Zero to Two: Logical Relations in Time
(Logic and Natural Language III)
Reason is never alone.
(Just try leaving it alone and watch what happens…)
If we begin again (always again, as what is prior already exists (as what pre-exists always already exists) ‘in medias res’, so to speak) comparing the limit problems of logic to our temporal embodiment, reducing our key logical categories to our experience of time, then founding a (kind of) logic based upon temporal experience, on the limits of our temporal being, on the eternal present and its relationships to its others… on our temporal intuitions, including one not even semi-present, but (one) entirely absent -eternity- absent yet continually referred to - on these ‘grounds’, then, such an attempt, such a temptation, seems worthy. Reducing, founding, or… finding; finding the parallels that already seem to exist between our experience of time and the problems of logic… and what they might imply for the idea of a temporal logic - such a project, such a thought-experiment, now begins to seem feasible.
If first we read all sentences as true (as if true, until proven otherwise) then we infer a meta-set (we begin with truths; a preexisting, guiding assumption). Conversely, if we read them as (if) false (as waiting to be disproved) we infer a negative meta-set, such that all sentences are treated as false (including this starting point itself) and so self-contradiction results; then we are into the realm of self-reference, or inner, downward, contradiction as we chase down levels (if true then false, if false then true, etc… and so on to infinity). Yet the way of the meta-set as the assumption of truth, also evinces a contradiction-producing blind spot (all possible, all makeable, sentences are not true, only some), so requiring a further meta-set asserting a new axiom, a new outside, or frame. And as each new set becomes ‘inside’, so a new ‘outside’ is called for; for its own self-justification depends upon moving up a level to assert a new staring point, axiom or a priori (putative universal). And so we witness ever more layers piling-up ‘outside’ to support the assertion… All such ‘levels’ are of course prior; they permit the construction of the levels that follow on the foundations of what went before… (and so on backwards to infinity). The contradiction-producing nature of the axiom of falsifiability as the default starting point for reason, the given priority, is bad news on an absolute level (‘all’) for Popper’s methodology of ‘falsifiability’: but not invalidating its usefulness in particular cases (‘some’).
So it that we find two kinds of logical contradiction or limit: infinite outwardness and infinite inwardness; infinite extension and infinite intension; infinite repetition as addition or accretion of layers and infinite repetition as alternation or circling; the notching up of the meta-set and the downwards spiral of self-reference. The repeated extension of the external view (always another step ‘out’) and the repeated intension of the internal alternation of viewpoint (always another step back ‘in’). For example as in (‘up’ =) a god is needed to create … but then who created god (then who created… etc) and so a potentially infinite succession of meta-sets ensues (A=>Ax (such that ~Ax=>~A)). Or as in (‘down’ =) all is in language… but how do we know; we need a viewpoint outside of language to know this, but for humans all is expressed in language, (but we need a viewpoint outside…) and so neither ‘in’ nor ‘out’ will do and we are referred from one to the other in an infinite or ‘vicious’ circle (A=> (A^~A)).
The assumption of ‘truth’ or meaningfulness is our habit in language, the default (or ‘unmarked’) position in linguistic experience (Grice, Pragmatics); is what comes first (unless the context or co-text, inclines us to suspicion… to the assertion of metaphor or some other ‘second meaning’ – again a relation of priority, to choose non-sense or another form of sense, the ‘tenor’ or second meaning of the phrase or proposition in question). What comes first and what second is a matter of priority, the definition of priority, a matter of ordering, in time, a matter of temporality. (As we witness in numerous aspects of our daily life: the importance of the place on the page, the implications of left/right positioning or movement, of the old/new information order or focus in unmarked and marked sentences, all these involve the signaling of priority; finding a difference and asserting a priority.) It is this assumption, that of ‘truth’ or meaningfulness, that validates rhetoric and its employment of all the creative means available to natural language, and frees it from the invalidating requirements of logic (alone). Innocent until proven guilty.
