Logic and layers… the logic of layers / logic and language.
(Logic and Language I)
Always two; never only one.
(Reason alone… language alone… perhaps only one, language, but in language one is never alone, and with many languages… many reasons).
Logic within the bounds of language alone.
If natural language allows us to think anything, subject to restrictions of grammar, word order, well-formedness (all honoured in the breach by the creative use of language, in poetry and rhetoric), does it not nevertheless need further restriction (logic, semantics, pragmatism): the rigour of an artificial and specially created (formal) language to eliminate all trace of fancy from our thought?
Is there one such that can replace the vague amorphousness of natural language？
No one (apologies to Parmenides). Nothing (sufficiently) unitary. Singular. Self-consistent.
No ‘one ‘. Only two. Directly from zero to two. But what kind of counting is this? The logic of layers.
Philosophy is made of layers or fields, we might even say viewpoints, each with their own apposite set of concepts, initially read as dominant in their particular field, but whose field of operations are often expanded into other fields, claiming priority, as it were, over their neighbours, over other levels. Imperialising.
So at one extreme, the concepts of metaphysics claim priority over logic et al as too narrow, as excluding too much. And at the other extreme, logic claims the last word, relegating all else to the dustbin of the incoherent, the un-sayable, mere untidiness of thought. In between lie the intermediate zones peopled by such pretenders as ontology, epistemology, phenomenology (which three are usually regarded as ‘metaphysics’ by practitioners and opponents alike), and then we have the sociology of knowledge, history of ideas, philosophical history and genealogies, and assorted specialisms such as the philosophy of mathematics, logic, the philosophy of religion, political philosophy, and of course, ethics, together with its various applications (justice, rights, ecology, fairness, etc).
If logic reduces gibberish, then metaphysics increases scope. If the latter asks the big questions, then the former questions the big questions. Descriptive modes show evolution and contextual factors, while a variety of other approaches proffer various starting points as privileged (from phenomenology to psychologism, from rhetoric to ethics, from ecology to economics (and from Gödel to God)). Pick your own poison.
So each level in effect prioritises itself and de-prioritises the others… finding the source of fault in the others to lie in their starting positions. Viewpoint, position, level, each proposing an implicit hierarchy. Each vaunting its own advantages.
Metaphysics accuses logic of only being able to produce tautology (so only being able to pronounce on highly restricted areas - where not limited to producing quantitative forms of knowledge; but not any other kind … not least its own justification, which must lie elsewhere…).
Logic would deny metaphysics its scope; maintaining that this range is self-created and self-justifying, producing imaginary entities without limit (language or reason, left to itself, left to ourselves, will produce anything we want it to – if only we are clever enough…). Sophism.
The other regions of philosophy, all have their own rationale for excluding or demoting the others in favour of themselves. Each announcing itself as the privileged staring point… yet what else can they do, for to do otherwise would immediately plunge them into self-contradiction, via the paradox of the Liar (by deferring the privilege of being their own starting point, they declare their own insufficiency). As in giving the Other the last word, the paradox at the heart of the Romantics’ valourisation of Nature (as opposed to Culture or Language), as of our dependency on an a-historical ‘outside’ for our ‘foundations’ (be they heaven, eternity, or universals). So each level, each region, evinces its own particular strengths… and its own weakness. A case of blindness and insight perhaps?
Logic, to be able to be rigorous, to be able to elide the errors that arise in the use of natural argumentation or natural rhetoric, uses axioms (the same is true of any artificial system of language from mathematics to computer languages). These axiom must protect the system from error production by making its workings transparent and providing a foundation which is unassailable.
