Second-Order Logic and the k-order Ladder;Second-Order Axiomatized Arithmetic;

Gödel’s “God Theorem” & Speedup Theorem

Rensselaer AI & Reasoning (RAIR) LabDepartment of Cognitive ScienceDepartment of Computer Science

Lally School of Management & TechnologyRensselaer Polytechnic Institute (RPI)

Troy, New York 12180 USA

Intro to Logic3/28/2019

Selmer Bringsjord

FOL

FOL√

FOL√Epistemic + FOL

BdBvBdV v

FOL√Epistemic + FOL

BdBvBdV v

(for coverage of “killer” robots)

FOL√Epistemic + FOL

BdBvBdV v√ (for coverage of “killer” robots)

FOL√Epistemic + FOL

BdBvBdV v√

TOL9X[X(j) ^ ¬X(m) ^ S(X)]

(for coverage of “killer” robots)

FOL√Epistemic + FOL

BdBvBdV v√

TOL9X[X(j) ^ ¬X(m) ^ S(X)]

(for coverage of “killer” robots)

FOL√Epistemic + FOL

BdBvBdV v√

?TOL

9X[X(j) ^ ¬X(m) ^ S(X)]

(for coverage of “killer” robots)

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

Double-Minded Man

9X[X(joseph) ^ ¬X(m(harriet , joseph)) ^ Sleazy(X)]

Double-Minded Man

9X[X(joseph) ^ ¬X(m(harriet , joseph)) ^ Sleazy(X)]√

Climbing the k-order Ladder

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.

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.

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.

ZOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

ZOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

ZOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

ZOL

FOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

ZOL

FOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

SOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

SOL

Things x and y, along with the father of x, share a certain property; and, x R2s y, where R2 is a positive property.

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

SOL

Things x and y, along with the father of x, share a certain property; and, x R2s y, where R2 is a positive property.

9x, y 9R,R2[R(x) ^ R(y) ^ R2(x, y) ^ Positive(R2) ^ R(fatherOf (x))]

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

SOL

Things x and y, along with the father of x, share a certain property; and, x R2s y, where R2 is a positive property.

9x, y 9R,R2[R(x) ^ R(y) ^ R2(x, y) ^ Positive(R2) ^ R(fatherOf (x))]TOL

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.

There’s some thing which is a llama and likes b (which is also a llama), and whose father is a llama too.

9x[Llama(x) ^ Llama(b) ^ Likes(x, b) ^ Llama(fatherOf (x))]

Things x and y, along with the father of x, share a certain property (and x likes y).

9x9y9R[R(x) ^ R(y) ^ Likes(x, y) ^ R(fatherOf (x))]

ZOL

FOL

SOL

Things x and y, along with the father of x, share a certain property; and, x R2s y, where R2 is a positive property.

9x, y 9R,R2[R(x) ^ R(y) ^ R2(x, y) ^ Positive(R2) ^ R(fatherOf (x))]TOL

…

“Leibniz was right that Descartes was right that … : God exists, necessarily.”

Gödel’s “God Theorem”

Gödel’s “God Theorem” (formalized, machine verified)

(1) 8P [Pos(¬P ) $ ¬Pos(P )] premise(2) 8P1 8P2 {Pos(P1) ^⇤8x [P1(x) ! P2(x)] ! Pos(P2)} premise

) (3) 8P [Pos(P ) ! ⌃9x P (x)] theorem(4) 8x [G(x) $ 8P [Pos(P ) ! P (x)] definition(5) Pos(G) premise

) (6) ⌃9x G(x) corollary(7) 8P [Pos(P ) ! ⇤Pos(P )] premise(8) 8x8P {Ess(P, x) $ [P (x) ^ 8P 0 (P 0(x) ! ⇤8y(P (y) ! P 0(y))} definition

) (9) 8x [G(x) ! Ess(G, x)] theorem(10) 8x {NE (x) $ 8P [Ess(P, x) ! ⇤9y P (y)]} definition(11) Pos(NE ) premise

) (12) ⇤9x G(x) (a.k.a. “Necessarily, God exists.) theorem<latexit sha1_base64="hUU4PNGXS59ajQJpc4+6CJe8DZI=">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</latexit>

PA = Z1

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.

PA = Z1

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.

1

PA = Z2A1 ⇥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)

New!

PA = Z2A1 ⇥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)

New!

8X([X(0) ^ 8x(X(x) ! X(s(x))] ! 8xX(x))Induction Axiom

PA = Z2A1 ⇥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)

New!

8X([X(0) ^ 8x(X(x) ! X(s(x))] ! 8xX(x))Induction Axiom

9X(8xX(x) $ �(x)) where �(x) 2 CComprehension Schema

G1 G

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

G1 G

EA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Gödel’s Speedup Theorem

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Gödel’s Speedup Theorem

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Then there is an infinite family F of ⇧01 formulae such that:

Gödel’s Speedup Theorem

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Then there is an infinite family F of ⇧01 formulae such that:

Gödel’s Speedup Theorem

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Then there is an infinite family F of ⇧01 formulae such that:

1. 8� 2 F , Zi ` �; and

2. 8� 2 F , if k is the least integer s.t. Zi+1 `k symbols �, then Zi 6`f(k) symbols �.

Gödel’s Speedup Theorem

G1 GEA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Then there is an infinite family F of ⇧01 formulae such that:

1. 8� 2 F , Zi ` �; and

2. 8� 2 F , if k is the least integer s.t. Zi+1 `k symbols �, then Zi 6`f(k) symbols �.

Gödel’s Speedup Theorem

G1 G

Can you build an AI that can prove this??

EA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?

Let i � 0, and let f be any recursive function.

Then there is an infinite family F of ⇧01 formulae such that:

1. 8� 2 F , Zi ` �; and

2. 8� 2 F , if k is the least integer s.t. Zi+1 `k symbols �, then Zi 6`f(k) symbols �.

Gödel’s Speedup Theorem

G1 G

Can you build an AI that can prove this??

Yes. Somehow …

EA/BA`

EA/BA|=

Q`

Q|=

TRUEA/QTRUEA/BA

√√

PA`

PA|=

ACA0√

TRUEA/I

? G

?

…

? GT

PA`II

PA|=II= TRUEA/II

?