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Second Law
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๐ ๐ป โ ๐ ๐ฟ = ๐
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๐ ๐ป โ ๐ ๐ฟ = ๐
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็ๅๅญธ Logic ๅบๆฌ่ฆ็พฉ ๅฝ้ก statement, proposition ็ฐกๅฎๅฝ้ก simple statement๏ผ๐ใ๐ ็ญ ่คๅๅฝ้ก composite statement ้ฃ่จ conjunction ๐โง๐ ๐ ไธ ๐ ๐ and ๐ ้ธ่จ disjunction ๐โจ๐ ๐ ๆ ๐ ๐ or ๐ ๅฆๅฎ negation โผ๐ ้ ๐ not ๐ ่ๅซ implication ๐โน๐ ่ๅซๆ่จฑๅค็จฎๆๅญ่กจ่ฟฐ๏ผ ๐ ่ๅซ ๐ ๐ implies ๐ ่ฅ ๐ ๅ ๐ if ๐ then ๐; ๐ if ๐ (ๅ่ฃๅฅ)
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่ๅซๅฝ้กๆไธ็จฎ่ฎๅ๏ผ้ฃๅๅๅฝ้กไธๅ
ฑๆๅ็จฎ๏ผ
ๅๅฝ้ก (original)๏ผ ๐โ๐ ้ๅฝ้ก (converse)๏ผ ๐โ๐ ๅๅฝ้ก (inverse)๏ผ ~๐ โ ~๐ ้ๅๅฝ้ก (transposition)๏ผ ~๐ โ ~๐ ๆๅธธ่ฆ็ๆ่ฌใไธๅ้่ผฏใๅฐฑๆฏ่ชคไปฅ็บ้ๅฝ้กๆๅๅฝ้ก็ญๅๆผๅๅฝ้กใ ๅๅฝ้ก๏ผๅๅฐๆฏ็ทๆง ้ๅฝ้ก๏ผ็ทๆงๆฏๅๅฐ ๅๅฝ้ก๏ผไธๆฏๅๅฐๅฐฑไธๆฏ็ทๆง ้ๅๅฝ้ก๏ผไธๆฏ็ทๆงๅฐฑไธๆฏๅๅฐ
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๐โ๐ original proposition ๐โ๐ converse ้ๅฝ้ก
Antecedent ๅ้ณ Consequence ็ต่ซ Sufficient Condition ๅ
ๅๆขไปถ Necessary condition ๅฟ
่ฆๆขไปถ ๐โ๐ original proposition ๐โ๐ converse ้ๅฝ้ก ๅๅฝ้กๆ็ซ๏ผๅ
ถ้ๅฝ้กไธไธๅฎๆ็ซใ
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๐๏ผใๅ
ฉไธ่งๅฝขไธๅฐๆ้็ธ็ญใ ๐๏ผใๅ
ฉไธ่งๅฝขไธ่งๅฐๆ็ธ็ญใ ไธๅฐๆ้็ธ็ญ๏ผไธๅฐๆ่งๅฟ
็ธ็ญใ ๐โ๐ ไธๅฐๆ่งไธ็ญ๏ผไธๅฐๆ้ๅฟ
ไธ็ญใ ~๐ โ ~๐ ไธๅฐๆ้ไธ็ญ๏ผไธๅฐๆ่งๆชๅฟ
ไธ็ญใ ~๐ โ ~๐ ไธๅฐๆ่ง็ธ็ญ๏ผไธๅฐๆ้ๆชๅฟ
็ธ็ญใ ๐โ๐ ๅ ็บ้ๅๅฝ้ก่ๅๅฝ้ก็ญๅ๏ผๅฏไปฅ็จ้ๅๅฝ้ก็่ญๆไฝ็บๅๅฝ้ก็่ญๆใ ้็จฎ่ญๆๆนๆณ็จฑ็บใๅ่ญๆณใๆใๆญธ่ฌฌ่ญๆณใ(reductio ad absurdum)ใ
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ๅ่ญๆณ ๐ ๏ ๐ T F ๏พ q ๏ p F T ๅๅฝ้ก(ไธ)ๆ็ซ๏ผๅ
ถ้ๅๅฝ้กๅฟ
(ไธ)ๆ็ซ ่ญๆ้ๅๅฝ้ก็ญๅๆผ่ญๆๅๅฝ้กใ
