RE: Making a Computer that runs on Magic

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RE: Making a Computer that runs on Magic

Post by Lambuzhao » 06 Nov 2019 01:02

I just saw this on Because Science

https://www.youtube.com/watch?v=pdmODVYPDLA

Reminds me of Elemtilas' use of Dwimmery in his conworld. Very interesting to see!

Could give one a different source of (theoretically workable) ideas to approach the conjuration of a Turing Machine, and therefore computer, by the use of the Black Arts, White Arts, Blood Magic, Wicca, or what have you, or any combination of what-have-yous.

Here's the link to the original paper:

https://arxiv.org/abs/1904.09828


I suppose using a Ouija Board as a sort of router wouldn't be out of the question.
Buffering charges would surely be hell to pay [}:D] :roll:

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Re: RE: Making a Computer that runs on Magic

Post by Tanni » 06 Nov 2019 09:52

Lambuzhao wrote:
06 Nov 2019 01:02
Could give one a different source of (theoretically workable) ideas to approach the conjuration of a Turing Machine, and therefore computer, by the use of the Black Arts, White Arts, Blood Magic, Wicca, or what have you, or any combination of what-have-yous.
For defining the Halt-problem, you need to define a certain type of Turing machine. You need a Goedel-coded input, the program to investigate if it halts or not. If I remember correctly, the Halt-problem is not solvable. A Turing machine is a theoretical computer model of theoretical computer science. There is nothing magic with it, barring a one-side or to-side unlimited band, the store. It cannot be used in whatever Magic, you cannot even run normal programms on it. It is just a means of theoretical investigation, to find out what is in principal computable, regardless of the size of your computer memory. That's why a Turing machine lacks an address decoder, which would limit the addressable memory space.
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Re: RE: Making a Computer that runs on Magic

Post by Reyzadren » 06 Nov 2019 12:12

So, basically any regular computer that can run on magic? That's pretty much a common thing in my conworld as well as some others' [:P]
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Re: RE: Making a Computer that runs on Magic

Post by lsd » 08 Nov 2019 19:39

I thought that all computers work on magic...
few people know how they works ( the sorcerers) how to build one (the magicians)...
all users believe in their magic and only know the tricks they are autorized to used on the computers they bought already witched...
they only could change of gurus and selling they soul to windows chapel or of apple sorcerers or african butus... but whatever, all their mind activity or privacy will be aspirated by the algorithm demons and inclosed on magical caves called datacenter that enslave humaity forever...
the better trick was to name this magic, "science" and to make believe their is no spells inside even if no one now could understand nothing of the artefacts they brought us and make us dependent ot them and replace nature... we are definitively all bewitched...

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Re: RE: Making a Computer that runs on Magic

Post by sangi39 » 09 Nov 2019 03:38

STP/Discworld's Hex, although as usually tends to be the case with the Discworld, exactly how Hex works isn't really explained, although it's mostly a lot of wordplay and stuff based around computer-related jargon, like Hex's power source being powered by a waterwheel inside a sheep's skull (RAM), and the "Out of Cheese Error" message indicating that the biological mouse has stopped working (referencing a broken computer mouse), but overall Hex seems to have a database-like knowledge and almost search-engine-like capabilities when it comes to answering some questions, but almost a child-like nature in other respects (again, I assume, parodying modern attempts at AI).

So, yeah, Hex probably runs on magic in the same way the most people might as well just think computers run on magic [:P]
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Re: RE: Making a Computer that runs on Magic

Post by Micamo » 09 Nov 2019 22:27

In principle, if you can use it to build a NAND gate, you can make a whole computer out of it. You can make simple computers out of dominoes, marbles, even by just having people hold up flags of different colors. The main factors for viability are size, speed, and cost. It's difficult to name any magic systems from popular fiction that you could use to make a viable computer, competitive with modern transistors. One could purpose-build a magic system to do it, of course, or more realistically you could use transistors for 99% of it, then have some magical component at a critical point that would allow for superior performance in some specialized domain. Some examples:

1. An "oracle chip" that can divine the answers to NP-hard problems in constant time. (Even if that constant time is hours, being able to crack arbitrary encryption or falsify arbitrary signatures would be massively disruptive.)

