copyright © 2003-2015 curt hill a brief history of logic some background

Copyright © 2003-2015 Curt Hill

A Brief History of Logic

Some Background


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


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


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?


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)


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


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


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


Pythagorean Society• The society made contributions to

many areas:– Music theory– Number theory– Astronomy– Geometry

• However, they proved themselves to be a contradiction


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


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


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


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


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


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


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


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


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


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


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 P2S2 P1


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


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


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



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


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


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


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


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


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


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


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


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 a brief history of logic some background

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