{"ok":true,"protocol":"maclain_hunter_q_number_item_v1","qid":"Q1109","item":{"qid":"Q1109","slug":"hunter-negation-of-prior-order","path":"/q/Q1109","subdomain":"q1109.4p-engine.com","schemaType":"CreativeWork","recordOrigin":"authored_source","authoredSource":{"author":"Maclain Hunter","text":"\n\nNegation of Prior Order by Maclain Hunter:\n\nAbstract: I will, in the form of a well formed formal logical proof, walk readers through the reasoning that explains why the apparent “Simpson Paradox” is not actually a paradox - since it is not contradiction in the same order or stage of reasoning and data interpretation/integration, or a direct contradiction of the data with itself in the same context (ie - a true contradiction). This unwinds the “paradox”.\n\nSimpson Paradox Restated:\n\nThe Simpson’s Paradox occurs when a trend that appears in several groups of data disappears or reverses when the groups are combined. This happens because the relationship between variables is influenced by a lurking confounding factor. In such cases, aggregated data can lead to misleading conclusions that contradict the data’s subgroup trends. The paradox highlights the importance of considering hidden variables and the structure of data before making interpretations. It demonstrates how context and grouping can significantly impact statistical analysis and conclusions. In the context of machine learning, many unsupervised learning algorithms infer patterns different training datasets that result in contradictions when combined across the board.\n\nPremises: \n\nP: The so called Simpson Paradox proposes a false premise. Namely, it ignores that it is not necessary for a common trend in various systems to combine in a manner that balances and flattens the original trends (in subgroups) in a logically coherent (and balanced manner). This does not necessarily contradict Logic and real world examples of where this makes perfect sense, is in audio engineering where you could take the same sound and have the overall waveform and “trends” look virtually identical, while adjusting phase to create phase cancelation. The simplest example, not the most complex feasible example (of how many inputs you could feasibly phase cancel with) would be two copies of the same audio 180 degrees out of phase with each other. That would lead to no sound coming out of the speakers even while the meters on both tracks show the amplitude of the original audio on each channel. The master channel will be silent and have no amplitude or activity on the loudness meter. This principle applies in other fields and contexts according to logical translation rules. \n\nQ: The Simpson Paradox is a contradiction of Logic. It does not need to be resolved since it is arrived at through misconceptions about how concepts interact logically, not a true contradiction in the way these concepts are reacting (ie - phase cancellation). \n\nR: The trend in the subgroups being negated logically by the combination and reasoning of those subgroups into the (main) group is not the same as a contradiction of a concept or data set (or trend in the data) that contradicts itself. Therefore while it can be said that the group trend contradicts the subgroup trend that is not an invalid logical contradiction, it is a contradiction between group and subgroup trend, plain and simple - a different and valid kind of contradiction compared to paradoxical and/or self defeating logical contradiction, in the same context and sense at the same time. \n\nS: Assume the premises here are false. That leads to logical contradiction and the Simpson Paradox. Logical interpretation of data and the proper understanding of how different orders and groupings of data interact allow the avoidance and thus resolution of the Simpson Paradox, and the reasonable acceptance of the premises and this proof are a way to understand the paradox as resolved. \n\nLogical Proof: \n\n{P, P <-> Q, Q <-> R, R <-> S} |= (P & Q) & (R & S)\n\nConclusion: {P, Q, R, S}: \n\nThe Simpson Paradox arises from a misunderstanding of how subgroups can