Quarry

Rationale for the Quarry
The role of the Quarry is to transcribe in the game world the ideas expressed by Burke in the first windmill video. For those who haven't really listened to what Burke says in this video and believe it's about technology being better than religion or this kind of simplistic idea, you're wrong. The video is about the forces that drive changes in our societies, and about how knowledge and philosophies provide ways and reasons for these changes. (To make the link between this video and the quarry, note how an optional puzzle in the control room of the concrete factory looks a lot like the one on the door of the vault giving that video).

The reason why we see traces of different societies on the island — from antique to modern or slightly futuristic ones — is to make us reflect on what were the knowledge and philosophies of these civilizations, i.e. what forces drove the changes on the island (Jon Blow talked about that). And the quarry is one of the places where this is the most obvious, with the church being transformed into a concrete factory when religion became an outdated idea. The drawings in the mountain show many studies and variants of some areas, but the ones concerning the quarry show the history of that place and how it evolved from its natural state to its current state (one of these drawings, with a different layout: here). And the architects who have worked on the game always talk about the quarry and how it has been transformed by the different civilizations (on their website or in the gamasutra post).

So the quarry is about progress, in a broad sense. It's about ideas that evolve, structures that get obsolete and get reused to better fit the needs of civilizations, and how it transforms these civilizations and these structures. It's actually more general than that, since it also encompasses the evolution of human beings, as can be seen with the statues in the concrete factory that reproduce the famous March of Progress painting.

This kind of progress has two representations that are used as design principles in the quarry: timelines and stairs. Progress is also characterized by two operations that are also used as design principles in the quarry: reuse and transform obsolete material, and removing obstacles to progression.

Timelines
Timelines is the idea that you can represent time by a line and that progress corresponds to a change along that line. Of course, the March of Progress is a timeline. Another realization of the timeline principle is the production line: products are created by advancing on the production line where they are transformed as they advance. In the quarry there are two production lines: one for the concrete and one for the wood. Interestingly, these two production lines goes in the same direction than the river flows, and the river is another linear representation of time (a lot of elements follow that direction in the quarry: the window of the concrete factory, the wood planks making the walls of the sawmill, the grid on the concrete factory chimney, etc.) The quarry cliff is itself a timeline, the most recent excavation being near the dam, while antique excavations were done close to the sea. The time evolution of the quarry cliff is shown in a drawing inside the mountain, which is used in Luis Antonio's GDC talk The Art of The Witness to illustrate how the landscape architects contributed in building a narrative for the island.

Stairs
Stairs is another representation of progress that highlights the idea that progress is made of discrete steps, each step allowing to reach an higher level. There are a lot of stairways in the quarry (each building has one inside, plus the one coming from the dam and the one going to the roof of the concrete factory). The quarry cliff itself looks like stairs. In the March of Progress you can also see a vertical progression as the human being stands upper as he evolves.

And notice how in the two buildings you need to access catwalks at 3 or 4 different altitudes, always getting higher.

And finally, closer to your concern, the cancellation symbol introduces a two-step checking of puzzle solutions: first, all symbols have to be checked as usual, then a cancellation symbol and an invalid symbol have to be cancelled so that the puzzle is solved.

Reuse and transformation
Reuse and transformation is obvious in the way the church has been refurbished as a concrete factory, but also in the way one of the walls of the former church is actually the quarry cliff. Less obvious is that the sawmill is also built on top of a previous building (that was probably a water mill, according to the shape of the parts that we see below the sawmill floor and to the drawings in the mountain). Transformation is also the goal of these buildings: transform stones and trees into concrete and wood planks. And the March of Progress is a depiction of Darwinian evolution, which assumes that nature reuses unused features of living organisms when they need to adapt to new conditions.

More interestingly, to progress in the two buildings in the quarry, you need to walk on parts that are not made for this purpose but are rather dedicated to the materials being transformed. In doing so, you reuse those parts.

Finally, the cancellation symbol works by reusing symbols that failed (it does not only cancel one, it consumes one). In that sense, the remark that the cancellation mechanism requires you to make errors is wrong: it requires you to reuse inappropriate symbols in an intelligent way to progress, and thus it transforms invalid symbols into valid ones. Or if you prefer it uses symbols as a raw material to produce valid ones.

Removing obstacles
Removing obstacles is also an important process for progress. Transforming the church into a concrete factory would not have been possible if religion was still an important matter to that civilization. The production lines use a lot of machines to remove the obstacles that are travel distance and weight of materials (quarry elevator, inclined planes, conveyor belts...). And the riverbed has been cemented so that wood trunks could be carried on water.

When you progress in the two buildings, you need in each building to first cancel the obstacle of altitude difference, using an inclined plane. Then, you need to solve a few puzzles. In the concrete factory it simply activates a control panel, but in the sawmill it removes an obstacle, allowing you to reach an already activated control puzzle. That control panel (or puzzle) will allow you to move on a platform in order to pass another obstacle (in the concrete factory, that obstacle is another altitude difference, but in the sawmill it's another machine). Note that in both cases this step affects the first inclined plane (in the quarry it inclines it, in the sawmill it translates it), which is a kind of reuse as this plane is now obsolete. Then a (fixed) inclined plane allows you to reach the catwalks (with a panel/puzzle that will give you an easier way to get back on the first floor). Solving a long set of puzzles is the last obstacle. (Another similitude between the two buildings: they both provide access to another path from inside the building: the boat dock in the sawmill and the stairway to the quarry top and profiles tree in the concrete factory.) And all this will not be enough to activate the laser: it only removes a lock on the laser puzzle. And these locks are only obstacles that prevent you to draw the solution of the puzzle.

Now, the principle of removing an obstacle is exactly how the cancellation symbol works: it cancels the constraint remaining after another symbol failed.