Artificial Intelligence

1
Artificial Intelligence Introduction to Formal
Logic Jennifer J. Burg Department of
Mathematics and Computer Science
2
What are the goals in the study of formal logic?

• To lay out a formal system whereby we reason.
• To make an abstraction of the reasoning process.
• But why?

3
But why?
So that we can understand human reasoning
processes better. To give reasoning ability to a
computer so that it can solve problems for us.
4
Where did it all begin?
Aristotle (384-322 B.C.) Descartes
(1596-1650) Leibnitz (1646-1716) George Boole
(1815-1864) Gottlob Frege (1848-1925) Bertrand
Russell (1872-1970) and Whitehead Alfred
Tarski (1902-1983) Kurt Godel (1906-1978) Alan
Turing (1912-1954)
5
Aristotle (384-322 B.C.)
Developed an informal system of syllogisms for
proper reasoning. With this system, you can
mechanically generate conclusions, given initial
premises.
6
What is a syllogism?
Major premise Every mammal has a spine. Minor
premise A dog is a mammal. Conclusion A dog
has a spine.
7
Descartes (1596-1650)
Emphasized the distinction between mind and
matter. Advocated a scientific method where we
doubt something until, through reason, we
establish it to be indubitable. The first
indubitable truth -- Je pense, donc je suis.
8
Leibnitz (1646-1716)
Introduced the first system of formal
logic Constructed machines for automating
calculation. Built a mechanical device intended
to carry out mental operations.
9
George Boole (1815-1864)
Introduced his formal language for making logical
inferences in 1864. His work was entitled An
Investigation of the Laws of Thought, on which
are founded Mathematical Theories of Logic and
Probabilities His system was a precursor to the
fully developed propositional logic.
10
Basic Questions
How expressive is propositional logic? How many
operators do we need for a complete set? How hard
is it to compute satisfiability? How hard is it
to determine validity? What assurance do we have
that we can be successful in proving validity?
What basic inference rules and axiom schemata do
we need? Would one inference rule suffice?
11
What CANT we do with propositional logic?
All horses are animals. Therefore, the head of a
horse is the head of an animal. Can you deduce
this in propositional logic? asked DeMorgan. No!
12
What CANT we do with propositional logic?
Say we have the expression a lt b b lt c a lt
c Then can we reduce this to a lt b b lt c But
we cant deduce this with propositional logic.
If we let p represent a lt b and q represent b lt
c and r represent a lt c, can we conclude p ? q ?
r ? p ? q (NO!)
13
Frege (1848-1925)
He did a comprehensive exploration of
propositional logic. Then he went on to develop
predicate logic. The formal system he developed
is essentially the same predicate logic we study
today. His language was intended to be a language
for describing mathematics. His notation was
awkward.
14
Tarski (1902-1983)
Introduced a theory of reference that shows how
to relate the objects in a logic to objects in
the real world. Worked in the area of semantics.
15
Russell (1872-1970) and Whitehead
Goals was to derive all of mathematics through
formal operations on a collection of
axioms. Theorem-proving would be mechanical. No
intuition would be involved. Strict syntax and
formal rules of inference.
16
Godel (1906-1978)
Incompleteness Theorem In any logical language
expressive enough to describe the properties of
the natural numbers, there are true statements
that are undecidable -- their truth cannot be
established by any algorithm.
17
Turing (1912-1954)
The validity of first order logic is not
decidable. (It is semi-decidable.) If a theorem
is logically entailed by an axiom, you can prove
that it is. But if it is not, you cant
necessarily prove that it is not. (You may go on
infinitely with your proof.)
18
Terminology
propositional logic (propositional
calculus) atomic symbols connectives propositions
conjunction disjunction antecedent
consequent well-formed formulas (wffs)
19
Terminology
syntax semantics interpretation inference
rules modus ponens satisfiable (consistent) unsati
sfiable (inconsistent) valid (a
tautology) sound complete
20
Terminology
resolution clause axiom (proper
axiom) theory axiom schema (schemata in the
plural) worst-case complexity NP-complete
21
Terminology
predicate logic (predicate calculus) universal
quantifier existential quantifier unification Skol
emization most general unifier Horn
clause semi-decidable