Hence the entirety of all logical possibilities is expressible, or perceiveable,
in a minimally space capable system.
Another way to say the same thing is that position is independent of
condition. You might not have many conditions you can detect, or
positions you can distinguish, but for space to be useful then the
positions are at least somewhat independent of the conditions. Which
means all their combinations are possible, and "all" means we can
enumerate them according to Logic.
IT(HERE) AND IT(THERE), or perhaps the same thing, HERE(IT) AND
THERE(IT), (Henceforth I'll write (HERE,IT) etc.) means that the
condition "IT", applies in both locations, and therefore has an EXTENT
or SIZE greater than could be captured by only one of the locations.
Hence AND means EXTENT or SIZE when applied to space.
Does this require an enhancement of the minimally spatial system?
It requires that the categorization system applies to multiple
locations and that there is a comparison capability that can say the
categories in the two locations are the same. This is some deep
But an organism that can't tell the size of anything is pretty
constrained in its use of space, so I'm comfortable saying if it
USEFULLY has space then it also has size or extent. Because space
enables you to say there is MORE of something. And that generates the
capability of the logical AND: MORE = SOME AND SOME (more).
Next let's do XOR. XOR over space seems to mean:
HERE, IT XOR THERE, IT
Spatially, XOR means: IT is HERE or THERE but not in both places.
This could be true over time as in a movement of something from
one place to another, which therefore is either here or there but not
in both places at once (barring AND, which we have already dealt
with). XOR could be an assertion of identity, the thing, IT, is
the same IT as might be in the other location, whether through
movement or uncertainty. Finally XOR could be an assertion of a size
limitation: whatever IT is, it's not so big that it extends into all
the locations I am able to represent (in this case, both locations).
Am I permitted to conclude that an organism which has any of the following
has the functional, logical, ability to compute and use XOR relative to space?:
I think so. Each of those requires an XOR operation applied over
distinguishable spatial locations. Are they required for a minimally
space-capable organism? No. But do they provide greatly enhanced,
shall we say, baseline, utility for spatial cognition? Yes.
- Any ability to sense a limitation in the size or extent of something.
- Any sense of object identity; otherwise how can it say it is here
or there, without any limit on its location. A thing without
limits on its location is no thing at all, but the Void itself,
or perhaps the Water to our Fish: IT is in every place, and
therefore not usefully an object separately handleable.
- Any sense of the identity of something over time, for example in
the tracking of a moving object. Predators would be selected for
this capability, over those without. Prey, ditto. Browsing
"P OR Q" canonically means P or Q or Both-P-and-Q.
Since we already have NOT and AND, we can express P OR Q as NOT (NOT P
AND NOT Q). Hence a system with categorization and AND already has
OR has the functional utility of representing partial knowledge. For
example, uncertainty as to position or size of some IT which
definitely occupies at least one of our perceivable, distinguishable
locations. Any organism capable of partial knowledge must implement
an OR operation in its systems. If it is capable of first knowing
something is out there somewhere, and then later that it is
specifically THERE, then it had the equivalent of an OR at the first
phase: (IT,THERE) OR (IT,HERE). In any hierarchy of knowing or
responding, the ability to focus more, to pay attention, or to learn,
all these require this cognitive characteristic of an
improvement-of-knowledge which amounts to having an OR operation at
the start: an uncertainty of knowledge, a partly-but-not-completely
specified piece of knowledge.
Returning to predators and prey, their job (survival) depends on the
OR operation. Perception being imperfect yet improveable, their
detectors are set on maximum inclusiveness and minimum specificity,
and when anything is detected in the inclusive-nonspecific initial
jolt, that is an OR: I see something HERE OR THERE, followed up
quickly by a focussed attention in the general direction, until enough
detail is known that action can be triggered, whether chase or escape,
respectively. Because any prey or predator that fails to implement
this logic will soon be selected out, certainly as compared with any
that somehow miraculously do acquire this, I would say,
baseline-useful, enhancement of spatial capability.
IMPLIES is a (usefully, proper) subset relationship between categories.
P IMPLIES Q means that P is inside of Q (and if proper, Q is bigger than P).
If the subset relationship is not proper (because P is the same as Q)
then the two qualities may be differently sensed aspects of some
sameness, thus just adding richness to one's knowledge about a single
category. Multiple kinds of sensory detectors might separately spark
off on the same kind of object, and sometimes this finds it, sometimes
the other, and there is benefit to the redundant detectors operating
simultaneously. So it's still useful to have equality in the subset
relationship, which one might express as P IMPLIES Q AND Q IMPLIES P.
But if it is a proper subset relationship where one includes the other
as well as some stuff that is NOT the other, then IMPLIES further
yields the capability of operating with a hierarchy of knowledge,
e.g. leggedness might imply edibility to an apex land predator.