logically lead to negation of subgroup trends in coherent groups and logical systems, as we observe in easily replicated fashion with the phase cancellation between sounds in audio engineering. \n\nThe correct understanding of this concept helps us to avoid seeing this “paradox” as any sort of oddity in the first place. Concept A can lead to a concept B that negates Concept A. This is not necessarily and contradiction, therefore to call it a paradox is itself incoherent. \n\nAssumption of the negation of the premises leads to contradiction and/or the illogical preservation of the Simpson Paradox. In fact, the Law of Identity leads obviously, as the Rule that A is A, to knowing ALL must be ALL. \n\nThis means that all concepts must be encompassed and united in the Universal Logos (also understood as saying “Reality must be Oneness”), all while in finite realities and contexts there must be at least one context where each thing is each thing and where each thing is not each thing. \n\nThis means that when we find a “valid form of incoherence”, while that is generally a logical contradiction, we can find contexts, like phase cancellation or the (not truly paradoxical) “Simpson Phenomenon”, where it is not actually a “logical contradiction” but merely a valid contradiction between concepts that don’t include Logic (or its substantive equivalent), like two discrete groupings of data. \n\nTruth Table Analysis: \n\nvalid\n     ↓  ↓   ↓   ↓       ↓      \nPQRS⎪P P↔Q Q↔R R↔S╞(P&Q)&(R&S)\n―――――――――――――――――――――――――――――――\nTTTT⎪T TTT TTT TTT  TTT T TTT \nTTTF⎪T TTT TTT TFF  TTT F TFF \nTTFT⎪T TTT TFF FFT  TTT F FFT \nTTFF⎪T TTT TFF FTF  TTT F FFF \nTFTT⎪T TFF FFT TTT  TFF F TTT \nTFTF⎪T TFF FFT TFF  TFF F TFF \nTFFT⎪T TFF FTF FFT  TFF F FFT \nTFFF⎪T TFF FTF FTF  TFF F FFF \nFTTT⎪F FFT TTT TTT  FFT F TTT \nFTTF⎪F FFT TTT TFF  FFT F TFF \nFTFT⎪F FFT TFF FFT  FFT F FFT \nFTFF⎪F FFT TFF FTF  FFT F FFF \nFFTT⎪F FTF FFT TTT  FFF F TTT \nFFTF⎪F FTF FFT TFF  FFF F TFF \nFFFT⎪F FTF FTF FFT  FFF F FFT \nFFFF⎪F FTF FTF FTF  FFF F FFF \n\nQ.E.D. 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Phase cancellation illustrates coherent negation across combinations: group/subgroup disagreement need not contradict Logic or Reality under the same context and sense.","text":"\n\nNegation of Prior Order by Maclain Hunter:\n\nAbstract: I will, in the form of a well formed formal logical proof, walk readers through the reasoning that explains why the apparent “Simpson Paradox” is not actually a paradox - since it is not contradiction in the same order or stage of reasoning and data interpretation/integration, or a direct contradiction of the data with itself in the same context (ie - a true contradiction). This unwinds the “paradox”.\n\nSimpson Paradox Restated:\n\nThe Simpson’s Paradox occurs when a trend that appears in several groups of data disappears or reverses when the groups are combined. This happens because the relationship between variables is influenced by a lurking confounding factor. In such cases, aggregated data can lead to misleading conclusions that contradict the data’s subgroup trends. The paradox highlights the importance of considering hidden variables and the structure of data before making interpretations. It demonstrates how context and grouping can significantly impact statistical analysis and conclusions. In the context of machine learning, many unsupervised learning algorithms infer patterns different training datasets that result in contradictions when combined across the board.