Do
note that throughout these chapters there will be small red flags -- -- by some sentences. These are intended to alert you to
ideas that should be given special attention because of typical
difficulties students have had in the past, and
more is noted about them at the end of the chapters.
1. GETTING STARTED
Socrates' prison in Athens
Welcome to the course. In the weeks to come we are going to learn
something about what is called formal logic. Specifically, we are
going to do some work with what is called symbolic logic.
So what does this mean? To see where all this begins we need to
go back to the days of Plato and Aristotle in ancient Greece.
Their city of Athens had developed a political system that saw power in
the hands of those who were citizens (well, those who were males and
not slaves or born elsewhere) rather than in the hands of one man or a
small group of men. They called this "democracy" (literally,
people power), and it worked through having people vote. While we think
of voting mostly in terms of elective office, the Greeks also saw
voting as the way to resolve disputes. Have a complaint against a
neighbor? Set up a trial and assemble a jury and in one sitting
present your case and let the listeners decide who should win. Of
course, it is understood you are entirely on your own here (no lawyers,
and no judges as we would use them to decide what evidence is
acceptable).
A good idea? It did have its problems, and it is because this
method could also be used to decide whether someone had acted against
the law that Western philosophy got its start. Plato's teacher
was a man named Socrates who was charged with what today we might call
subversive activities, and in a one-day trial he made an unsuccessful
case for his innocence, was convicted, and then sentenced to
death. Plato
left Athens
after Socrates was executed but returned
years later to found what can be thought of as the world's first
university. He presented his own ideas through a dramatic form in
which Socrates would carry on debates with many of the important
figures of his time, and in these Dialogues we have a lasting model for
philosophy. The philosopher Alfred North Whitehead (one of the
key figures in the development of symbolic logic as we use it today)
once commented that all philosophy is a footnote to Plato, and in other
courses you take at Pierce you will be able to see how true this is.
Plato was not a friend of Greek democracy. A key problem, as he
saw it, was that the harsh reality of having to convince an audience of
everyday people that you are right and your opponent is wrong put too
much emphasis on using the tricks of the trade for persuasion.
Citizens who wanted to get ahead began engaging experts to train them
or their young heirs in how to make a good-sounding case. Okay,
we still do the same thing today for getting ahead politically in our
own democratic system, but you have to imagine the desperation
individuals might feel when these same techniques were required to
survive in a courtroom. Often enough there was a strong sense of
discomfort with these experts, collectively called Sophists (from the
Greek word for someone who was wise), in part because they were
typically foreigners who were seen as weakening the traditional values
of Athenian society by encouraging a certain relativism. What
came to matter was not whether someone really was right but whether he
could present a case that would sound better than it actually
was. Plato's Dialogues can be seen as a prolonged attack on the
Sophists, and the irony is that Socrates died because in effect he was
portrayed as just another Sophist himself (one of the most famous plays
of the period is a burlesque by Aristophanes in which Socrates was
represented as a money-hungry guru encouraging his young students to
disrespect their parents).
Quickly enough the task Plato and then his own student Aristotle set
themselves is to determine when a case might meet certain standards for
being convincing. Let's look at a couple of examples.
Suppose we say that some students are ambitious and anyone who is
ambitious will be successful, so it should follow that some students
will be successful. Seems reasonable, right? But suppose we
make a minor switch here and say that all students are ambitious and
some individuals who are ambitious will be successful. Do you
think it follows that some students will be successul? Careful,
don't think about whether you already know that some students are
successful. Just look at the pattern here and ask yourself
whether you can imagine a scenario in which only non-students manage to
be successful. (If it helps, draw some circles to show what
happens. In our first example we have intersecting circles for
students and ambitious individuals, but you have to draw the circle for
those who are successul so that it includes everyone who is ambitious
and so it will necessarily include some students. In our second
example we can draw the circles differently so that while the circles
for those who are ambitious and those who are successful do interesect,
we can still have the circle for students inside the circle for those
who are ambitious while not being being part of the
intersection.) If you can, you should be able to see that the
second example does not work the way the first does.
Aristotle in particular attempted to organize how different statements
of the kind we have been using could be put together (we call these
syllogisms, from the Greek for saying things together). Some
patterns are perfect, others are not. When
the pattern is perfect
(meaning, that the conclusion or what we're trying to prove could not
be wrong if the statements leading up to it--the premises--are correct)
we call the argument deductively
valid. That does not
mean that
the conclusion really is true, since that depends on whether the
premises in fact are true. For instance,
this is a
pattern-perfect (deductively valid) case: everything black is
sweet and salt is black, so salt is sweet. Of course, salt really
is not sweet but then both the statements we start with are not really
true. And here is a case that is not pattern-perfect, even though
you know that everything in it is actually true as a general statement:
some teenagers are poor drivers and some poor drivers get into
accidents, so some teenagers get into accidents. It is not
pattern-perfect (imagine a very small town in which a couple of
teenagers are poor drivers but the only poor drivers who do get into an
accident are a couple of old guys), and it is an example of a
misleading pattern--what we call a formal fallacy (a fallacy or mistake
in the form or pattern).
Now what we have just been looking at is only part of an overall study
of logic. Aristotle also began the process of organizing
instances of poor reasoning in which it is not the pattern that is the
problem but the way in which evidence is presented. In Philosophy
6 you would look more closely at what we call informal logic, and there
you would find examples of leading informal fallacies. For
instance, saying that you should buy something I'm selling because you
cannot show me why you shouldn't is an example of what is called the
argument to ignorance. It may be a leading salesman's trick (keep you
on the defensive until I wear you down), but it reverses what should be
the role of salesman and customer. This is the principle behind
the courtroom rule that in a criminal case an individual must be
presumed innocent until proven guilty (imagine I accuse a student of
cheating but instead of presenting evidence that this is so I make it
the student's responsibility to prove she didn't).