The implication may be toward action, or toward knowledge.
Toward action, clearly cue response is enabled by IMPLIES: IF CUE THEN
Generally we are talking about locational information, HERE vs THERE.
(HERE,IT) might IMPLY a response different from the response to
(THERE, IT). Certainly predation would select for any development of
this capability, since chases are more effective when the prey is near
and ineffective predation yields death. H. More generally,
instrumental action, such as contacting something, holding or eating
it, even walking and knowing where to put your feet, requires an
IMPLIES relationship: IF hole THEN adjust stride; a spatial logic
which every coordinated ambulator must implement.
Toward knowledge, if we know something about a place, for example,
that prey drink THERE, that is an IMPLIES relationship between a
location and a condition, known by the (predatory) organism. This kind
of knowledge seems much more abstract and high-level than the
cue-toward-action kind of knowledge. One could say that systematic
action cognizant of conditions is itself knowledge, logically
speaking. So I'm happy to rest my argument there.
Another use of space and IMPLIES is in searching and finding.
Assuming bounded spaces,
IT is HERE IMPLIES IT is NOT THERE.
In finite spaces, the logic is more favorable, and known as the
Process of Elimination: If IT is NOT HERE then IT must be THERE.
But in a moveable frame or an infinite-space-representing system,
the logic of search seems to be: if IT is NOT HERE then IT could be THERE OR NOT THERE.
I don't see that a hunting organism needs the cognitive capability of
IMPLIES to continue to search fruitlessly, though the implication is
that the searcher at least hopes that its goal is somewhere. A
searcher that searches systematically, that is, not returning randomly
or equally to the various already-examined locations, would seem to
have cognitively implemented the Process of Elimination. But this
seems not nearly as powerful and primary a selection force on
predation as the others.
I hoped to justify the radical claim that, cognitively, space = logic.
Instead I walked a line of expanded capabilities, starting without
space, then adding no more than a minimal space-representing
capability, then adding some quite obviously useful, indeed
evolutionarily selected-for, fundamental capabilities that "having
space" would enable.
First, before space even exists in the cognitive capabilities of a
system or organism, we already have processes which can trigger or
not, thus we have categories.
Second, since to have a category that may
trigger or not, means we also have its negation, then we have
Third, we add a minimal spatial capability: as if, at least two, spatial
encodings, that can be handled differently.
Fourth, we add that some category can apply to one or the other or both or neither.
Fifth, we add that the system can tell (and act differently) if the
category applies to both or not.
Then, the AND connective of logic falls out, because the system can respond differently
to IT HERE AND IT THERE versus the alternative.
Sixth, we add size limitation, identity-detection, or movement tracking,
and XOR falls out.
Seventh, we add partiality of knowledge, and OR falls out.
Last, we add cue response, and IMPLIES falls out.
So I didn't achieve my goal, that space = logic.
But I came to nearly as tight a conclusion: if you have a minimal
spatial capability (at least two distinguishable locations) PLUS
common categorization across locations, PLUS comparison (across
locations), PLUS target limitation (among locations), PLUS the
ability to make use of partial (spatial) knowledge, PLUS cue response
(connected to location), THEN you have all the components of logic.
It seems to me that these amount to a definition of a minimum "useful
Does a passively-floating jellyfish require all these? Some, it seems, but not all.
But clearly any effective, mobile predator, or any prey animal which
has evolutionarily flourished using an escape strategy in the context
of effective predators, must by then have evolved all these cognitive
capabilities associated with spatiality.
This amounts therefore to an assertion that the prerequisites for
logical cognitive capability were evolved by the time predation -- and
escape -- were on the scene, that is, by the Cambrian Explosion, which
gave us the Animal, its many Body Plans, and their ability to Move,
which enhanced their ability to Eat, and eventually to eat not just
immobile plants but Other animals.
This assertion is hardly arguable: that predatory ability depends on
spatial cognition, incorporating all these components. That animals
should have later evolved the abilities to carry out logical cognition,
given these logical foundations, is no impossibility.
To return to the deep and fundamental problem posed by the Crease of
Logic, which this essay is an attempt to solve, it seems to me
that the certainty with which we perceive stuff in space is identical
to the certainty with which we view logical "certainties". I claim,
then, that perceptual, spatial certainty, is the foundation of, and
identical to, "Logical" certainty. And it is because we evolved from
predators with a necessary, a deeply ingrained willingness to bet our
lives on what we spatially perceive, and with the logical operations
and certainties of these six or eight elements that give us the very
utility of our spatial-cognitive systems, it is these
organismally achieved cognitive capabilities that make us certain,
when we are certain, about the logical, and perhaps mainly or even
always tautological, concepts that we consider and statements that we
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