Copyright © Curt Hill A Brief History of Logic Some Background

Copyright © Curt Hill Greece and the beginnings The Greek legal system had some similarities to ours with juries and lawyers –Juries were much larger –Less screening There was much more dependence on what was reasonable Less on codified laws

Copyright © Curt Hill How to win The arguments of the lawyer are much more important Rhetoric becomes an important science –Citizens who were not particularly wealthy could be their own lawyers Philosophy was also quite important and depended on rhetoric –Socrates and Plato among others

Copyright © Curt Hill The Sophist as Lawyer The sophist could argue that right was wrong –A lawyer is not looking for justice, but for the client to win So how do we tell if the speech is good but the argument flawed?

Copyright © Curt Hill Mathematical progress Some important names we will consider Thales of Miletus ( BC) Pythagoras ( BC) Zeno of Elea(early fifth century) Aristotle ( BC)

Copyright © Curt Hill Thales of Miletus ( BC) Wealthy merchant –Became rich by cornering the olive oil market Prior to Thales geometry was mostly concerned with surveying –Techniques on how to accomplish a practical thing He chose several statements on geometry –These were well known as practical facts

Copyright © Curt Hill Statements –A circle is bisected by any of its diameters –When two lines intersect the opposite angles are equal –The sides of similar triangles are proportional –The angles at the base of an isosceles triangle are equal –An angle inscribed within a semcircle is a right angle However, Thales showed that they could be derived from previous statements This is the precursor of the idea of a proof –He founded the Ionian school of thought

Copyright © Curt Hill Pythagoras ( BC) The most famous of the Ionian school A lot of myth has grown up about him because of his impact on mathematics His followers formed a secret society with mysticism, worshipping the idea of number and the hoarding of knowledge He was the first to assert that proofs were based upon assumptions, axioms or postulates – things that were given and in their own right not provable He also was the first to offer a proof about sizes of sides of right triangles

Copyright © Curt Hill Pythagorean Society The society made contributions to many areas: –Music theory –Number theory –Astronomy –Geometry However, they proved themselves to be a contradiction

Copyright © Curt Hill The Contradiction One of their fundamental assumptions that the integer was the basis of all truth One of their members proved the existence of irrational numbers –Numbers that are not the ratio of two integers They took him in a boat out to sea and drowned him They suppressed the knowledge for some time, but ultimately he had disproved one of their fundamental principles

Copyright © Curt Hill Zeno of Elea (early fifth century) Student of Parmenides They believed: –Motion and change are only apparent –Everything is one – no multiplicity He produced several paradoxes that nobody could resolve This was an affront to the whole notion of a proof and opposed to Pythagorean reality

Copyright © Curt Hill Line Segment If we assume that a line segment is composed of a multiplicity of points We can always bisect the line Each of the resulting segments can itself be bisected We can do this ad infinitum We never come to a stopping point so lines must not be composed of points

Copyright © Curt Hill Achilles and the Tortoise Achilles and a tortoise are in a race where the tortoise is given a head start Whenever Achilles catches up to where the tortoise was, the tortoise has advanced Thus Achilles can never catch the tortoise

Copyright © Curt Hill The arrow Assume that the instant is indivisible An arrow is either at rest or moving in any instant An arrow cannot change its state in an instant Therefore an arrow at rest cannot move It turns out that neither of these paradoxes can be handled until the calculus is introduced with its notion of limits

Copyright © Curt Hill Aristotle ( BC) Tutor of Alexander the Great Greatest mathematician and scientist of the day Wrote a number of works in philosophy and science His science works were not usually superseded until the Renaissance –About 17 centuries of pre-eminence

Copyright © Curt Hill Logic Contributions Four types of statements, each denoted by a letter –Universal affirmative All S is P A –Universal negative No S is P E –Particular affirmative Some S is P I –Particular negative Some S is not P O

Copyright © Curt Hill Four types (continued) In each of these statements: –S which is the subject –P is the predicate All or no have obvious meanings Some means one or more

Copyright © Curt Hill Syllogism Aristotle's main form was a syllogism Each syllogism consisted of two premises (a major and minor) and one conclusion The premises and conclusion are of one of previous four statement types

Copyright © Curt Hill Syllogism Example –All cats eat mice –Felix is a cat –Therefore Felix eats mice Statement types –First is universal affirmative –Second is a particular affirmative –Third is a particular affirmative

Copyright © Curt Hill Example continued Subjects –Cats (all) for major premise and Felix for minor Predicates –The set of items that eat mice for major and conclusion –Is a cat for minor The form: –S 1  P 1 S 2  P 2 S 2  P 1

Copyright © Curt Hill Discussion Subjects identify an item or group of items Predicates state a property The conclusion –Has a subject and predicate that are each only used once in the premises –However there is a middle term used in the premises that is not used in the conclusion –The major premise contains the conclusions predicate –The minor premise contains the conclusions subject

Copyright © Curt Hill More discussion There should be three items in these two premises The conclusions subject, the conclusions predicate and a middle term The major premise should contain the conclusions predicate The minor premise should contain the conclusions subject

Copyright © Curt Hill Combinatorics There are four different ways to arrange the S, P and M into a syllogism There are four different statements that can be plugged into the three statement This give 4^4 = 256 syllogisms However, not all of these are valid What Aristotle did is identify (some of) the valid syllogisms and some of the invalid syllogisms Some of these received names, which will be mentioned as we re-encounter them

Archimedes 250 BC Seems to have figured out the paradoxes of Zeno Very close to inventing both Calculus and the underpinning idea of limits The work did not get out and was lost for centuries Killed in Roman siege of Syracuse Ranked as one of top mathematicians along with Newton and Gauss Copyright © Curt Hill

Gottfried Liebniz Invented calculus Postulated the concept of balance of power Postulated that there was a universal characteristic –A language in which errors of thought would appear as computational errors –This part of his work was ignored –However this is a long standing goal of logic

Copyright © Curt Hill George Boole ( ) Almost single handedly moved logic from philosophy to mathematics What we now know as a Boolean algebra stems from his work Separated the logical statements from their underlying facts Once this occurred the gates opened and a number of people joined in

Copyright © Curt Hill Boole’s Successors –Jevons –DeMorgan –Peirce –Venn –Lewis Carroll –Ernst Schröder –Löwenheim –Skolem –Peano –Frege –Bertrand Russell –Alfred North Whitehead –Hilbert –Ackermann –Gödel The early ones corrected Boole's work and the later ones extended it

Two of note Many of this above list will be considered in the course of this class but the following two bear more comment now David Hilbert Kurt Gödel Copyright © Curt Hill

David Hilbert An extraordinary leader in the mathematical community –The dominant mathematician from about 1885 to 1940 List of career accomplishments could be a course itself –Geometry –Number theory –Physics In 1900 he published a list of 23 problems that needed to be solved in the 20 th century Copyright © Curt Hill

The 23 problems Some have been solved Some are too vague to solve Many are still in process The second is relevant today –Prove that the axioms of arithmetic are consistent Seems like a good goal Copyright © Curt Hill

Kurt Gödel Proved the first and second incompleteness theorems –1931 or so There is considerable belief that this is the death knell of problem 2 –The second states that a proof of the consistency of arithmetic cannot be from within arithmetic itself Copyright © Curt Hill

First Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory Copyright © Curt Hill

Second For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent Copyright © Curt Hill

So? Among other things these two state that no formal system of axioms can prove the validity of itself If this were a three hour course of logic we would be compelled to study these two theorems As it is, this is as close as we will come However, these theorems do not disprove the usefulness of axiomatic systems Copyright © Curt Hill