Анисов А.М. Современная логика
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Ïðèìå÷àíèÿ
1
Åñëè ÷üÿ-òî èíòóèöèÿ ìîë÷èò, òî íè÷åãî ñòðàøíîãî. Ýòî òðàäèöèîííàÿ
ëîãèêà âî ìíîãîì èíòóèòèâíà, â òî âðåìÿ êàê ñîâðåìåííàÿ ëîãèêà ìîæåò
áûòü èçëîæåíà áåç ññûëîê íà èíòóèöèþ.
2
Ñì. ïðåäûäóùåå ïðèìå÷àíèå.
3
Ñîãëàñíî ýòîìó «çàêîíó», óòâåðæäàþùèé S äîëæåí îáîñíîâàòü (äîêàçàòü)
S.  îòëè÷èå îò ïðåäûäóùèõ çàêîíîâ, çäåñü ìû èìååì äåëî íå ñ óòâåðæäå-
íèåì, à ñ òðåáîâàíèåì. Îòíîñèòü òðåáîâàíèÿ ê çàêîíàì ëîãèêè íåëåïî,
äàæå åñëè ýòè òðåáîâàíèÿ ðàçóìíû. Íàïðèìåð, òðåáîâàíèå «Ñîáëþäàé çà-
êîíû ëîãèêè!» âïîëíå ðàçóìíî, íî îíî âñåãî ëèøü áëàãîå ïîæåëàíèå, à
âîâñå íå çàêîí.
4
Íàïð., èìåþòñÿ ñîâðåìåííûå ðåêîíñòðóêöèè ñèëëîãèñòèêè. Íî ýòè ðåêîí-
ñòðóêöèè ïðîâîäÿòñÿ ìåòîäàìè è ñðåäñòâàìè ñîâðåìåííîé ëîãèêè è ïîòî-
ìó íå ìîãóò ñ÷èòàòüñÿ ÷àñòüþ òðàäèöèîííîé ëîãèêè.
5
Ó÷¸ò ôàêòîðà âðåìåíè òðåáóåò áîëåå èçîùð¸ííîé ëîãè÷åñêîé òåõíèêè, î
êîòîðîé ðå÷ü âïåðåäè. Ïðè èçó÷åíèè óïîðÿäî÷åííûõ ñîâîêóïíîñòåé ìû
äàäèì îäíó èç âîçìîæíûõ èíòåðïðåòàöèé îáîðîòà «À è çàòåì ».
6
Ïðàâèëà âûâîäà äàþòñÿ ðàíüøå, ÷åì ôîðìóëèðóåòñÿ ïîíÿòèå âûâîäà, ïî-
ýòîìó ïîðî÷íîãî êðóãà çäåñü íåò.
7
Ñî ñðåäíåâåêîâûõ âðåì¸í è äî ñèõ ïîð îáñóæäàþòñÿ ïðèíöèïû èíäèâèäóà-
öèè, ò.å. òåîðåòè÷åñêèå êðèòåðèè, íà îñíîâàíèè êîòîðûõ âûäåëÿþòñÿ èí-
äèâèäû.
8
Ñì. äèàëîã Ïëàòîíà «Åâòèäåì».
9
Ñì. äèàëîã Ïëàòîíà «Ãèïïèé Áîëüøèé».
10
Íå ïóòàòü ñ èíäóêöèåé êàê ïðàâäîïîäîáíûì ðàññóæäåíèåì, âåäóùèì îò
÷àñòíûõ ïîñûëîê ê îáùåìó çàêëþ÷åíèþ áåç ãàðàíòèé èñòèííîñòè çàêëþ-
÷åíèÿ.
11
Íà ñàìîì äåëå ýòî íåäîêàçàííûé ôàêò, â èñòèííîñòè êîòîðîãî, îäíàêî,
ìíîãèå óâåðåíû, ïîñêîëüêó íèêîìó ïîêà íå óäàëîñü íàéòè ñïîñîá óñòðà-
íèòü ýêñïîíåíòó â çàäà÷å «âûïîëíèìîñòü» (ýòî îäèí èç àñïåêòîâ çíàìåíè-
òîé P-NP ïðîáëåìû).
