The Physical Ground of Logic

[updated 13 January 2024 to add a links at the end for additional information]
[updated 14 January 2024 to add note on semantics and syntax]
[updated 17 January 2024 to add note on universality of NAND and NOR gates]

The nature of logic is a contested philosophical question. One position is that logic exists independently of the physical realm; another is that it is a fundamental aspect of the physical realm; another is that it is a product of the physical realm. These correspond roughly to the positions of idealism, dualism, and materialism.

Here, the physical basis for logic is demonstrated. This doesn't disprove idealism, dualism, or materialism; but it does make it harder to find a bright line of demarcation between them and a way to make a final determination as to which might correspond to reality.

There are multiple logics. As "
zeroth-order" (or "propositional logic") is the basis of all higher logics, we start here.

Begin with two arbitrary distinguishable objects. They can be anything: coins with heads and tails; bees and bears; letters in an alphabet, silicon and carbon atoms. Two unique objects, "lefty" and "righty" are chosen. The main reason is to use objects for which there is no common associated meaning. Meaning must not creep in "the back door" and using unfamiliar objects should help that from inadvertently happening. A secondary engineering reason will be demonstrated later.



LeftyRighty
\|

These two objects can be combined four ways as shown in the next table. You should be able to convince yourself that these are the only four ways for these combinations to occur. For convenience, each input row and column is labeled for future reference.






$1$2
C0\\
C1\|
C2|\
C3||

Next, observe that there are sixteen ways to select an object from each of the four combinations. You should be able to convince yourself that these are the only ways for these selections to occur. For convenience, the selection columns are labelled for future reference. The selection columns correspond to the combination rows.






S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15
C0||||||||\\\\\\\\
C1||||\\\\||||\\\\
C2||\\||\\||\\||\\
C3|\|\|\|\|\|\|\|\

Now the task is to build physical devices that combine these two inputs and select an output according to each selection rule. Notice that in twelve of the selection rules it is possible to get an output that was not an input. On the one hand, this is like pulling a rabbit out of a hat. It looks like something is coming out that wasn't going in. On the other hand, it's possible to build a device with a "hidden reservoir" of objects so that the required object is produced. But this is the engineering reason why "lefty" and "righty" are the way they are. "lefty" can be turned into "righty" and "righty" can be turned into "lefty" by rotating the object and rotation is a physical operation on a physical object.

Suppose we can build a device which has the behavior of S7:

S7

We know how to build devices which have the behavior of S7, which is known as a "NAND" (Not AND)1 gate. There are numerous ways to build these devices: with semiconductors that use electricity, with waveguides that use fluids such as air and water, with materials that can flex, neurons in the brain, even marbles.2 NAND gates and NOR gates (S1) are known as universal gates, since arrangements of each gate can produce all of the other selection operations. The demonstration with NAND gates is here.

John Harrison writes
3:

    The correspondence between digital logic circuits and propositional logic has been known for a long time.









Digital designPropositional Logic
logic gatepropositional connective
circuitformula
input wireatom
internal wiresubexpression
voltage leveltruth value

The interesting bit now is how to get "truth values" into, or out of, the system. Typically, we put "truth" into the system. We look at the pattern for S7, say, and note that if we arbitrarily declare that "lefty" is "true", then "righty" is false, then we get the logical behavior that is familiar to us. Because digital computers work with low and high voltages, it is more common to arbitrarily make one of them 0 and the other 1 and then to, again arbitrarily, make the convention that one of them (typically 0) represents false and the other (either 1 or non-zero) true.

This way of looking at the system leads to the idea that truth is an emergent behavior in complex systems. Truth is found in the arrangement of the components, not in the components themselves.

But this isn't the only way of looking at the system.

Let us break down the what the internal behavior of each selection process has to accomplish. This behavior will be grouped in order of increasing complexity. To do this, we have to introduce some notation.

~ is the "rotate" operation. ~\ is |; ~| is \.
= tests for equality, e.g. $1 = $2 compares the first input value with the second input value.
? equal-path : unequal-path takes action based on the result of the equal operator.
\ = \ ? | : \ results in |, since \ is equal to \.
\ = | ? | : \ results in \, since \ is not equal to |.

The first group of selectors simply output a constant:




SelectionBehavior
S0|
S15\

The second group outputs one of the inputs, perhaps with rotation:






SelectionBehavior
S12$1
S3~$1
S10$2
S5~$2

The third group compares the inputs for equality and produces an output for the equal path and another for the unequal path:












SelectionBehavior
S8$1 = $2 ? $1 : |
S14$1 = $2 ? $1 : \
S1$1 = $2 ? ~$1 : |
S7$1 = $2 ? ~$1 : \
S4$1 = $2 ? | : $1
S2$1 = $2 ? | : $2
S13$1 = $2 ? \ : $1
S11$1 = $2 ? \ : $2
S6$1 = $2 ? | : \
S9$1 = $2 ? \ : |


It's important to note that whatever the physical layer is doing, if the device performs according to these combination and selection operations then the internal layer is doing these logical operations.

The next step is to figure out how to do the equality operation. We might think to use S9 but this doesn't help. It still requires the arbitrary assignment of "true" to one of the input symbols. The insight comes if we consider the input symbol as an object that already has the desired behavior. Electric charge implements the equality behavior: equal charges repel; unequal charges attract. If the input objects repel then we take the "equal path" and output the required result; if they attract we take the "unequal path" and output that answer.

In this view, "truth" and "falsity" are the behaviors that recognize themselves and disregard "not themselves." And we find that nature provides this behavior of self-identification at a fundamental level. Charge - and its behavior - is not an emergent property in quantum mechanics.

What's interesting is that nature gives us two ways of looking at it and discovering two ways of explaining what we see without, apparently, giving us a clue as to which is the "right" way to look at it. Looking one way, truth is emergent. Looking way, truth is fundamental. Philosophers might not like this view of truth (that truth is the behavior of self-recognition), but our brains can't develop any other notions of truth apart from this "computational truth."

Hiddenness

How the fact that the behavior of electric charge is symmetric, as well as the physical/logical layer distinction, plays into subjective first person experience is
here.

Syntax and Semantics

In natural languages, it is understood that the syntax of a sentence is not sufficient to understand the semantics of the sentence. We can parse "'Twas brillig, and the slithy toves" but, without additional information, cannot discern its meaning. It is common to try to extend this principle to computation to assert that the syntax of computer languages cannot communicate meaning to a machine. But logic gates have syntax and behavior. The syntax of S7 can take many forms: S7, (NAND x y), NOT AND(a, b); even the diagram of S7, above. The behavior associated with the syntax is: $1 = $2 ? ~$1 : \. So the syntax of a computer language communicates behavior to the machine. As will be shown later, meaning is the behavior of association, this is that. So computer syntax communicates behavior, which can include meaning, to the machine.

More elaboration on how philosophy misunderstands computation is here.

Notes

An earlier derivation of this same result using the Lambda Calculus is here. But this demonstration, while inspired by the Lambda Calculus, doesn't require the heavier framework of the calculus.



[1] An "AND" gate has the behavior of S8.
[2] There is a theory in philosophy known as "multiple realizibility" which argues that the ability to implement "mental properties" in multiple ways means that "mental properties" cannot be reduced to physical states. As this post demonstrates, this is clearly false.
[3] "
A Survey of Automated Theorem Proving", pg. 27.

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