Artificial Intelligence Introduction to Formal Logic

Jennifer J. BurgDepartment 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 (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)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 (384-322 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 (1596-1650)

Introduced the first system of formal logic

Constructed machines for automating calculation.

Built a mechanical device intended to carry out mental operations. Leibnitz (1646-1716)

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 ProbabilitiesHis system was a precursor to the fully developed propositional logic.George Boole (1815-1864)

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 CANT we do with propositional logic?

Say we have the expressiona < b && b < c && a < cThen can we reduce this to a < b && b < cBut we cant deduce this with propositional logic. If we let p represent a < b and q represent b < c and r represent a < c, can we concludep q r p q (NO!)What CANT 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 (1848-1925)

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 (1902-1983)

Russell (1872-1970) and WhiteheadGoals 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 (1906-1978)

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.)Turing (1912-1954)

propositional logic (propositional calculus)atomic symbolsconnectivespropositionsconjunctiondisjunctionantecedent consequentwell-formed formulas (wffs)Terminology

syntaxsemanticsinterpretationinference rulesmodus ponenssatisfiable (consistent)unsatisfiable (inconsistent)valid (a tautology)soundcompleteTerminology

resolutionclauseaxiom (proper axiom)theoryaxiom schema (schemata in the plural)worst-case complexityNP-complete

Terminology

predicate logic (predicate calculus)universal quantifierexistential quantifierunificationSkolemizationmost general unifierHorn clausesemi-decidableTerminology

Boole was interesting in thinking about thought. He wanted to develop a formal language for logical computation.

He took an algebraic approach. His algebra was based on 2 values and three operators -- and, or, and not. The operators have meanings by definition, and their definition can be represented by truth tables. He defined what expressions mean in his algebra in terms of functions. You take in input in the form of truth values for the propositional variables, and return TRUE or FALSE. We can show commutative, associative, and distributive laws for and and or. 1 is the identity for and. 0 is the identify for or. 0 is an annihilator for and. But there are ways in which and and or are not like multiplication and addition. 1 is an annihilator for or. Both and and or are idempotent. Precedencenotnandnorandor->equivalenceBoole was interesting in thinking about thought. He wanted to develop a formal language for logical computation.

He took an algebraic approach. His algebra was based on 2 values and three operators -- and, or, and not. The operators have meanings by definition, and their definition can be represented by truth tables. He defined what expressions mean in his algebra in terms of functions. You take in input in the form of truth values for the propositional variables, and return TRUE or FALSE. We can show commutative, associative, and distributive laws for and and or. 1 is the identity for and. 0 is the identify for or. 0 is an annihilator for and. But there are ways in which and and or are not like multiplication and addition. 1 is an annihilator for or. Both and and or are idempotent. Precedencenotnandnorandor->equivalenceThis results concerns not predicate logic as we have discussed it, but rather a specialization of this logic that lets us talk about integers and the usual operations on integers. In particular, we have to modify predicate logic to introduce predicates for the arithmetic operations, such asplus(X,Y,Z)times(X,Y,Z)less(X,Y)

Further, we need to restrict the domain in interpretations so that the values appear to be the nonnegative integers.

The important consequence is that there is a limit on our ability to answer questions about math. If we have a math system as complex as the integers, there is no way we can distinguish true statements from false ones. If we prove a statement is true, we know it is true. But our search may go on forever.