Yet the axioms that logic uses to protect itself from the production of rational unintelligibility may themselves be used against itself. The two taboos that guard the gates of reason are the prohibition on any reference to a meta-set and to any form of self-reference (the varieties of axiomatisation available are pretty much all reducible to these two). In effect we may not make sentences which involve reference to a final or limit case, ‘meta-sets’, nor may we employ any reference by an element in a set to itself (the paradox of self-reference or self-grounding) - both of which risk leaving us with both A and not-A, or infinite repetition or regress (‘bad infinity’) in both ‘directions’ (‘out’ and ‘in’, ‘outer’ limit case and ‘inner’ self-reference). Yet it must be admitted that this restriction does bar a spectacular amount of nonsense from our argumentation. Unfortunately it also defines as non-problems many, even most, of the things we find worth arguing about…
Furthermore, it appears that (an axiomatised) logic damns its others (other ‘logics’ as well as other arguments) for breaking the rule of ultimate exterior or intimate self-reference whilst itself committing this error in excluding them (founding exclusion, definition against, and only permitting ‘truth’ to emerge through reference to itself alone ((self-reference as transgression)). Paradoxically, it is precisely by excluding all that is outside of itself that it finds itself left alone in an incestuous self-engendering, so subject to the paradox of infinite regress.
One problem with logic is that everything outside of itself can function as its meta-set (it relies, finally, upon ‘the rest’ of language, which came first); so exclusiveness again functions as if a reference to an element in a set; so again the self-reference taboo appears to have been broken. Also logic’s insistence on itself as the only valid starting point (reason alone) is prior to its own logical operations, so also an act from outside… a deus ex machina. But one which must not be referred to… for if it is, infinite regress again results… as the exterior element is ‘taken in’ so leaving ‘something else’ to further play the role of ‘outside’, of an infinite extension of context. This kind of thinking is best left to natural language, whose job it anyway is…. And perhaps the use of natural language to philosophise (think, construct or criticise) is the best description of what it is to do ‘metaphysics’?
However, in the case of metaphysics, there is no exclusion, rather the total inclusivity of metaphysics is its problem, which already means that the taboo of self-referentiality has been breached… everything is always already inside. But if everything is always already inside then, in envisioning it as such, we are talking at the extreme outer limit, at the margin, even at a god-like point in our ‘totality’ of perception, so ‘outside’ the putative totality… (but once ‘said’ it is, in turn, ‘brought in’, etc). Again reference to a meta-set is inevitable (as is infinite repetition ‘outwards’ in dealing with this, as we keep taking up a position ‘outside’ the previous one…). So are the two alike; our two taboos, the two key or fundamental axioms, are they two facets of the same problem, two faces of the same knot, twins begat of the same womb? A case of words needing a fixing point outside, or an autochthonous birth, a rooting which (the slide-ability of definition inside…) language denies… so always revealing a ‘fault’ somewhere later along the line…? Or is this rather a question of limit as the problem of the unlimited potential of language, leaving any artificial or second order language always (already) in contradiction with the first order language which supports it; but which it, in an act of matricide (or patricide), sought to supplant….
So no ‘grounding’, no ‘rooting’, or ‘beginning’ point; rather only endless tautology. A saying of that which we already know? (Plato would have approved of this…) So on yet closer examination even these limitations, this sacrifice made for clarity, the gift of (in both senses) tautology to the world, this too appears on closer examination to fall into contradiction when we examine further its relation to its origins, its context, and (crucially) to its inner consistency.
In fact it would appear that any attempt to seek justification (for rational domination over things and language) from the ‘outside’ results in a contradiction (the reference to an exterior, anterior, to a meta-set); just as any attempt at total rigour only leads to ever greater contradiction, as self-referential claims become self-undermining. The logics of ‘external’ or, indeed, ‘eternal’ justification are dealt with by deconstruction and post-foundationalism; just as the limits of internal coherency mapped by logic as source of the ’last word’ meets all the contradictions as listed from Kant to Gödel, as well as the problem of excluding context and the embedding of discourse owed to life (in the later Wittgenstein’s critique, later Habermas, the problem of infinite con- or co-text in ‘deconstruction’, etc.).
So the axioms we use to make logic work (as a second order language, along with other formal languages such as grammar and math, number theory and programming languages) also apply to our choice of realm, or region of philosophy or thinking. Reference to an outside and (an equally infinite) self-reference, both operate equally well as a critique of logic itself (as opposed to being its saviour) as of the field of thought in general. Reminding us of the limits of ‘beginnings’ in a world that has (always) already begun…
A field whose paths are tortuous, self-undermining, prone to leaps and loss, all natural, even necessary; source of our creativity. Not to be feared if we only remember to check (in both senses of the word) their potentially infinite meanderings in terms of their reference or in an empirical moment (which is how theoretical physics works). And for some human requirements cohesion alone will do.
