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).
Such that:
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