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- P1. Doubt everything (knowledge, trustworthiness of senses, Logic, external world, etc)
- P2. Doubt thoughts
- C1. Thoughts cannot be denied (the act doubting thoughts is a thought.)
- P3. Certainty is an occurrence which cannot be doubted or denied. (Denial is occurrence of thought)
- C2. thinking and experience exist (certainly, at least within my observation)
- P4. Let T = {X1, X2, X3, ..., Xn} be the set of specific thoughts, where each Xi represents a different thought (for example, X1 = "I hate ice cream", X2 = "I hate pizza", X3 = "I want to go for a walk").
- P5. X1 occurs.
- C3. The content of X1 is not contained in T - {X1}.
- C4. Therefore, X1 = T−(T−{X1}) ⇒ −−X1 ⇒ X1 (Law of Identity)
- C5. Therefore, X1 ≠ T - {X1} (Law of Non-Contradiction)
- P6. repeat P5-C5 for Xi to Xn.
- C6. IF a proposition about Xn ≠ Xi, then it must = -Xi ("NOT-Xi")
- C7. Therefore, Xn = Xi OR Xn ≠ Xi (Law of Excluded Middle)
- C8. the truth of any proposition belonging to the set T is determinately TRUE or NOT-true
- P7. there are n Xi that i am not certain of.
- C9. Knowledge is limited.
- P8. but the truth is/was determined already. (C4, C5, C7, and C8)
- C10. Therefore, Truth exists even when i don't certainly know it.
- P9. For something to be true it must be possibly true (able to be true or NOT-true).
- C11. If a proposition Xi violates C4, C5, and/or C7 (for example, the proposition "A square-circle exists" which violates C5, and C7 by extension) it must be NOT-true.
- P10. The proposition "Does certain truth(s) exist outside of 'me'" occurs
- C12. IF NO then all certain truth(s) must be contained in "me, which implies i know all available certain truths. But this cannot be, since (C8, C9, C10), and there are truths of which i am not certain (including this very proposition). IF YES then certain truth(s) must exist outside of "me." and determined already (P8)
- C13. Therefore, there must be another that contains certain truth(s)
- P11. Let Knower = one that possesses an understanding of truth(s)
- P12. Let Knowledge = the determinate truth-value of a given proposition.
- C14. Therefore, to have knowledge you must be a knower (of the truth).
- C15. Therefore, knowing is the relation where a Knower possesses a determinate truth
- C16. Therefore, Knowledge = the determinate truth possessed by a Knower.
- C17. Therefore, all truth is known by a knower as knowledge.
- C18. Therefore, For all knowers Kx and Ky, there exists a knower Kz such that Kz knows all truths that Kx and Ky know. (we know that the Xi in Set T are determinately TRUE or NOT-true, say a knower K1 only knows X1 and K2 only knows X2. then there must be a knower (K3) which knows X1 + X2 labeled here as X3, because the content of X3 ≠ X1 OR X2)
- C19. Therefore, there exists a Knower which knows all truths in Set T (X∞)
- C20. Therefore, The Knower who knows all truths in Set T knows that there exists a knower ('me') who knows some subset of T.
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