In part it is the logic of the starting point that is the problem; we always have to begin with the language we have -and then try to customize it- or pretend we have escaped its ‘problems’ when it persists as a substratum to which the formal language can be reduced, but which this formal language can not, itself, subsume, leaving the natural (received) language as having the ‘final word’ – in practice a ‘final word’ endlessly deferred (see Russell and Whitehead with respect to the attempted mathematisation of language: in fact only the reverse procedure turned out to be possible; the natural language is the ultimate meta-set - so not reducible to a more limited range of expressions… we may tidy-up ‘bits’ but they are only ‘bits’ and not the whole, the latter is always a moving target). And so not only perpetually deferred, but perpetually subverting… as the natural language is always ‘more powerful’ than all attempts to customise it… (as a restricted set artificial languages must defer to natural language for their explication and starting point, so subverting their claim to logical priority (they are ultimately dependant on that which they seek to overcome)).
So we are left observing from within – in transgression of the taboo on self-reference. Logic, by contrast would try to view language from without, an existential taboo (playing god, condition of enunciation) as well as a logical taboo (reference to meta-set).
reference to a meta-set also points to a banned logical topology ‘outside’ (a
taboo on the name of god, on gazing upon the face of god) where the infinite
reach of the exterior zone, its permanent ‘otherness’, or un-reachability,
echoes the infinite repetition of the logical operation ‘upwards’ where one
zone once reached implies another without. Whilst mirror-wise we find that
self-reference, as self-definition, is suggested by any reference to an
incestuous inside… Twin references to the horizons of sense, twin surpassings
of sense, as the mapping (of the limits) of sense.
Family resemblances: Let us look again at our two limitations and their parallel operations in other areas of discourse. The first, reference to a meta-set, resembles the act of grounding outside, the ‘logic of the outside’, anchoring outside, the rhetoric of eternity, transcendence, deus ex machina… (by the act of looking out, looking up, for a timeless place for our eternally true first cause, dooming ourselves to the eternal repetition of this operation). The view from without (language).
The second, self-reference, or reference to an element in a set as part of that element’s definition, resembles all attempts at self-grounding, the issues of embeddedness and relativism, of immanence, of autochthonous-ity and self-given teleology. Always assuming what we should have previously proved… (by the act of looking in, looking down, to found, to seek ever deeper for the foundation, dooming ourselves to the eternal repetition of this operation). The view from within (language).
The first, a psychological habit, putting the foundation outside; the second, our permanent condition within language…
…of ones self within language, but then to position one self without, outside, language, (leaving one self within) impossible - but necessary; one self in and one out, two… (source of every paradox, within language and in logic).
As a staring point both axioms position themselves on either side of a relationship to language: one notes the contradiction at the heart of positioning oneself ‘outside’ language, yet how to operate without some such point of ‘reference’ (humans appear to need some such point, no matter how fictional, illusory or provisional, not least to ‘ground’ or ‘guarantee’ ethics and values); the other notes the impossibility of any absolute starting point within language itself (the restriction the application of the axioms would involve, only restrict the sayable to an even narrower set of sentences within language).
An ever receding ‘outside’ as each becomes an ‘inside’: an ever narrower set of sentences…
Which may not be such a bad thing. For the resort to tautology after axiomatics opens up the path to second order or artificial languages. Actually very useful (akin to ‘bracketing-out’ in phenomenology and the elision of the complexities of process in formalist or structural analysis): especially if further tempered by reference and empiricism. But is not the same true for ‘pure’ reason with its potentially infinite spread - also requiring an (external) empirical moment. Reason as careful reason (not too metaphysical) or a looser logic (not too excluding) in alliance with a sense of (illogical) reference or an empirical moment. But the necessity of Reference and/or Empiricism already denies Logic and Metaphysics their dominance (‘Reason alone’) and acts as a corrective to both modes of thought in their moments of excess. (‘Cohesion’ is the other path of ‘tempering’ available to us; a path that returns us to natural language and (all) its resources – yet this move itself may also be seen as requiring tempering, by reference or empiricism as appropriate).