2. A magical battery that allows the device to operate perpetually.

3. A magic "router" that allows connection to a server without the need for electromagnetic signals, possibly also allowing instantaneous communication over arbitrary distances. We're already encountering serious issues with running out of room in the useful parts of the electromagnetic spectrum: Replacing all wireless devices with sympathetic magic would be incredibly useful. Of course, depending on the side effects of such magic, we may well just be replacing one issue for another.
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Re: RE: Making a Computer that runs on Magic

Post by svld » 11 Nov 2019 07:04

The first application comes to mind is:

4. A magical basic unit of information which is more than the binary 0 or 1.

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Re: RE: Making a Computer that runs on Magic

Post by Tanni » 11 Nov 2019 08:31

svld wrote:
11 Nov 2019 07:04
The first application comes to mind is:

4. A magical basic unit of information which is more than the binary 0 or 1.
There are in fact such systems. The speakers of Aymara are known to use such a system, which is a cause of severe misunderstandings between Aymara people and native speakers of other languages, see here.
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Re: RE: Making a Computer that runs on Magic

Post by Salmoneus » 11 Nov 2019 15:36

svld wrote:
11 Nov 2019 07:04
The first application comes to mind is:

4. A magical basic unit of information which is more than the binary 0 or 1.
That doesn't require magic! Remember, binary computing is the invention, non-binary is the default. Binary computers were invented in the 17th century, but weren't seriously implemented until the middle of the 20th - even the early electric computers (like ENIAC) were non-binary in conception, even though they were mechanically binary.


----------------------------

I think there's at least four ways to answer this question, with increasing closeness to modern computers: non-mechanical computers, mechanical non-linear computers, linear non-binary computers, and binary linear mechanical computers like the ones we use today.

First:

Computers don't have to be mechanical

Fantasy is full of magical computers. These include:
- asking a question, killing a wildfowl, and finding the answer encoded as a patten of dark spots in its entrails
- praying, until an angel brings you a sealed envelope with the correct answer inside
- asking a large group of rats, and then reading the answer from the complicated shapes the rat-swarm makes with its congregated bodies
- keeping an omniscient demon (incapable of lying) in a small box and poking it with sticks until it gives you the answer.

If the omniscient demon is relatively small, and its complaints relatively muted, the demon-in-a-box is functionally equivalent to a modern computer (with an encyclopaedia and surveillance system attached for good measure).

If you have magic, and don't have modern philosophy/mathematics, then this sort of oracle-computer will probably be more intuitive, and sooner-invented, than a modern mechanical computer. Why bother with circuits when you can just conjure up a demon?

Now, the demon in a box has several features:
- natural-language commands ("tell me the square root of two to nineteen figures or I'll poke you with this stick!")
- occult (pun intended) microprogramming (whatever goes on in the demon's brain)
- physical but non-mechanical hardware (the demon, which runs the microprogram)

But magical computers can change any of those features. For instance, imagine that you can write down an equation, or a propositional argument, then say a spell that incorporates fixed formulas providing the microprogramming to convert these things into binary and conduct the operations, and the ink then magically rearranges itself to tell you the solved variables or the truth validity of the argument. This would be a computer with ideal-language commands, overt microprogramming, and non-physical hardware. There is, as it were, computing, but no actual computer (or: the computer is magic itself).

Indeed, in some magic systems, all magic works like that, as though magic is a giant computer. In others, all magic works like praying to a god. In others, it's somewhere in between. It depends, really, on how smart you want your universe to be...

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Re: RE: Making a Computer that runs on Magic

Post by Tanni » 11 Nov 2019 17:42

Salmoneus wrote:
11 Nov 2019 15:36

Computers don't have to be mechanical
Indeed, modern computers rely on electricity.
Fantasy is full of magical computers. These include:
- asking a question, killing a wildfowl, and finding the answer encoded as a patten of dark spots in its entrails
- praying, until an angel brings you a sealed envelope with the correct answer inside
- asking a large group of rats, and then reading the answer from the complicated shapes the rat-swarm makes with its congregated bodies
- keeping an omniscient demon (incapable of lying) in a small box and poking it with sticks until it gives you the answer.