So the assumption of meaningfulness and its utility, this insight, would appear to be true of contradictions too. What works for ‘ordinary’ sentences and propositions appears to work for language at full stretch where analysis seems to teach more about thought and language than about the objects they purport to represent; it is certainly the case that paradox can be found to illuminate the problems it poses (the prior article, on Epimenides’ famous paradox, ‘Logic & Language II’, was all about this). The whip of contradiction is required to send us into search mode…
The result reinforces the importance of priority or ordering, already evident in our starting point, the choice made for a default assumption of truth or falsity. The move from first to second, level or layer, stage or step, constructs a directionality, even a causality… such that there is an irreversibility from outer to inner, from upper level to lower level (as from assumption to inference in order of discovery if not in temporal precedence) in the limits of reason as exemplified by axioms concerning the use of the meta-set and the virtual taboo on self-reference. So we may posit two axioms, such that two kinds of paradox result: a ‘default’ paradox and its alternative; ‘almost’ mirror images of one another, but not quite, so not reversible; existing in a relationship of priority, such that one, the default, comes first (in all matters that count, that can count, also a matter of priority, of ‘ordination’, or ordinality). In this way making these terms relatable to our human temporality… also unidirectional, irreversible. Indeed it is in our experience of temporality, and perhaps definitively so, where directionality also implies (the possibility of) causality – what we usually call ‘the arrow of time’.
(This brings us the stage where we can take up the argument where we left it in the article on Epimenides, ‘Logic & Language II’, it is, however, not necessary -though possibly helpful- to have read that article first).
Only one starting point is habitual, is ‘default’; the other is its alternative; there is a question of priority, of directionality, indeed of temporality…but of cause and effect…? Indeed the assertion of the meta-set does imply a nesting process … or bad infinity, as we move to ‘higher’ levels. Is the contrary the case? Indeed the arrow of time does appear to point ‘down’ (as in Chinese from shang上 to xia下). The question is : is this directionality reversible (‘ illusory’ as in certain kinds of physics) or irreversible as in the entropy implied by the second law of thermodynamics? And does the arrow of time itself have any connection with our basic, intuitive, sense of temporality; our situation in past, present and future? The past or prior (level) is implied in the present as memory; we might say that the present implies a prior state such that certain non-present, or semi-present images are posited as past (if not this then they are fantasy, or a projection forwards, the future). The ‘after’ that comes after the present (level), the future, projection or prediction, is also an implication based upon the past (the reverse seems untenable, thus the dissymmetry of the two ‘directions’ or semi-presences that point away from the present). Both of these other levels, or ‘directions’, of course, may be designated as potentially infinite. An infinity which may be temporal, reaching backwards or forwards without end, or it may be based upon the accretion of levels.
Matching intuitions. We are haunted by the semi-presences of past and future, as if they were ghosts forever trapped in the present. Are these ‘other’ secondary levels of human (temporal) experience like the sense of levels in general? Or are they perhaps the model for such, their template? And as with other levels, we may take their standpoint, that is, we may enter them, they then may become the present (level), in our imaginations… but not in reality, where we are bound to one level, the present (one). Which does not stop us from experimenting, from building on this intuitive foundation in the ‘other’ realms of human thought…
Is the intuitive basis of our idea of ‘levels’ based upon our experience of temporal ‘levels’; the next (proximate) levels to this (present) one; the past and the future, the sense of a ‘before’ and an ‘after’; ‘windows’, tags or the presence of that certain ghostly flavour that indicates that our current mental image or perception is not part of the present? These temporal basics of our experience, our most fundamental temporal intuitions, are they the model for certain basic elements of logical thought (‘up’ and ‘down’ levels, meta-set, self-reference, the problems of ‘inside’ and ‘outside’, even ‘all’ and ‘some’)? Is their intuitive, indeed foundational, difference, the basis of our sense of axioms, sets and elements?
