Last
time we talked about Wittgenstein’s early thought, the Tractatus, and
proving axioms with Wittgenstein’s truth table method. Now we will look
at how Wittgenstein’s thought changed in his later notebooks, the most
famous being the Philosophical Investigations. Wittgenstein went from
believing logic is an airtight system in his early period to believing
logic is a messy incomplete set of games in his later period.
First
we will consider how Wittgenstein’s thought changed since he formulated
the truth table method in the Tractatus. Second we will read passages
of the Philosophical Investigations. Third, we will continue to study
truth tables, using them to prove several tautologies that are
fundamental rules of substitution in logic (including Modus Tollens).
THE MIDDLE AND LATER THOUGHT OF WITTGENSTEIN
In
the Tractatus, Wittgenstein put forward a system of tautologies
structured by truth tables, arguing that basic elements like ‘true’,
‘and’ and ‘or’ have a necessary structure but do not mean anything in
themselves while propositions that have meaning are always contingent,
dependent and possible. In this early period, Wittgenstein believed
that ‘and’ always has exactly the same structure but no meaning in
itself, while a proposition like ‘I have an apple and a pear’ has a
meaning but is only true if, and is conditional on, I have both an apple
and a pear. This is why Wittgenstein became upset with positivists,
like his teacher and friend Russell and the Vienna Circle, who tried to
use his system to discover primary and necessary facts in the world.
At
first Wittgenstein thought that he had solved all the problems of
philosophy with the Tractatus truth table system (we do still use it
today to teach formal logic), so he left philosophy and Cambridge
behind, went to war, had many experiences, and then later decided that
his earlier thinking contained horrible problems. He no longer believed
that logic could be crystallized in the head as a truth table matrix,
but rather it existed as a complex out in the world, as arrangements of
people, thoughts, symbols, and objects. He continued to work on
notebooks, progressing in his thought until his death, after which his
notebooks were published.
In
the early thought of his Tractatus tried to lay the ground for a
complete system of logic, such that ‘and’ always has the same, perfectly
defined meaning. In his later thought, Wittgenstein argued against
this view. Just as he argued against positivists that there are no
necessary facts in the world, he came to argue that no individual thing,
not even ‘and’ or ‘true’ in logic, can have a single or necessary
meaning. He came to see that he had attempted to provide singular
meanings for things that merely have common uses. Thus, there is no
more a single necessary meaning of ‘and’ or ‘true’ in the head than
there is a single correct scientific description or picture of the
world. He came to argue that ‘and’ and ‘or’ are not exclusively in the
head, but real things involved in the irreducibly complex games we play
in the world. He called these situations ‘language games’ and ‘forms of
life’.
Consider
how we have seen that the IF>THEN operation in logic is capable of
working in many different ways, and the truth table method chooses a
regular way for it to work. Consider that OR can be inclusive or
exclusive, and the truth table method chooses to use OR inclusively.
Consider that AND can be used like OR as a general inclusion of several
things, but we may agree that the AND is true even if every last thing
is not present. Wittgenstein came to argue that any system of logic,
like his truth tables, is a tool that is a thing in the world like any
other. Because logic is a tool, it does not need to be perfectly
defined or entirely consistent any more than an apple needs to be
perfectly defined or entirely consistent. What is important is that
logic and the world are relatively consistent such that we can use and
describe things.
There
is an excellent passage from Lectures and Conversations that
illustrates the turn nicely. This work was taken not from
Wittgenstein’s notebooks but from the notes of his seminar students in
the years leading up to his work on the notebooks which would be
published after his death as the Philosophical Investigations. At this
time Freud’s ideas had stormed onto the academic scene, infuriating
Wittgenstein who now had come to hate the idea that things in the world,
even logical operators and systems, can be boiled down to a single
essential element or factor like sex, power or truth. It is this
skepticism, which can be called the “problem of essences”, which marks
the turn from his earlier thinking to his more influential later
thought.
