origins of logic l3
TRANSCRIPT
-
8/12/2019 Origins of Logic L3
1/24
Origins of Logic
Greek mathematics
Rhetoric: Eristic and Sophistry
Core Logic 2007/08-1ab p. 2/
-
8/12/2019 Origins of Logic L3
2/24
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
Core Logic 2007/08-1ab p. 3/
-
8/12/2019 Origins of Logic L3
3/24
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.
Core Logic 2007/08-1ab p. 4/
-
8/12/2019 Origins of Logic L3
4/24
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)
Core Logic 2007/08-1ab p. 5/
-
8/12/2019 Origins of Logic L3
5/24
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
Core Logic 2007/08-1ab p. 6/
-
8/12/2019 Origins of Logic L3
6/24
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.
Core Logic 2007/08-1ab p. 7/
-
8/12/2019 Origins of Logic L3
7/24
Theoria et Praxis (1).
The School of Athens (Raffaello Sanzio; 1509)
Core Logic 2007/08-1ab p. 8/
-
8/12/2019 Origins of Logic L3
8/24
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.
Core Logic 2007/08-1ab p. 9/
-
8/12/2019 Origins of Logic L3
9/24
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.
Core Logic 2007/08-1ab p. 10/
-
8/12/2019 Origins of Logic L3
10/24
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
Core Logic 2007/08-1ab p. 11/
-
8/12/2019 Origins of Logic L3
11/24
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)
Core Logic 2007/08-1ab p. 11/
-
8/12/2019 Origins of Logic L3
12/24
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
Core Logic 2007/08-1ab p. 12/
-
8/12/2019 Origins of Logic L3
13/24
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.
Core Logic 2007/08-1ab p. 13/
-
8/12/2019 Origins of Logic L3
14/24
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
SocratesCore Logic 2007/08-1ab p. 14/
-
8/12/2019 Origins of Logic L3
15/24
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.
Core Logic 2007/08-1ab p. 15/
-
8/12/2019 Origins of Logic L3
16/24
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.
Core Logic 2007/08-1ab p. 16/
-
8/12/2019 Origins of Logic L3
17/24
Matter & Form.
Categories/De anima: There are three kinds ofsubstance: matter,formand the compound of the two.
Matter ispotentiality;formisactuality.
Core Logic 2007/08-1ab p. 17/
-
8/12/2019 Origins of Logic L3
18/24
The most famous syllogism.
Every man is mortal.Socratesis a man.
Socratesis mortal.
Proper name / Particular substance
Core Logic 2007/08-1ab p. 18/
-
8/12/2019 Origins of Logic L3
19/24
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
Core Logic 2007/08-1ab p. 19/
-
8/12/2019 Origins of Logic L3
20/24
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
Core Logic 2007/08-1ab p. 20/
-
8/12/2019 Origins of Logic L3
21/24
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.
Core Logic 2007/08-1ab p. 21/
Y t th i il d
-
8/12/2019 Origins of Logic L3
22/24
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.
Core Logic 2007/08-1ab p. 22/
A fi i
-
8/12/2019 Origins of Logic L3
23/24
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.
Core Logic 2007/08-1ab p. 23/
B k D ii d D i i
-
8/12/2019 Origins of Logic L3
24/24
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
Core Logic 2007/08-1ab p. 24/