I assert that an organism able to internally, usefully, represent space is an organism that is capable of logic.

Specifically, six elements comprise logic: real categories and the five logical operations NOT, AND, XOR, OR, IMPLIES. We will build up a minimum space-representing system and see how it becomes a logical machine, and what its capabilities become then.

As this work constitutes cognitive science, please consider the argument here as tracking in parallel and applying to logic, to biology, to evolution, and to the design of more or less intelligent computing machines.

Category

First, and even before biology comes into the picture, we have the reality of categories. A real category implies a categorization system, a means of deciding if something is in the category or not. Every physical, chemical, biological, etc., process constitutes exactly such a categorization system. When its input requirements are met, the process applies, and yields its outputs. Applies: In the category. Does not apply: Outside the category. Anything that happens systematically can be considered as a category. It could even be probabilistic and still amount to an effective system; it has to detect-and-do ENOUGH to make a difference, that's all we need. (Fuzzy logic is still logic.)

NOT

NOT is an operation already implicit in every category. Whatever \(x\) is in the category, say, \(p\), is labelled \(p(x)\), great; whatever is NOT in the category can be equally labelled NOT\(p\).

Minimum Capability for a Space-Representing System

(Some earlier work on the logical properties of space and time as represented within cognitive systems can be found in Veatch 1991.)

Consider as a minimum that an organism or computer has structures and processes capable of operating "as if" it has "differential response capacity" over "spatially encoded or represented" "conditions", for "at least two" spatially-differentiated and organismally-differentiated "locations".

• "as if": I don't care if it calls its systems "spatial"; the code documentation is immaterial. It could be an electrical resistance proportional to some physical dimension, and the system only actually handles electrical properties, but it still operates as if it knows about distances. That's fine. I just care that it handles different locations differently.

• "differential response capacity": if you can't respond differently to stuff based on WHERE it is, then space provides you no utility: you have no useful capacity for representing space. Because it's not useful if you can't respond differently.

• "conditions": I don't even want to specify that the condition is the presence of a thing or of some stuff or of any kind of process or event or happening or even none of the above. "Condition" is a term that is the general supercategory of thing vs stuff vs process vs nothingness: if any of those is in a location, then that is a condition that applies to that location. Allow me to use the word, "IT" to generically refer to any such condition.

• "spatially encoded or represented": the condition may apply to some location in space, fine, but from the perspective of the organism operating with space, that condition must be spatially encoded or represented, that is, inside the organism and separate from the actual location that the actual condition applies to. Allow me to use the word "WHERE" to refer to the particular encoding of a place as represented within the organism.

• "at least two locations": normally space is continuous and extended but a minimized capability to represent space should at least be able to distinguish between two locations or positions or spatial encodings, whatever those might be. Allow me to use "HERE" and "THERE" to refer to the (at least two, distinguished) locations, as technical terms not even to particularly imply relatively near vs far, but just as convenient names for any Location1 and Location2.

Hence the minimally space-capable system, organism, computer can respond differently based on "WHERE" "IT" is, whether "HERE" or "THERE".

Can the minimally space-capable system carry out the operations of logic?

Its condition-detection systems constitute one categorization system. Its location-assignment systems constitute a second categorization system. It seems that the possibilities in a truth table are necessarily asserted and exhausted by these two systems:

HERE	IT
HERE	NOT-IT
NOT-HERE (=THERE)	IT
NOT-HERE (=THERE)	NOT-IT

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.

AND

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 logic.

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).

XOR

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?:

• 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 vegetarians, ditto.

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.

OR

"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.

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

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 RESPONSE.

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.

etc.

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.

Conclusion

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 negation.

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 spatial capability".

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 make.

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