In Philosophy 9 we definitely limit ourselves. First off, we are
interested only in the statements that can be thought of as definitely
true or false, even if we may not know what the truth really is.
For instance, we can work with statements such as "all students are
ambitious" but not with statements such as "it might rain
tomorrow." If we say that all students are ambitious then right
away we are ruling out the possibility of even one students who is
not. However, when we say that it might rain tomorrow we cannot
rule out saying that it might not. Next, we actually work in a
somewhat unreal manner by assuming that in the context we have in mind
(call it the universe of discourse) our categorical statements (ones in
which everything is black or white) can be really true or false.
We are interested in how we do combine these categorical statements
even though in real life we know that we are constantly working with
probabilities rather than certainties.
We are interested, then, in what cases are pattern-perfect. We do
this primarily by using letters to stand for statements or parts of
statements. For instance, we'll just say "F" for the complete
statement "logic is fun" or "Sa" for "Alice is a student" and "(x)Ax"
for "everyone is ambitious." We will want to see how we can test
patterns for deductive validity (we have a couple of ways to do this)
and we will want to be able to show how, given certain statements as
premises, we could work our way step by step to an intended
conclusion. We are interested, then, in symbolization, in testing
for deductive validity, and in what we call proofs or
derivations. You will see more of what we mean as we begin moving
through the course.
By now you might have noticed that symbolic logic does seem to have a
mathematical look about it, especially when we think about
symbolization (think about the delights of algebra) and proofs
(remember the fun you had in geometry class). Well, without going
too much more into the history of things let's just note that when a
century ago Whitehead partnered with Bertrand Russell to present
symbolic logic as we know it today a key goal was to see how to reduce
mathematics itself to logic. The idea was that in principle you
should be able to decide in advance whether particular propositions in
number theory could ever be proven. That idea proved to be a
mistake, but with the advent of the computer and complicated electronic
circuits it came to be recognized that symbolic logic can be used not
just for working with language but with electric currents (think how
everything has to be on or off so that whatever we were doing for true
or false applies here as well). This is why symbolic logic is
typically a requirement for computer science majors as well as for
philosophy majors.
TERMS WE USE
Arguments are
combinations of statements in which one or more are used as the basis
for saying another statement is true. We say the statements used
this way are the premises
while the statement that is supposed to be proved by them is the conclusion. In another
course (Philosophy 6, for instance) we might examine longer chains of
reasoning in which the same statement can be the conclusion from one
set of premises and itself be a premise for still another conclusion,
but in this course we limit ourselves to more simple patterns.
Also, in an analysis of everyday arguments we need to pick out what
statements actually make up part of an argument and which statements in
the same sentence or paragraph are not really either premises or
conclusions. In the same way, in another course we would be
interested in what we call implied premises or conclusions--things not
said but clearly understood. In symbolic logic, however, we work
only with what is expressed.
All arguments are said to be either deductively valid
(pattern-perfect) or not. An argument that is not deductively
valid can still be judged in terms of the reasonableness (the
probability) of the conclusion being true given that the premises are
true. When the probability is high enough we say the argument is inductively strong and
otherwise it is inductively weak.
Keep in mind that we work this through without going into whether the
premises are actually true. When they are true in a deductively
valid argument then we say it is sound.
When they are true in an inductively strong argument then we say it is cogent. Using these
terms carefully does lead to some unexpected points: an argument that
is deductively valid is not strong, while an argument that is
inductively strong is not valid.
This should serve as a
reminder
that "valid" is a technical term in logic that applies only to
arguments while "true" applies only to the individual statements.
We do call conclusions valid or not but we do this only as a quick way
of saying not that they are true or false in themselves but that they
would have to be true provided all the premises are true. (Still
another unexpected point is that when we have inconsistent
statements--statements which could not both be true at the same
time--as premises the argument is considered valid by default. In
other words, we would not meet the condition for saying an argument is
not valid: all true premises but a false conclusion. This will be
a key to some of the techniques we work with later as we move through
the course.)
EXERCISES
Look at each of the following,
then decide whether you have an argument
at all. If not, explain why. If you do have an argument,
decide whether it is pattern-perfect (deductively valid).
1. Logic is interesting
because it is fun. 2. Logic must not be easy,
because it is not fun. 3. If logic is easy then
it is interesting. 4. If logic is easy then
it is fun. Logic is not easy, so
it must not be fun. 5. If logic is easy then
it is fun. Logic is not fun, so it
must not be easy. 6. All students are
ambitious, but some students are lazy. 7. Only someone ambitious
will succeed, so some students will
succeed. 8. No one who is lazy is
successful, but no honor students are
lazy, so all honor students will be successful. 9. Today is Thursday,
therefore today is Thursday. 10. Alice is lazy but
Alice is not lazy, so Alice will be a
success.
Write out your answers then go
to the
answer key to see how you did. Go back and review as
necessary.
Did
you see
the red flags above? They were to warn you about a very common mistake
made by those just beginning logic. This is the tendency to call
an argument valid or not depending on whether someone agrees with the
statements making it up. An argument in which everything is false
(saying for instance, that since anything black is sweet and salt is
black it would follow that salt is sweet) can still be valid.
This happens when there is something about the pattern that means it
would be impossible to deny the conclusion if in fact you accept the
premises. One of the main tasks ahead of you is to learn how we
can tell whether an argument does show this kind of pattern.
-- by some sentences.