The elements or logical operations that take place in the present offer no problems (‘and’/^; ‘or’/v; ‘implies’/’if x then y’/=> taken as non-temporal); yet already we can see that ‘a or b/ a v b’ implies that ‘a and b/ a ^ b’ can not be true at the same time. Logical operations that cross levels are most obviously affected by our intuitive sense of temporal difference or levels (‘not’/~ as possibly pointing elsewhere [not ‘now’, but not ruling-out another time]; ‘implication’/=> as possibly temporal/causal; ‘All’ and ‘Some’, as temporal regarding their adverbial implications [often left unexplored: ‘all’ or ‘some’ as possibly true at ‘all times’ or at ‘some times’, so ‘in eternity’ or in history; all x always, some x always, etc; the unrestricted implication of ‘All’ is all cases, at all times…]). So offering parallels with the semi-present ‘other’ levels, of past and future, in their relationship to the present – and of temporality’s complementary other, eternity. Posing the question: the intuitions that offer other levels, such as those of meta-set and self-reference, are they linked to this intuitive semi-presence? The crossing of a level from present to semi-present: otherwise put; a presence which is present in the present as semi-presence. Becoming, in our formal, artificial languages, as if (the) present (themselves) when we have stepped into one or the other - stepping ‘up’ or ‘down’ a level - exactly as if into the past, into the future.
Into the past… including everything that went before, as the intuitive, or natural, meta-set (or infinite sequence of nesting meta-sets, each ‘giving way’ to the one before). The past as prior, originary, on which the ‘moment’ of the present floats (an element in the set of the past, in the set of ALL our experience); the past of the re-construction of causality; the past as ‘last word’. The present, however, is actually a (or better, for each of us, ‘the’) ‘Eternal Present’; our ‘eternal’ (while we are conscious) sense of the present; which leaves us with the, ‘our’, Eternal Present as the final meta-set, the final frame of our experience (the place or level from which we start, and which we can not leave). With the exception of this all important foregoing proviso, then, a succession of nestling meta-sets offers a ‘backward’ temporal directionality (A=>Ax (=>Ax2 (…3))).
Or into the future… which has not happened, is not, and so is … here, as the intuitive, or natural, field of self-reference (a potentially infinite sequence of circularity, each step suggesting the one that follows). The future is (like the past) in the present; but the past is supposed to come from the past, whereas the future can only come from the present (actually a projection of the past as repetition/variation of this previous experience). So not really in the future, in actuality always in the Eternal Present; it appears to point elsewhere, but actually only points back to the same (so offering a confusion of element and set). Otherwise put, the future is ‘not yet’, is not, is a negation which is also posited as ‘true’, therefore self-reference (A=~A) as self-contradiction.
So leaving us with the conclusion that level-crossing is necessary; but inevitably contradiction forming… but that the contradictions may be treated as productive in certain cases.
Furthermore, as we have seen, in actuality, escape from the Eternal Present is impossible, whence the (semi)fictionality of past and future (as in the posited existence of places where we are not, which are not present). In this (and only in this) sense their eternal semi-presences are ‘real’ only in the imagination, the thought experiment, or in artificial, formal languages. Whence the contradictions which must beset them, as natural language, in the sense that the Eternal Present always (must) return to undermine their foundations, their axioms, as unnatural, as excessive, as closed… Whence the root of problems of relations of natural to artificial languages, foundational in the openness of natural language, rooted in temporal experience and its tension between layers (present and not present (past, future)). With ‘Eternity’ as ‘not not present’, we are returned to the present on a ‘higher’, but non-existent, level – describable, temporally and logically, but apparently not just a (culturally-specific or logical) construction: actually structural to human being… to human thought… Whether manifested as meta-set, universal, a-historical, eternal, heaven or hell, or as Absolute Other, where we form the Absolute Outside, the absolute other (of) time as ‘Eternity’, as a grandiose repetition of the Eternal Present. Platonic shadow thrown on the wall of the cave of the present indeed. Shadow of the pretender, so often taken as prior.
The question of a temporal logic, of assuming that some at least of our intuitions concerning logic come from our fundamental experience of time, and so deserving of an examination, a ‘thought experiment’ creating a ‘temporal logic’, applicable, to be sure, to a narrow range of uses, such a speculation will be further explored in what follows…
Meta-sets and self-reference appear on either side of a ‘present’ (level, sentence, proposition) – whose presence, the present, what is presented to us, is intuitive. Not exchangeable, not interchangeable, they (meta-set and self-reference) offer a model for past and future (or, indeed, I would like to suggest, may be ultimately modeled on, that is based upon, the intuitions of ‘past’ and ‘future’ in their relation to the present; on the intuitive sense we have of the difference of ‘pastness’ and ‘the future’ over and above their difference from our sense of the present; their existence on ‘either side’… of where we are). Isomorphic, perhaps even (because) genetic (again, perhaps, in both senses of the word).
