The History of Logic
Scott T. Cella
Obvious Existence of Logic
Rise in Greek Mathematics

Greeks sought to replace empirical
methods with demonstrative science.

Answers the question “Why?”

The Greeks are the one’s to blame for
our High School Geometry struggles!
Pythagoras (More than a Triangle)

Pythagoreans started a system of math
containing proof.

Three principles of geometry
◦ I. Certain propositions must be accepted as
true without proof
◦ II. Every proposition is proven through these
◦ III. Derivations of propositions must be formal
Plato (428 – 327 BC)
Plato’s Contribution (We Think)

No surviving logic work remains.

Plato is credited with the following
important contributions:
◦ What can be called True or False?
◦ What is the nature of the connection
between the assumption of a valid argument
and it’s conclusion?
◦ What is the nature of definition?
Aristotle (384-322 BC)
The Grand-daddy of ‘em All
Aristotle’s Impact on Others
Aristotle is credited as the first thinker of
a logical system.
 The following were adopted from
Aristotle:

◦
◦
◦
◦
Universal definition found in Socrates.
Reductio ad Adsurdum in Zeno.
Propositional structure and negation in Plato.
Body of argumentative techniques found in
legal reasoning and geometric proof.
The Power of Syllogism

Syllogism: A logical argument in which
one proposition is inferred from two or
more others of a certain form.
Aristotle’s Organon
The Six Parts of the Organon
The Categories
 The Topics
 On Interpretation
 The Prior Analytics
 The Posterior Analytics
 Sophistical Refutations

These form the earliest formal study of logic
that have come down to modern times.
Book 1: The Categories

Specifies all possible types of things which
can be subjects and predicates of a
proposition.

Elaborates on Substance, Quantity,
Quality, Relevance, Where, When, Beingin-a-Position, Condition, Action, and
Affection.
Book 2: The Topics
A treatise on the art of dialectic.
 A topic (topos) is a general argument
which is sort of a template from which
many individual arguments can be
constructed.
 Doesn’t necessarily deal with forms of
syllogism, but contemplates the use of
topics as places from which dialectical
syllogisms may be derived.

Book 3: On Interpretation


Deals with relationships between language
and logic in a comprehensive, explicit, and
formal way.
Analyzing simple propositions and draws a
series of basic conclusions on routine issues
(negation, quantities, etc.)
1. "Every tree has leaves” ("x)
2. “Not every tree has leaves” (Ø"x)
3. “Some trees have leaves” ($x)
4. “No trees have leaves” (Ø$x)
Book 4: The Prior Analytics
Work on deductive reasoning (specifically
syllogism).
 Contains first formal study of logic (study
of arguments).
 Identifies valid and invalid forms
 Aristotle’s three claims:

◦ 1) P belongs to S
◦ 2) P is predicated of S
◦ 3) P is said of S
Aristotle’s Notation
a = belongs to every
e = belongs to no
i = belongs to some
o= does not belong to some

Categorical sentences may then be abbreviated as
follows:
AaB = A belongs to every B (Every B is A)
AeB = A belongs to no B (No B is A)
AiB = A belongs to some B (Some B is A)
AoB = A does not belong to some B (Some B is not A)
The Three Figures
Depending on the position of the middle term,
three syllogisms can be formed:
First Figure
Second Figure
Third Figure
Predicate - Subject Predicate - Subject Predicate - Subject
Major Premise
A-B
B -A
A-B
Minor Premise
B-C
B-C
C-B
Conclusion
A-C
A-C
A-C
The First Figure: AaB and BaC, therefore AaC
AeB and BaC, therefore AeC
AaB and BiC, therefore AiC
AeB and BiC, therefore AoC
The Figure Chart
Figure
First Figure
Second Figure
Third Figure
Major
AaB
AeB
AaB
AeB
MaN
MeN
MeN
MaN
PaS
PeS
PiS
PaS
PoS
PeS
Minor
BaC
BaC
BiC
BiC
MeX
MaX
MiX
MoX
RaS
RaS
RaS
RiS
RaS
RiS
Conc
AaC
AeC
AiC
AoC
NeX
NeX
NoX
NoX
PiR
PoR
PiR
PiR
PoR
PoR
Mnemonic Name
Barbara
Celarent
Darii
Ferio
Camestres
Cesare
Festino
Baroco
Darapti
Felapton
Disamis
Datisi
Bocardo
Ferison
Book 5: The Posterior Analytics
Deals with demonstration, definition, and
scientific knowledge.
 In the previous book, syllogistic logic
considers formal aspects. This book
considers the logic’s matter.
 The form may be plausible, but the
propositions which it is derived from may
not.

