The slim volume (less than eighty pages) sets forth a complete philosophical system that may be construed as the completion of Bertrand Russell's early philosophy of "logical atomism." The book comprises a system of short, vadic utterances, numbered to demonstrate their nested interrelations, as:
(as the last line in the book; 7 has no supplemantary propositions).
(Here should follow a more detailed account of the book)
| Table of contents |
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2 The Truth-Functions, and their C-language equivalents 3 Reception and Influence of the Work 4 External links |
Proposition 5.101
From the point of view of a computer or electrical engineer, Proposition 5.101 is a pioneering insight. To demonstrate this, below, we transcribe the 5.101 notation into (C language) and Electrical Engineering boolean logic notation, where "&&" means AND, "||" means OR, "!" means NOT. The notation "=>" can be read as IMPLIES.
We then re-sort the truth-functions into numerical order 0 to 15 decimal, or 0 to f hexadecimal. This yields the following table of truth functions X of 2 binary variables, 'a' and 'c' (with their C programming language equivalent). Note that variable 'c' takes the successive True-False values TTFF, and that variable 'a' takes the successive values TFTF. To use the table, take the values of c and a, 1 column at a time, and read the Truth Function value at the corresponding row and column.
Thus for example, Truth Function e emerges from the truth table row labelled 'e', and e(a=T,c=T) yields T, but e(F,F) yields F.
In Electrical Engineering terms, e(a,c) is a boolean OR logic gate. Note that 1(a,c), the NOR gate, is a valid implementation, along with 7(a,c), NAND, of the Sheffer stroke symbol.
The 16 possible Truth-Functions of 2 binary variables follow:
In other words, Wittgenstein demonstrated that bit-patterns, such as "TFTT" can correspond directly to word concepts, such as "If C then A". Note that C and A are logic predicates, shorthand for sentences like "Socrates is a man", and "Socrates is mortal"
From the perspective of eight decades, it is remarkable that these Truth-Functions arose in the mind of a philosopher, well before Emil Post's machine (1936), before Alan Turing's machine (1936), before Walther Bothe's coincidence circuit (1924), before the Atanasoff-Berry computing circuits (1938), before the Mauchly-Eckert computer (1946), before Claude Shannon's Boolean switching circuits (about 1936), 50 years before the C programming language, 60 years before Programmable Logic Arrays, but a half century after George Boole, and a decade after the Principia Mathematica.
Wittgenstein himself concluded that with the Tractatus he had resolved all philosophical problems; he upon its publication he retired to become a schoolteacher in Austria.
Meanwhile the book was translated into English by C. K. Ogden with help from the Cambridge mathematician and philosopher Frank Ramsey, then still in his teens. Ramsey later visited Wittgenstein in Austria. The Tractatus also caught the attention of the philosophers of the Vienna Circle, especially Rudolf Carnap and Moritz Schlick. The group spent many months working through the text out loud, line-by line. Schlick eventually convinced Wittgenstein to meet with members of the circle to discuss the Tractatus when he returned to Vienna (he was then working as an architect).
Wittgenstein would not meet the circle proper, but only a few of its members, including Schlick, Carnap, and Waissman. Often, though, he refused to discuss philosophy, and would insist on giving the meetings over to reciting poetry with his chair turned to the wall. He largely broke off formal relations even with these members of the circle after coming to believe Carnap had used some of his ideas without permission.
Nonetheless, it was conversations with Schlick during this period that were largely responsible for drawing Wittgenstein back to philosophy. He began to doubt both the ideas and methods of the Tractatus, and in 1929 returned to Cambridge. He worked extensively but published nothing for the next twenty years. Shortly after his death in 1951 his second magnum opus, Philosophical Investigations was edited and published by his executors. Much it is given over to critiquing the ideas of the Tractatus.
The Truth-Functions, and their C-language equivalents
X(a,c) :
c (TTFF)
a (TFTF)
0 (FFFF) (a, c) Contradiction. False. (0)
1 (FFFT) (a, c) In words : Neither a nor c. !(a || c)
2 (FFTF) (a, c) In words : a and not c. (a && !c)
3 (FFTT) (a, c) In words : Not c. (!c)
4 (FTFF) (a, c) In words : c and not a. (c && !a)
5 (FTFT) (a, c) In words : Not a. (!a)
6 (FTTF) (a, c) In words : a or c, but not both. (a != c)
7 (FTTT) (a, c) In words : Not both a and c. (!(a && c))
8 (TFFF) (a, c) In words : c and a. (c && a)
9 (TFFT) (a, c) In words : a is c. (a == c)
a (TFTF) (a, c) In words : a
b (TFTT) (a, c) In words : If c then a. (c => a), (!c || a)
c (TTFF) (a, c) In words : c
d (TTFT) (a, c) In words : If a then c. (a => c), (!a || c)
e (TTTF) (a, c) In words : a or c. (a || c)
f (TTTT) (a, c) Tautology. True. ( !(0) )Reception and Influence of the Work