272
Îãëàâëåíèå
Ââåäåíèå .................................................................................................................... 3
×àñòü I. Èñòîêè ëîãèêè .......................................................................................... 10
ÃËÀÂÀ 1. ÄÎËÎÃÈ×ÅÑÊÎÅ ÑÎÇÍÀÍÈÅ .......................................................... 10
§1. Áèêàìåðàëüíûé ðàçóì ...................................................................................... 10
§2. Ðåöåïòóðíàÿ ìàòåìàòèêà .................................................................................. 13
§3. Ïîÿâëåíèå ðàññóæäåíèé .................................................................................. 15
§4. Ñîôèçìû è ïàðàäîêñû ..................................................................................... 18
ÃËÀÂÀ 2. ÒÐÀÄÈÖÈÎÍÍÀß ËÎÃÈÊÀ ............................................................... 23
§1. Ïîíÿòèÿ ............................................................................................................. 23
§2. Ñóæäåíèÿ ........................................................................................................... 26
§3. Óìîçàêëþ÷åíèÿ ................................................................................................. 32
×àñòü II. Îñíîâû ëîãèêè ......................................................................................... 42
ÃËÀÂÀ 3. ÇÍÀÊÈ È ÇÍÀÊÎÂÛÅ ÑÈÑÒÅÌÛ ................................................... 42
§1. ×òî òàêîå çíàê? ................................................................................................. 43
§2. Âèäû çíàêîâ ...................................................................................................... 46
§3. Åñòåñòâåííûå è èñêóññòâåííûå ÿçûêè ........................................................... 51
ÃËÀÂÀ 4. ËÎÃÈÊÀ ÂÛÑÊÀÇÛÂÀÍÈÉ .............................................................. 57
§1. ßçûê ëîãèêè âûñêàçûâàíèé ............................................................................ 57
§2. Òàáëè÷íàÿ ñåìàíòèêà ....................................................................................... 67
§3. Çàêîíû, ïðîòèâîðå÷èÿ, ôàêòóàëüíûå âûñêàçûâàíèÿ .................................... 74
§4. Î ñîâìåñòèìîñòè âûñêàçûâàíèé ..................................................................... 83
§5. Ëîãè÷åñêîå ñëåäîâàíèå .................................................................................... 94
§6. Ïîëíûå ñèñòåìû ëîãè÷åñêèõ ñâÿçîê ............................................................ 104
§7. Àëüòåðíàòèâíûå èíòåðïðåòàöèè ÿçûêà ëîãèêè âûñêàçûâàíèé ................. 120
§8. Àêñèîìàòè÷åñêîå èñ÷èñëåíèå âûñêàçûâàíèé .............................................. 127
§9. Íàòóðàëüíîå èñ÷èñëåíèå âûñêàçûâàíèé ...................................................... 138
ÃËÀÂÀ 5. ËÎÃÈÊÀ ÏÐÅÄÈÊÀÒÎÂ ................................................................... 159
§1. Èìåíà è ïîíÿòèÿ ............................................................................................ 159
§2. Ñâîéñòâà è îòíîøåíèÿ ................................................................................... 168
§3. Êâàíòîðû è óíèâåðñóìû ðàññóæäåíèé ......................................................... 177
§4. ßçûê ëîãèêè ïðåäèêàòîâ................................................................................ 184
§5. Îïðåäåëåíèå ïîíÿòèé .................................................................................... 192
§6. Îïðåäåëåíèå îáúåêòîâ .................................................................................... 196
§7. Íîìèíàëüíûå è ðåàëüíûå îïðåäåëåíèÿ ....................................................... 198
§8. ßâíûå è íåÿâíûå îïðåäåëåíèÿ ..................................................................... 202
§9. Îïåðàöèè äåëåíèÿ è êëàññèôèêàöèè ........................................................... 206
§10. Îòíîøåíèÿ ìåæäó ïîíÿòèÿìè .................................................................... 209
§11. Íàòóðàëüíîå èñ÷èñëåíèå ïðåäèêàòîâ ......................................................... 211
×àñòü III. Ëîãèêà â ïðèëîæåíèÿõ ......................................................................... 226
ÃËÀÂÀ 6. ÀÊÑÈÎÌÀÒÈ×ÅÑÊÈÅ ÒÅÎÐÈÈ ..................................................... 226
§1. Òåîðèè ýêâèâàëåíòíîñòè è ñõîäñòâà ............................................................. 227
§2. Àêñèîìû ðàâåíñòâà ......................................................................................... 233
§3. Òåîðèè ïîðÿäêà ............................................................................................... 238
§4. Òåîðèÿ ðîäñòâåííûõ îòíîøåíèé .................................................................. 242
ÃËÀÂÀ 7. ÂÛ×ÈÑËÈÌÎÑÒÜ È ËÎÃÈÊÀ ....................................................... 246
§1. Íàòóðàëüíûé óíèâåðñóì è êîäèðîâàíèå ...................................................... 246
§2. Ëîãè÷åñêèé êîìïüþòåð .................................................................................. 249
§3. Âû÷èñëèìûå ôóíêöèè è ðàçðåøèìûå ïðåäèêàòû ...................................... 253
ÃËÀÂÀ 8. ÂÎÏÐÎÑÍÎ-ÎÒÂÅÒÍÛÅ ÑÈÒÓÀÖÈÈ .......................................... 257
§1. Âîïðîñû è îòâåòû ........................................................................................... 257
§2. Íåêîððåêòíûå âîïðîñû .................................................................................. 260
§3. Ïðîáëåìà ðàçðåøèìîñòè âîïðîñîâ ............................................................... 264
Ïðèìå÷àíèÿ ........................................................................................................... 271
Íàó÷íîå èçäàíèå
Àíèñîâ Àëåêñàíäð Ìèõàéëîâè÷
ÑÎÂÐÅÌÅÍÍÀß ËÎÃÈÊÀ
Óòâåðæäåíî ê ïå÷àòè Ó÷åíûì ñîâåòîì
Èíñòèòóòà ôèëîñîôèè ÐÀÍ
 àâòîðñêîé ðåäàêöèè
Õóäîæíèê Â.Ê.Êóçíåöîâ
Òåõíè÷åñêèé ðåäàêòîð À.Â.Ñàôîíîâà
Êîððåêòóðà àâòîðà
Ëèöåíçèÿ ËÐ ¹ 020831 îò 12.10.98 ã.
Ïîäïèñàíî â ïå÷àòü ñ îðèãèíàë-ìàêåòà 26.03.02.
Ôîðìàò 60õ84 1/16. Ïå÷àòü îôñåòíàÿ. Ãàðíèòóðà Òàéìñ.
Óñë. ïå÷. ë. 17,06. Ó÷.-èçä. ë. 15,76. Òèðàæ 500 ýêç. Çàêàç ¹ 004.
Îðèãèíàë-ìàêåò èçãîòîâëåí â Èíñòèòóòå ôèëîñîôèè ÐÀÍ
Êîìïüþòåðíûé íàáîð àâòîðà
Êîìïüþòåðíàÿ âåðñòêà: Þ.À.Àíîøèíà
Îòïå÷àòàíî â ÖÎÏ Èíñòèòóòà ôèëîñîôèè ÐÀÍ
119992, Ìîñêâà, Âîëõîíêà, 14