Wittgenstein
attacked Freud’s psychoanalysis and dream interpretation for boiling
everything down to sex. In the Lectures and Conversations (20-21),
Wittgenstein proposes a thought experiment for consideration. If we
cook a human being down to carbon ash in an oven, are we left with the
essence of the human being? A human being is a “carbon-based” life
form, so carbon is a dominant element. Consider that we could cook a
human down to water in the same oven, and claim that because humans are
3/5ths water we have the essence of the person.
Would
it be correct to say that humans are essentially ashy, or essentially
wet? Why not? We would not say that a human is essentially carbon or
water, nor would we say ashy or wet, because the human being is a
complex situation that is not reducible to a single element. The
properties of carbon or water do not in themselves explain how humans
behave or what they mean to us. If we cooked people down to ashes or
water, we have destroyed the situation and can no longer investigate how
they work. In the same way, a person is not merely their DNA. While
carbon, water and DNA have very important, even necessary roles to play
in any person, they are not exclusively the essence or meaning of the
complex that is a human individual.
In
the same way, Wittgenstein had come to believe that neither facts in
the world nor logic in the head can be reduced to a single element or
necessary structure. Facts and logic are not true in themselves, but
true in real situations of the world which are irreducibly complex.
Wittgenstein says in the Lectures and Conversations that we have to
avoid the “lure of the secret cellar”, the urge to boil situations down
to a single element like Freud tried to boil human relations, meaning
and the mind down to sex or Wittgenstein himself had tried to boil logic
down to its simple structure.
The
task of philosophy, logic and science is not to fully or completely
explain anything, but to investigate things. Thought never fully
defines things but rather describes and re-describes things. If science
is thinking about the world, then science has endless work to do
describing and re-describing things. Consider whether we fully
understand apples, or whether we ever need to entirely understand them
in order to continue to understand them and use them a great deal.
Likewise, if philosophy and logic are ‘thinking about thinking’, and if
thinking is merely a possible description of things, philosophy and
logic have endless work to do describing our descriptions, describing
and re-describing the ways that we describe things. Much insight can be
gained even if no subject is entirely explained or exhausted.
One
good way to approach this is to describe how cultures of thought,
perspectives, facts and models are gathered together and lived in
institutions. Thought ceases to be completely abstract, but is rather a
culture and situation in the real world that involves people, buildings
and money. The cryptanalysis of algebra worked so well as a modeling
language that we came to believe that the mathematics was not in our
practices and text books but rather sewn into the fabric of the world
itself. As we look over the history of thought in the wake of
Wittgenstein’s later work, it becomes evident that mathematics and logic
are tools and lenses, not the hidden structures of things operating at
secret levels out of our immediate sight. In scholarship today,
particularly the history of religion, philosophy and science, it has
become popular to consider a system of thought as a real lived situation
rather than an abstract set of beliefs and ideas. A religion or
science is in fact a situation of human beings who never have to have
entirely the same set of beliefs as long as they can generally and
relatively cohere as a culture.
THE PHILOSOPHICAL INVESTIGATIONS
In
this monumental work, one of my favorites of modern European thought,
Wittgenstein argues for a middle way between two extreme positions,
between the position of scientific positivism (objective truth is facts
in the world) and the position of psychological skepticism (subjective
truth is meanings in the mind). He presents each position in quotes
again and again, and then argues against both in a three stage process.
First he states a position (either that there are facts given in the
world or meaning is all in the head), then shows situations in which the
position works, then shows situations in which the position does not
work. He shows us that taking either position to extremes would be
understandable given particular ways we think and act, but neither
position explains all the ways in which we think and act. We live in
complex ways that depend both on there being a coherent world and there
being human perspectives. Wittgenstein argues that we can take both
positions rather than determine one to be the actual and the other to be
the illusion.
