On the relationship between logic, computation and thought

1. What does it mean to say the mind is a computer?

2. How the study of logic assisted the creation of computation

Philosophers and others concerned with human reasoning long ago identified normatively acceptable patterns of reasoning. For example, consider the syllogism:

(S) 1. Socrates is a man.

2. All men are mortal.

3. Therefore Socrates is mortal.

This argument is valid in that if the premises are true, the conclusion must be true. One way to think of validity is that the truth of the premises is preserved in the conclusion.

Here's another valid argument:

(F) 4. If you are born in Utah, then you can ski.

5. Ed was born in Utah.

6. Therefore, Ed can ski.

This argument, again, is valid. Notice, however, that it contains a false premise. 4 is not true, for some people who are born in Utah cannot ski (consider, to begin with, newborns). So, an argument can be valid even though it has false premises.

Argument F exhibits a rule of reasoning called modus ponens. The rule can be stated in the following general way:

(MP) If P, then Q.

P.

Therefore, Q.

The rule is general in that any replacement of the letters P and Q by propositions (or sentences) produces a valid argument. Such a statement of modus ponens abstracts away from particular subjects (e.g. Socrates or skiing) instead focusing on general properties of arguments. Modern symbolic logic abstracts further using a formal notation.

P &or Q

P

&there4 Q

In symbolic logic, we simply manipulate symbols on a page without worrying what they stand for, because we know that the rules by which we manipulate them preserve truth.

Another, less symbolic example.

(N) 7. Nikta is a glurp.

8. All glurps are bitata.

9. Therefore, Nikta is bitata.

One can appreciate that the conclusion must be true if the premises are, even without knowing what the premises mean.

3. Syntax and semantics:

Symbolic logic thus teaches that it is possible to divorce the syntax of argument from its semantics.

Syntax is the name for the physical shapes we use as placeholders, symbols, and so forth. Syntax is, in language, somewhat arbitrary.

Semantics is the assignment of meanings to symbols.

Consider the syntax/semantics distinction for math.

Consider the syntax/semantics distinction for chess.

4. From logic to computation:

The capacity to make up rules governing syntax that preserves truth relations paved the way for computation. For all we needed to due was design a machine that would follow valid rules for manipulating items (or tokens) of physical syntax, and we can then assign meanings to the tokens as it is useful to do so.

Consider a pocket calculator.

5. From computation to cognition:

Computation thus provided a model for thinking about how a physical object could be structured in such a way that it respected reasons. And that was a breakthrough in solving (at least part) of the mystery of the relation of mind and body.

It does not, of course, show that the mind is a computer. But it shows how it might be that a physical object might produce one of the most puzzling features of cognition.