trying to talk of water as distinct from the hydrogen and oxygen which comprise it. So since time and the universe absolutely coincide, are inseparable, then it makes no sense to talk of the universe having a beginning in time. This is to try and suggest that there is a field of time and then the universe is introduced at some point into this field – to talk of the universe or world, and time as distinct phenomena. ‘The world’ is used here in the sense of what is. Time cannot exist as an autonomous element outside of what is. If one removes what is or the world, then along with it goes time.

Another point in tandem is that since time and the world are inseparable then any point within the field of existence is simply another point within this field. One cannot talk of any such point as the beginning, it is instead a point like any other (though the reality of any ‘point’as distinct from the field as a whole is illusory).

To talk of a finite length to some system is necessarily to require a point of observation outside of that system to observe its finitude. Since in the case of life or the universe this ‘external’ point must be part of what is or the world, then such a point cannot after all be an external point. There can only be what is, in other words: ‘the world’.

To try and argue otherwise someone might say if one pictures oneself walking along a timber plank with nothing else in view, then from one’s point of observation on the plank one can come to a clear end to itself, and similarly if one traverses the plank’s length in the other direction,; thus observing the finite nature of the plank. Since this is the only object, then this is comparable to the absolute system of ‘the world’ from within which one can speak of it being finite.
However, for the plank to come to an observable end is for the plank to exist within something extending beyond itself, namely space, and so the plank is not ‘the world’ but merely an aspect of a broader world. And so the plank is not an absolute system but merely dwells within a greater one. Similarly to talk of a beginning in time is to set limits to an absolute system.

It is obviously quite a difficult matter to convey and understand but to extend it further, treating it again as a matter of language. To say there is a beginning in time is also to necessarily say or imply that ‘Before this point nothing was.’ Again this is to talk of a point of observation external to the system from which one can observe the finite length. From all necessary points of observation within what is – the world – such an observation point cannot exist. However we can also now look at the logic of the language statement:

Before this point nothing was.

The very notion of there being no time before creation is self-contradictory. "Before" is a word dependent on things existing within some sequential order, and here that order is time. It then makes no sense to place a word whose specific context is within time, and inseparably the world, in a context you declare to be before existence. "Before" can only have its rightful place within the world of time.
In the very same way this translates to the world as a spatial structure. To talk of this world as being finite is to try and say that beyond certain points there is nothing. Beyond's necessary context is within space and so it nakes no sense to try and talk of nothing beyond a certain point. The only thing that can be beyond is more space.

Not only that but it is logically impossible to talk of the existence of nothing, as in ‘Nothing was’. To do so is to talk of the existence of non-existence, which is nonsense. To talk of a finite universe or existence is to attempt to talk of existence bracketed by non-existence. Since non-existence tautologically doesn't exist, this 'framing of existence by non-existence' is meaningless. There can be no end to existence since existence is all there is, and so what is is infinite.

So all in all, as stated previously, the ‘truth’ of this antinomy is merely an intellectual hallucination, and to say that life is finite and has a beginning in time is a matter of meaningless language. If I were to delve into the alleged proof for this illogical language statement of the world having a beginning in time, then these proofs would of course be no such thing, identically to the illusory nature of proofs for 2+2=5.

One last point which might be necessary is that someone might dispute my use of ‘What is’ and ‘The World.’ Someone might talk of for example the existence of Heaven as a realm distinct from ‘the world,’ or as might be qualified: ‘this world ‘ Such use of language is as above to try and set limits on what is. ‘What is’ includes all of what is, and again just as water cannot be treated as it were distinct from hydrogen and oxygen, any labels like ‘heaven’ and ‘this world’ are false divisions of reality or what is. This would again be to try and talk of an absolute system, ‘the world’, as dwelling within something beyond itself, and so in other words, as with the plank in space, not as an absolute system.

Later edit: I said I'd look at the antinomies, meaning all of them, but having glanced at the others, they're so inane I don’t see any point. There’s no substance there to get your teeth into!

Kant Again: Antinomies, Noumena

I wrote a couple of posts on Kant's Antinomies, which antinomies are essentially paradoxes or logical inconsistencies, and I stated that language producing a meaningless conclusion or result can only be the result of meaningless language. Just as with mathematics, if language is used correctly or meaningfully in the given situation then it cannot but produce a meaningful conclusion. Here it says that Kant "resolved the four antinomies by drawing a distinction between phenomena (things as they are known or experienced by the senses) and noumena (things in themselves; see noumenon). Kant insisted that we can never know the noumena, for we can never get beyond phenomena." So in short, Kant attempted to resolve the illogicality of the antinomies by insisting that there are areas of reality to which language has no access, and thus is error produced when language steps into the illegitimate territories. So Kant claims to produce sensible intellectual arguments which yield illogical, contradictory conclusions and then invents a further unifying theory to explain the by definition intellectual nonsense he has produced. Thus, I would say, an intellectual hallucination grows.