If the omniscient demon is relatively small, and its complaints relatively muted, the demon-in-a-box is functionally equivalent to a modern computer (with an encyclopaedia and surveillance system attached for good measure).
This reminds me somewhat on the Deus ex machina.
If you have magic, and don't have modern philosophy/mathematics, then this sort of oracle-computer will probably be more intuitive, and sooner-invented, than a modern mechanical computer. Why bother with circuits when you can just conjure up a demon?
Well, demons don't work for nothing ... you have to pay them!
Now, the demon in a box has several features:
- natural-language commands ("tell me the square root of two to nineteen figures or I'll poke you with this stick!")
- occult (pun intended) microprogramming (whatever goes on in the demon's brain)
- physical but non-mechanical hardware (the demon, which runs the microprogram)
Can you explain to me what you mean by "occult", as I didn't get the pun.
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Re: RE: Making a Computer that runs on Magic

Post by Salmoneus » 11 Nov 2019 18:52

Tanni wrote:
11 Nov 2019 17:42
Salmoneus wrote:
11 Nov 2019 15:36

Computers don't have to be mechanical
Indeed, modern computers rely on electricity.
REALLY!?!?

However, electricity is still mechanical in the sense intended here.


Well, demons don't work for nothing ... you have to pay them!
Not if you have a sharp stick
Can you explain to me what you mean by "occult", as I didn't get the pun.
"Occult" typically means having to do with magic, witchcraft, secret arts, the supernatural, superstition, etc. Demons are in this sense 'occult'.

However, "occult" is also an old-fashioned word meaning 'concealed from view'. The workings of a demon's mind are in this literally sense 'occult', because we can't see them. They are occulted by the demon's skin and skull (and the mind-body problem).

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Re: RE: Making a Computer that runs on Magic

Post by Tanni » 11 Nov 2019 19:18

Salmoneus wrote:
11 Nov 2019 15:36
Well, demons don't work for nothing ... you have to pay them!
Not if you have a sharp stick
Then, there is no story.
Salmoneus wrote:
11 Nov 2019 18:52
Can you explain to me what you mean by "occult", as I didn't get the pun.
"Occult" typically means having to do with magic, witchcraft, secret arts, the supernatural, superstition, etc. Demons are in this sense 'occult'.
That's what people typically think what it means, but what it is NOT.
Salmoneus wrote:
11 Nov 2019 18:52
However, "occult" is also an old-fashioned word meaning 'concealed from view'.
That's what it actually means! See Mark Passio: Anarchy & The Occult at 6:56 for the definition of occult.

But I still don't get the pun.
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Re: RE: Making a Computer that runs on Magic

Post by Salmoneus » 12 Nov 2019 00:08

Second:

Computers don't have to be linear

We think of computers as running a sequence of commands. But these, of course, are equivalent to simple logical propositions, which is why computer people still use terms like "logic gate" - a logic gate, as it were, enacts a logical operator.

For instance, let's define eight data positions: A, B, C and D are input variables, E, F and G are output variables, and H is just used for calculation purposes. Data is binary. We can then say:
- set E, F, G and H to 0
- if B is 1, set G to 1
- if G AND D are now 1, set H to 1
- if G OR D are now 1, set G to 1
- if H is 1, set G to 0 and F to 1
- set H to 0
- if A AND C are now 1, set E to 1
- if A OR C are now 1, set H to 1
- if E is 1, set H to 0
- if H AND F are 1, set E to 1 and F to 0
- if H AND E are 1, set H to 0
- if H is 1, set F to 1
- set H to 0


Alternatively, we can simply say:
A^B^C^D⊢E^F^G
A^B^C^¬D⊢E^F^¬G
A^B^¬C^¬D⊢¬E^F^G
etc etc

And we can rephrase that as ⊢[[A^B^C^D]>[E^F^G]]^[[A^B^C^¬D]>[E^F^`G]], etc etc.
Which by the magic of the ampheck can be rephrased as... ⊢((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))*, etc etc etc etc


The command sequence above can be regarded therefore as in some way a way of 'spreading out' these propositions over time. Which makes intuitive sense, because after all the proposition itself is linear, a two-dimensional sequence of symbols, and it's very intuitive to convert a line into a sequence of events, because that's what we do when we read, or calculate.

But the proposition isn't linear, that's just a representation of it. We could alternatively depict the proposition as a three-dimensional pattern, as Peirce did - here's a Peircian proposition (a theorum by Leibniz, appropriately enough).

