Glances Ahead: More to Think About
III. Logic and Natural Language
The logical tools found at this website
can easily convince us that the following
syllogism is a valid argument:
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
We might then want to ask what form the argument takes, and see how
we can represent it symbolically. We let the statement variable p stand
for the first premise, the variable q stand for the second premise, and
the variable r stand for the conclusion. None of these statements is a
conjunction, implication, disjunction, or negation, so it seems as though
our translation into statement variables is the simplest translation we
can find. However, the translation tells us that our original valid argument
is one of this form:
3. Therefore, r.
We can clearly see that this argument
is not valid. Then what has gone wrong? The problem is that the original
sentences had a structure to them, and that structure is lost when they
are translated into statement variables. Note that Socrates is the subject
of both the second premise and the conclusion. Also, both the first premise
and the conclusion talk about something being mortal. Finally, both premises
involve men - either making a statement about men in general, or telling
us that someone is a man.
The validity of this argument must
lie in the fact that all three sentences are talking about the same three
things: Socrates, being a man, and being mortal. Translating these three
clearly related sentences as p, q, and r is definitely inadequate for
our purposes. We want to be able to show that this argument is valid.
However, the logical tools we have so far do not tell us how to retain
the structure of the sentences in our translation. For this, we must turn
to predicate logic.
In predicate logic, we can distinguish
two kinds of variables. Predicate variables are generally uppercase letters,
and object variables are generally lowercase. For example, using M to
stand for the predicate ‘being a man’, and s to stand for
Socrates, we would symbolize the sentence “Socrates is a man”
as M(s). So if we were to use a to stand for Aristotle, translating the
sentence “Aristotle is a man” as M(a), we can notice that
there is something that both Socrates and Aristotle have in common. This
fact would not be captured if we had used our previous technique of allowing
a single letter to stand for an entire sentence.
A full translation of the syllogism
into predicate logic would involve some symbolism that hasn’t been
introduced yet - existential and universal quantifiers. The former allows
us to represent existence claims, such as “There is a dog in the
room”, while the latter allows us to express general claims, such
as “All dogs have four legs.” I won’t go into too much
more detail than this here, but I will provide a translation of the original
argument. Let s represent Socrates, M represent the predicate “being
a man”, and L represent the predicate “being mortal”.
We then have the following:
1. All men are mortal. 8x(M(x) ! L(x))
2. Socrates is a man. M(s)
3. Therefore, Socrates is mortal. L(s).
Even predicate logic does not capture
all the structure of our ordinary English sentences, though. Often, even
when it seems obvious how we translate a sentence, something can be lost
in the translation. Here’s an example of a sentence that seems easy
to translate: ”I went outside and opened my umbrella.” We
don’t need predicate logic to translate this sentence, so we’ll
stick to our original method for now. Let p stand for the statement ”I
went outside”, and q stand for the statement ”I opened my
umbrella”. It seems clear, then, that the translation would be p
^ q. We’ll see why it’s not quite so clear-cut.
A simple fact about conjunctions
is that p ^ q is always equivalent to q ^ p. You can verify this using
truth tables if you like. But switching the conjuncts around in our original
sentence gives us this: ”I opened my umbrella and went outside.”
This last sentence doesn’t seem to mean quite the same thing as
the original, though. We translated the “and” in my first
sentence as a simple conjunction, but it actually suggested something
more than that. It seemed to suggest a temporal order to the events. First,
I went outside. Then, I opened my umbrella. However, the second translation
seems to suggest that I did those things in the reverse order. So even
cases that seem simple can be deceptively complicated, as ordinary sentential
logic doesn’t have the means to represent a temporal order to events.
Now, the fact that there are some
things about natural language that logic can’t quite capture doesn’t
mean that logic is somehow inadequate. It’s a valuable tool, but
in using it, we should be aware that it has some limitations. So I’ll
conclude with some things you might want to think about with respect to
this issue. What we just saw is that there is structure to natural language
that logic either does not capture or captures with difficulty. How much
do we want logic to imitate the structure of natural language? We probably
don’t want it to be a new natural language. After all, the rigid
structure and precision of logic is what makes it so useful. Yet one thing
it is used for is to analyze our natural language arguments. Where does
the balance lie?
I. Introduction to Symbolic Logic: Using
II. Philosophical Analysis
IV. The Law of the Excluded Middle