So it is that logic also appears to be, at once, caught in incestuous self-reference, and, at the same time (like metaphysics), is found to be sitting firmly on the ‘foundations’ provided by reference to the infinite realm of the meta-set. Perhaps this is what natural language (our language) naturally does. Perhaps, indeed, this is what our minds do – the lack of limitation being our cue to creativity, to imagine the moving of mountains, to thinking afresh… (as well as producing mountains of gibberish). So perhaps a non-normative, but a ‘descriptive logic’, as used in natural language (with all possible contradictions), would seem to be the most useful thing. Just as we use both A=A AND A=not A (especially when it, ‘A’, becomes B, or when we cross a level, ‘up’ or ‘down’), according to context… and still manage to understand what we mean.
As evinced in ‘contradictions’ as banal as the Zeno paradox, that of the tortoise and Achilles, where the finitude of a distance is contrasted to the potential infinity of its subdivision; however what should be contrasted is the finite time it takes to cross the given distance and the potentially infinite subdivision this distance may be subject to. The latter (level) in no way affects the former, so no contradiction…
Axiomatised tautology is already a problem (tautologous system plus limits, Wittgenstein’s last word in the Tractatus, the ‘last word’ of the ‘early Wittgenstein’, demonstrates the irrelevance of pure reason in its most perfect recent incarnation, the propositional calculus). But after Gödel demonstrated the insufficiency of all systems, does this not mean that even axiomatised tautologies are open-ended… (so in-practice imitating their parent, natural language). And does this strange and unwelcome fact not mean that they are even less reliable than thought, despite their surrender to tautology and the insurance policy of axiomatisation? Or does it mean that they always imitate natural languages in the (distant) last resort… and so may be more useful for being so…?
So the choice once was: An axiomatised tautology versus … that which the axioms (attempt to) forbid: a starting point rooted ‘outside’ of itself (a reference ‘up’ to its meta-set – which may, in turn, become infinite); or the equally infinite digging ‘down’ (another ‘bad infinity’) of self-reference as a starting point; a rooting ‘inside’ of itself. Yet according to Gödel, all axiomatised languages ‘unwind’ at a certain point; contain a ‘hole’, a ‘lacuna’. So returning the formal languages to the state of natural languages (even if at one ((useful)) step removed). An uncanny doubling back not unlike the return to the arrow of time of thermodynamics, into a physics previously thought to be (theoretically) reversible (as in the potential shrinking of the universe after its expanding); an irreversibility courtesy of chaos theory (as part of the pattern evinced by the ‘strange attractor’). Or by the arrow of time implied in our differing perception of past and future within the experience of the eternal present? So it is that both unaxiomatised ‘natural’ logic (‘metaphysical’ or taboo-breaking) and post-axiomatised logic (those logics which have ‘unraveled’), both forms of reason, all return us to natural language, and so are finally open-ended…evolving as (or with) natural language itself. The conclusion is that reason in natural languages can have no end; is open-ended; is infinite (given sufficiently wide-ranging and interesting material). Where we decide to ‘stop’ the argument is occasioned by where we are and who we are (as is the case in limiting the numbers after the decimal point). And that is were the argument is… (in perpetuity…).
(Indeed the ‘axiomatisation’ of formal languages is precisely the limitation that prevents their use in the examining of themselves, their origin and context of use (their outside as meta-set) and their self-justification (their inside as self-reference as ’bad infinity’). Whence the tautology… the ‘closed’, or formal aspect… which when recognised as such (as a tool made for a task within a wider field, as provisional restriction of range is a tool of many sciences, not least linguistics) is fine. But the abrogation of the whole field, including the denial of (the existence off (the question of)) its origins and context is perhaps the true transgression of an axiom… Its virtue, that of appropriateness and limitation to the task it was designed for - or can do without stretching credulity or denying the obvious).
