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I'm trying to find out who invented truth-tables. Here is what I have so far.

Leibniz 'invented' binary arithmetic, or at least is the first one recognized to have codified and explained a base 2 system for arithmetic (there's no doubt that there are vague priors from China; Leibniz himself mentions 'Fohy' and goves a figure that looks like 'I Ching').

Boole, in his Laws of Thought, essentially created boolean algebra (notice that the name of the thing is not the same as the thing).

Peirce and Frege separately introduced symbolism that treated boolean things functionally (that's a roundabout way of saying they were the first to use the notation of functions on booleans, i.e. boolean functions).

Schroeder and later Post also are mentioned in connection with truth tables, but I'm having a hard time substantiating that more than just by repeating that other people collocate those names with 'truth table'.

Wittgenstein, in Tractatus Logicus-Phiosophicus, chapter 5, discusses boolean functions using that terminology and -graphically- expresses these functions in tabular format, calling them 'truth tables' (in German).

Shannon is famous for introducing boolean logic in his master's thesis (1936) for use in design of computers. (his primary references being Couturat, The Logic of Algebra (1905), and Whitehead, Universal Algebra (1898).

It is taken by philosophers that Wittgenstein is the inventor of truth tables: (Stanford Encyclopedia of Philosophy entry on Wittgenstein). But with a background more in cs and math, I find that it doesn't fit with my preconceived notions of intellectual culture - I just wouldn't expect scientific/mathematical types to trace the usage history of something so technical back to someone (Wittgenstein) so humanities-oriented (to be frank, I can't believe that people who built the first digital computers learned truth tables directly or indirectly from TLP).

It is the introduction of a tabular format called 'truth table' that I am looking for the provenance. Since boolean functions were well known before then, it is simply the distinctive visual device of laying out the function in a table with all the possibilities for the arguments next to the value of the function for those arguments. To me this seems like who invented 'FOIL' for teaching how to multiply binomials, in that a name was given to something that people had been doing anyway already (or not even bothering to do).

So, really, where did the use of truth tables really come from, and is there any indication in the history that follows, where the mathematicians and engineers got it from? So maybe Wittgenstein 'popularized' it among the philosophers (I don't have any doubt there), but maybe he was (or maybe wasn't) the source for later use by the engineers.

So, whoever was the first to have invented truth tables (Peirce is definitively the first one there is evidence for), who popularized the use of truth tables in elementary mathematical logic? Wittgenstein certainly popularized it among philosophers, but he (and the Wiener Kreis by extension) doesn't have much influence on mathematicians, so I find it unlikely that mathematicians learned it there.

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It is hardly appropriate to frame Wittgenstein as "humanities oriented"! –  amWhy Jun 3 '11 at 15:55
@amWhy: any particular reason? in terms of humanities and sciences as broad categories, philosophy is usually located in the humanities. –  Mitch Jun 3 '11 at 16:02
@Mitch: Wittgenstein is sui generis. –  TonyK Jun 3 '11 at 19:39
Wittgenstein worked with Russell on the mathematical foundations of logic. That's pretty "hard" science. –  Raskolnikov Jun 3 '11 at 19:39
Even in his later work, where he broke away from logical positivism, Wittgenstein continued his exploration in Mathematical Foundations (Remarks on the Foundations of Mathematics, etc). The most recent publication of the journal *Philosophia Mathematica" (Series III, Volume 19, no. 1, 2011), the very first article is entitled "Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms" by Simon Friederich, 1-19. –  amWhy Jun 3 '11 at 23:12

4 Answers 4

The popularization question I find hard to answer sufficiently. What culture are we talking about? English people didn't necessarily read American textbooks, and Americans didn't read German textbooks did they? At least, not usually, I would believe. That said, I will say this. A famous textbook is Lukasiewicz's "Elements of Mathematical Logic" Elementy logiki matematycznej He has 3 logical matrices in it for verifying the independence of axioms, and a 4th logical matrix to indicate how truth values might work in a 3-valued logic at variance with the standpoint of classical propositional logic he adopted in the rest of the book. In my opinion, the textbook is particularly excellent in that I think it has strong advantages for students both new to the study of classical logic and those experienced in it.

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Yes, the specific strand of culture is difficult to specify, because there are language/nation separations (in addition to the faculty differentiation), but academic borrowing, too. Also, the 20th century has seen a lot of changes in communication, which complicates the matter. If forced, I'd have to limit it to English speaking culture (where things might be likely to have been borrowed from German or French, and doubtfully from Russian or Polish (but that would be very interesting!)) –  Mitch Jun 19 '13 at 15:51

If you do a little research, Peirce WAS a scientist. Wittgenstein's work was primarily in the philosophy of mathematics and in the philosophy of language after having studied mechanical engineering and Russell and Frege were logician/mathematician's. They had large followings from academics in all departments. They were the originators, the popularizer's and the teachers, not the other way around.