\n\nPremises: \n\nP: The so called Simpson Paradox proposes a false premise. Namely, it ignores that it is not necessary for a common trend in various systems to combine in a manner that balances and flattens the original trends (in subgroups) in a logically coherent (and balanced manner). This does not necessarily contradict Logic and real world examples of where this makes perfect sense, is in audio engineering where you could take the same sound and have the overall waveform and “trends” look virtually identical, while adjusting phase to create phase cancelation. The simplest example, not the most complex feasible example (of how many inputs you could feasibly phase cancel with) would be two copies of the same audio 180 degrees out of phase with each other. That would lead to no sound coming out of the speakers even while the meters on both tracks show the amplitude of the original audio on each channel. The master channel will be silent and have no amplitude or activity on the loudness meter. This principle applies in other fields and contexts according to logical translation rules. \n\nQ: The Simpson Paradox is a contradiction of Logic. It does not need to be resolved since it is arrived at through misconceptions about how concepts interact logically, not a true contradiction in the way these concepts are reacting (ie - phase cancellation). \n\nR: The trend in the subgroups being negated logically by the combination and reasoning of those subgroups into the (main) group is not the same as a contradiction of a concept or data set (or trend in the data) that contradicts itself. Therefore while it can be said that the group trend contradicts the subgroup trend that is not an invalid logical contradiction, it is a contradiction between group and subgroup trend, plain and simple - a different and valid kind of contradiction compared to paradoxical and/or self defeating logical contradiction, in the same context and sense at the same time. \n\nS: Assume the premises here are false. That leads to logical contradiction and the Simpson Paradox. Logical interpretation of data and the proper understanding of how different orders and groupings of data interact allow the avoidance and thus resolution of the Simpson Paradox, and the reasonable acceptance of the premises and this proof are a way to understand the paradox as resolved. \n\nLogical Proof: \n\n{P, P <-> Q, Q <-> R, R <-> S} |= (P & Q) & (R & S)\n\nConclusion: {P, Q, R, S}: \n\nThe Simpson Paradox arises from a misunderstanding of how subgroups can logically lead to negation of subgroup trends in coherent groups and logical systems, as we observe in easily replicated fashion with the phase cancellation between sounds in audio engineering. \n\nThe correct understanding of this concept helps us to avoid seeing this “paradox” as any sort of oddity in the first place. Concept A can lead to a concept B that negates Concept A. This is not necessarily and contradiction, therefore to call it a paradox is itself incoherent. \n\nAssumption of the negation of the premises leads to contradiction and/or the illogical preservation of the Simpson Paradox. In fact, the Law of Identity leads obviously, as the Rule that A is A, to knowing ALL must be ALL. \n\nThis means that all concepts must be encompassed and united in the Universal Logos (also understood as saying “Reality must be Oneness”), all while in finite realities and contexts there must be at least one context where each thing is each thing and where each thing is not each thing. \n\nThis means that when we find a “valid form of incoherence”, while that is generally a logical contradiction, we can find contexts, like phase cancellation or the (not truly paradoxical) “Simpson Phenomenon”, where it is not actually a “logical contradiction” but merely a valid contradiction between concepts that don’t include Logic (or its substantive equivalent), like two discrete groupings of data. \n\nTruth Table Analysis: \n\nvalid\n     ↓  ↓   ↓   ↓       ↓      \nPQRS⎪P P↔Q Q↔R R↔S╞(P&Q)&(R&S)\n―――――――――――――――――――――――――――――――\nTTTT⎪T TTT TTT TTT  TTT T TTT \nTTTF⎪T TTT TTT TFF  TTT F TFF \nTTFT⎪T TTT TFF FFT  TTT F FFT \nTTFF⎪T TTT TFF FTF  TTT F FFF \nTFTT⎪T TFF FFT TTT  TFF F TTT \nTFTF⎪T TFF FFT TFF  TFF F TFF \nTFFT⎪T TFF FTF FFT  TFF F FFT \nTFFF⎪T TFF FTF FTF  TFF F FFF \nFTTT⎪F FFT TTT TTT  FFT F TTT \nFTTF⎪F FFT TTT TFF  FFT F TFF \nFTFT⎪F FFT TFF FFT  FFT F FFT \nFTFF⎪F FFT TFF FTF  FFT F FFF \nFFTT⎪F FTF FFT TTT  FFF F TTT \nFFTF⎪F FTF FFT TFF  FFF F TFF \nFFFT⎪F FTF FTF FFT  FFF F FFT \nFFFF⎪F FTF FTF FTF  FFF F FFF \n\nQ.E.D. 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