Now, Peirce separates out his procedures from his notation, so he works by graphically altering these pictures step-by-step in accordance with a small set of rules. But of course, those altered diagrams (change in time) can be considered to be a sequence of parts of a larger diagram (change in space). Each part would be related to adjacent parts by a permitted type of transformation.

In fact, imagine a space with infinite dimensions. Each 'panel' of the larger diagram could be surrounded by every panel that is linked to that panel by a certain form of transformation, an each of those panels linked to other panels, and so on. Effectively, you'd have an infinite wallpaper - a fractal wallpaper, in fact - of subtly and regularly repeating/transforming patterns. [It's even more literally a fractal if we include our deductive procedures as true facts within our diagrams, because then the relationship between any two adjacent panels must be similar to the relationship between two particular adjacent areas within a single panel - and if we use our infinity to replace each variable with a list of every possible value for that variable, then each pair of panels will be a magnification of a section of each panel, forming a perfect fractal image!]

But such an infinite-dimension fractal cannot comprehensibly be understood, at one time, in two- or three-dimensional representation. Which is why with have deductive procedures. If you think of a proposition as a Peircian pattern, rules of deduction are simply rules that allow us to generate a finite segment of that wallpaper at will, because we can't simply consult the wallpaper as a whole. Those rules, as it were, get around the limitations of space by operating through time.

Except...

...with magic, we don't need to do that, do we?

Imagine something like Borges' Library of Babel. We're in a large room, polygonal room with somewhere between three and twenty walls, with various patterns on the walls (or, if we prefer, a certain selection of books on its shelves). Each wall has a door, which takes us into another room of the same sort, with a similar but slightly different pattern (/book collection). That room leads us to another room, and so on. Now, this sort of infinite library is equivalent to the complete world of deducible propositions given a certain premise, which means it's also equivalent to the operation of a computer program.

So, if you want to run a computer program, you don't have to build a computer. You just need to draw a picture, conjure up an infinite library of this sort, and then send a fast-running scenthound through the library, with a red ribbon tied to your hand and its collar, trained to stop and bark when it finds a room with a certain type of pattern on the wall (in most cases, this will be a room with a simpler pattern than any of its adjacent rooms have).


* assuming I've not gone wrong - which I almost certainly have, "⊢((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))" can be interpreted as the fact that 4+4=8, expressed via binary.











----------------


It occurs to me that i MAY not have explained that as well as I might have. Well, moving swiftly on!

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Re: RE: Making a Computer that runs on Magic

Post by Tanni » 12 Nov 2019 14:13

Salmoneus wrote:
12 Nov 2019 00:08
Second:

Computers don't have to be linear
What does that mean?

You probably mean that computers does not have to be digital. Yes, there are the so-called analog computers.
They usually compute a single function. These function does not need to be linear, of course, it might e. g. an exponential function.
Salmoneus wrote:
12 Nov 2019 00:08
We think of computers as running a sequence of commands. But these, of course, are equivalent to simple logical propositions, which is why computer people still use terms like "logic gate" - a logic gate, as it were, enacts a logical operator.
That's a twisted argumentation. From the point of view of a hardware developer, a logic gate performs a logic function. That's why its called a logic gate. Is there a reason why you shouldn't call it like that? Because "we" think of computers as running a sequence of commands?
Salmoneus wrote:
12 Nov 2019 00:08
For instance, let's define eight data positions: A, B, C and D are input variables, E, F and G are output variables, and H is just used for calculation purposes. Data is binary. We can then say:
- set E, F, G and H to 0
- if B is 1, set G to 1
- if G AND D are now 1, set H to 1
- if G OR D are now 1, set G to 1
- if H is 1, set G to 0 and F to 1
- set H to 0
- if A AND C are now 1, set E to 1
- if A OR C are now 1, set H to 1
- if E is 1, set H to 0
- if H AND F are 1, set E to 1 and F to 0
- if H AND E are 1, set H to 0
- if H is 1, set F to 1
- set H to 0
You could call that an algorithmic description, even so it is not easy to understand what it does.
I'm not inclinde to analyse it or make a truth table from that.
Salmoneus wrote:
12 Nov 2019 00:08
Alternatively, we can simply say:
A^B^C^D⊢E^F^G
A^B^C^¬D⊢E^F^¬G
A^B^¬C^¬D⊢¬E^F^G
etc etc
Does the ^ mean an AND operation? What does the etc mean here? You just have 8 variables!
Salmoneus wrote:
12 Nov 2019 00:08
In fact, imagine a space with infinite dimensions.
Can we imagine that?
Salmoneus wrote:
12 Nov 2019 00:08
So, if you want to run a computer program, you don't have to build a computer. You just need to draw a picture, conjure up an infinite library of this sort,
If I only could do that ...
Salmoneus wrote:
12 Nov 2019 00:08
* assuming I've not gone wrong - which I almost certainly have, "⊢((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))((((AA)(BB)(CC)(DD))((AA)(BB)(CC)(DD)))((EE)(FF)(GG)))" can be interpreted as the fact that 4+4=8, expressed via binary.
You at least could all variables XX replace just by X according to the law of idempotence, as 1 AND 1 = 1 and 0 AND 0 = 0 or 1 OR 1 = 1 and 0 OR 0 = 0.
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Re: RE: Making a Computer that runs on Magic