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Interesting. 1) it is unclear who you are saying did what (who is 'they'?). Is it Russell/Frege? Truth Tables are not done in BR or GF sources, but they are in Wittgenstein, therefore my question. 2) Can you cite any references which support this? I'm curious about the train of provenance. –  Mitch Nov 30 '12 at 18:30

In addition to the Stanford encyclopedia of Philosophy's attribution to Wittgenstein, Wikipedia's discussion of Wittgenstein's masterpiece Tractatus Logico-Philosophicus also credits Wittgenstein as the inventor of truth tables.

"Wittgenstein is to be credited with the invention of truth tables (4.31) and truth conditions (4.431) which now constitute the standard semantic analysis of first-order sentential logic.[7] The philosophical significance of such a method for Wittgenstein was that it alleviated a confusion, namely the idea that logical inferences are justified by rules. If an argument form is valid, the conjunction of the premises will be logically equivalent to the conclusion and this can be clearly seen in a truth table; it is displayed. The concept of tautology is thus central to Wittgenstein's Tractarian account of logical consequence, which is strictly deductive.

5.13 When the truth of one proposition follows from the truth of others, we can see this from the structure of the propositions.

5.131 If the truth of one proposition follows from the truth of others, this finds expression in relations in which the forms of the propositions stand to one another: nor is it necessary for us to set up these relations between them, by combining them with one another in a single proposition; on the contrary, the relations are internal, and their existence is an immediate result of the existence of the propositions.

5.132 If p follows from q, I can make an inference from q to p, deduce p from q. The nature of the inference can be gathered only from the two propositions. They themselves are the only possible justification of the inference. "Laws of inference", which are supposed to justify inferences, as in the works of Frege and Russell, have no sense, and would be superfluous."

The book (Tractatus) itself is worth the read!

Update: In The Development of Logic (1962), Kneale and Kneale argue that Boole, Frege, Peirce, Jevons, and Venn all contributed to the essentials popularized by Post and Wittgenstein in 1920 (420 and 531).

In making their argument, Kneale and Kneale cite Peirce's paper "On the Algebra of Logic: A Contribution to the Philosophy of Notations" (American Journal of Mathematics 7 [1885], 180-202):

[T]o find whether a formula is necessarily true substitute f and v for the letters and see whether it can be supposed false by any such assignment of values (CP 3.387).

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My question is getting at the problem with claims you present in your answer that Wittgenstein 'invented' truth-tables, because there is evidence that others had done the same thing before him and that he was only reproducing such things without attribution in TLP (read his introduction where he himself expressly disavows that anything is particularly original in TLP). –  Mitch Jun 4 '11 at 20:46

Well, Aristotle qualifies as quite "humanities oriented" according to the modern way of classifying subject areas, but up until recently plenty of usage history can get traced back to him. Actually, there still exists some usage history traceable back to him.

That said, you might want to see this page, which indicates that perhaps no real inventor existed. Shosky tried to argue that Russell did so. Anellis, though, if his evidence is correct clearly enough indicates that C. S. Peirce had them before that. So, unless historians have missed something in Frege, Peirce gets the prize here:

... the discovery by Zellweger of Peirce’s manuscript of 1902 does permit us to unequivocally declare with certitude that the earliest, the first recorded, verifiable, cogent, attributable and complete truth-table device in modern logic attaches to Peirce, rather than to Wittgenstein’s 1912 jottings and Eliot’s notes on Russell’s 1914 Harvard lectures

One might also ask here, "who first used numbers for truth-values in the context of truth tables?" I'm not so sure here, but I'd think Lukasiewicz did that first, though when he wrote a truth table he wrote them horizontally instead of vertically.

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nice links (how did you find them? google didn't help me). So those links nicely address the nominal question 'Who was first?'. I am also looking for the pathway of popularization. I think it is undoubted that Wittgenstein popularized it (whether he invented it or not) among philosophy. But How did it become popular among the engineers? What was the pathway from WIttgenstein (or Post or Peirce) to those later? (I added to the end of my question to reflect this). –  Mitch Jun 3 '11 at 17:33
@Mitch Wittgenstein certainly could well have publicized it's use. He was part of what is known as the "Vienna Circle"...worked closely with Russell, Carnap, and others. Most of his work was in mathematics and logic...It is well known in philosophy/logic, that Wittgenstein conceived of truthtables while working as a teacher of mathematics and logic, and later in his work and correspondences with colleagues, who may then have also adopted its use. –  amWhy Jun 3 '11 at 21:57
Excellent links...they go directly to the 'inventor' question. –  Mitch Jun 4 '11 at 21:18

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