In
his earlier thought, the world and the head are separated by a Kantian
gulf between objectivity and subjectivity. In his later thinking, the
world and our heads work together seamlessly as a complex situation.
One cannot remove either the head or the world to get the bedrock or
anchor of meaning and truth without resulting in absurdities. The clean
and ideal side of logic, math and grammar mislead us into thinking that
meaning must be anchored entirely on one side, either exclusively in
the head or exclusively in the world, but we gain much more ability to
think and describe our heads and our world if we stop looking for
meaning and truth to be entirely in one place rather than the other. In
the same way, rather than determine which facts are absolute truths to
the exclusion of their opposites, or refuse to identify coherence of
belief as it is all ‘mere theory’, we can better understand the facts
and theories we can share by recognizing that they are one and the same
viewed from opposite sides. If we can continue to determine relative
fact from relative fiction, there is no need to entirely separate the
two. To use the tool analogy, as long as we have decent tools we do not
need perfect or eternal ones. Note that this is very similar to Mill’s
Utilitarianism, and a far cry from Russell’s atomic facts which he had
hoped Wittgenstein would save from inconsistency.
Games
and rules gather people together, but individuals can also variously
interpret rules and meanings. ‘Rules can always be variously
interpreted’ is a central idea of the Philosophical Investigations.
This is not to say that they always are or they should be, but the
window remains open (Wittgenstein writes, “It is like locking a man in a
room but leaving the window open”). Both are only what they are
together as a form of life. Notice that this thinking has much in
common with Laozi’s thinking on the wheel as both empty and solid at the
same time, and much like the Zen koans of a rock being not a rock and a
rock and the sound of one hand clapping.
Consider
the ‘child at the blackboard’ metaphor used in the text. It is always
possible for a child to misunderstand rules and demonstrate this
misunderstanding by making mistakes, even after you show the child and
explain. Let us say that you then write a new ‘inner’ set of rules to
further explain the rules when the child fails. What if the child does
not understand those? Is there a bedrock set of rules that the child
can not possibly misunderstand? If the child can misunderstand rules,
then no set of rules within rules within rules is ever perfectly
airtight. We have an infinite regress unless we can find a set of rules
that can never be misinterpreted.
Just
like Wittgenstein had been seeking the fundamental inner workings of
logic and mathematics, as Frege and Russell had tried to do before him,
if people do math decently there need not be any inner rules aside from
the explanation and demonstration that, if repeated, children can often
follow. Notice we are considering mathematics as a culture, learned by
children and taught by adults, not as an abstract set of necessary or
immutable rules. It turns out that there are no inner workings to
mathematics. Mathematics works as it does openly, on the board in front
of one’s face, without the need of a ‘secret cellar’. If mathematics
on a board is not entirely secure, then no inner set of rules could or
must be either for it to function.
When
we try to explain mathematics, as Frege had tried to do with set
theory, Russell had tried to do with his theory of types, and
Wittgenstein had tried to do with his truth tables, mathematics is not
being simplified, being stripped down to its core, at all. It is the
opposite: we are making it more complex as we try to simplify and
explain it. The rules are not being discovered in the thing, but being
added to the thing. When we explain things, we are not whittling away
the unnecessary but adding our descriptions to it. In the same way,
when we do science to explain apples or human beings, we are adding our
descriptions and the situations of our descriptions, not uncovering the
simple truth of the thing. The simple truth of a thing is the simple
thing itself. The simplest truth of an apple is the single apple. An
explanation of where apples come from is very useful, but it puts the
apple in a complex with many other things (such as trees, farms, stores,
trucks). An apple is not merely its DNA, but is in a infinitely
complex situation with its DNA and ourselves attempting to isolate
components with the tools and technology we bring into the situation.