Firstly to look at the unifying theory of the noumena: if we can never know this noumena, then how can Kant possibly know he can never know the noumena? By his own definition since he has no access to the alleged phenomenon, then in the absence of information emanating from this wholly inaccessible region, we can only take it that he is just making the noumena up out of thin air. The thing-in-itself or noumena are merely words which are entirely self-referential; they are words with reference to nothing to which one can point or state anything. So the thing-in-itself truly is 'the thing-in-itself' - that is to say it consists of those very words and nothing else.

Logic has been abandoned in favour of what one might call magical linguistics. One permits oneself to introduce phenomena without proof of their existence and about which one can say nothing. That which suggests their existence one declares unreal. This is all little other than a restating of the negative gnostic position of Plato where the world of the senses is declared to be a fallen shadow world, and the truth some region with absolutely nothing to do with this world of the senses declared to be as said fallen, spiritually corrupt. It is the result of the individual shrinking from life in fear of its perceived awful depths and resorting to a spiritual world that is by contrast undefiled and perfect.
As in this post on Plato and the notion of the void http://wwwinabstentia-andrewk.blogspot.com/2009/08/greek-void.html,

the world of the senses has been decided to be unreal, and so what is most real should partake least of all of the sensory world, and what partakes least of all being apparently an idea. Ideas are stated to be the purest of substances, and the most pure of these substances is an idea which is utterly self-referential and distinct from the debased world of external reality. And so the void: a pure self-contained idea without reference to the debased world of sense perception. Hence through the ages, and still, the exaltation of the imagined holy landscape of Pure Reason.

Kant, as a manifestation of the same inner turmoil and fleeing from reality but further along the historical path, has to try and deal a little deeper with the perceived fallen world of matter and in a more convoluted comprehensive fashion declare matter's and sense-perceptions ultimate non-existence; thus the phenomena are delcared unreal and the noumena, just like Plato's Void and World of Ideal Forms, real, but again inaccessible to us, except, it is declared, to our intellectual senses. In the historical interval matter has been raised a little in value compared to Plato's version- the thing-in-itself being allegedly matter in its true form - but just as with Plato this is a substitution of life for an idea of it,  for both the noumena and the world of idea forms are both purely language constructs with reference to nothing beyond themselves. And the identical point is made that since the existence of the void or noumena can only be inferred by the intellect - since there is by definition nothing to suggest its being, sensory experience having been declared unreal and it being wholly absent from this reality - then there is no basis to make the inference even if the substance made intellectual sense.

It is also clear that while Kant has had to engage himself more comprehensively than Plato with the fallen world of matter in order to dismiss it, he also has in tandem moved further from the spiritual reality to which Plato clings. The spiritual inner world has become less of an inner reality and more of an unexperienced idea of itself - which is the inevitable process once ideas like Plato's work their faulty logic out in the world of time. The reality becomes ever more lost and one is ever more left with merely an idea of it, and in turn the very idea of it will also fade, and one is left with atheism. 

Reluctant as I've been to have to devote some time to the tedium of looking at the actual antinomies themselves, having merely declaring or showing them to be false simply on principle, I'll have a closer look soon. Here as it transpires

Language, Antinomy - Again

In a post on Kant's Antinomies I said the truth of any language statement rests upon the inescapable foundation of language's truthfulness or meaningfulness, and this is why the antinomy is a false notion simply on principle. One knows this to be so without having to look at whatever particular antinomy may be in question, as this seeks to declare mutually incompatible language statements to be capable of co-existence, i.e. that they can contradict each other. A commenter sought to explain two plus two equals four being true not because it is a consistent statement of language but because it reflects the external world. However mathematics is an autonomous language that needs no reference to a world 'beyond itself'. The very thought of a world beyond itself has no place within the language of mathematics; it is a meaningless intervention within that mathematical language, the same as it would make no sense within the score for a piece of music. That the world or language of mathematics does reflect or correspond to the forms of the natural world, even at the most abstract, complex levels is because this intelligence imbues all of existence, the human mind included. Life and the intelligence intrinsic to life animates all within life. Life is not divided against itself. And so ordinary language that is self-contradictory is merely a false use of language that will be found to be simply erroneous.