However it is the present, ‘of course’ (obviously), that is the final frame for all (that is ALL) finally (intuitively, experientially). The putative meta-set of the past is therefore answerable finally to this definitively (experientially) final meta-set (in the sense of, in actual experience, always being contained by it). Final, because we escape from it completely only in death. Is the self-referential sequence therefore allied to the future as on the other or opposite side of the present? This ‘final’, indeed we might as well say, ‘originary’, meta-set, the present, may be its (the future’s) beginning, and all is framed therein, but the self-referential type (of -infinite- extension) does not have the same, or similar relation to the present (as final frame) as does the meta-set type of infinite extension that we call the past; so we may posit it (the future) as ‘unreal’ (the future, possible, conditional, subjective, etc…any way ‘not yet’). That is to say, all past sets (memories of references to the past) have as their final meta-set, their final frame, the present; but that there is no such mirror relation for the self-referential extension which relies on, or begins in, the present, but does not imitate it. So (as in cause and effect) the past tries to be the final meta-set for all that follows, but fails, being returned to the present as such (it is as the implied ‘prior present’ of the ‘present present’ that it owes its sense of a meta-set; from its point of view it is the meta-set, but when we return to the actually experienced present, this then becomes the natural or default meta-set of all, lapsing only when we need to assume another point of view). Whilst the future has no such pretentions… having only to refer back to itself to continue ‘forwards’. It does not ‘come’ from the future as the past ‘comes’ from the past, from memory (rather it is ‘projected’ from the present). Self-referentially relies on previous levels in order to move, or posit, ‘forwards’… as these levels must ‘build’ on some thing… so they are built on the present (and underground, as it where, ‘behind’ it, the buried experience of the past). From the point of view of the future, all else is the past – but we have no access to this point of view which is not at the same time fictional – a work of the imagination (so both ‘so’ and ‘not so’). Self-reference then is like the future because the future does not exist (is not), but is a projection of the present, which it continually refers back to (or where the present, continually refers back to itself…). As something projected from the present (repeating, with variation, situations from the past), it is a self-reference to its own point of origin in the present – and so a reminder of its fictional nature, as projection…
The element that is the future is always an element in the set of the present. But this it should not be; it should be elsewhere. Perhaps only via cause and effect, by a, as yet unaccomplished, chain of causation, is it implied as an element of the present. (It is asserted, but it is not (yet)).
The element that is the past is always an element in the set of the present. And as it has already happened, and its presence is a memory, something that has already caused the present, as its prior stage, then this is where it should be. (It remains, but it is not (any longer)).
Temporally, from the point of view of the human experience of time, it would appear that these two types of logical extension (accumulating meta-sets, and circle of self-reference) are both suited to (if not derived from) our species temporality insofar as they both, in their own way, move ‘away’ from the present, as away from the present level (or sentence or proposition), in a relation of levels, of nested levels or implications, and not as part of a totalized sequence (which latter would imply an outside or external point of view – often useful but not finally precise, when we need to relate the matter back to our basic experience). So these forms of logical extension would appear to be isomorphic with human time, such that past, present and future, are parallel to the logical levels, ‘upper’, present and ‘lower’ (or, meta-set, present level of propositional production or presence, and self-reference).
(Perhaps the syntax of the present may be likened to a syntagmatic relation, where the present is the default tense, or, the presence of the sentence, its form and being is such that it may, temporarily, be filled by a given temporality (there are only two), by the past or the future, and so with the tenses allied to them (or, in different phrases of the same sentence, the past and the future, although not at the same time; because the sentence exists in time; and ‘because the sentence exists in time’ so the past and the future are also in the present, they occur within the frame of the present, of what is present, is presented…). These latter (the past(s) or the future(s) available to grammatical tense) being in a paradigmatic relation, so occupying the -verbal or adverbial slots in the- sentence, either in part or in whole).