We
are always essentializing (like when we say, ‘We are always
essentializing’). Meaning, grammar and logic do separate things into
parts. They are useful for doing so. However, Wittgenstein is arguing
something quite revealing: There are no final explanations, only complex
descriptions. Any explanation is a partial human description, which
can then itself be described and explained. The task of philosophy,
science and mathematics is not to reduce things to simple truth, but to
generate more meaning and situations.
Let
us turn to the text. In the preface to the Philosophical
Investigations, Wittgenstein says that he now sees grave mistakes in his
Tractatus. He says that this new book is not to spare thinking, but to
stimulate thinking. Notice that the Tractatus had the opposite goal:
to put an end to the problems of logic and philosophy. Now Wittgenstein
does not believe that anything can or need be fully solved or closed,
but should rather be opened up and made complex.
In
section 7, Wittgenstein says that language games are actions and
language interwoven as forms of life. He considers that words such as
‘this’ and ‘there’ are learned interwoven with gestures such as
pointing. ‘This’, ‘now’, and other simple words get their meaning from
their use by gesturing human beings. They can not be described fully in
language alone, nor do they represent specific objects consistently.
Rather, they are demonstrated to children and translated into other
languages of cultures that share similar gestures. We know from
neuroscience that in the brain the centers of language and control of
the hands are next to each other, likely because language developed in
apes along with gestures. There are also similarities in the basic
gestures of humanity, including extending the arm to indicate the
direction of sight and attention. Infants only months old will read eye
direction to try to see what others are seeing, and if they can not see
what is being looked at they will check the eyes again. The arm,
whether a culture uses the index finger or not, provides an extension of
the line of sight so that it can be easily recognized and followed by
others. Notice that ‘this’ is as simple as it gets in itself, and to
further explain it we need to bring arms and neuroscience into the
picture.
In
section 11, he says language is like toolbox, a complex set of tools
that have no absolute necessity but are useful as a set. Consider that a
hammer, a screwdriver and glue are a decent set of tools. Are they
absolutely right or necessary tools? We could invent others, but they
work decently well for putting things together and taking things apart.
We do not need an absolute screwdriver any more than we need to
completely understand the relationship between the hammer and
screwdriver. We need only use them. In the same way, we do not need to
understand the word ‘and’ such that it is always used exactly the same
way, nor do we need to completely understand its relationship to ‘or’.
When we use them relatively, not exclusively, they are interchangeable.
At a buffet, I could equally say, “You can have eggs or salad or
steak” or “You can have eggs and salad and steak”. The two can
sometimes be used differently and sometimes similarly, as long as we are
decently consistent in all of our interconnected uses. Indeed, the
words, like tools such as hammers and screwdrivers, are more useful
being often but not always used in particular and exclusive ways (ex:
one can use a screwdriver to open paint cans). ‘And’ and ‘or’ are more
useful when we can use them oppositely, but also identically. In his
earlier thinking, with the truth tables, Wittgenstein was trying to
secure them merely opposite and different meanings, which does not give
us their full usage and meaning.
In
section 12, Wittgenstein uses the metaphor of a locomotive cabin to
further illustrate the same point. There are many levers and switches
that function in various ways. Words and language function in various
ways that form a complex with their environment. We could always
redesign the train cabin, but it works well enough as it is.
In
section 15, he uses another simple tool-oriented metaphor to make
language and meaning physical rather than ideal: naming is compared to
attaching a label (or sticker, with “Hello My Name Is”). The
association of a word with a particular thing is like attaching a label
for ease of use and identification.
In
section 17, he says that we can classify words as we do things, but our
classifications will depend on our situation and purposes. He uses the
metaphor of chess pieces. If you want to move a great distance, you
would classify chess pieces one way, but if you want to break through or
jump over the enemy lines (with a knight or a bishop) you would
classify them another way. The pieces, like words and other things, do
not have classifications and meanings in themselves, but in how they can
be used in concrete situations. This is again much like Mill.
