Symbolic%20Logic%20Foundations:%20An%20Overview%201/22/01

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Symbolic Logic FoundationsAn Overview1/22/01

• Logic (Programming) AI
• Selmer Bringsjord
• selmer_at_rpi.edu
• www.rpi.edu/brings

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Logic Programming Two Perspectives

• Logic Programming as arising from Herbrands
  Theorem, etc.
• Logic Programming as using a logical system (in
  mathematical sense of this phrase)
• I will take second perspective, which subsumes
  first
• E.g., completeness theorem for first-order logic
  (LI) allows one to affirm Herbrands Theorem
• This theorem fully done in LCU
• What you need to know to understand second
  perspective is precisely what you need to know to
  understand first

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Logical Systems(Are Programming Langs in here?)
Name Alphabet Grammar Proof Theory Semantics Metatheory
LPC Propositional Calculus p, q, r, and truth-functional connectives Easy Fitch-style and natural deduction Truth tables! Sound, complete, compact, decidable
LI First-Order Logic Add variables x, y, and ? ? Easy Fitch-style and natural deduction Structures and interpretations Sound, complete, compact, undecidable
LPML Add box and diamond for necessity and possibility Wffs created by prefixing new operators to wffs Add necessitation, etc. Possible worlds Same as PC
LII New variables for predicates Pretty obvious New adapt quantifier rules Quantification over subsets in domain allowed Sound but not complete
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Readings

• AIMA
• Natural Deduction on Pollocks web site
• OTTER manual
• Logic and AI Divorced, Still Married
• http//kryten.mm.rpi.edu/COURSES/ILOGPROG/lai.ed2.
  pdf
• LCU
• http//www.rpi.edu/faheyj2/SB/LCU/lcu.driver.pdf

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LPC (Propositional Calculus)

• Where we left off Logic Theorist problems in
  OTTER
• Ad lib in HYPERPROOF
• Some problems
• NYS 1, NYS 2, NYS 3, J-L 1
• Semantics of Propositional Calculus Truth
  Tables
• Boole
• Via HYPERPROOF
• Full formal view LCU

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NYS 1
Given the statements ?a ? ?b b c ? a which one
of the following statements must also be
true? c ?b ?c h a none of the above
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NYS 2
Which one of the following statements is
logically equivalent to the following statement
If you are not part of the solution, then
you are part of the problem. If you are part of
the solution, then you are not part of the
problem. If you are not part of the problem,
then you are part of the solution. If you are
part of the problem, then you are not part of the
solution. If you are not part of the problem,
then you are not part of the solution.
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NYS 3
Given the statements ??c c ? a ?a ? b b ? d ?(d ?
e) which one of the following statements must
also be true? ? e ?c e h ?a all of the above
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J-L 1
Suppose that the following premise is true If
there is a king in the hand, then there is an
ace in the hand, or else if there isnt a king
in the hand, then there is an ace. What can you
infer from this premise?
There is an ace in the hand.
NO!
NO!
In fact, what you can infer is that there isnt
an ace in the hand!
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Proof Theory of LI (First-order logic)

• Ad lib in HYPERPROOF
• Syllogisms in OTTER
• Dreadsbury Mansion Mystery
• The Bird Problem

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The Dreadsbury Mansion Mystery
Someone who lives in Dreadsbury Mansion killed
Aunt Agatha. Agatha, the butler, and Charles
live in Dreadsbury Mansion, and are the only
people who live therein. A killer always hates
his victim, and is never richer than his victim.
Charles hates no one that Aunt Agatha hates.
Agatha hates everyone except the butler. The
butler hates everyone not richer than Aunt
Agatha. The butler hates everyone Agatha hates.
No one hates everyone. Agatha is not the
butler. Now, given the above clues, there is a
bit of disagreement between three (competent?)
Norwegian detectives. Inspector Bjorn is sure
that Charles didnt do it. Is he right?
Inspector Reidar is sure that it was a suicide.
Is he right? Inspector Olaf is sure that the
butler, despite conventional wisdom, is innocent.
Is he right?
Can you get it, prove it?
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The Bird Problem
Is the following assertion true or false? Prove
that you are correct.
There exists something which is such that if its
a bird, then everything is a bird. ?x(B(x)
??yB(y))
Good litmus test for mastery of proof theory in
FOL
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Metatheory for PC and FOL

• Soundness
• If you start with true propositions in an agents
  knowledge base, deduction from that kb will
  always yield a true conclusion.
• Completeness
• If something intuitively follows from a given
  kb, then the agent can prove it from the kb.
• Decidability undecidability
• If Dec There is a decision algorithm which can
  tell you whether a given formula is a theorem.
• Compactness
• Not today
• Herbrands Theorem etc.
• Not today
• Godels Theorem
• Not today
• LII not complete
• Not today
• Lindstroms Theorems
• Already characterized intuitively at start of
  lecture