So the past resembles, and may be the basis of, our intuition of the meta-set (the chain of priority), with the future standing in a similar relation to matters of self-reference (self-contradiction); but with the significant exception that, experientially, the present returns as the final meta-set, so rendering the two ‘other’ levels (past and future) opposite but not mirror images, they must be asymmetrical, to be differentiated, to prove an arrow of time, to offer the possibility of uni-directionality…
So past and future both must refer finally back to the present: the past as to the really, final meta-set that frames all (and so is ALL, so constituted as self-contradictory); the future as to its point of origin in the present, self-grounding and self-referring (and so as A=~A, and so constitutionally self-contradictory).
Again it is worth repeating that these three levels are based upon a sense of being within the structures about which we talk: the view from without – as onto them ‘as if’ from ‘above’, is either unreal, that is ‘false’ or imaginary, a ‘thought experiment’, useful but, in the final analysis, inaccurate, or without experiential foundation (its proofs must be found elsewhere).It is because of our constitutional existence, or consciousness, in the Eternal Present, our human temporality, that we find it so hard to ‘finally’ prove the existence of the sea in which we swim, reality.
So both forms of extension, chains of levels or processes of nestlings: stepping ‘back’, or ‘up’ through a sequence of meta-sets; or ‘forwards’ and ‘down’ through a sequence of self-referential steps, contradiction driving each new twist; both ‘directions’ and their form as spirals or infinities (of levels or layers), in fact turn back to (are finally framed by) the present, where they begin… (and where they are in fact framed, or anchored). They are, however, differentiable. One, the past, finds itself more similar to the present (as ‘real’, as once upon a time, real, if the memory is accurate (and the final proof for this must come from without…)). Whereas the other level or layer, the future, extends out, but does not finally evince the same ‘family resemblance’. Whence the asymmetry of past and present, and whence the use of the term, now with a logical, as well as experiential aspect, the Eternal Present. (Yet in relation to the present they, past and future, are both ‘semi-present’; it is, in this sense, that they are, experientially speaking, in the final analysis, en-famed, reliant upon, or anchored in the present).
The thought of this final frame as ‘final’ is, of course, itself contradictory BOTH because of its positing as a meta-set AND because self-referential…
So perhaps the kind of extended use of self-reference used in these arguments may be found to be ‘illegal’ in terms of a strict application of the axiomatic ban on self-reference: however they appear more than ever necessary to any starting point – not least one self-aware of the inherently paradox producing nature of logic itself (see articles, Logic &Language I and II). However, this restriction may in any case have been superseded by the use of such in computing and other forms of mathematical modeling (fuzzy-set theory, etc.),where such fecundity has become normal. (Perhaps outdating the refined or received use of logic as ‘pure logic’, a pure reason limited to a tautology (whose putative hermetic efficacy was anyway called into question by Gödel)).
Over and above verbal tense, language offers the relation of time and adverbials; but how can this relation offer insight into the logical categories of ‘all’ and ‘some’? This is where we need to go to next…
Adverbials of time and logical relations (ALL/SOME).
Adverbials… Temporally speaking the terms ‘All’ and ‘Some’ actually mean, or imply, ‘always’ (at all times, true) and ‘sometimes’ (not at all times, true) unless specified otherwise; with the further refinement -which leaves them amenable to truth tables- such that all cases (that are the case) are the case always, or all cases (that are the case) are the case sometimes (and some cases are the case always and some cases are the case sometimes).So suggesting that we treat them as adverbials – or at least pay attention to the temporal (or extra-temporal) aspects of their meaning in any particular context. So of ALL x, SOME x, we can ask, do they imply or restrict time as well, in the sense of is the ALL or SOME always or sometimes true (as in a tautology or as historically, contextually, restricted). For example, ‘all men are mortal’ (although incomplete, and so unprovable) is read as (if) always true; and ‘some people live on the mountain’, implies sometimes, in the sense of not always being the case, of only being the case now (unless further specified such that,’ there are always (some) people living on the mountain’). Do ALL and SOME if not spelt out in terms of proposition and adverbial, imply the adverbial, as itself also ALL or SOME (as always, all times or some times…)? Some adverbial element seems (always) inescapable here.