Book 6: Sophistical Refutations

Talks about 13 Fallacies
◦ Six are verbal fallacies
◦ Seven are material fallacies
The Other Logicians
The Stoics were another
school in Greek times,
tracing it’s roots back to
Euclid of Megara.


Like Plato, there is currently no existing
work from the Stoics, so historians rely
on accounts from other sources.
Stoic’s Contribution 1: Modality

There is no distinction between potentiality
and actuality.
◦ Possible: That which either is or will be.
◦ Impossible: That which cannot be true.
◦ Contingent: That which either is already, or will
be false.
Diodorus claimed that these propositions are
inconsistent in his ‘Master Argument’:
◦ “Everything that is past is true and necessary.”
◦ “The impossible does not follow from the
possible.”
◦ “What neither is nor will be is possible.”
Stoic’s Contribution 2:
Conditional Statements

A true conditional is what could not
possibly begin with a truth and end with
falsehood
T T
TF
F T
FF
(good)
(bad)
(good)
(good)
Stoic’s Contribution 3:
Meaning of Truth

The biggest difference between Stoic and
Aristotelian logic is that Stoic deals with
propositions rather than terms; hence it is
closer to modern propositional logic.

According to the Stoics, three things are
linked together: that which is signified,
that which signifies, and the object.
Skip a Few Hundred Years…

Logic spread through several civilizations,
such as India, Asia, Islam, and several
European countries in Medieval times.

Fields of Psychology and Philosophy
benefited from advancements in logic.

However, from the 14th Century to the
19th Century, much of logic’s work was
neglected.
Skip a Few More Hundred Years…
The marriage between logic and
mathematics was formed in the midnineteenth century.
 The rise in "symbolic" or "mathematical"
logic is considered one of the greatest
achievements in logic history.
 Modern logic is fundamentally a calculus
whose rules of operation are determined
only by the shape and not by the meaning
of the symbols.

0.b
What is logic?
Logic = Science about correct reasoning.
As such, it is only interested in the form rather
than content.
Every hemin is melin
Solik is a hemin
---------------------------Solik is melin
Every H is M
S is an H
---------------------------S is M
It’s Okay to Fail… at First

Universal acceptance played a key role in
the rise of modern logic
◦ Ex: Pierce noted that even though a mistake
in the evaluation of a definite integral by
Laplace led to an error concerning the
moon's orbit that persisted for nearly 50
years, the mistake, once spotted, was
corrected without any serious dispute.
Constructive vs. Abstractive

Constructive: Builds theorems by formal
methods, then looks for an interpretation
in ordinary language.

Abstractive: Formalizing theorems
derived from ordinary language.

Modern Logic is constructive and entirely
symbolic.
The Five Modern Day Periods

The embryonic period (Leibniz )
Logical calculus was developed

The algebraic period (Boole & Schröder)
Greater continuity of development.

The logicist period (Russell & Whitehead)
aimmed to incorporate the logic of all mathematical
and
scientific discourse in a single unified system.
The Five Modern Day Periods

The metamathematical period (Hilbert,
Gödel, and Tarski)
combination of logic and metalogic. Also had Gödel’s
Incompleteness Theorem.

The period after World War II (Cella &
Japaridze)
Rise of model theory, proof theory, computability theory and
set theory