However the language of ordinary discourse unlike that of mathematics is not entirely autonomous; that is, all that is referred to cannot all be contained on the page. However, many language statements cannot be said principally to point to some external reality, and are a matter of the correct working of language. An example below from an earlier post:

An idea extremely prevalent in the current era is that everything is equally valid. For instance you cannot, you will be told, say that one work of art is greater than another because of this equal validity of everything. So a quick look at the logic at play here.
Everything is equally valid. Therefore all statements are equally valid. And an example of one such statement within the totality of equally valid statements is that:
Everything is not equally valid.
And so if everything is not equally valid, then the first statement that everything is equally valid is false. So the statement of all being equally valid contains within itself its own disproof.

So the truthfulness of this was purely a matter of language and the equal validity of all was shown to be self-contradictory and so false. The idea of all being equally valid violates the necessary internal logic of language. Were there no such intrinsic order or intelligence to language, then no statements could be said to make any sense. However there are uses of language whose truth or falsehood rests also on their correctly referring to the world they allegedly point to. So, 'I am now standing on my head reciting a nursery rhyme' is a false statement as I am not doing those things. However as a matter of language it is fine. No such false statements can exist within mathematics.Statements within that field are purely autonomous. If it is true as a matter of mathematical language then it is true.

Someone might then think that statements of ordinary language are not primarily a matter of the internal logic or intelligence of that language, and so for one to say that 'I am wearing black shoes,' is true solely on the basis of whether it reflects that particular case. If he is wearing black shoes then it is true, and language has nothing to do with it. However any statement of language is existentially a statement of language, and if that person were to say instead, 'Black I wearing shoes am', then that would be not a reflection of anything but simply gibberish. It fails as a matter of language. So even the simple statement of wearing black shoes must first be a sensible matter of language. All language in the sense of a truth tool inescapably just so rests on the internal meaningfulness and logic of that language, which again can lead us back to the antinomy or paradox as simply a wrong use of language.

Kant's Antinomies

Back to The History of Western Philosophy again, and within is written of Immanuel Kant's antinomies, of which Kant alleges four principal ones, which are imagined to be examples of mutually contradictory statements being simultaneously true; and this being known by the method of thesis, antithesis, synthesis, which is apparently of prime importance for the thought of Hegel also, and onwards to Karl Marx amongst others. What Kant's particular antinomies are is irrelevant; it is the principal or notion of mutually contradictory statements being true that is the essence of the matter. So to look at what this involves.

Two plus two equals four.
Two plus two does not equal four.

The first statement we describe as true, because meaningful language rests upon a foundation of being true, a foundation which does not even need formulating since it is the necessary and natural faith inescapably bound up with the use of language. And so the second statement is false. The two statements cannot both exist as truths. However Kant and his successors claim otherwise; that statements can contradict each other. How is this possible? Upon what would this idea of language rest?

It rests upon language and truth not being inseparable, and so the "contradictions" are not in fact contradictions but equally valid, since there is no truth which they contradict. It is to treat words as lumps of matter which can be placed in whatever order one wishes and the results are equally valid, all equally sensible or senseless. This would imply and necessitate the demolition of the entire notion of language as meaningful, since something and its contrary are alleged to both be capable of meaningful co-existence. But this meaningful co-existence is dependent on language not being meaningful but meaningless, since if it is meaningful then one cannot have coherent contradictions within that language. Language cannot be used in a self-contradictory manner and remain an instrument of truth. Such contradictions are violations of the nature of language, and will be found to be merely an erroneous use of that language. Also Kant's whole notion of the antinomy is entirely self-contradictory: an attempt to be a true statement of language, while the very statement inescapably implies that language is not true. If the antinomy is true then language is not true, and so the antinomy is not true. It is a perfectly senseless, and so unreal, use of language.

The notion of building a philosophy of truth upon the notion of the non-existence of truth is clearly ludicrous, where according to the implied logic of any sequence of words being as good or bad as any other, one could build this entire philosophy and then with a final flourish claim that the contrary to all this is also true, i.e. that it is not true - gibberish having been sanctified. However for this notion of language's meaninglessness to be sensibly be formulated in the first place requires the acceptance that language is meaningful; one is trying to use language meaningfully. So all in all Kant and his successors are trying to build an edifice upon completely self-contradictory and delusional grounds. How one could use the truth tool of language all one's life while remaining ignorant as to its essence is particularly lamentable for a philosopher.

I look at Kant's first antinomy, The World Has a Beginning in Time, here.

Parodox Parody?

Below might seem a parody of the recent debunked paradoxes, but is in fact entirely faithful to the spirit and standard of reasoning in Zeno's and Epimenides' famously alleged paradoxes, where the intelligent understanding of words and what they stand for breaks down.