And conversely, a temporal ‘ALL’ would imply a universal, eternal or a-historical range of operation (‘always’); is this what we mean…?
So ‘sometimes’ suggests that we should pay attention to the temporal element of any given sentence’s context, its relationship to actually-experienced human time (its restriction in time). To the ‘now’ in which we live, and to it’s others… From this point of view there are some times (several), or we might count, one to two (same and other).
Or better, zero to two… as ‘one’ already implies a view from without, a totalizing picture from elsewhere, a step outside (and so a change of levels… the shift into a putative meta-set). With the implication that the next number is ‘two’, comprising viewer and viewed; as well as their infinite alternation as ‘we’ ‘picture’ ourselves and that which we previously viewed (‘we’ and ‘the previously viewed’… etc). ‘Two’ also may refer to the entities that are the other of the present, or, more precisely, the others’ in the present; the past and future – numbering two. So temporally we might say that we count (directly) from zero to two.
To return to logical (and adverbial) terms: ALL in this case gives us the infinite and eternal: SOME offers the finite (on all counts), if unknown… In its non-temporal aspect, in non or extra-temporal terms, ALL must refer to universals or Eternity, the inaccessible, but positable, ‘Outside’ of human experience. However in temporal terms, in terms of our actual experience of time, the only sense of the eternal we have is the Eternal Present, final set or meta-frame of our experience (so the intuitive origin of the former). Infinity (ALL) in this sense, applies only to the ‘Now’ or ’Eternal Present’ of our temporal experience (our ‘zero’ degree) – there being no ‘without’ or ‘outside’, we can not see the limitations of experience, can not quantify it, but feel it as infinitely (even as we know it is not indefinitely) extendible. Leaving SOME, then, to be found in the other temporal valences (past and future); the ‘other’ temporality that gives ‘two’, but is itself divisible according to its postulation, implying ‘before‘, or ‘after’ the ‘now’ (so only present, or better semi-present, in the present, SOME of the time). So reversing the order of non-temporal logical relations (which exist ‘nowhere’, ‘now where?’, such that the eternal meta-set is the absolute outside and the sense of the present is its pale echo -Plato might have said, ‘shadow’- persisting in our subjectivity) and in this way offering a third case (a kind of ‘not, not ALL’) ‘eternity’. From the temporal point of view this would be an (over) generalization and projection of the ‘now’ moment, beyond our experience; a projection onto an ‘always-outside’, an elsewhere, an ALL, always, outside of history, place of myth, heavens, foundation of religions and ideologies… and universals (of values, reason and mathematics). The Other (outside of temporality); unreachable (by definition) so not increasing our number - we ‘count’, zero, two and (…); or ‘Same’, ‘other’ and ‘Other’, to use the terms current in Continental Philosophy (zero, as we noted above is first but not ‘one’). The (…) in the previous sentence indicates the given term’s un-definability because of its un-reachability, non-location, or existence as a purely sublime representation (a representation of the sublime, or unrepresentable). If unreachable, then ‘Absolute’; the would-be top level or ultimate meta-set. An ALL outside of ALL. (An ALL to ‘cap it all’! A … ‘not ALL’ (~ALL)). Interestingly a formulation we can use… but not find. Apparently, equally fictional and necessary… that is, both fictional and necessary to human being. So, both true and not-true, both true and false… Pragmatically and existentially, experientially, ‘true’. As in, useful. Or (in the case of its being part of our species-being, or ‘hard-wiring’) unavoidable…
Otherwise ‘not not ALL’, would, in reverse, equal our consciousness, the illusion of eternity in the persistence of the present; human as machine, viewed, but by whom, from without… leaving ALL as applying to the ‘outside of history’ as meaning either ‘always’ true of history (a-historical as in unaffected by history) or only true elsewhere (Absolute Outside).This later seems to me to be the only true realm of universals etc.
