Number Theory & Astrologic; Second-Order Logic and the k-order Ladder; Second-Order Axiomatized Arithmetic; Gödel s Speedup Theorem; Time Travel
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1 Number Theory & Astrologic; Second-Order Logic and the k-order Ladder; Second-Order Axiomatized Arithmetic; Gödel s Speedup Theorem; Time Travel Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Lally School of Management & Technology Rensselaer Polytechnic Institute (RPI) Troy, New York USA Intro to Logic 4/2/2018
2 Number Theory & Astrologic; Second-Order Logic and the k-order Ladder; Second-Order Axiomatized Arithmetic; Gödel s Speedup Theorem; Time Travel Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Lally School of Management & Technology Rensselaer Polytechnic Institute (RPI) Troy, New York USA Intro to Logic 4/2/2018
3 Q (= Robinson Arithmetic) A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x)
4 Q (= Robinson Arithmetic) A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x)
5 PA (Peano Arithmetic) A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) where (x) is open w with variable x, and perhaps others, free.
6 PA (Peano Arithmetic) A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) 1 where (x) is open w with variable x, and perhaps others, free.
7 PA (Peano Arithmetic) A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) 1 where (x) is open w with variable x, and perhaps others, free.
8 Astrologic (Aliens & Angels on the Same Race Track ) TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
9 Astrologic (Aliens & Angels on the Same Race Track ) An intellectual steeple for human persons! TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
10 Astrologic (Aliens & Angels on the Same Race Track ) An intellectual steeple for human persons! TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
11 Astrologic (Aliens & Angels on the Same Race Track ) An intellectual steeple for human persons! TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
12 Astrologic (Aliens & Angels on the Same Race Track ) An intellectual steeple for human persons! TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
