Symbolic logic

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Lewis Carroll (1863)

Symbolic Logic is a popular science textbook on elementary logic by Lewis Carroll . It was originally planned as a three-volume work. The first volume, Elementary , was published in 1896. Before the completion of the other volumes, Carroll died. The existing parts of the second volume Advanced were only published posthumously in 1977 by William Warren Bartley , the third volume should have been called Transcendental .

content

Symbolic Logic is divided into eight books. After some general definitions, it is shown how to bring natural language statements into a standardized form. Carroll then introduces a diagram similar to a four-field table in order to divide a basic quantity into the four possible combinations with two possible properties. To indicate which objects actually exist, Carroll uses red and gray tokens that could be purchased separately. He then extends the diagram to use it with three properties. He uses these diagrams to work with syllogisms . Finally, he introduces an algebraic notation in order to be able to handle syllogisms without the diagrams, he uses a cross † for “ and ”, a mirrored paragraph mark ⁋ for “ implies ”, a subscript 1 for “it exists”, a subscript 0 for “It does not exist” and an apostrophe for the negation. The conclusion is the treatment of chain links .

Carroll provides a few examples for all definitions and methods and provides many exercises, to which the solutions and the solution are attached. The tasks often contain the nonsense typical of Carroll . So he sets the task of drawing the right conclusion from the following three statements:

  1. No ducks waltz;
  2. No officers ever decline to waltz;
  3. All my poultry are ducks
  1. No duck waltzes.
  2. No officer refuses to waltz.
  3. All of my poultry are ducks.

So it turns out that there is no officer among my poultry.

In the appendix, he defends a number of decisions regarding the structure of the work and presents the advantages of his method over conventional methods, for example using Venn diagrams .

Diagrams

Exemplary use of the diagrams to solve logical problems

What is special about Carroll's approach are the diagrams, which give the solution of logic tasks a playful approach. The example from the frontispiece shows how the procedure works:

The two premises are given : "Your story of how you once met a sea serpent always makes me yawn." And "I never yawn, unless I hear a story that is completely uninteresting."

The diagram divides all possible stories in different ways: in the upper half is the story with the sea serpent, in the lower half all the others. In the left half are the completely uninteresting stories, in the right half the interesting ones. After all, on the inside are the stories that make you yawn, on the outside all the others.

The first premise now says that the story about the sea serpent makes you yawn, so that there is a story that lies in both the top half and the inner area. However, it is not clear whether it is interesting or not, so it can be in the left or right half. Carroll indicates this by placing a red token on the border between these two areas.

In addition, it follows from the first premise that there is no story that lies both in the upper half and in the outer area. Carroll uses a gray token to indicate that the two L-shaped areas are empty.

In a second diagram, he now represents the second premise analogously. This says that there are no stories that lie both in the inner area and in the right half. As for the first premise, he marks this with two gray stones in the two corresponding areas.

In the next step, Carroll combines the two diagrams. The second diagram shows that the area on the top right inside is empty (indicated by the gray piece), so the red piece from the first diagram must be in the left half.

Finally, to draw the conclusion , Carroll abandons the distinction between inside and outside. The upper left area receives a red token because the inner of the two sub-areas contained one, the upper right area receives a gray token because both sub-areas contained a gray stone. No statements can be made about the other areas, so they are not marked with stones.

From this last diagram you can now see that there is a story in the upper left area as it contains a red stone. So the story about the sea serpent is completely uninteresting.

reception

Symbolic Logic was one of the first books on logic that was designed for the general public; according to the preface, the target group already included children from the age of twelve. The work was a great success, and the fourth edition was available after just one year.

However, Carroll's methods and notations could not prevail. To the modern reader, they seem unnecessarily complicated. On the other hand, his tasks on the chain links are still partly quoted and used in other popular scientific works. In his 1980 review , Irving M. Copi criticized the stereotypical image of the Jew that appears in some of the chain links.

classification

While Carroll's diaries show an early interest in logic, his first mathematical publications concerned Euclidean geometry . His publications on logic began in 1886. Carroll was in the tradition of Aristotle's logic , but like many of his contemporaries, he was looking for new methods. The extent of his major work on logic form The Game of Logic in 1887 and Symbolic Logic , both books were published under the pseudonym Lewis Carroll and are aimed at a general audience. Already in The Game of Logic he introduced his diagrams, which appear earlier in his diaries. To what extent these are influenced by Venn diagrams is controversial. Some historians see it as a further development, others as an independent new development. The main difference between Carroll's diagrams and Venn diagrams lies in the symmetry: while in Venn diagrams there is a clear contrast between inside and outside, i.e. between a property and its negation, and consequently only objects that do not have any of the properties examined have their place implicitly outside the diagram, a property and its negation are on an equal footing with Carroll and are symmetrically opposite each other.

The algebraic notation that Carroll first introduced in Symbolic Logic can also be found earlier in his diaries. While the diagrams offer a playful, but in practice rather cumbersome, access to logical problems, the symbolic notation is very concise and can be used according to formal rules.

Despite the sales success, Carroll's approaches could not prevail. A few years after Carroll's death, the Principia Mathematica appeared, a work that had a decisive influence on the further development of logic.

literature

  • Amirouche Moktefi: Lewis Carroll's Logic. In: Dov M. Gabbay, John Woods (Eds.): British Logic in the Nineteenth Century. Elsevier, 2008. ISBN 978-0-08-055701-4 .

expenditure

Individual evidence

  1. ↑ Example: Ian Stewart : Professor Stewart's Mathematical Treasures. Rowohlt, 2012. ISBN 978-3-498-06415-0 . P. 235.
  2. ↑ Irving M. Copi: Review: Symbolic Logic by Lewis Carroll, William Warren Bartley, III. In: The British Journal for the Philosophy of Science. Vol. 31, No. 1 (March 1980), pp. 81-85. ( JSTOR 687254 )