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?

So that we can understand human reasoning processes better. To give reasoning ability to a computer so that it can solve problems for us. But why?

Aristotle ( B.C.) Descartes ( ) Leibnitz ( ) George Boole ( ) Gottlob Frege ( ) Bertrand Russell ( ) and Whitehead Alfred Tarski ( ) Kurt Godel ( ) Alan Turing ( ) Where did it all begin?

Developed an informal system of syllogisms for proper reasoning. With this system, you can mechanically generate conclusions, given initial premises. Aristotle ( B.C.)

Major premise: Every mammal has a spine. Minor premise: A dog is a mammal. Conclusion: A dog has a spine. What is a syllogism?

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. Descartes ( )

Introduced the first system of formal logic Constructed machines for automating calculation. Built a mechanical device intended to carry out mental operations. Leibnitz ( )

Introduced his formal language for making logical inferences in 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. George Boole ( )

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? Basic Questions

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!) What CAN’T we do with propositional logic?

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. Frege ( )

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. Tarski ( )

Russell ( ) 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.

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. Godel ( )

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.) Turing ( )

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 Terminology