Post by Ser » 12 Nov 2019 20:26

Tanni, I... don't know what to say. It's amusing how Sal's posts are going completely over your head, although amusing in a rather bad way. Let's see if I can do any better. If I fail to convey the meanings as well, then I'd advise just giving up...

1. He wrote a thought experiment about a non-linear way of finding the outcome of a calculation or algorithm by using magic and and an infinite Borges-style library. In such a world, you'd state the premises and the scent-hound would find the deducible proposition (output statement) in the infinitely-dimensional library that encompasses all possibilities. Maybe you're completely unfamiliar with thought experiments of this sort?

2. Yes, ^ stands for the AND operator. It's a standard logic symbol, used especially commonly in mathematics and philosophy. ⊢ is called a "turnstile" and means "proves/yields". The "etc etc" stood for all other possible combinations of negation in the variables A, B, C, D.

3. This is not Boolean algebraic notation, so "AA" must mean something other than "A AND A", so idempotency doesn't apply here. I don't know what repeating a variable like that means though.

4. The word "occult" includes the meanings "hidden" and "demonic", and the pun he made used both meanings ("hidden/unknown microprogramming" ~ "demonic/magical microprogramming"). A pun is a phrase where you get away with saying two things at the same time.

By the way, arguments about the True Meaning of a word based on etymology, as that weird Mark Passio guy does in the video, are worse than stupid. Words can perfectly have more than one meaning, and words change their meanings all the time. For example, "pants", as in the rather loose article of clothing for one's legs,
- originally meant "completely lion-like",
- which was then adapted as a human name,
- which was then associated with an ancient Christian martyr and saint,
- who centuries later was associated with the medieval Italian city of Venice,
- including the people of that town,
- later giving his name to the Old Man of Venice (a theatre character that stood for Venetians in Renaissance Italian comedy),
- so the name then stuck for the tight trousers that actors playing the character would wear,
- and then leg tights became popular in early modern France and people called them that,
- and as time passed the meaning gradually switched to looser types of legwear that today are called "pants" in English,
- all while spinning off the diminutive variant "panties" for women's underwear.
Are you now going to call muscular men with long wild hair like Jason Momoa "pants men"? (Or "panties men".) Look at the "True Meaning" of "pants", and hey, the guy looks like a lion after all.

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Re: RE: Making a Computer that runs on Magic

Post by Salmoneus » 13 Nov 2019 00:16

Ser wrote:
12 Nov 2019 20:26

2. Yes, ^ stands for the AND operator. It's a standard logic symbol, used especially commonly in mathematics and philosophy. ⊢ is called a "turnstile" and means "proves/yields". The "etc etc" stood for all other possible combinations of negation in the variables A, B, C, D.
Specifically, both the algorithm and the partial premise set are basically ways of defining addition of 1- or 2-digit numbers in binary. A and B are the potential digits of one number, and C and D are the potential digits of a second number (since classical logic is already binary, you don't have to say 'A=1', you can just take 'A' to be equivalent to '1 in A-position' and '~A' to be equivalent to '0 in A-position'). I'm defining each equation one-by-one in these examples - so 11+11 = 111, etc. Of course, you could do it systematically instead (c.f. Whitehead and Russell, etc - they don't use a place-value number system, but I'm sure you could define one from their work), which would let you add numbers of any number of digits, but for small numbers it's probably quicker just to define directly. Addition of numbers of up to two digits in binary is also what that algorithm is meant to do, but it was off the top of my head so there's presumably an error of some sort there...

