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A Brief History of Logic
Some Background
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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
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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
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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?
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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)
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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
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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
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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
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Pythagorean Society• The society made contributions to
many areas:– Music theory– Number theory– Astronomy– Geometry
• However, they proved themselves to be a contradiction
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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