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Origins of Logic

Greek mathematics

Rhetoric: Eristic and Sophistry

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Greek mathematics.

Pre-greek mathematics was not primarily concernedwith proof, but more with computation. (Egyptians,Babylonians)

Geometry = measurement of the earthThales of Miletus (c.625-c.546 BC): the first proofDmitriPanchenko, Thales and the Origin of Theoretical Reasoning,Configurations1

(1993), p. 387-414

Pythagoras (c.569-c.475 BC)

Mathematics built on proof:

Theaetetus (c.417-c.369 BC); student of SocratesEuclid (c.325-c.265 BC); compilation ofmathematical knowledge

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Mathematical techniques.

Proof by contradiction

Claim.

2is not a fraction of integers.

Suppose it were, then there are integers nand mwithoutcommon divisor such that

2 =

n

m.

But then

2m2

= n2.

In particular, nmust be even. But then n2 must be divisibleby 4, and so mmust be even. Contradiction.

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Informal logic.

The Dialectic method.

Proof by contradiction in mathematics.

Zeno of Elea (c.490-c.425 BC)

Socrates (469-399 BC;elenchus,diairesis)

Argumentation in everyday life

Sophists

Public disputations as part of democratic life

Plato,Euthydemus

Aristotle,TopicsandRhetoric

Megarians (next week)

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Plato.

Plato (c.427-347 BC)

Student and follower of Socrates until399 B.C.

399-387 BC:Plato travels widely,including Italy and Sicily

387 BC:Plato founds theAcademy

367 BC:Plato is invited to Sicily byDionysios II.

347 BC:Plato dies and is succeeded bySpeusippus

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The Platonic Academy.

387 BC 526 ADAcademiawas a public garden named afterAcademos.

DavidFowler, The Mathematics of Platos Academy: A New Reconstruction

JohnDillon, The Heirs of Plato: A Study of the Old Academy (347-274 BC), Oxford, 2003

Members. Speusippus(347-339),Xenocrates(339-314),Polemo(314-276),Crates, Crantor,Arcesilaus(268-240),Lacydes, Evander, Hegesinus, Carneades, Clitomachus,andPhilo... andAristotle.

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Theoria et Praxis (1).

The School of Athens (Raffaello Sanzio; 1509)

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Theoria et Praxis (2).

[Uestium Philosophiae] in extremo margine Graecum, in supremo uero legebatur

intextum atque inter utrasque litteras in scalarum modum gradus quidam insigniti uidebantur,

quibus ab inferiore ad superius elementum esset ascensus.

Bothius, Consolatio Philosophiae

Book 1, Prosa 1

On the lowest border of [the garments ofPhilosophia] a Greek was embroidered, while on

the highest a could be read, and between both letters an ascent could be seen in the

manner of stairs, by which you could move from the lower to the higher element.

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Aristotle.

Aristotle (384-322 BC)

367 BC: Aristotle joins the Academy.347 BC: Plato dies, Aristotle leavesAthens.

343-336 BC: Aristotle works at thecourt of Macedonia.

335 BC: Aristotle founds theLyceuminAthens (Peripatetics).

323 BC: Alexander the Great dies, Aris-totle retires to Chalcis.

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Esoteric / exoteric.

Aristotle:

Esoteric works: lecture notes and textbooks, designedfor use within the Lyceum.

Exoteric works: dialogues (modelled after the Platonicdialogues), designed for the general public.

Plato AristotleExoteric survive lost

Esoteric ? mostly survive

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Esoteric / exoteric.

Aristotle:

Esoteric works: lecture notes and textbooks, designedfor use within the Lyceum.

Exoteric works: dialogues (modelled after the Platonicdialogues), designed for the general public.

Platos unwritten doctrine:

Neoplatonism: Plotinus (204-270 AD)

Porphyry (c.232-c.305 AD)

[St. Augustine (354-430 AD)]

Proclus (411-485 AD)

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Aristotles work on logic.

TheOrganon.

Categories: Classification of types of predicates

On Interpretation(De interpretatione): Basics ofphilosophy of language, subject-predicate distinction,Square of Oppositions

Prior Analytics: Syllogistics

Posterior Analytics: Correct reasoning in general

Topics: Valid reasoning; probable conclusions

On Sophistical Refutations(De Sophisticis Elenchis):Fallacies

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The square of oppositions.

Aristotle,De interpretatione

Every B is A. CONTRARIES No B is A.

SUBALTERN

CONTRADICTORIES

SUBALTERN

Some B is A. SUBCONTRARIES Some B is not A.Contradictorypropositions cannot both be true and they cannot both be false.

Contrarypropositions cannot both be true but can both be false.

Subcontrarypropositions cannot both be false but can both be true.Asubalternmust be true if itssuperalternis true, and the superalternmust be false

if thesubalternis false.

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The Categories.

Aristotle,Categories:The ten categories(1b25).

Substance When

Quality Position

Quantity Having

Relation Action

Where Passion

The two ways of predication.

essential predication: Socrates is a human being;human I S S A I D O F Socrates

accidental predication: Socrates is wise; wisdom ISIN

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Essential predication.

essential: You cannot deny the predicate withoutchanging the meaning of the subject.

animal I S S A I D O F human.

human I S S A I D O F Socrates.

I S S A I D O F is a transitive relation.

Related to the category tree:Genus. Animal.

Species. Human.

Dog.

Individual. Socrates. Aristotle. Lassie. Boomer.

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Substances.

Universal substances Universal accidents

human,animal wisdom

Particular substances Particular accidents

Socrates, Aristotle

Plato(). Theuniversal substancesarethe (only) real things.

Aristotle (). Without the particular sub-stances, nothing would exist.

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Matter & Form.

Categories/De anima: There are three kinds ofsubstance: matter,formand the compound of the two.

Matter ispotentiality;formisactuality.

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The most famous syllogism.

Every man is mortal.Socratesis a man.

Socratesis mortal.

Proper name / Particular substance

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A more typical syllogism.

Every animal is mortal.Every man is an animal.

Every man is mortal.

Every B is an A.Every C is a B.

Every C is an A.

Barbara

a valid moodmood =modus

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Another valid mood.

Every philosopher is mortal.Some teacher is a philosopher.

Some teacher is mortal.

Every B is an A.Some C is a B.

Some C is an A.

Darii

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A similar but invalid mood.

DariiEvery B is an A.Some C is a B.

Some Cis an A.

Every Ais a B.Some C is a B.

Some C is an A.

Every philosopher is mortal.Some teacher is mortal.

Some teacher is a philosopher.

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Y t th i il d

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Yet another very similar mood.

Darii

Every B is an A.

Some C is a B.

Some Cis an A.

The invalid mood

Every A is a B.

Some C is a B.

Some Cis an A.

Datisi

Every B is a A.

Some B is a C.

Some Cis an A.

Some C is a Band Some B is a C

are intuitively equivalent.

Every B is an Aand Every A is a Barent.

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A fi i

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A first conversion rule.

This yields a simple formal (syntactical) conversion rule:

SomeX

is aY

can be converted to

Some Y is an X.

This rule isvalidity-preservingandsyntactical.

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B k D ii d D i i

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Back to Dariiand Datisi.

Darii

Every B is an A.

Some C is a B.

Some Cis an A.

Dati si

Every B is a A.

Some B is a C.

Some Cis an A.

Simple ConversionSome X is a Y Some Y is an X

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