1. Symbolic Logic Foundations: An Overview 1/22/01
Logic (Programming) & AI
Selmer Bringsjord

2. Logic Programming: Two Perspectives
Logic Programming as arising from Herbrand’s 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 Herbrand’s 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

3. 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  
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

4. Readings AIMA “Natural Deduction” on Pollock’s web site OTTER manual
“Logic and AI: Divorced, Still Married…”
LCU

5. 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

6. “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

7. “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.

8. “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

9. 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 isn’t 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 isn’t an ace in the hand!

10. Proof Theory of LI (First-order logic)
Ad lib in HYPERPROOF
Syllogisms in OTTER
Dreadsbury Mansion Mystery
The Bird Problem

11. 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 didn’t 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?

12. Good litmus test for mastery of proof theory in FOL
The Bird Problem
Is the following assertion true or false? Prove that you are correct.
There exists something which is such that if it’s a bird, then
everything is a bird.
x(B(x) yB(y))
Good litmus test for mastery of proof theory in FOL

13. Metatheory for PC and FOL
Soundness
If you start with true propositions in an agent’s 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
Herbrand’s Theorem etc.
Godel’s Theorem
LII not complete
Lindstrom’s Theorems
Already characterized intuitively at start of lecture