So back to ‘reason alone’ as ‘language alone’, but (this is another issue) requiring empirical/referential correctives (so another kind of reference to an outside…). Lesson: whatever mode of argument is used; reference to fact, testability of hypotheses, test of language in context, pragmatics of use; all or some (one) of these modes will be required at some point.
(Yet by language alone… clearly implies circularity within language, a hermeneutics, like deconstruction, or, more broadly, historically, the circulation of intellectual fashions, of ‘positions’ that reappear periodically under a different name; the argument is where we stop the definitions and argument; each ‘stop’ however (necessary) risking an exclusion that leaves something important unsaid…. So the process of debate (which we call philosophy) circles on (or perhaps that should be ‘spirals on’ ((‘upwards’))…).
Gödel has shown that all attempts at consistency lead to contradiction; just like inner self-reference, self-grounding is but another name for the internal definition of axioms. Likewise the meta-view is also implied in any overview with the aim of consistency… the consistency of the … whole – but from whence do we see this? Which position is outside (of ourselves) just as which place is prior (to ourselves). Or better, which point of view is outside of all of our points of view (but which we can nevertheless know) and which thought is prior to all our thought (but which we can nevertheless think)). Damned if we do and damned if we don’t (the promise of tautology frays if we demand consistency…). In plain English, the usefulness of such logical operations is limited to areas apposite to their (largely quantitative) mode of mapping. And both types of axiomatisation (and their impossible avoidance) require a place which is not ‘this’ place (not this level) require another level such that we can think it and not be able to think it. And so A becomes not-A when two such levels are collapsed into one (proposition). So again, one limit turns into the other. One becomes not-one. Always (already) two.
Zero to two; the ‘eternal present’ and its Other (others…). Court on which the game is played.
So the state of play is one where we recognize out dependency on that which we can not believe (its insufficiency). If axioms once gave us formal languages which restricted meaning and field of operations (above all supporting the quantitative), so differentiating itself from ‘metaphysics’, then now such languages are shown in the last analysis also to be liable to the same contradictions and aporia, self-confusions and under-minings, as natural language (naturally used reason, logic or rhetoric).Which if they were a part of natural language, which they are, is precisely as we would expect it to be…
(None of which is a problem if we regard formal languages as man-made, as ‘formalisms’ eternally indebted to their parent, natural language: if we regard all our logic as the result of an ‘hardwired’ foundation, or ’intuitionism’ (perhaps closest to Kant) then we are bound on an endless search for the best verbal formulas for this internal (and unreachable, indeed infinitely regressing, Grail). But if we require that the efficacy of logic, of numbers, is found to rest on an objective exterior, their ‘Truth’ (aside from immediately suffering from the paradox of the reference to a meta-set) then we find ourselves quickly in the realm of faith alone, a latter day Platonism… blind belief (a return to blind assertion by any other name)).
So finally back to two; two layers. ‘Ours’ and the ‘other one’, the one we are now ‘inside’ of , and the one posited ‘outside’; indeed the ‘now’ position and its other, to which we may move, or may have moved through, an other which may be ‘before’ or ‘after’ – in another uncanny parallel with our basic, indeed inescapable, levels of temporal experience; the ‘eternal present’ and its windows opening onto the different levels of the past and future. Together with the generalisation of the ‘eternal present’ into the ‘Absolute Outside’ of ‘eternity’, offering us the basis of the binary sub-divisions that dominate (and often disfigure, that is, configure) our fundamental ways of thinking…
layers: If ‘O’ (’zero’) gives a monism (as viewed from within) as viewed from
within there is no ‘one’ else (not ‘
No Parmenadian One, only none (No-one) none; or two.