"I am here. You are there."
"No, I am here. You are there."
"No, I think you'll find that I am here while you are most certainly there."
"Bullshit. I am definitely here, while you are there."
"Are you calling me a liar? I am most certainly here, while you are without a sliver of doubt over there."

And so on.

Perhaps the Final Paradox Piece

As said at some point in the earlier posts, the mental urge towards the existence of the paradox or/and the worldview arising from belief in its existence amount to gnosticism, reality divided against itself, and as Borges says, false, though the very word 'false' should have told Borges all he needed to know about the falseness of this vision. Ultimately a schizophrenic universe. But as shown all the given paradoxes are not paradoxes at all, but are either illogical at source or seen as paradoxes as a result of false views of the reality mirrored by the logic of a mental set-piece.
The paradox is itself, obviously enough, an exercise in language, and here the senselessness of the very concept of the paradox becomes clear. The entire basis of the correct use of language is its meaningfulness, and as a reasoning tool all language properly used must be meaningful. The paradox seeks to meaningfully use language to create a meaningless result. This is self-evidently impossible, and makes as much sense as to imagine one can get a wrong answer within mathematics. And if you do apparently get a wrong answer, then that's the problem. You've gone wrong.

And knowing there must simply be an error, the given paradoxes were shown to be childishly easy to unravel. The error people make is to leap aboard what is imagined to be the ensuing logical train, rather than looking closely at the alleged paradox precisely as an existential language construct. If an example of the simply meaningless construct, there is no way out of the resulting train of logic. Being meaningless it contains no meaning, and one is lost within the enclosed loop of the labyrinth.
The most elegant 'paradox' I've come across and the most subtle to unravel is below, and its elegance of construct and deconstruction lies in its being so purely a matter of language.

A famous idea or paradox is that God could not conceive of something of which he could not do. God of course being used here in the sense of an all-powerful entity, and so since he could not do the above, then he cannot be all-powerful.
I have no idea to what extent this notion has been examined and what excursions of reason it has involved, but a possibly unexamined view is that the thought is being looked at from the wrong linguistic angle. The usual emphasis being that God could not conceive of something of which he could not do.
However, the same sentence can be looked at from a slightly different perspective offering a radically different significance.
God could not conceive of something of which he would not be capable, and he achieves this by not conceiving of it. Thus an all-powerful entity's absolute mastery is maintained.

Again, the error we make is to rush to examine the logic, while it is in the linguistic structure where the truth lies.
Though come to think of it, this will probably not be the last paradox post, since as reason, or what is imagined to be reason, is usually set in motion in the service of already imagined and desired 'truths,' then why this gnostic divided worldview occurs in the first place should be looked at.

The Epimenides Paradox/ Liar Paradox

A re-post of what lies beneath:

On the continued theme of the paradox, I saw mentioned last night in a television programme on mathematics the alleged paradox "This statement cannot be proved." And this apparently a tangent of what is called the Epimenides Paradox, which seems to go in something like its pure form:

This statement is false.

So do we have here what Borges described as a crack in the architecture of reason where we see that the world is false? The supposed logic going, "Well, if it is true that it is false, then it is true. Which means it is false. Which means it is true." And so on. A unsolvable paradox. Logic has been breached. We are free!

The vital essence of language in the form of reason, rather than say poetry, is that at every level what one says is reasonable- makes sense. And here, as should be immediately obvious, is where this paradox gets in trouble. "This statement is false." What statement? There is no statement here, simply the referring to one which does not appear. The sentence, without the accompanying statement, is linguistically meaningless. And so it is of course meaningless to describe a non-existent statement as true or false.

"My dog is black" is a statement which may be true or false, depending on the colour of my dog, and in this correct understanding of what a statement is, no paradoxes occur. The statement exists- refers to something. "This statement" is not however a statement, and so the supposed paradox is simply resting on a foundation of gibberish. And so, alas, no paradox.
Identically- "This statement can't be proved." What statement?

Even if, and here I am giving far more respect and time to this nonsense than it merits, "This statement is false" wasn't meaningless and actually made some kind of sense, the moment it is false it makes no more sense to apply logic to it. It is illogical to treat illogical statements as logical- you don't build and form deductions from a foundation of "Two plus two equals five." Logic doesn't apply. You don't apply rational conclusions to irrational statements. That is simply, and literally, insanity: take meaningless nonsense and run with it. Logic applies to the logical implications of true statements, not false ones.

So summing up, "This statement is false" is linguistic nonsense as it isn't a statement, but in any case a logical train should not follow illogical statements. Or as written in a separate post, "Just because words combined may make what appears a proper sentence doesn't mean the structure is a legitimate one, i.e. language isn't simply a matter of structure but of course meaning also, and here the meaning is absent."

