Katabasis
“No—the reason paradoxes trouble us is that their absurd conclusions make us rethink all of our premises. A paradox is like a staircase, in which each step leads inexorably to the destination. But you get to the top, and the destination is impossible; you’ve stepped off into empty air.”
Katabasis
Katabasis is an allegory of an already existing allegory.
It takes the structure of Dante’s Inferno and adds a power system called “magick” which is another name for “linguistics”. Specifically, linguists who study “symbolic systems” or “grammar”, including both math and English.
The plot is good, but I only got through it because of the random references scattered throughout the book. I wrote this post to talk about the referenced philosophical paradoxes and the resolutions people have proposed to them.
Paradoxes
M.C. Escher
The book cover of Katabasis is a reference to the work of M.C. Escher. What is so special about Escher’s works is his ability to play with perceptual reasoning. It is an interesting showcase of how unconscious and opaque our reasoning systems can be.
The paradox is resolved by recognizing how our visual systems are processing the image. The paradoxical effect exists because our visual language uses the junctions to infer movement. The problem arises when we integrate the junctions and the deduced movement, which can be crafted to contradict each other.
Importantly, Escher is not playing with our reflective systems; he’s playing with our “pre-reflective” “visual-language” system.
Sorites’ Paradox
1,000,000 grains of sand are a heap. Take away a single spec of sand, and it is still a heap. Repeat, and you will end up with 0 specs and a heap. The paradox arises from and is resolved by asking for the definition of a “heap” or any “predicate”.
There is a simple solution: defining the predicate quantitatively. A heap is only a heap when there are at least X grains. The problem is we don’t necessarily have an “objective” definition of a heap. There is an intrinsic “vagueness” to some words1.
Zeno’s Paradox
Zeno’s paradox is described with Tortoise & Hare interlocutors. Consider the following statements. If it rains, it will be wet outside. It rained. So it is wet outside. It is an evocation of the “modus ponens” schema. The Tortoise, however, asks why we are justified in believing in modus ponens.
The Hare adds another premise by stating If the first premise is true and the second premise is true, then the conclusion is true. Then again, why should the Tortoise agree with the newly added premise?
The resolution here is realizing how the grammatical rules of a system are not defined within the system. When people do formal logic, “modus ponens” isn’t in the “object language” of formal logic. It’s in plain English or its own grammar.
Curry’s and Russell’s Paradox
I grouped these two because they are mechanically the same. These two paradoxes arise from the use of “names” versus the actual content of the “name”.
Curry’s paradox goes like this. S is the sentence: “If S is true, then X is true.” By assuming the sentence S and because of the self-reference, anything you put in the implication is true.
Looking at the Wikipedia for Curry’s, it shows how the paradox can be made to be a variant of Russell’s Paradox. Russell’s Paradox in natural language is this: “Does the set of all sets that do not contain themselves, contain themselves?” or R = { x: x not in x}.
The question here is how the name R is resolved. Does it resolve to contain itself or not? This is what causes the paradox. The resolution to these problems is not too bad. Add grammar rules about how names are resolved and how they can’t be treated in a self-referential way.
Liar’s Paradox
This sentence is false.
or
The next sentence is false. The previous sentence is true.
There isn’t a satisfying resolution besides choosing a different system or banning it.
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Some words escape vagueness. Being bald necessarily means a person has no hair on their head. ↩