Wittgenstein’s 1913 “On Logic, and How Not to Do It”

Because I wanted nicer electronic versions of Wittgenstein’s not-exactly-nice book review, I have digitized it (https://landondcelkind.com/on-logic-and-how-not-to-do-it/). Also, the text is now publicly available in PDF and Markdown formats on the Internet Archive (https://archive.org/details/wittgenstein-1913-review-coffey) and in MP3 format on LibriVox (https://librivox.org/short-nonfiction-collection-vol-080-by-various/). It is also reproduced below for good measure. Feel free to share, and enjoy!


“On Logic, and How Not to Do It”

Ludwig Wittgenstein

  • Review: The Science of Logic: an inquiry into the principles of thought and scientific method. By Peter Coffey, Ph.D. (Louvain), Professor of Logic and Metaphysics, Maynooth College. Longsman, Green, & Co 1912. (link to Coffey)

In no branch of learning can an author disregard the results of honest research with so much impunity as he can in Philosophy and Logic. To this circumstance we owe the publication of such a book as Mr Coffey’s Science of Logic: and only as a typical example of the work of many logicians of to-day does this book deserve consideration. The author’s Logic is that of the scholastic philosophers, and he makes all their mistakes—of course with the usual references to Aristotle. (Aristotle, whose name is taken so much in vain by our logicians, would turn in his grave if he knew that so many Logicians know no more about Logic to-day than he did 2,000 years ago). The author has not taken the slightest notice of the great work of the modern mathematical logicians—work which has brought about an advance in Logic comparable only to that which made Astronomy out of Astrology, and Chemistry out of Alchemy.

Mr Coffey, like many logicians, draws great advantage from an unclear way of expressing himself; for if you cannot tell whether he means to say “Yes” or “No,” it is difficult to argue against him. However, even through his foggy expression, many grave mistakes can be recognised clearly enough; and I propose to give a list of some of the most striking ones, and would advise the student of Logic to trace these mistakes and their consequences in other books on Logic also. (The numbers in brackets indicate the pages of Mr Coffey’s book—volume I.—where a mistake occurs for the first time; the illustrative examples are my own).

  1. [36] The author believes that all propositions are of the subject-predicate form.
  2. [31] He believes that reality is changed by becoming an object of our thoughts.
  3. [6] He confounds the copula “is” with the word “is” expressing identity. (The word “is” has obviously different meanings in the propositions—“Twice two is four”and “Socrates is mortal.”)
  4. [46] He confounds things with the classes to which they belong. (A man is obviously something quite different from mankind.)
  5. [48] He confounds classes and complexes. (Mankind is a class whose elements are men; but a library is not a class whose elements are books, because books become parts of a library only by standing in certain spatial relations to one another—while classes are independent of the relations between their members.)
  6. [47] He confounds complexes and sums. (Two plus two is four, but four is not a complex of two and itself.)

This list of mistakes could be extended a good deal.

The worst of such books is that they prejudice sensible people against the study of Logic.

March 6th, 1913


originally published in The Cambridge Review 34 (1912–13), p. 351; reprinted in:

  1. 1 January 1970, E. Homberger, William Janeway, and Simon Schama, The Cambridge Mind, London: Jonathan Cape: pp. 127-129
  2. 1 January 1988, Brian McGuinness, Wittgenstein: A Life, Young Ludwig: 1889-1921, Berkeley & Los Angeles, University of California Press: pp. 169-170
  3. 1 June 1993, Ludwig Wittgenstein, Philosophical Occasions, 1912-1951, Indianapolis, Hackett Publishing Company: pp. 1-3

“Principia” Notation in LaTeX

I recently had occasion to use the Peirce arrow, NOR (↓), in LaTeX! But I wanted to use Peirce’s notation from 1902. This takes the form of the symbol for strict implication, , turned 270 degrees so that the arrow’s head faces down, like so: ⥿. I was helped in typesetting this by Richard Zach’s helpful post. I have worked on Principia quite a bit, and I have needed to make ad hoc commands to rotate letters and other symbols to match the logical symbols used there. So I thought to share my work in case it saves others some time! I will update this as I devise further ad hoc commands.

All this requires is the package graphicx. With it, you can then use the command \text{\rotatebox[origin=c]{[DEGREES]}{[TEXT]}}. The reason for the \text{…} on the outside is to avoid conflicts in math mode and similar environments.

With that ado, here is LaTeX code I use for:

  • \Runi for the universal set, V (an upside-down Lambda):
    \newcommand{\Runi}{\text{\rotatebox[origin=c]{180}{$\Lambda$}}}
  • \Dom for the domain, D’, of a relation:
    \newcommand{\Dom}{$\text{D}`$}
  • \CDom for the converse domain of a relation (a ‘D’ rotated 180 degrees):
    \newcommand{\CDom}{\text{\rotatebox[origin=c]{180}{$\text{D}`$​}}}
  • \Parrow for the Peirce arrow NOR in Peirce’s 1902 notation (⥿):
    \newcommand{\Parrow}{\text{\rotatebox[origin=c]{270}{$\strictif$}}}

No doubt there will be more to come!

References for foundations of math!

If you are not interested in the foundations of mathematics (and of computer science, I think), then you may disregard this post. For those of you that are interested but are unsure where to start: I hope this will help you get started with the foundations movement at the turn of the twentieth-century!

  • From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Jean van Heijenoort, editor)

This source book is definitely worth purchasing for your own library and use. It contains highly influential primary texts in logic by, among others, Cantor, Peano, Frege, Russell, Zermelo (yes, the one of ZF fame), von Neumann, Hilbert, Brouwer, and, of course, Gödel. It is sorely missing Tarski and Church and Marcus, but you can make up for that elsewhere. Frege’s Begriffsschrift alone justify purchasing this, as do Gödel’s papers.

  • The Search for Mathematical Roots, 1870-1940 (Ivor Grattan-Guinness, author)

Grattan-Guinness, sadly now deceased, was a highly regarded historian of mathematics (he also wrote a book on the history of calculus). This book offers a good sketch of the history of the foundations of mathematics movement. It is not exhaustive, but it covers a good enough selection to contextualize whatever else you go on to read.

A combination of those two should put you in a position to pursue whatever else you find interesting about the foundations of mathematics movement at the turn of the 20th century. I am, of course, happy to recommend further items. Just get in touch!