Logic is a game (uh huh,
sure...)
Well, think about games
for
a minute. There is not really one common characteristic that
defines
a game as such (a point emphasized by the Austrian philosopher Ludwig
Wittgenstein).
Rather, there are "family resemblances" so that we might imagine
overlapping
groups of things we definitely consider to be games. Things like
football and chess and solitaire and paintball.
So let's narrow what we
mean
to games in which a score is important. There are going to be
rules
for scoring and winning, and we expect something about these rules.
-
They should be consistent,
so that it cannot happen that by looking at one section of a rulebook
we
find that Jack wins but by looking at another section we find that Jill
wins.
-
They should be complete,
so that we do not come up with a situation in which we cannot find a
rule
that tells us what to do.
-
They should allow the
score
to be decidable, meaning that we should
not
have a situation in which we couldn't tell whether Jack wins or Jill
wins.
Arithmetic (or number
theory,
to be more fancy) is an example of a game in which we have two out of
three
of these characteristics. We have consistency and we have
decidability,
but, as someone named Gödel showed us, number theory is not and
cannot
be complete. By this we mean that there are more things true than
we could ever prove to be true.
Symbolic logic is an
example
of a game in which, up to a point, we do have all three characteristics
(we can come up with some relationships about which we could say we
cannot
decide whether they would be true or not).
Now the reason I ask
you to
be thinking about symbolic logic as a game is this:
We are working with
techniques
for mapping one system on to
another.
This means we have to have ways of translating one kind of organization
into another. This sounds exotic, but whenever we read something
out loud we are mapping a system of visual symbols on to a system of
sounds.
Whenever we do this by reading into a tape recorder, we have a machine
mapping the sounds on to a system of electrical impulses that in turn
maps
these sounds on a piece of magnetic tape.
Here is an example of
the
mapping we work with. We have the sentence "Some students read
every
book" that can be symbolized as (Ex)[Sx &
(y)(By
-> Rxy)]. What we have
done, if
you think through this with me without running from the screen in
terror,
is rethink the ideas so that the sentence has become "there is an
unknown
someone such that this someone is a student and for every thing else,
if
that thing is a book then that someone is reading that
something."
We will learn how to do this close to the end of the course, so do not
panic. Instead just think how clever you will be if you stick
with
this. And think how much fun it is going to be to amaze people
with
your skill. Right...
Well,
we do not stop here. We might go on to think about what would
happen
if we also said that anyone who reads every book is a genius.
Would
it also have to be true that we have at least one student who is a
genius?
(The answer, by the way, is "yes.") Now, can we also "prove" that
this has to be so? Here is where we work with techniques for
showing
that when we know certain things already, other things follow.
Symbolic
logic, then, has three components to it:
-
symbolization
(as in the example above)
-
testing
for validity (showing whether or not something does have to be
"true")
-
derivations
(showing why this has to happen)
We
are
going to go through all this step by step. Your task is to learn
the game, but like any game requiring a measure of skill (poker and
basketball
as well as chess), practice is very important. And it may even be
fun as well.
To
get
a better idea of the kind of game we are playing,
I am
going
to ask you to work with a pretty simple game that in several ways is
like
what we do in symbolic logic. We are going to use four letters
from
the alphabet:
-
a
(which
comes early and is a vowel)
-
b
(which
comes early but is not a vowel)
-
d
(which
comes late and is not a vowel)
-
e
(which
comes late and is a vowel)
In
this
game we have what we call well-formed formulas (wff's) that must
obey these rules:
-
a
late
letter may not be followed by a different late letter (I could
have
dd or ee
but not
de or
ed)
-
a
wff
must have at least one letter but cannot have more than six letters
In
order
for something to happen, we are going to start off with two wff's that
are given as axioms: abd
and ead
Here
are the rules for changing one wff into another, but it is understood
that
they cannot be applied in any way that would violate the above
conditions
for a wff :
-
Rule
1:
Any two letters may be reversed (abd
could
become bad)
-
Rule
2:
An initial letter may be removed (abd
could become bd)
-
Rule
3:
Any wff can be attached to the end of any other wff (abd
could become
abdbad)
Now
this is the game: we want to see what wffs can be derived from the two
axioms above. We need to show every step of the derivation.
Here is an example. Given abd
we want to derive bead.
1.
abd axiom
2.
adb 1 rule 1
3.
ead axiom
4.
adbead 2,3 rule 3
5.
dbead 4 rule 2
6.
bead 5 rule 2
How
many other English words are wff's that could also be derived with
these
axioms and these rules?
Are
there any English words that are wff's that could not be derived from
these
axioms and these rules?
What
I am asking you, if you think about it, is whether the game I've
proposed
can be described as having the property of completeness.