But in no case replacing the Eternal Present (O) as the real, unavoidable, inescapable meta-set within which we use or pose or believe in the pretender we posit as (…) elsewhere.
Other logical terms and their relations to our ‘fundamental’, that is, experiential temporality.
Our basic logical terms, ‘and’/‘or’; ‘not’; ‘implication’ (’if x then y’), treated temporally, might well yield the following:
‘And’ (^) and ‘or’ (v) as present, operating on the level of the present, on what is present (on the same level), offering addition or subtraction, accretion or choice; with ’and then’ as offering causal sequence, offering a future, with ‘or then’ as offering alternative futures (or, in both cases, as in the past, as offering a reconstruction of a past trajectory). ‘Or/v’ as we have seen, implies, or allows that the term not present may be present at another time… In relation to our experience of time SOME/ALL become temporal, that is to say in their relation to the eternal present as ALL and past/future as the place of SOME). In other words only in relation to events and our experience in time do ‘and’ and ‘or’ need awareness of temporal restrictions… otherwise business as usual… (applying to tautologous, quantitative knowledge only, with the proviso that real things, in time, are subject to being in excess of measure, such that, to enable calculation, we must limit the numbers after the decimal point at some stage – all numbers are rounded up or down or averages, are ‘unreal’). Only ‘and/^’ and ALL then have a special relationship to the present.
With respect to quantitative relations, number, second order languages: mathematics, geometry, grammar, logic; these tautologies are so many folds that we lay over the object, place or topology, that we require to describe, measure, or model. (Could it be that a sequence of meta-sets, or the repetition of self-contradictory steps be taken as numbers, cases, as well as levels?)
‘Not’ (~) as temporal; offering what is ‘not present’, not present ‘now’ but (not yet) ruling out other times, so possibly present elsewhere; as once present or as potentially present in the future. If the ‘not’, the ‘not present’, as temporal therefore suggests SOME, as in sometime not present ‘now’ (not now, but maybe later!) as not present now (but possibly (but not necessarily) present at another time, past or future), then the sense of ‘not’ as ALL, as ‘never’, suggests something that has never existed, a kind of ‘not at ALL’. A further, very useful form of the negative, would be the double negation, the ‘not not’ (~~), which may be read as, ‘always not’, ‘not ever’, ‘not anytime’. But as still present elsewhere; an Absolute Other. Eternity (together with its sacred and secular surrogates). Therefore, if ‘not’ read as SOME is taken as A (= not now, but at another time perhaps), then ‘not’ read as ALL or, nowhere, at no time (keeping the emphasis on temporal being), may be read as B, as a kind of ’not not’, as elsewhere (as nowhere in time); or better as outside time (as pointing to a ‘presence’ outside time). Perhaps as simultaneously ‘not’, but ‘necessary’; therefore the double negation, both present AND un-present, returning ‘it’ to present use, but not as ‘real’; having no objective status, but only subjective status as a performative (if we believe it, so it is) a collective mental or cultural entity true of and for human existence; the subjunctive becomes indicative in the performative (much as the subjective becomes objective in the performative) in speech acts and ritual performance. This brings us to the category, (…), which followed zero (0) and two (2), as discussed above under the definition of ALL (or ‘Same’, ‘other’, ‘Other’).
