The History of Logic

1. The History of Logic Scott T. Cella

2. Obvious Existence of Logic

3. 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!

4. 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

5. Plato (428 – 327 BC)

6. 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?

7. Aristotle (384-322 BC)The Grand-daddy of ‘em All

8. 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.

9. The Power of Syllogism • Syllogism: A logical argument in which one proposition is inferred from two or more others of a certain form.

10. Aristotle’s Organon

11. 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.

12. 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, Being-in-a-Position, Condition, Action, and Affection.

13. 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.

14. 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” 2. “Not every tree has leaves” 3. “Some trees have leaves” 4. “No trees have leaves”

15. 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

16. 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)

17. The Three Figures 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, thereforeAoC

18. The Figure Chart Figure Major MinorConc Mnemonic Name First Figure AaBBaCAaC Barbara AeBBaCAeCCelarent AaBBiCAiCDarii AeBBiCAoCFerio Second Figure MaNMeXNeXCamestres MeNMaXNeXCesare MeNMiXNoXFestino MaNMoXNoXBaroco Third Figure PaSRaSPiRDarapti PeSRaSPoRFelapton PiSRaSPiRDisamis PaSRiSPiRDatisi PoSRaSPoRBocardo PeSRiSPoRFerison

19. 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.

20. Book 6: Sophistical Refutations • Talks about 13 Fallacies • Six are verbal fallacies • Seven are material fallacies

21. 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.

22. 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. Diodorusclaimed 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.”

23. 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)

24. 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.

25. 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.

26. 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.

27. 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

28. 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.

29. 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.

30. 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) aimmedto incorporate the logic of all mathematical and scientific discourse in a single unified system.

31. 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