AND two… (where 0 and 2 equal two… as in two forms of temporality, past and future, or two levels, ‘above and below’ or’ in and out’ or ‘in-front and behind’… )
Despite the various models and forms, in mathematics and in logic, of sets, axioms, propositional calculus et al with all their careful working out and construction and reconstruction of formal languages, all problems finally resolve into, return to, can not avoid the problem of: the reliance on the other, the requirement that there be two, a minimum of two layers. With the concomitant problem; reference to the top limit and to inner self reference, to tautologise or not to tautologise (and so fall into contradiction… but it now seems that even tautologised systems must do this, must be incomplete). So perhaps these twin problems are both one, an expression, two expressions, of their difference; (adding a level up and adding a level down, meta reference and self reference, are they both aspects of one problem, the adding of a level, the final reliance on another level) the adding of a level by focusing upon one level alone (‘inner’, or ‘outer’); the problem occasioned by the movement between two levels, across a field of difference that is nevertheless required (in turn) to make the logic or other formal system work.. And un-work (not work). At the same time…
So self-reference does not remain on the ‘same’ level, but immediately pushes one down to another level (‘down’ as opposed to ‘up’, as ‘up’ is more apposite to references ‘out’) yet we do not think of going in ‘up’ to another level for foundation or ground (intuitively ‘foundation’ or ‘ground’ should be ‘down’… yet we use ‘up’ because of its sense of outside or eternal verity - a sense inherited from myth, religion and our sense of the descent of the vertical, part due to light, part to gravity). Going ‘up’ or ‘down’ a level suggests symmetry: yet is going ‘down’ a level for grounding, as opposed to going ‘up’ one for such, really symmetrical (or like the arrow of time with respect to the past and the future, asymmetrical, regardless of our imaginary wonderings to and fro…)? Perhaps the key point is that if one level is said to support another level it must be prior (that it equals the past, like the assumption of the truth of a sentence at first sight, awaiting contradiction) implying something must be a next stage means it must follow (that it equals the future, like the assumption of falsehood in a sentence which has been found wanting, the second or following step). Yet is it that in other readings, other examples, both ‘up’ and ‘down’ directions may include layers that may be read as prior? The present as ever-present event horizon suggests all (else) either as past, as logical implication points back to causes, or as future, in the move, imaginary, but also implied, to an event that has not occurred - to results (with non-reversibility, perhaps taken as an axiom, as to the position of the present as moving only ‘forward’, away from what was to what has only been imagined, or supposed, often incorrectly…). Either/or: present only in what is non-present. (Semi-present). Framed by the present. Either/or: could it be that this pair is the product of this (temporal) relation or intuition?
Asymmetrical: the usefulness of this image is: as clearly informing us that there is no place to hide (the foundation). Both this and an(any)other level all are fallible (are not final). All places (levels) are liable to infinity or contradiction.
Symmetrical: mirror images; so reducible to one difference (with two references (or a difference further divided) on either side, up and own, in and out). Finally one kind of infinity or contradiction?
Or (again) asymmetrical, two kinds on either side of a difference, two because different in kind, not (finally) isomorphic. Their difference finally offering two kinds of infinity or contradiction?
Two layers, one as containing the other, upper ‘meta’ limit as contradiction (infinite progression upwards or ‘out’), and inner as self-preferential contradiction (infinite progression inwards or ‘down’). If in parallel, then no relation. So one must contain, ‘nest’, the other… whence the origin of these axioms or contradictions….
But strangely irreducible to one, the same difference, the ‘same difference’, a contradiction in terms, the importance of binary relationships to human thought…? A bit like the intuitive sense of non-collapsibility of past and future (and vice-versa) of temporal experience. Both are semi-present in our experience or time, but one ‘has been’ and one ‘never arrives’ with a line ‘connecting them’ drawn through the ‘eternal present’, a line that is uni-directional… So both are finally not reversible or reducible to one. Like inner and outer limits? Or does the popular confusion of subjunctive and indicative, of what we wish for and what actually happened, indicate a zone of over-lapping?
