Artificial Minds - Spring 2009

Class notes for Thursday, January 29 (week 2)

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DISCLAIMER:  These are my own working lecture notes, which I'm posting here in
case they're useful.  I'll try to post my notes on a more or less regular
basis (assuming that I have some to post), but I make no guarantees as to
their completeness or comprehensibility.  They're also not too pretty, unless
you're a fan of plain text and ASCII art, so I'd recommend using them as a
supplement rather than a replacement for your own note-taking.  In other words,
you're still responsible for the material we cover in class, whether or not
that material is included here.
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Mechanizing computation and thought processes
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George Boole's book "The Laws of Thought" (1854) introduced a mathematical
formulation of the fundamental processes of reasoning

P & Q => P

"Roses are red and violets are blue"

Conclusion: "Roses are red"

P => P | Q

"Roses are red"

Conclusion: "Roses are red or pigs can fly"

~(~P) => P

"It is not the case that pigs cannot fly"

Conclusion: "Pigs can fly"

P & Q => Q & P
P | Q => Q | P
P | (Q & R) => (P | Q) & (P | R)
P & (Q | R) => (P & Q) | (P & R)
(P => Q) & ~Q => ~P
(P => Q) & P => Q

"If it is sunny then it is not raining"
"If it is not sunny then it is cloudy"
"It is raining and it is cold"

a = "It is sunny"
b = "It is raining"
c = "It is cloudy"
d = "It is cold"

a => ~b
~a => c
b & d

Rule: P & Q => P

b & d => b

Conclusion: "It is raining"

Rule: (P => Q) & ~Q => ~P

(a => ~b) & b => ~a

Conclusion: "It is not sunny"

Rule: (P => Q) & P => Q

(~a => c) & ~a => c

Conclusion: "It is cloudy"

Augustus de Morgan's laws:
~(P & Q) => ~P | ~Q
~(P | Q) => ~P & ~Q

"It is not the case that it is cloudy and cold"

Conclusion: "It is not cloudy or it is not cold"

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Formal systems
  - syntactically defined tokens
  - axioms: starting configurations
  - finite set of rules
  - theorems: new strings derivable from the axioms by the rules
  - proof: the sequence of steps of a derivation

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The PQ-system

Symbols:  P  Q  -

Axioms: xP-Qx- is an axiom, whenever x is composed of hyphens only

Example:
--P-Q---
x: --

Rule: Suppose x, y, and z all stand for particular strings containing only
hyphens.  And suppose that xPyQz is known to be a theorem.  Then xPy-Qz- is
also a theorem.

Example:
x: --
y: ---
z: -
Then if --P---Q- turns out to be a theorem, then so will --P----Q--

Well-formed strings:
(group of hyphens) P (group of hyphens) Q (group of hyphens)

Interpretation 1:
  P = plus
  Q = equals
  - = 1
 -- = 2
--- = 3
etc.

--P---Q-----    <===>   "two plus three equals five"

Isomorphisms (one-to-one mappings) induce *meaning*

The meaning of the symbol P is "plus"

WARNING: We cannot actively use the meaning of symbols in any way.
--P--P--Q------   <===>  "two plus two plus two equals six"
This is NOT a theorem. It's not even a well-formed string.

Other meanings may exist, if other isomorphisms are found

Interpretation 2:
  P = equals
  Q = taken from
  - = 1
 -- = 2
--- = 3
etc.

--P---Q-----   <===>  "2 equals 3 taken from 5"

This interpretation, where the meaning of P is "equals", is just as valid.

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The TQ-system

Axioms:   xT-Qx is an axiom, whenever x is a hyphen-string.

Rule:  Suppose that x, y, and z are all hyphen-strings.  And suppose that xTyQz
is an old theorem.  Then, xTy-Qzx is a new theorem.

Example:
--T-Q--        axiom
--T--Q----     using line 1 as the old theorem
--T---Q------  using line 2 as the old theorem

This system captures multiplication formally.

Composite numbers (i.e. non-primes)

Rule 2: Suppose x, y, and z are all hyphen-strings with at least 1 hyphen.
If x-Ty-Qz is a theorem, then Cz is a theorem.

Example:  --T---Q------   (derived previously)
          C------

Note: C-- cannot be a theorem, because Rule 2 does not apply to -T-Q--

The theorems beginning with C represent facts about composite numbers:

C----         "4 is composite"
C------       "6 is composite"
C--------     "8 is composite"
C---------    "9 is composite"
...etc...

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