the history of logic
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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!. - PowerPoint PPT PresentationTRANSCRIPT
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 OrganonThe 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 FerioSecond Figure MaNMeXNeX CamestresMeNMaXNeX CesareMeNMiX NoX FestinoMaNMoXNoX BarocoThird 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