‘Implication’ or ‘entailment’ (=>), ‘IF…THEN…’, as temporal on two counts: one ‘obvious’ in terms of exposition or the perception of a record or representation in real time; the other as ‘implying’, ‘pointing to’, a prior or following stage which is temporal and not merely tautologous. In the first, something follows, or … precedes, something, as one proposition stands in relation of priority to another; so not temporal in the sense of the progression of referents or stages in life: but in the sense of logic; of logical priority, of priority in the thought process (but in actuality a-temporal as in tautologous). In the second, something follows or precedes something in real time; but with the added observation that the ‘IF’ moment or stage, or proposition is in the present, with the following ‘THEN’ clause or proposition as in the past or the future; as implying a previous stage or prerequisite, as cause; or as an effect, as the next thing, that which follows, consequence, or result. ‘IF… THEN…’ (=>), then may point in two ‘directions: a ‘pointing to’ what must have happened (first), the past, or to what must happen next, the future. As in ‘if x, then y follows in time’: the future equivalent. Or ‘if x, then y before x’ as pointing to a prior, prerequisite, event in the past: the past equivalent. These two aspects offer: a logical versus a temporal deixis; that is the a-temporal (although viewed or reconstructed ‘in time’) purely logical relationship of wholes and parts, sets and elements, as opposed to a logical exposition of ordinality, sequence, or causality. The contrast of an ‘at the same time’ relation (equality, expositional or purely logical priority) versus a ‘cause and effect’ relation (priority in fact, not just in thought or in written text). WE might add that ‘IF…THEN…’, implies ALL, in all cases, so at all times, unless restricted by SOME…. (so ALL is abstract, is a-temporal, all universals, all ‘all’, are like this, axioms as a-historical, so applicable to formal-tautologies only). So, if ALL ‘IF… THEN…’ is the case, then it must be abstract, or outside time; an a-temporal relation is indicated (outside, Eternity or (…)). As a real action can only happen once, its relation of similarity or mimesis to other relations (x is like y) is an assertion that differing actions at different times have a family relation (similar effect implying similar causes and vice versa, this relation would, of course, have to be ‘tested’ or, as we are dealing with unrepeatable history, subject to the irreversible arrow of time, researched and documented thoroughly). SOME ‘IF …THEN…’ may imply that actual phenomena are being dealt with (SOME may be once, unique, an event with a time and a date).
ALL ‘Same’ = present (0) (tautology) ‘first but not one’; embodied as individual: unless we chose the meaning as ALL of the same, ALL capable being in, of carrying the Eternal Present, human beings and maybe animals too…as seen from within (only possible in one (‘s own) case, otherwise supposed, inferred…). Whence, the ‘We’ of species being, or, by an extension of identification, the ‘We’ of sentient life. Collective? By extrapolation.
ALL ‘other’ = past and (^) future (2); the semi-present (tautology); only two.
ALL ‘Other’ = Eternity (…) the unpresent-able (tautology); as such.
So we count; 0, 2, (…). An
omnipresent level, or frame (‘
SOME ‘Same’ = discernable matter, organic and inorganic (quality and quantity); their existence, describability and measurability as signs (virtual).
SOME ‘other’ = past or (v) future; one only of the semi-present (one only can be chosen).
SOME ‘Other’ = an element of (…) unknowable: but posit-able… (we have just posited such). Universals, absolutes, infinities, eternity, the supernatural, immortals, angels on the head of a pin, Heaven and Hell, God.
Regarding our experiential temporality and its externality. Otherwise put, the (my) Eternal Present and its relation to all that is ‘outside’ (that is outside ‘our’ or better ‘my’ temporality; but actually if imaginable, or representable to self, so now ‘inside’ our temporality, within the Eternal Present).So while ALL that is outside us may be posited as an equally (finally) unknowable absolute ALL if including us (in and out = ALL, unlimited and final meta-set…) Yet experientially we are ALL (our temporal being as experienced as such, from within, the Eternal Present) to the SOME (the elements that represent other things).The two points of view, actually-experienced inside and posited ‘outside’ put together, give both starting points as ALL to the SOME of the other, what remains or comes next, what is contained by the meta-set of the ALL; put together give ALL = SOME; set and element are the same; A=~A (so making impossible the ‘true’ or ‘greater’ ALL of their unity, as if from a third point of view… the next meta-set ‘up’). So if both points of view, both starting points (in and out), are made identical, then A=~A. And yet they are the same in the sense that they interpenetrate, one contains the other; either as our consciousness as part of all matter; or as all matter (that is the matter, that is perceived) as part of our consciousness. Reversible: both with their uses… (The contradiction, or aporia of subjective and objective may be ‘resolved’ in a similar way, indeed may be read as the same, as is the aporia of thought and things, representation and represented, etc…)
The application of our experience of time to logic thus alters, restricts or even reverses the usual meanings of basic logical terms.
But perhaps renders them more amenable to practical use.
Copyright Peter Nesteruk, 2013