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่ฅ ๐โน๐ ๆ็ซ๏ผ ๐ ็จฑ็บ ๐ ็ๅ
ๅๆขไปถ (sufficient condition)๏ผ ๐ ็จฑ็บ ๐ ็ๅฟ
่ฆๆขไปถ (necessary condition)ใ
ๅ
ๅๆขไปถ๏ผๆๅๅฟ
็ถ๏ผ็ผบๆชๅฟ
ไธ็ถใ ๅฟ
่ฆๆขไปถ๏ผๆๆชๅฟ
็ถ๏ผ็ผบๅๅฟ
ไธ็ถใ ไฟ่ฉฑ๏ผใ้ข้่ฌ่ฝ๏ผ็ก้ข่ฌ่ฌไธ่ฝใใ ๆฌฒ่ญ ๐โน๐๏ผๅฏ็จ็ด่ญๆณ่ๅ่ญๆณใ ็ด่ญๆณไน่ญๆ ๐ ็บ ๐ ไนๅ
ๅๆขไปถ๏ผ ๅ่ญๆณไน่ญๆ ๐ ็บ ๐ ไนๅฟ
่ฆๆขไปถใ
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Equivalence of KP (Kelvin-Planck) statement and C (Clausius) statements
KP ่กจ่ฟฐ่ C ่กจ่ฟฐ้ฝๆฏ็ถ้ฉๆ่ฟฐ๏ผ็กๆณๅพ็ๅๅญธๆจๅฐๅบไพใ ๆๅ่ฝ่ญๆ็ๆฏ้ๅ
ฉ็จฎ่กจ่ฟฐๆฏ็ญๅ็๏ผ KPโC and CโKP ไฝๆฏ้ๅ
ฉๅ่ๅซๅฝ้กๆฌ่บซ้ๆฏไธๅฎนๆ็ดๆฅ่ญๆ๏ผ ๅ ๆญค็จๆญธ่ฌฌ่ญๆณ(ๅณๆฏๅ่ญๆณ)่ญๆ๏ผ ~C โ ~KP and ~KP โ ~C
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๐พ: Kelvin-Planck statement of 2nd law
โ๐พ : negation of Kelvin-Planck statement ๐ถ : Clausius statement of 2nd law โ๐ถ : negation of Clausius statement
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KPโC and CโKP ~C โ ~KP and ~KP โ ~C KP โ C and C โ KP ~C โ ~KP and ~KP โ ~C ~Cโ~KP๏ผ ่ฅๆๅฎ็พ็ๅทๅปๆฉ๏ผๅๅฟ
ๆๅฎ็พ็็ฑๆฉ๏ผไฝๆฏๆๅฎ็พ็ฑๆฉ้ๅ็ฌฌไบๅพ๏ผไบๅฏฆไธๆฒๆๅฎ็พ็ฑๆฉใ ๅ ๆญคๆฒๆๅฎ็พ็ๅทๅปๆฉใ ~KPโ~C๏ผ ่ฅๆๅฎ็พ็็ฑๆฉ๏ผๅๅฟ
ๆๅฎ็พ็ๅทๅปๆฉ๏ผไฝๆฏ ๆๅฎ็พ็ๅทๅปๆฉ้ๅ็ฌฌไบๅพ๏ผไบๅฏฆไธๆฒๆๅฎ็พ็ๅทๅปๆฉใๅ ๆญคๆฒๆๅฎ็พ็็็ฑๆฉใ
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Final remark: ไปฅไธ่ซ่ญไธฆ้่ญๆ็ฑๅๅญธ็ฌฌไบๅพ ่ๆฏ่ญๆ็ฌฌไบๅพ็็ฑๆฉ่กจ่ฟฐไปฅๅๅทๅปๆฉ่กจ่ฟฐ็บๅฝผๆญค็ญๅ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ Arrow of Time
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ ๅ็
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ ๅ
ๅพ ไธๅ็
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็ฌฌไบๅฎๅพ่ๆ้ไน็ฎญ
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