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Artificial Intelligence
“Introduction to Formal Logic”
Jennifer J. Burg
Department of Mathematics and Computer Science
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?
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.
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)
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.
What is a syllogism?
Major premise: Every mammal has a spine.
Minor premise: A dog is a mammal.
Conclusion: A dog has a spine.
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.
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.
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.
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?
What CAN’T 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!
What CAN’T we do with propositional logic?
Say we have the expression
a < b && b < c && a < c
Then can we reduce this to
a < b && b < c
But we can’t deduce this with
propositional logic. If we let p represent
a < b and q represent b < c and r represent
a < c, can we conclude
p  q  r  p  q (NO!)
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.
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.
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.
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.
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 can’t necessarily prove that it is
not. (You may go on infinitely with your
proof.)
Terminology
propositional logic (propositional calculus)
atomic symbols
connectives
propositions
conjunction
disjunction
antecedent
consequent
well-formed formulas (wffs)
Terminology
syntax
semantics
interpretation
inference rules
modus ponens
satisfiable (consistent)
unsatisfiable (inconsistent)
valid (a tautology)
sound
complete
Terminology
resolution
clause
axiom (proper axiom)
theory
axiom schema (“schemata” in the plural)
worst-case complexity
NP-complete
Terminology
predicate logic (predicate calculus)
universal quantifier
existential quantifier
unification
Skolemization
most general unifier
Horn clause
semi-decidable