Reja predikatlar va kvantlar. Predikatlar hisoblash formulasi haqida tushuncha. Tor predikat hisobining aksiomatik tasviri. Tabiiy tor predikatlar hisobi. Aristotel sillogistikasining tor predikat hisobiga singdirilishi

Isbot:

1. 
   F(y) faraz
   

x F(x) B : 1
(37) formulasini isbotlaymiz:


p x (p F(x))


Isbot:

1. 
   p Faraz
   
2. 
   p F(x) ID: 1
   

x p F(x) B : 2
Endi (38) formulani isbotlaymiz:

x F(x) x F(x)

Isbot:

1. 
   x F(x) faraz
   

2) F(y) Y : 1
x F(x) B : 2


5. Aristotel sillogistikasining tor predikat hisobiga singdirilishi.

11- asr oxirigacha asosiy rol A, E, I, O kategoriyali deb nomlangan to'rt turdagi hukmlarga berilgan. A ramziy taklifi " Hammasi S - P " quyidagicha yozilgan:

x ( S(x) P(x)) (39)

E hukmi "Yo'q S emas P ":
x (S(x) P(x)) (40) yoki boshqacha x ( S(x) (x)) (40 ¹ )
Hukm I "Some S are P ":
x (S(x) P(x)) (41)
Hukm O "Ba'zi S P emas ":
x (S(x) (x)) (42)
Keling, to'g'ridan-to'g'ri xulosa chiqarishning ba'zi usullarini isbotlaylik.
ASP ISP rejimi , (39)-(42) dan foydalanib, biz quyidagicha yozamiz:
x ( S(x) P(x)) x (S(x) P(x)) (43)


Isbot:

1. 
   x ( S(x) P(x)) Faraz
   
2. 
   S(y) P(y) Y : 1
   
3. 
   S(y) faraz
   
4. 
   P(y) PO: 2.3
   
5. 
   S(y) P(y) VK: 3.4
   

x (S (x) P (x)) B : 5
ESP OSP rejimi yana (39-42) yordamida quyidagicha yoziladi:
x ( S(x) (x)) x (S(x) (x)) (44)
Isbot:

1. 
   x ( Sx (x)) Faraz
   
2. 
   x ( S(x) (x)) (S(y) (y)) aksiomaga almashtirish e)
   
3. 
   S(y) (y) PO: 1.2
   
4. 
   S(y) faraz
   
5. 
   (y) PO: 3.4
   
6. 
   S(y) (y) VK: 4.5
   

x S(x) (x) B : 6
ASP IPS rejimi quyidagicha yozilgan:
x ( Sx (x)) x (S (x) (x)) (45)
Isbot:

1. 
   x ( S(x) (x)) Faraz
   
2. 
   x ( S(x) (x)) (S(y) (y)) aksiomaga almashtirish e)
   

3. 
   S(y) (y) PO: 1.2
   
4. 
   S(y) faraz
   
5. 
   (y) PO: 3.4
   
6. 
   S(y) (y) VK: 4.5
   

x S(x) (x) B : 6

To'g'ridan-to'g'ri xulosa chiqarishning boshqa usullari ham xuddi shunday tarzda yozilgan va isbotlangan.

Keling, sillogizmlarning ma'lum usullarining to'g'riligini isbotlaylik.
AMP ASM ASP sillogizmining birinchi figurasining birinchi rejimini quyidagicha yozamiz:

x (M(x) P(x)) x (S(x) → M(x)) → x(S(x) → P(x)) (46)

Isbot:

1. 
   x (M(x) P(x)) x (S(x) → M(x)) Faraz
   
2. 
   x (M(x) P(x)) Buyuk Britaniya: 1
   
3. 
   x (S(x) → M(x)) Buyuk Britaniya: 1
   
4. 
   M(y) R(y) Y : 2
   
5. 
   S(y) M(y) Y : 3
   
6. 
   S(y) P(y) (29): 4.5
   

x(S(x) → P(x)) B : 6
Sillogizmning ikkinchi figurasining birinchi modasining to'g'riligini isbotlaylik


EPM ASM→ESP.

(39)-(42) dan foydalanib, uni quyidagi shaklda yozamiz:

x (P(x) M(x)) x (S(x) → M(x)) → x(S(x) → P(x)) (47)

Isbot:

1. 
   x (P(x) M(x)) x (S(x) → M(x)) Faraz
   
2. 
   x (P(x) M(x)) Buyuk Britaniya: 1
   
3. 
   x (S(x) → M(x)) Buyuk Britaniya: 1
   
4. 
   R(y) M(y) Y : 2
   

5. 
   S_ (y) M (y) Y : 3
   
6. 
   M(y) R(y) (30): 4
   
7. 
   M(y) R(y) (9): 6
   
8. 
   S (y) P (y) (29): 5.7
   

x(S(x) → P(x)) B : 8
Nihoyat, sillogizmning uchinchi figurasining birinchi uslubini isbotlaymiz