(and to dot the t's, the turnstile being used to the right of a proposition means that the proposition follows from anything/nothing. Strictly speaking it means that it's provable, so arguably I should have used the double turnstile instead, which is closer to 'is true' than 'is provable', but let's not get caught up in metalogical pedantry...)
3. This is not Boolean algebraic notation, so "AA" must mean something other than "A AND A", so idempotency doesn't apply here. I don't know what repeating a variable like that means though.
Ah, sorry, I skipped ahead too quickly there! [And I'll explain here, because it'll actually be relevant to my 'suggestion three' post, which hopefully will make more sense and be more historically relevant than my last suggestion]

Most calculus in classical logic uses four operators - disjunction ('or'), conjunction ('and'), negation ('not') and material implication ('if...then...'), although an endless number of other ones can be defined if you want (in particular, these four are all either unary (negation) or binary (the others), but of course you cand define ternary, etc, operations for convenience).

The material implication can be converted, however, into a combination of the other three: A→B is synonymous with ~AvB. (the fact that 'if A, then B' in English is actually NOT synonymous with 'either not A, or else B' was one of the main motivations for the development of non-classical logics). So you don't actually need to have the concept of material implication to do classical logic, so long as you have the other three operators.

Now, eventually Charles Sanders Peirce (yes, he of the "hey, we could notate all this with two-dimensional pictures!" idea) discovered that ALL three/four common operators can be done away with, if you replace them all with what Peirce called "ampheck", and which has since been called "Peirce's arrow" or "Quine's dagger", or in common language simply 'neither-nor'. So:
~A = A↓A, [neither A nor A]
A^B = (A↓A)↓(B↓B) [neither neither A nor A nor neither B nor B, = neither not A nor not B = both A and B]
AvB = (A↓B)↓(A↓B) [neither A nor B nor neither A nor B, = not both not A and not B = either A or B]

However, since we're now using a system in which ampheck is the ONLY operator, we don't need to mark it at all, we only need to mark the order in which items are operated on. The most straightforward way to do that is to drop the ampheck but keep the brackets - so "either A or B" can be written (AB)(AB). There are also other, less intuitive ways to indicate operation orders without explicit operator symbols.

The makers of early binary computers quickly realised that, just as logicians could do without non-ampheck operators, so too their computers could do without any state-transforming element ("gate") other than the gate that corresponds to ampheck, which they called, simply enough, "NOR".

As it happens, however, it was also quickly realised that ampheck is one of TWO possible operators in a single-operator system. The other is called the Sheffer stroke, named after Henry Sheffer, who discovered the ampheck*. The Sheffer stroke equates in computing to a "NAND" gate, and for some reason at some point in the early days of digital computers everybody switched from using NOR gates to using NAND gates. I'm not sure if this is because it enables more intuitive programming, or just because early NAND gates were physically cheaper to make than early NOR gates...

Anyway, none of this matters for this, rather conceptual suggestion, but it will become relevant in my next two, more concrete suggestions...

EDIT: sorry, I intended to recognise explicitly the fact that Micamo had mentioned NAND gates and their sole-sufficiency already, but I forgot. Sorry, Micamo!

*yes. See, the ampheck was discovered by Charles Sanders Peirce, but being Charles Sanders Peirce, he never actually told anyone about it, and it wasn't until many, many decades later that people realised he'd known about it all along. This allowed Henry Sheffer, in the early 20th century, to independently re-invent the ampheck, which he indicated by means of a stroke, /. He also discovered what we now call the Sheffer stroke, but pretty much ignored it. Later, however, either due to misunderstanding or bloodymindedness, someone else decided to use Sheffer's stroke symbol for what we now use it for, and the usage stuck, which conveniently allowed Peirce's original dagger symbol to be discovered in his papers and used to indicate the ampheck (which Sheffer had used a stroke for), although not before Quine had already decided to use a dagger symbol for it. It's all, suitably enough, rather complicated.

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