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.
THE TRACTATUS
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.
WITTGENSTEIN’S TRUTH TABLE METHOD
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:
NOT, AND, OR, IF-THEN and EQUIVALENT (or TAUTOLOGY)
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) ~
AND (TFFF) ^
OR (TTTF) v
IF-THEN (TFTT) >
TAUTOLOGY (TFFT) =
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)
