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Formal Systems and a Language of Thought

"All our reasoning is nothing but the joining and substituting of characters, whether these

characters be words or symbols or pictures, ... if we could find characters or signs appropriate for

expressing all out thoughts as definitely and as exactly as arithmetic expresses numbers or

geometric analysis expresses lines, we could in all subjects in so far as they are amenable to

reasoning accomplish what is done in Arithmetic and Geometry." Leibniz (1677)

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.

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 systemhas 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.

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. Asymbol 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 thatwe 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 differentsets 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 = aswell 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 tocompose expressions from the

elements of the vocabulary. +, = and space in the caseof sentences are used to represent a composition of

elements of a set. For example,

2is a number

(expression)

3is a number

(expression)

2 + 3is a number

(expression)

2 + 3 + 2is a number

(expression)

2 + 3 + 2 + 3is a number

(expression)

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 offollowing 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.

This last point is important ... formal systems are also often called syntactic systems to contrast themwith 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.

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.