| Judging from the quote above,
it appears that Leibniz was quite certain that thinking of the
mind as a formal system is a useful way to view reasoning. The
explicit idea of a formal system is pretty much an intellectual
product of this century, but Leibniz uses areas of mathematics
as examples of what he has in mind. These areas certainly qualify
as examples of what we would now refer to as a formal system. |
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The interest in the idea of a formal system arises from
the intuition that there is a kind of "language of thought."
Indeed, a naive assumption is that the language of thought is
determined by the natural language that we have learned as a
child. And, certainly it is hard to escape the intuition that
the language we speak is intimately related to our thought.
But we needn't resolve this issue now, because in this century
the idea of a formal language or system has been well-defined.
In one sense, this is simply an abstraction of the idea of a
natural language. And, as such it provides a clear presentation
of some of the basic properties of a "language." We
could be very careful and exact in defining the class of things
that we call formal systems, but at this point we just want to
get the idea out there so you can use it to help think about
the issues that were, and still are argued about, by people who
study human reasoning.
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A formal system consists first of all of a set of things,
usually we think of this set of things as a set of symbols. A
symbol is something that someone "dreams up"
as opposed to something that nature provides on its own. To capture
this idea, it is often said that symbols are 'arbitrary.'
For example, the letter 'A' is not a phenomenon of nature...someone
decided to adopt this set of conventions to make this form and
treat it as - the letter 'A'. And, a symbol doesn't automatically
refer to anything other than itself...you and I had to learn
that the letter 'A' could be used to refer to the sound-A. Another
property of symbols is that we usually try (or are taught to
try) to make the symbols unambiguous...if you are writing
an 'A' you try to write it in such a way that it won't be confused
with any other symbol that is in the set of symbols you are using.
The numbers used in mathematics, letters used to write down
a natural language, notes used to write down music, are just
some of the familiar examples of differing sets of symbols. Notice
that the letters of our alphabet and the notes used in music
are each finite in number...there are 26 letters in the English
alphabet. But we can use these finite sets to create sets of
things, expressions; and the set of possible expressions
is not really bounded in size....the set is infinite.
In a technical sense, there are an infinite number of sentences
in the English language and we can use the alphabet to express
each of these. A similar claim could be made about the number
of musical expressions.
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In the picture on the left, I have used three different
sets of symbols - integers {3,2,5...}, a 'stick' {|},
and the English alphabet {B,a,n,...}. In each of the gray
boxes I have grouped these symbols in particular ways to serve
as examples for this discussion. First, note that I added some
symbols - for example, + and = as well as a blank space and and
period (.) in the case of the sentences.
These symbols seem to be a bit different and they are. Recall
that we have only a finite number of symbols but we want to be
able to create an infinite set of things that we call expressions
from this finite set. Well, the only way in which to obtain an
infinite set of expressions from a finite set of expressions
is to define ways in which to compose expressions from
the elements of the vocabulary. +, = and space in the case of
sentences are used to represent a composition of elements of
a set. For example,
| |
| 2 |
is a number (expression) |
| 3 |
is a number (expression) |
| 2 + 3 |
is a number (expression) |
| 2 + 3 + 2 |
is a number (expression) |
| 2 + 3 + 2 + 3 |
is a number (expression) |
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and so on.
So how does this help us think about the mind. Well, perhaps
the mind has a finite vocabulary of "basic ideas"....perhaps,
it has a finite set of ways of composing these ideas into well-formed
expressions (complex ideas)...and perhaps it has a set of syntactically
defined rules of inference. Perhaps, then there is a sense in
which we have an infinite set of ideas (how do we fit them into
our brain then?) And, perhaps the mind can imagine syntactic
expressions that are false or describe a completely imaginary
world such as Alice's Wonderland. Could this be possible without
a language of thought?
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But as soon as we allow ourselves to string elements
of our set together, we need to introduce the idea of following
rules for stringing them together. We call these rules, syntactic
rules....they are rules that define the way in which we form
expressions using our basic set of symbols. In the figure above,
the first two sentences in the lower box are syntactically correct.
The last sentence, shown in red is not syntactically correct.
A more general term that is often used to refer to this distinction
is to say that syntactically correct expressions are well-formed
expressions or formulae.
Now we can create an infinite set of expressions from a finite
set of elements. Can there be more? Well, yes. We would like
to able to say something more about these expressions...more
specifically, we would like to be able to say something about
possible relations between elements of these expressions. Note,
that I have exemplified the "commutative law" in the
equations in the upper right. Now, if the commutative law holds;
then if we have the expression '2 + 3 = 5,' then we can infer
or derive the expression '3 + 2 = 5' using the commutative axiom.
This represents a rule of inference and rules of inference
are another component of a formal system. I used a similar type
of rule with the "Bacon and Eggs" phrase to derive
the lower sentence from the first. Note that the rule of inference
says something about how to modify one expression to yield another...and
technically, that is all it says.
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This last point is important ... formal systems are
also often called syntactic systems to contrast them with
systems where the expressions are intended to refer to something
outside the system...to have an associated semantics.
Now, this can get really tricky but the intuitions are familiar.
"Bacon and Eggs have high Cholesterol." is simply a
well-formed expression and nothing more from the syntactic point
of view. But, of course, these words refer to something outside
the syntactic system and in addition to being syntactically correct,
the sentence may be semantically correct....it may make a true
statement about the things that the word 'Bacon' and the word
'Eggs' and the word 'Cholesterol' refer to in the world.
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| So how does this help us think
about the mind. Well, perhaps the mind has a finite vocabulary
of "basic ideas"....perhaps, it has a finite set of
ways of composing these ideas into well-formed expressions (complex
ideas)...and perhaps it has a set of syntactically defined rules
of inference. Perhaps, then there is a sense in which we have
an infinite set of ideas (how do we fit them into our brain then?)
And, perhaps the mind can imagine syntactic expressions that
are false or describe a completely imaginary world such
as Alice's Wonderland. Could this be possible without a language
of thought? |
| This idea that the mind possesses
a "language of thought" in this formal sense is, more
or less, the rationalist position. This stands in contrast
to the empiricist position that relies on the "world outside
our mind" to populate our mind with ideas. |
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