More on the Paradoxes

Another mentioned paradox by Borges is that if one continuously halves the distance between oneself and the object approached, then one will never arrive at one's destination. Again, however, there is no paradox here. This is simply the internal logic of this mental set-piece. It is simply another way of saying, If you continually get closer to something, you will never reach it. Nothing is being contradicted as the mental framework simply refers to the dynamics of itself.
If movement were made to mirror the given thought of halving teh distance between one point and another, then, yes, the destination would not be reached. But this is movement made to follow the logic. This logic does not mirror movement where the object point is actually reached, but refers to a situation where the object is not reached; and so there is no paradox here. Just childish reasoning.
It is in no sense a rational representation of ordinary external movement for the same reasons given earlier; ie by the specifics of time and distance, as in a simple car speedometer where one travels at fifty miles per hour.

Zeno's Paradox, Borges

We have dreamt the world...but in its architecture we have allowed tenuous and eternal crevices which tell us it is false.

Borges is writing here of the implications of logical paradoxes where reason betrays its connection to reality; irrefutable logic leading to conclusions contradicting the reality logic purports to reflect.
Borges doesn't quite seem to understand the implications of the paradox correctly here. If logic leads to falseness then the entire edifice of language as an instrument of truth falls apart. There can be no cracks in 'truth', else there is no truth. Nothing is left us which can be intelligently said, including 'the falseness of the edifice.' In other words, nihilistic chaos.

Borges mentions 'perhaps the most elegant of all' paradoxes is where William James denies that 14 minutes can pass, "as first seven minutes must pass, then three and a half minutes, then half this figure, and so on until the invisible end, through tenuous labyrinths of time."

So it is imagined that there is a transgression of truth here, as time- specifically fourteen minutes- most certainly does pass. The 'paradox' here is all rather embarrassing, however. Time passes at the rate of itself, and of course will effortlessly pass all demarcations within the segment being measured. James and Borges are simply confusing - and misusing - the organ of measurement for that which is being measured. All the 'paradox' amounts to is that if one were to count all the imaginable fractions between one whole number and another, this could and would literally take forever as the possible demarcations are infinite. This is the nature of numbers. It bears no relationship, however, to the passage of time. So all this 'paradox' amounts to is a meaningless misuse of logic or reason.

Earlier Borges mentions Zeno's Second Paradox, where Achilles runs ten times faster than a tortoise, giving it a start of ten metres. It proceeds: For every measurement the tortoise travels, Achilles travels ten. Achilles runs ten metres, the tortoise one; Achilles a metre, the tortoise ten centimetres; Achilles ten centimetres, the tortoise one; a centimetre to a millimetre, and so on to infinity without the tortoise ever being overtaken. There is always a gap that cannot be overcome. But in external reality, Achilles does overtake the tortoise, and so logic is seen to be inconsistent with reality.

Is this an intelligent question? Is it an intelligently framed mirror of reality? To measure comparative speed, it is useless simply to say that one is travelling ten times faster than another. It is essential to be specific, and the moment this scenario is specific about the distance travelled, then we find the paradox to fall apart. For if every second, or specific time segment, the tortoise travels one centimetre and Achilles travels ten, then Achilles will, by simple mathematics, overtake the tortoise. And of course the same will occur if the length travelled is a millimetre to ten, or any specific distance. Zeno however keeps changing the terms of reference, leaping from metres to centimetres and so on, with again the more or less identical statement being made as in James' example; ie that one can if one wishes keep splitting the numerical demarcations into smaller and smaller fragments. Again, all the 'paradox' amounts to is faulty reason; the unintelligent framework of a scenario mistakenly imagining the internal logic to refer to something beyond itself.

Borges' position is that of gnosticism or at least gnosticism in the negative sense where, I would say for reasons of timidity before frightening immense and messy reality, a negative value judgement is imposed on life, and this when closely examined, as in the examples, simply amounts to flawed reasoning.

All-Powerful

A famous idea or paradox is that God could not conceive of something of which he could not do. God of course being used here in the sense of an all-powerful entity, and so since he could not do the above, then he cannot be all-powerful.
I have no idea to what extent this notion has been examined and what excursions of reason it has involved, but a possibly unexamined view is that the thought is being looked at from the wrong linguistic angle. The usual emphasis being that God could not conceive of something of which he could not do.
However, the same sentence can be looked at from a slightly different perspective offering a radically different significance.
God could not conceive of something of which he would not be capable. God is capable of not conceiving of that which he is incapable, and he achieves this by not conceiving of it. Thus an all-powerful entity's absolute mastery is maintained.