In
section 18, he says that a language is like an old city, with side
streets and squares. It is interesting to compare San Francisco to Salt
Lake City here. San Francisco can be a nightmare to drive, while Salt
Lake City is almost entirely a perfect grid surrounding the Mormon
Temple. In San Francisco, you never know which way a street might turn
due to hills or other intersecting streets. European cities are
similarly often much older than the automobile, and do not lend
themselves to outsiders’ easy navigation. A language, like an old city
or old growth forest, is the result of a long process of many layers of
evolution and development. We can understand meanings just as we can
navigate streets, but things are not always clear cut as Salt Lake City.
In
section 57 and 58, he asks about where ‘red’ exists. He suggests that
‘red’ as a color does not exist in itself, but as an association of many
red things having been experienced, the word ‘red’ being said by others
and oneself associated with these things, as well as the
imagination/projection of the color red in the mind. None of these
things needs to be exclusively present for there to be red. We could
even say the word without any red things or thinking of red, and it
still means the color in our language and culture. We could likewise
see a red thing, and it is red without using the word or seeing red in
the imagination. Thus, the color red is not essentially any red thing,
or the word, or the color in the mind, but the complex of all these.
The color red is not simply a subjective concept, nor it is an
objective fact, but it is a conception and association of many things
and words, both in the head and the world together.
In
section 65, Wittgenstein argues that there is no form common to
language games or forms of life, such as the color red or the use of a
tool. The common element is merely form, or association. In the same
way that Hegel argues that Being itself can not be qualified (green,
good, necessary) or quantified (much, fourteen, half), Wittgenstein
argues that there is no common form to meaning. It is a complex
association that can take any possible form. This is similar to the
organic forms of fractal geometry: No two trees are identical, but they
share what Wittgenstein calls a ‘family resemblance’. In the next
section (66), he uses the example of games. There is no rule common to
all games, but they resemble each other as a family.
In
the next section (67), he uses the metaphor of a thread (one could also
use a rope). The thread is strong not because there is one underlying
strand that runs through its entire length, but because many stands
overlap each other. Mathematics and the meaning/use of a word do not
need a single inner rule to make them entirely consistent. Rather they
are complex networks that are continuously reinforced by our
re-inscription as we use them daily. Derrida, the French
deconstructionist, argues that there is no language set in stone. Old
English, like Old French, drift slowly like tectonic plates. When we
use language, we are not using something already set. Rather, we are
resetting it, re-associating it, re-gluing it together with the only
glue it was ever fashioned from, each and every time we speak or write.
Likewise, mathematics such as algebra is not true in itself. We
continue to use it consistently, and this rebinds its consistent use.
When dividing by zero creates problems, we add additional rules and
then continue to lash them together with the system.
In
section 83, Wittgenstein gives another metaphor to describe the
emergence (a chaos-theory, fractal geometry term) of forms of life.
Imagine people in a field, playing with a ball. The ball is tossed
about, another person joins and kicks it back to another, who chases it,
then pegs the kicker with the ball, and then it is tossed about again.
A game has arisen, but it does yet does not seem to have consistent
rules. It is being made up as it goes along. Human beings are rule
making, association generating beings, who know how to follow and bend
rules as they see fit. This means that the only complete consistency is
both consistency and inconsistency, the only consistent rule is there
are yet there are not consistent rules.
In
section 85, he gives another metaphor to back up this conception. A
rule is like a signpost, such as a sign that points to the right and
reads, ‘San Jose’. Does the sign force you to go to San Jose? No. If
you go to the right, will you surely reach San Jose without getting lost
or running out of gas? No. We can even imagine that if you go to the
left, because you are afraid of San Jose, you could get turned around
and end up there anyway. In spite of these loose ends, we find signs
quite useful.
