Tuesday, April 13, 2010

Logic Lecture April 14: Wittgenstein's Tractatus and Truth Tables

BCC Logic
Eric Gerlach


Wittgenstein (1889-1951) is one of the most important thinkers in academics today. His early book, the Tractatus, and his later book, the Philosophical Investigations, are considered two of the most important influences for the American and British Analytic school of philosophy, the dominant school of philosophy in America.

In an end of the century poll in 2000, philosophy professors from America and Canada were asked to list the five most important books that influenced their own work. When all of the results were tallied up, the Philosophical Investigations was #1, and the Tractatus was #4. The Philosophical Investigations was cited far more frequently than any other book, was listed first on far more ballots, and crossed over more into many different disciplines and areas of study.

Wittgenstein’s thought can be divided into his early, middle and later work. His early work is the book the Tractatus, the book which gave the world truth table logic. This tool, as Wittgenstein later came to see it, remains the mathematical system taught as logic today. Just as Wittgenstein became famous for his truth tables, he switched positions in his thinking and came to reject his earlier work. He wrote in notebooks that were only published after his death, and the Philosophical Investigations is the most celebrated of these.

First, we will look at the life of Wittgenstein. Second, we will consider some of the important ideas in the Tractatus and Wittgenstein’s early thought. Third, we will begin learning the truth table method of logic Wittgenstein introduced in the Tractatus.

The Life and Thought of Wittgenstein:

Wittgenstein’s Father was the Austrian Carnegie, making a fortune in Steel. Though his father was Protestant, and his mother Jewish, Ludwig was baptized Catholic because of antisemitism at the time. In his early years, Ludwig was a proud atheist but by the time he was working on his Tractatus he had a mystical transcendental outlook which he kept for the rest of his life. Though never religious, and though he had to bribe Nazis later to smuggle his “Jewish” family from Austria, he was buried as a Catholic.

The Wittgenstein family was known for intense criticism, musical talent, depression, and suicide. Three of Wittgenstein’s four brothers committed suicide, and he himself considered suicide for awhile before launching into his late period. Unfortunately, suicide was considered romantic for Austrian elites at the time.

Wittgenstein was in Hitler’s elementary school, 2 days younger, but because he was put forward a grade and Hitler was held back a grade he was 2 years ahead. Both he and Hitler hated the school and the lessons.

He began studying at university in Berlin to become an engineer with an interest in flight (the Wright Brothers had recently invented the motorized glider, but flew it in France and Germany until 1907 as the US Army did not believe them). After failing in his attempt to build a better propeller, he began studying mathematical theory and philosophy of mathematics, becoming entranced with two thinkers who are along with Wittgenstein foundational for Analytical philosophy and logic: Russell from Britain, and Frege from Germany. Wittgenstein went to see Frege, who did not fully understand his questions and advised him to go see Russell, which he did in 1911.

He showed up unannounced to Russell’s room at Trinity College, impressed him with his intense and brilliant arguments. Russell became convinced that the young Austrian was going to carry his work forward and be his successor, solving the remaining problems of logic that Russell’s work on the foundations of mathematics had left open. As mentioned last lecture, Russell had shown there were contradictions unresolved in Frege’s work with set theory, but Russell had become frustrated trying to solve these contradiction with his theory of types.

Wittgenstein, an eccentric and difficult personality, was never fully comfortable at Cambridge, and often got into disagreements with Russell and threatened to leave many times before fleeing to Norway where he believed he could finish his work on Logic. While some still disagree, it is generally accepted that Wittgenstein was gay, developed a relationship with Pinsent, a young graduate student, and some believe that Russell encouraged the relationship if he did not introduce the two with the purpose of keeping the emotional and unstable genius with him at Cambridge.

When WWI broke out, he served for Austria, at the same time as he was developing the material for the Tractatus. Learning of Pinsent’s death in the war in Italy, he became suicidal, moved in with his uncle and finished the Tractatus which he dedicated to his ‘friend’ Pinsent. He tried to get it published, but no one would take it. Remember: this book went on to be the #4 influence in the US and Canada according to the poll, the book that gave modern logic truth tables, the method that replaced Aristotle’s syllogisms.