13
14 FOL
15 FOL
16 FOL Epistemic + FOL B d B v B d Vv
17 FOL Epistemic + FOL B d B v B d Vv (for coverage of killer robots)
18 FOL Epistemic + FOL B d B v B d Vv (for coverage of killer robots)
19 FOL Epistemic + FOL B d B v B d Vv (for coverage of killer robots) TOL 9X[X(j) ^ X(m) ^ S(X)]
20 FOL Epistemic + FOL B d B v B d Vv (for coverage of killer robots) TOL 9X[X(j) ^ X(m) ^ S(X)]
21 FOL Epistemic + FOL B d B v B d Vv (for coverage of killer robots)? TOL 9X[X(j) ^ X(m) ^ S(X)]
22 Double-Minded Man
23 Double-Minded Man
24 Double-Minded Man
25 Double-Minded Man
26 Double-Minded Man
27 Double-Minded Man
28 Double-Minded Man
29 Double-Minded Man
30 Double-Minded Man 9X[X(joseph) ^ X(m(harriet, joseph)) ^ Sleazy(X)]
31 Double-Minded Man 9X[X(joseph) ^ X(m(harriet, joseph)) ^ Sleazy(X)]
32 Climbing the k-order Ladder
33 Climbing the k-order Ladder a is a llama, as is b, a likes b, and the father of a is a llama as well.
34 Climbing the k-order Ladder Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
35 Climbing the k-order Ladder ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
36 Climbing the k-order Ladder There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
37 Climbing the k-order Ladder 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
38 Climbing the k-order Ladder FOL 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
39 Climbing the k-order Ladder Things x and y, along with the father of x, share a certain property (and x likes y). FOL 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
40 Climbing the k-order Ladder 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). FOL 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
41 Climbing the k-order Ladder SOL FOL 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
42 Climbing the k-order Ladder Things x and y, along with the father of x, share a certain property; and, x R 2 s y, where R 2 is a positive property. SOL FOL 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
43 Climbing the k-order Ladder 9x, y 9R, R 2 [R(x) ^ R(y) ^ R 2 (x, y) ^ Positive(R 2 ) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property; and, x R 2 s y, where R 2 is a positive property. SOL FOL 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
44 Climbing the k-order Ladder TOL 9x, y 9R, R 2 [R(x) ^ R(y) ^ R 2 (x, y) ^ Positive(R 2 ) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property; and, x R 2 s y, where R 2 is a positive property. SOL FOL 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
45 Climbing the k-order Ladder TOL 9x, y 9R, R 2 [R(x) ^ R(y) ^ R 2 (x, y) ^ Positive(R 2 ) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property; and, x R 2 s y, where R 2 is a positive property. SOL FOL 9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))] Things x and y, along with the father of x, share a certain property (and x likes y). 9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))] There s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too. ZOL Llama(a) ^ Llama(b) ^ Likes(a, b) ^ Llama(fatherOf (a)) a is a llama, as is b, a likes b, and the father of a is a llama as well.
46 PA = Z0 A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) where (x) is open w with variable x, and perhaps others, free.
47 PA = Z0 A1 x(0 = s(x)) A2 x y(s(x) =s(y) x = y) A3 x(x = 0 y(x = s(y)) A4 x(x +0=x) A5 x y(x + s(y) =s(x + y)) A6 x(x 0 = 0) A7 x y(x s(y) =(x y)+x) And, every sentence that is the universal closure of an instance of ([ (0) x( (x) (s(x))] x (x)) 1 where (x) is open w with variable x, and perhaps others, free.
48 G1 G
49 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II
50 G1 G TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = TRUE A/I ACA 0?? GT? G PAÌI? PA = II = TRUE A/II
51 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II
52 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II Gödel s Speedup Theorem
53 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II Gödel s Speedup Theorem Let i 0, and let f be any recursive function.
54 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II Gödel s Speedup Theorem Let i 0, and let f be any recursive function. Then there is an infinite family F of 0 1 formulae such that:
55 G1 TRUE A/BA EA/BA` EA/BA = TRUE A/Q ` Q Q = PA` PA = ACA0 TRUE A/I?? GT G? G PAÌI? PA = II = TRUE A/II Gödel s Speedup Theorem Let i 0, and let f be any recursive function. Then there is an infinite family F of 0 1 formulae such that:
56 TRUEA/BA G1 ` EA/BA EA/BA = TRUEA/Q Q` Q = PA` G TRUEA/I PA =? GT ACA0??G PA`II = PAII = TRUEA/II? Gödel s Speedup Theorem Let i 0, and let f be any recursive function. Then there is an infinite family F of 01 formulae such that: F, Zi ` ; and 8 2 F, if k is the least integer s.t. Zi+1 `k symbols, then Zi 6`f (k) symbols.
57 TRUEA/BA G1 ` EA/BA EA/BA = TRUEA/Q Q` Q = PA` G TRUEA/I PA =? GT ACA0??G PA`II = PAII = TRUEA/II? Gödel s Speedup Theorem Let i 0, and let f be any recursive function. Then there is an infinite family F of 01 formulae such that: F, Zi ` ; and 8 2 F, if k is the least integer s.t. Zi+1 `k symbols, then Zi 6`f (k) symbols. Can you build an AI that can prove this??
58 TRUEA/BA G1 ` EA/BA EA/BA = TRUEA/Q Q` Q = PA` G TRUEA/I PA =? GT ACA0??G PA`II = PAII = TRUEA/II? Gödel s Speedup Theorem Let i 0, and let f be any recursive function. Then there is an infinite family F of 01 formulae such that: F, Zi ` ; and 8 2 F, if k is the least integer s.t. Zi+1 `k symbols, then Zi 6`f (k) symbols. Can you build an AI that can prove this?? Yes. Somehow
59 Supererogatory Reflection
60 Supererogatory Reflection Prove `PA ; how long is your proof?
61 Supererogatory Reflection Prove `PA ; how long is your proof? Prove `PA ; how short can you make your proof?
62 And time travel
63 And time travel
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