In
section 90, he again speaks of the human instinct to look for a secret
meaning as if it is inside the thing. We open apples to find seeds, and
likewise we take the metaphor too literally and think that to know
something is to grasp an inner essence. Frege, Russell and Wittgenstein
each tried to find the inner essence of mathematics through
developments in logic, but this was the development of new things, not
the secret inner workings. Logic was a new form of mathematics that
made other forms more useful through interconnections. Similarly, a
description or explanation of a thing is not the final level within the
thing but the creation of a network between the thing, ourselves and
other things such that we can get more use out of the thing and its
newly complexified situation.
In
section 99, he comes to the previously mentioned ‘locking a man in a
room but leaving the window open’ metaphor. You can not lock someone in
a room such that they can never get out. There are no simply solid
substances. There are no things without cracks of any kind. Similarly
we can not secure meanings such that they will always stay exactly the
same, but we can secure them by locking them down repeatedly through use
and association. To give an entire account of how a meaning is secured
would be like “repairing a torn spider’s web with one’s fingers” (106).
Just like an old city, we can use things without ever being able to
fully describe them.
Criticizing
his earlier approach (107) he says that a pure crystalline logic would
be meaningless. It would be like trying to walk on slippery ice rather
than rough ground. If we want to use and describe things, we can not
use absolute ideals but rather messy associative concepts. This is very
much like Alice and the glass table, which we will look at in a few
weeks. We can not grasp things with timeless and space-less universals
(the sort Aristotle sought above the lunar sphere). Logic and thought
should be used in messy situations for un-guaranteed purposes.
Wittgenstein
writes, “We must do away with all explanation, and description alone
must take its place” (109)...”What we do is to bring words back from
their metaphysical to their everyday use” (116)...”What we are
destroying is nothing but houses of cards and we are clearing up the
ground of language on which they stand” (118).
Do
we base our own behavior, let alone the culture of mathematics, on
simple rules? He gives the example of being certain that a table will
resist one’s finger, that a fire will hurt one’s hand. It is not that
there is a simple rule set in an atomic language that reads, ‘Table is
solid’ or ‘Fire hurts hands’. Rather, “a hundred reasons present
themselves, each drowning out the voice of the others” (478). Like the
thread woven of many strands, our belief and certainty that objects are
solid and flame hurts us are woven out of many experiences that are then
woven together with the table and flame before us.
Wittgenstein
writes, “To say, ‘This combination of words makes no sense’ excludes it
from the sphere of language and thereby bounds the domain of language.
But when one draws a boundary it may be for various kinds of reason.
If I surround an area with a fence or a line or otherwise, the purpose
may be to prevent someone from getting in or out, but it may also be
part of a game and the players be supposed, say, to jump over the
boundary” (499). Wittgenstein seems to be thinking specifically of
humor and comedy, though it could also apply to modern and conceptual
art, forms of culture that break rules on purpose. Comedy and modern
art are games where the rule is to break the rules without breaking
anyone’s neck. Wittgenstein had an appreciation both for the Alice
books of Lewis Carroll as well as American slap-stick comedy which he
preferred to opera. At the end of the course, we will study the Alice
books, humor and modern art in light of the later work of Wittgenstein.
The
last image to examine, near the end of the book, is Jastrow’s
duck-rabbit, often called Wittgenstein’s duck-rabbit because the
psychologist Jastrow is far less famous. One can look at the figure
from the left, and it is a duck, and one can look at the figure from the
right, and it is a rabbit. Which is the single correct face?
Wittgenstein says we may have seen only the rabbit face our entire
life, and that does not prevent us or others from seeing the duck when
it is pointed out. This is much like the duel between objectivism
(truth is in the world) and subjectivism (meaning is in the head).
* * * * * * * * * * * * *
Now
let us turn to proving several of the fundamental substitutions of
logic with the truth table method. This is the subject of your second
assignment. Remember: to prove a tautology, you should get four T’s as
your final answer, and to prove something is NOT a tautology, you should
get something OTHER than four T’s as your final answer. Each of the
expressions below are tautologies, so you should get four T's as the
final answer for each using the truth table method.
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)