Russell intervened back in Cambridge, and had it published and wrote and introduction for it. This was the start of the end. Though Russell saw the work as genius, he did not completely understand much of it and his introduction reflected this. Wittgenstein read the introduction and realized Russell had great misunderstandings of his work. Believing that his Tractatus had solved all the problems of philosophy, Wittgenstein left Russell and Cambridge again and went to be a school teacher in Austria. He gave away his portion of the family fortune, anonymously to writers but also to his family. Since his family was already wealthy, he wrote in a letter, “they won’t be corrupted by it”. He left the school after a short while (not a good fit, and parents thought he was crazy). He became a gardener’s assistant, and then his sister had him design her a house.

While finishing the house, he was contacted by members of the Vienna Circle, positivists using Hegel’s logic and Wittgenstein’s Tractatus to give a solid foundation for science and mathematics. This was what Russell had hoped for, minus the Hegel who Russell hated. While Wittgenstein had been away, the Tractatus had become famous, and central to many already inspired by Frege and Russell. Many came to visit and discus and progressively Witt became disgusted. He began to realize that there were fundamental problems with his Tractatus and truth tables, and got into intense arguments with the Vienna Circle members, at one point turning his back on his guests and reading Tagore, an Indian transcendental poet out loud. For the rest of his life, Wittgenstein thought logical positivism (the analytic school of philosophy) misunderstood his Tractatus.

In his early period, Wittgenstein believed he had fully solved the problems of a complete system of logic. He saw it like Schopenhauer, a big early influence: logic is a perfect crystal tool of analysis, life is a messy chaotic ocean, and so logic is perfect but unfortunately never fits perfectly with life. This is like having the perfect tool for an impossible and continuous job. In conversations with positivists he started to change his thinking around and continued to write until he died. These writings were published after his death as the Philosophical Investigations and other books. In his later thought, Wittgenstein saw logic not as a perfect crystal castle in the sky but as rules and games that are imperfectly lived in the real world imperfectly and without complete definition. He no longer believed that logic could provide a foundation for mathematics, science or philosophy. He denied that contradictions are necessarily false, or disprove a mathematical-logical system.

In 1929, he decided to return to Cambridge to correct his thinking and teach. To his horror, when he arrived at the train station he was greeted by a vast crowd of intellectuals as the new hero, the author of the Tractatus, the work he now thought was exactly wrong.
The famous economist Keynes wrote to his wife: ‘Well, God has arrived. I met him on the 5:15 train’. Wittgenstein continued to lecture at Cambridge, developing his ideas.

In 1934, he defected to Soviet Russia, wanting to be a plumber or work with his hands. When he was told that according to the Soviet system he would be put to work as a philosophy professor in Moscow, he defected back to Britain.

In 1937, Hitler annexed Austria. Wittgenstein had to bribe Nazis to get his Jewish family passage out and spend the equivalent today of $50 million in gold and foreign currency. Since he had given away his own portion of the family fortune, he had to get much of this from his collegues at Cambridge and other admirers of his work.


In his early thought, expressed in the pages of the Tractatus, reality consists of atomic facts, states of affairs that are true. Thought, expressed grammatically in language, ‘pictures’ the world with these atomic facts. The world does not perfectly fit this atomic language, but because it is the way the head makes sense of the world we cannot understand things otherwise. Wittgenstein said that it is the part of the book that is unwritten that is important, the part where life itself goes beyond this logic and makes the world what it is. Of the world beyond logic, he wrote “Of what one cannot speak, one must remain silent”, which is in fact a quote from Confucius. Our logic and the world are two things that do not fit, yet mysteriously (and mystically) the two are one.

If we boil logic with truth tables down to its tautologies, the necessary and basic workings, and leave the rest open as the world which always is beyond our thoughts, we can have the perfect system of logic and grammar that we use to understand things spelled out even if it cannot perfectly predict the world or tell us how the world works. Think of logic as a set of reading glasses, and the world as something one looks at through the glasses. Wittgenstein believed that with the Tractatus he had spelled out the perfect crystal form of the glasses, and beyond this nothing can be said for certain.

