Trilemmas Define Syllogistics
By Nathaniel Hellerstein
Trilemmas Define Syllogistics
By Nathaniel Hellerstein
October 2019
Table of Contents:
0. Introduction 3
1. Trilemma, Troika, and Triad 4
2. Various Trilemmas 7
3. Inevitability of Trilemmas 13
4. Spoofing Classical Logic 16
5. Trilemma Deduction 26
6. Anti-Sorites 32
7. Loopless Loop Trilemmas 38
8. Arrow’s Trilemma 43
9. NP-complete Trilemmas 46
10. Exercises for the Student 49
11. Bibliography 61
Introduction
This essay is about Trilemmas, which are trios of statements, any two of which can be true but not all three. Therefore any two imply the negation of the third. That is a deductive system, here explored.
No doubt the reader will notice a Carrollian quality in many of the trilemmas listed here. You will find the Mad Hatter, and the Cheshire Cat, and Alice; these are my tributes to Lewis Carroll, master of logical nonsense. I also include angels, and aliens, and dragons, and cats, and bats, and Batman, and ghosts, and frog princes; and behind all of them, you will find none other than the Three Stooges; Moe, Larry, and Curly, whose endless quarrels yield all of this essay’s trilemmas.
Why the absurdism? For two reasons:
Because trilemmas are absurd, so they naturally provoke laughter. Each of these trilemmas has the formal structure of a joke; setup, development, punchline. Trilemmas are mathematical jests.
And also, to lighten presentation. To quote one of the trilemmas in the “Exercises” section:
This essay is about advanced Logic;
Any essay about advanced Logic will put you straight to sleep;
This essay will not put you straight to sleep.
I hope that this essay satisfies clause 3.
Fastest mastery of this material will be gotten by reading section 1, for the rules of the game, then working out all of the examples in section 10, where most of the jokes are.
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