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A Brief History of Logic Some Background Copyright © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 Curt Hill Mathematical progress • Some important names we will consider • Thales of Miletus (640-546 BC) • Pythagoras (570-500 BC) • Zeno of Elea(early fifth century) • Aristotle (384-322 BC) Copyright © 2003-2015 Curt Hill Thales of Miletus (640-546 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 © 2003-2015 Curt Hill Statements • 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 © 2003-2015 Curt Hill Pythagoras (570-500 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 Curt Hill Aristotle (384-322 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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: – S1 P1 S2 P2 S2 P1 Copyright © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 Copyright © 2003-2015 Curt Hill 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 © 2003-2015 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 © 2003-2015 Curt Hill George Boole (1815-1864) • 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 © 2003-2015 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 Copyright © 2003-2015 Curt Hill 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 © 2003-2015 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 20th century Copyright © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 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 © 2003-2015 Curt Hill