The History of LogicScott T. Cella

Obvious Existence of Logic

Rise in Greek MathematicsGreeks 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 OthersAristotle 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 SyllogismSyllogism: 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 CategoriesThe TopicsOn InterpretationThe Prior AnalyticsThe Posterior AnalyticsSophistical Refutations

These form the earliest formal study of logic that have come down to modern

times.

Book 1: The CategoriesSpecifies all possible types of

things which can be subjects and predicates of a proposition.

Elaborates on Substance, Quantity, Quality, Relevance, Where, When, Being-in-a-Position, Condition, Action, and Affection.

Book 2: The TopicsA 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 InterpretationDeals 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” 2. “Not every tree has leaves”3. “Some trees have leaves”4. “No trees have leaves”

Book 4: The Prior AnalyticsWork on deductive reasoning

(specifically syllogism).Contains first formal study of

logic (study of arguments).Identifies valid and invalid formsAristotle’s three claims:

◦1) P belongs to S◦2) P is predicated of S◦3) P is said of S

Aristotle’s Notationa = belongs to everye = belongs to noi = belongs to someo= 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

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

Depending on the position of the middle term, three syllogisms can be formed:

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 ChartFigure Major Minor Conc Mnemonic Name

First Figure AaB BaC AaC BarbaraAeB BaC AeC CelarentAaB BiC AiC DariiAeB BiC AoC Ferio

Second Figure MaNMeXNeX CamestresMeNMaXNeX CesareMeNMiX NoX FestinoMaNMoXNoX Baroco

Third Figure PaS RaS PiR DaraptiPeS RaS PoR FelaptonPiS RaS PiR DisamisPaS RiS PiR DatisiPoS RaS PoR BocardoPeS RiS PoR Ferison

Book 5: The Posterior AnalyticsDeals 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 RefutationsTalks about 13 Fallacies

◦Six are verbal fallacies◦Seven are material fallacies

The Other LogiciansThe 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: ModalityThere 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 (good)T F (bad)F T (good)F F (good)

Stoic’s Contribution 3:Meaning of TruthThe 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 mid-nineteenth 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.

What is logic?

Logic = Science about correct reasoning.

As such, it is only interested in the form rather than content.

0.b

Every hemin is melinSolik is a hemin----------------------------Solik is melin

Every H is MS is an H----------------------------S is M

It’s Okay to Fail… at FirstUniversal 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. AbstractiveConstructive: 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 PeriodsThe 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 PeriodsThe 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