THE HISTORY OF LOGIC

                                          THE HISTORY OF LOGIC

Ancient Period  (650 B. C.-400 A.D.)

        Among those Greek philosophers  preparing the way for the major developments in ancient logic are Pythagoras, Zeno of Elea, the Megarians (especially Eubulides), Socrates and Plato. 

      Pythagoras (580-497 B.C.) emphasized the intellect (rather than observation) as he searched for form and order in the cosmos, particularly through numbers and mathematics.  With Zeno of Elea (490-430) we find examples of  arguments and paradoxes, and the use of dialectic.  In his paradox of “The Arrow” Zeno suggests that we consider the apparent flight of the arrow and focus on a particular point in that flight, in which case we realize the flight could not really take place, since at a particular  point in the flight the arrow must be just where it is and not moving; and since it is at rest at that point it could never move. 

       With Eubulides the Megarian (ca. 390 B. C.) we find more paradoxes, including “the Liar” (if I tell you “I am lying” am I telling the truth or not?); “the Hooded Man” (you know your brother, but in comes someone in disguise ((presumably in a hood)) and you don’t recognize (know) him, and it turns out, its your brother; “the Horned Man” (What you have not lost you must still have, and did you lose your horn?); and “the Bald Man” (is a person with only 28 hairs on his or her head bald?  Yes or no?  27?)

       As described by Plato, Socrates (470-399 B. C.) displays an interest in definition, argument , truth, deduction and  dialectic.  Typically a Platonic dialogue finds Socrates in a search for a clearer definition and understanding of an important concept, such as justice, excellence or beauty.  Socrates seemed to suggest that through dialectic we might obtain the actual (non-relative) truth.

       At this time the Sophists also were active.  They were paid instructors or teachers who claimed that “truth is relative” and imparted skills in language and communication  to their students.  Among them were Protagoras (490-421 B. C.) who first distinguished different types of sentences,  Prodicus (460-399) who was interested in the correct use of words, especially synonyms, Gorgias (ca. 480-399) who emphasized rhetoric and persuasion, and Antiphon (ca. 475-400) who emphasized the importance of education and a good foundation.

        The key contributions to ancient logic are those of Aristotle and the Stoics.  Aristotle (384-322 B. C.) was the student of Plato.  The Stoics were influenced by the Megarians and Socrates.

        Aristotle’s collection of writings on logic is called the Organon (which meant “tool’ or “instrument” and suggests Aristotle’s view on the use of logic).  Included in the Organon are several books: Prior Analytics, Posterior Analytics, Topics, Categories, On Interpretation, and the Fallacies.  The first systematic treatment of logic occurs in the Prior Analytics.  This first system of logic is the theory of the syllogism. A syllogism is a type of argument such as this:

                                        All dogs are mammals,

                                        All collies are dogs,

                                        Consequently, all collies are mammals. 

Aristotle analyzed basic syllogisms involving 3 statements, 3 terms (such as dogs, collies, mammals) and quantifier words such as “all”, “no” and “some”.   He developed rules to prove that only a few (of the more than 250) basic syllogisms  are good or valid arguments.  (Currently the logic of Aristotle is referred to as “categorical” or “traditional” logic).

        The Stoics were not so interested in the categorical statements that Aristotle examined  but instead they analyzed  compound statements, such as conditionals (if/then), disjunctions (or), and conjunctions (and), and arguments containing them.

A typical example of a Stoic argument is this:

                                   If she is living then she is breathing.

                                   She is living.

                                   Therefore, she is breathing.

We can depict this argument as follows:

                                    If p, then q

                                    p

                                    Therefore,  q

“She is living” is represented by “p”, while “she is breathing” is represented by “q”.  The name give to this form of argument is “modus ponens”.  This and similar arguments intrigued the Stoics, including their leading logician, Chrysippus (280-206 B. C.).  Historically their work was generally  overlooked and had to be “rediscovered”.  The terms “propositional logic” and “truth-functional logic” currently refer to this type of logic.

       The logic of the Greeks, particularly Aristotle, was preserved and transmitted to some degree by various Roman philosophers and logicians, including Cicero (106-43 B.C.) who wrote “the Topica” and introduced the term “proposition”, and Galen the physician (129-199 A.D.) who wrote  “Introduction to Dialectic” and was familiar with both Stoic and Aristotelean logic.   Alexander of Aphrodisias (3rd century A.D.)  commented on and “clarified” Aristotle and used  the term  “logic” in the “modern” sense of distinguishing correct from incorrect reasoning,  the “Square of Opposition” appeared in a commentary by Apuleius of Madauros, while the skeptic Sextus Empiricus  spoke of “the futility of all system making”.  The inventing of Latin equivalents for Greek terminology was a key Roman contribution.  