Logic consists of things that are always necessary or impossible. For instance, if one knows A is necessarily true, then one also knows that it is impossible that A is false. Notice that in his early thought Wittgenstein follows Russell and Frege in believing the principles of non-contradiction and the excluded middle, and from these alone we can build the perfect interlocking system of necessary truths that is logic.

When it comes to facts in the world, however, everything is contingent on something else and is neither simply necessary or simply impossible. Logic is the necessary and impossible book-ends with which we interpret the world and its facts, but the world is always between the necessary and the impossible, is always somewhat necessary and somewhat impossible, which creates a gulf between our pure and necessary logic and the unpredictable world. This is a very similar position to Avicenna, and quite different from Aristotle, Averroes and Russell who want some facts in the world to be necessary truths themselves. For Wittgenstein, only logic and math can be sets of necessary truths, and this is because (as Avicenna and Mill believe) they are concepts and are ideal, unlike situations of real things in the world. Once we nail down the perfect tool of logic, we can use it to examine the world and all of its messy situations. Our examinations will never be perfect because of the gap between logic and the world, but at least the logic will be necessary and perfect.


The genius of Wittgenstein’s truth table method is it turns mathematics on its head. Normally, mathematics starts with something given as true, and then deduces additional necessary truths. This is the deduction that Aristotle and Russell believed was the true and sure method of logic, science and mathematics. Wittgenstein, however, did not believe that there are ever necessary truths in the world, because necessity is only in the head as thought and logic.

The truth tables do not assume that the beginning proposition (in the atomic mathematical form) is true. Rather, truth tables evaluate the proposition by giving us all possible states of the statement being true and false, and then determine if anything is additionally true given these possibilities. It is assumed that any element of the proposition could be true or false, and all of the possible combinations are given and then one looks for anything being always true given each and every possibility. Truth tables do not just show us what follows if the proposition is true (how mathematics typically works), but also shows us what follows if the proposition is false, or if some of it is true and the rest false.

For Truth Tables, we assume both the principle of non-contradiction and the principle of the excluded middle. This means that there are two truth values used, True (T) and False (F). Typically, p q and r are the three variables used in symbolic logic (possibly because ‘p’ is the first letter of ‘proposition’). These variables stand for atomic facts, so ‘p’ could mean, “The dog is green” or ‘q’ could mean, “Steve is a jerk”. So, if “The dog is green”, is either true or false this means that p has the value T or F.

There are five operations or operators used in symbolic logic:
Each of these has a basic truth table we construct that shows how these operations work. These basic truth tables are then the rules or “moves” which we use to solve more complicated truth tables.

In this class, we will only be working with two variable propositions. I will provide you with an additional print out and post on the truth tables and how they operate, as well as demonstrate these in class. The truth values for each operator are:

NOT (FT) ~

Remember that OR is always used inclusively, so if p and q are both true, the whole statement is true. Remember also to watch out for IF-THEN, because it is the hardest one and trips people up a lot. In truth table logic, if a statement cannot be proved false then it is considered true, which is why the last two values are T. This could work the other way (if it is not proved true, it is considered false) but this is not the way it is done.

Once you learn how to construct the basic tables, then you can construct more complicated tables. If you construct a truth table for the following propositions, you should get the values listed to the right:

If (p and q) then (p or q) TTTT
If (p or q) then (p and q) TTFF

The reason’s Wittgenstein’s truth tables were such a success is that they proved, for the first time, that many of the axioms logicians had discovered were necessarily true (tautologies) in a way that is simple to do and easy to see. Remember Modus Tollens from Kanada? Wittgenstein’s truth tables can demonstrate why it is always true!

Once you learn how to construct the more complicated truth tables, you should be able to prove each of the tautologies below by getting TTTT (true in all four cases) underneath the equals sign.

Transposition (Modus Tollens): (p > q) = (~ q > ~ p)

Implication: (p > q) = (~ p v q)

Disjunction: (p v q) = (~ p > q)

De Morgan’s A: ~ (p ^ q) = (~ p v ~ q)

De Morgan’s B: ~ (p v q) = (~ p ^ ~ q)