Back to natural language… as the quest for the holy grail of an artificial language (formal language) designed to offer fault-free calculation or calculus have foundered on the rock of tautology (expected of a formal language which is, by definition, closed) but also on the (Gödelian) joker in the pack of a ‘hole’ or contradiction (axioms and tautologies not withstanding) which emerges to sabotage the Holy Grail, with the impurity of an unavoidable inconsistency (turned positive by some, seeing in this ‘leaking’ of closed systems an approach to the open-endedness and so creativity of natural languages). Leaving the latter also as imitators of the ‘faults’ of the former. So when all the forays into number theory, logic and other formal languages (grammars, computer languages and Artificial Intelligence) have convinced us only of the secondary nature of such languages, their fictionality, contingency and provisionality (all tempered in terms of their use, which finally oversees the decision to use or reject) in sum their dependency upon the natural language which is required to understand them, and which they can not encompass (it possesses infinitely more power that any of them – together with the concomitant ability to create an infinite amount of nonsense too…)… So when all is said and done, we are returned, both for comprehension and for the general communication as to their worth, to the realm we have evolved over thousands of years and can all use with marvelous ability, our mother tongue (and for those fortunate enough, for the -truly- bilingual, a shadow language (which also implies being bi-cultural)). So it is that Gödel finally trumps Wittgenstein by showing that artificial languages are in the end like the natural languages (naturally enough as it is the latter that spawned them) with respect to the diseases that their formalized offspring were to be the cure. (Wittgenstein had, along with others, shown the limitations of formal languages as their strength, tautology; yet in so doing leaving all the really interesting questions of the human condition to a realm ‘beyond logic’…so echoing the findings of twentieth century pragmatic linguistics with respect to meaning). Yet in this Wittgenstein may be read as having the last word, as, in his latter philosophy, he finally leaves significance, or meaning, in the hands of our language and its contexts, its use - again echoing the findings of pragmatic linguistics with respect to meaning…
If all the fuss only results in tautology and self-contradiction, well then (Kant and Hegel had already dealt with this) thinking goes on - in need of a little training perhaps, and certainly in need of testing, but not replaceable by a narrower – putatively, error-free form…
Otherwise put: from the ‘inner secret’ to the ‘outer limit’; a range of human experience only subject to rarification (the path from mysticism to logic).
Therefore the solution to the ‘problem’ or maybe even ‘scandal’ of the impossible choice between using tautologised second-order languages or rejecting these self-same languages is the recognition that the incompleteness that, it now appears, is endemic to formalised thought, is, in fact, endemic to all forms of thought, to all kinds of language (formal and natural) and furthermore that, in terms of creativity, they may well be essential to all thought, but not to Truth (necessary but not sufficient). So therefore the most useful strategy would be to differentiate between the ‘good’ and ‘bad’, useful and non-useful, manifestations of the two main contradictions or aporia that result from such incompleteness, to differentiate between the ‘in and out’, or ‘up and down’, ‘bad’ infinities and other forms of open-endedness that effectively breach the gold standard of A=A, such that we can use them effectively (as we do in the use of ‘fuzzy logic’ in the programming of… domestic appliances – a not inconsiderable taming of the putatively iconoclastic ‘breaking of the bonds’, or twin transgressions, of logic…). Useful, because unavoidable.
Logic and layers within the bigger picture. The determinations of species–being + the determinations of culture = our experience, our perception. The foregoing + the open-endedness of natural languages and the creativity this permits (a permission guaranteed by the isomorphism between natural and artificial languages, by their open-ended nature) = the possibility of new thinking, of a ever-fresh conceptualization of a ever-changing world.
Last word on the non-isomorphism of the two sides (‘up’ and ‘down’, ‘out’ and ‘in’). Is any attempt to reduce them to a single difference doomed? Two viewpoints, or two languages, or two levels are required; but the difference between the two is never quite one (we are never quite locked in to the ‘prison-house’ of one); we are left showing two ‘directions’… which never quite permit collapse into one. Irreducibles, like (infinite) repetition, show us that we have ‘hit’ something, what that might be is quite another matter… (indeed a matter for matter itself, our constitution as mind or ‘spirit’ on the foundations of a prior material level…).
(So like the ‘arrow of time’ a part of our metal hardware, our way of dealing with, of forming, ‘difference’?)
Within (the bounds of) language alone.
Copyright Peter Nesteruk, 2012.