Medieval Period (400 A.D.-1600 A.D.):

        The 11th -15th centuries were the main time of activity regarding logic in this period.  The influence of Aristotle dominates the medieval logicians, who wrote commentaries on him and on others who had commented on him such as Boethius and Porporhy.   Among the important logicians were Peter Abelard (1079-1142) with 4 works on logic, William of Sherwood (ca. 1200-1271) who developed mnemonic verses as an aid in learning the syllogisms (Barbara, aka AAA, being the best known), William of Ockham (died 1349) (best known for Ockham’s Razor which suggests the importance of simplicity), and J. Buridan (mainly known for Buridan’s Ass, involving decisions in cases of equal preference, but which does not apparently actually occur in his writings).  The development and emergence of universites during this period is important for the study of logic.  Textbooks and manuals on the subject began to appear.  One of the more important textbooks was the “Port Royal Logic” of Arnauld and Nicole, appearing in 1662, in which logic is the “art of managing one’s reason right in the knowledge of things, both for the instruction of oneself and of others”.

Early Modern Period (1600-1850 A.D.):

          Leibnitz (1646-1716) is considered a great logician and his work exhibits a respect for traditional “Aristotelean” logic but also an interest in general theories of arrangements, plans for an “ideal” language, and general science of method.   The German philosopher Kant (1724-1804) made the distinction between types of statements a key to understanding his philosophy; he distinguished between analytic statements whose truth can be determined on the basis of the meanings of the words in the statements, and synthetic statements, which require a direct appeal to experience.   Bolzano (1781-1848) continued to examine the analytic-synthetic distinction in his chief work Wissenschaftslehre.

         

Modern and Contemporary Period (1850-present)

        The 19th and 20th centuries have involved great activity and discovery in logic, including the “rediscovery” of the Stoic type of logic or logic of propositions.  De Morgan  (1806-1871) discovered the theorems that  bear his name and that are now routinely part of the logic of propositions.  George Boole (1815-1864), considered the founder of symbolic logic, used symbols to depict arguments; he wrote the “Analysis of the Laws of Thought” and “Mathematical Analysis of Logic”, in which he argues that math is the basis of logic; and his use of numbers to    

express the truth values of compound statements (conjunctions, disjunctions, etc.)  directly influenced the development of computers.

        The British logician John Venn (1834-1923) developed circular diagrams used as a tool to test the validity of syllogisms.  J. S. Mill (1806-1873), another British philosopher, was particularly interested in inductive arguments and gave an account of methods for checking such arguments, known in fact as “Mill’s Methods”.  His countryman Charles Dodgson (1832-1898) wrote “Symbolic Logic” and “The Game of Logic”, but is better known under his pen name Lewis Carroll (“Alice in Wonderland”).  In the United States C. S. Pierce (1839-1914), an initiator of American pragmatism,  was the earliest influential logician.  Pierce stated that “few persons care to study logic, because everybody conceives himself to be proficient enough in the art of reasoning already.  But I observe that this satisfaction is limited to one’s own ratiocination, and does not extend to that of other men.”

        G. Frege (1848-1925) claimed that logic is the basis of math, and specifically aimed to reduce or derive arithmetic from logic; also, he developed the predicate calculus (quantification theory) which brought the categorical (Aristotelean) and propositional (Stoic) traditions together.  His greatest work is perhaps the Begriffsschrift. 

        Bertrand Russell (1872-1970)  and Alfred North Whitehead (1861-1947) continued the development of the predicate calculus in their Principia Mathematica (1910-1913).  Russell’s criticism of some of Frege’s ideas led to the development of a paradox involving the “set of all sets”, known in its more popular version as “the Barber paradox”: in a certain city a barber shaves the heads of all those people and of only those people who do not shave themselves, . . . but then who shaves the barber?   Reflections on this paradox led Russell to develop his “theory of types”. 

         Other contributions in this century have been from Wittgenstein (1819-1951), one of the developers of “truth tables”, K. Godel (1916-     ), known for his “ incompleteness theorem”,  and Lofti Zadeh (1916-     ), who is associated with the development and formalizing of “fuzzy” logic in 1965.  Rudolph Carnap (1891-1970), who defends a thesis of extensionality in his Logical Syntax of Language (1934) attempted to give precise definition to the distinction between analytic and synthetic statements and was associated with the philosophy of  logical empiricism and it famous verifiability principle, according to which a synthetic statement is meaningful only if it is verifiable.

Another of Carnap’s works is The Logical Structure of the World (1928) . The development of non-Euclidean geometry, many- valued logics, proof theory and systems theory, and of course computers and information technology have had far-reaching impact and significance for logic and critical thinking.