The History of Mathematics: A Brief Course

1. LOGIC 537

When we wish to give the idea of symbolical algebra... we ask,

firstly, what symbols shall be used (without any reference to mean-

ing); next, what shall be the laws under which such symbols are

to be operated upon; the deduction of all subsequent consequences

is again an application of common logic. Lastly, we explain the

meanings which must be attached to the symbols, in order that

they may have prototypes of which the assigned laws of operation

are true. [Quoted by Richards, 1987, pp. 15-16]

This set of procedures is still the way in which mathematical logic operates,
although the laws under which the symbols are to be operated on are now more ab-
stract than de Morgan probably had in mind. To build a formal language, you first
specify which sequences of symbols are to be considered "well-formed formulas,"
that is, formulas capable of being true or false. The criterion for being well-formed
must be purely formal, capable of being decided by a machine. Next, the sequences
of well-formed formulas that are to be considered deductions are specified, again
purely formally. The syntax of the language is specified by these two sets of rules,
and the final piece of the construction, as de Morgan notes, is to specify its se-
mantics, that is, the interpretation of its symbols and formulas. Here again, the
modern world takes a more formal and abstract view of "interpretation" than de
Morgan probably intended. For example, the semantics of propositional calculus
consists of truth tables. After specifying the semantics, one can ask such questions
as whether the language is consistent (incapable of proving a false proposition),
complete (capable of proving all true propositions), or categorical (allowing only
one interpretation, up to isomorphism).

In his 1847 treatise Formal Logic, de Morgan went further, arguing that "we
have power to invent new meanings for all the forms of inference, in every way in
which we have power to make new meanings of is and is not... ." This focus on
the meaning of is was very much to the point. One of the disputes that Peirce
overlooked in the quotation just given is the question of what principles allow us
to infer that an object "exists" in mathematics. We have seen this question in
the eighteenth-century disagreement over what principles are allowed to define a
function. In the case of symbolic algebra, where the symbols originally represented
numbers, the existence question was still not settled to everyone's liking in the early
nineteenth century. That is why Gauss stated the fundamental theorem of algebra
in terms of real factorizations alone. Here de Morgan was declaring the right to
create mathematical entities by fiat, subject to certain restrictions. That enigmatic
"exists" is indispensible in first-order logic, where the negation of "For every ÷, P"
is "For some x, not-P." But what can "some" mean unless there actually exist
objects x? This defect was to be remedied by de Morgan's friend George Boole.

In de Morgan's formal logic, this "exists" remains hidden: When he talks about
a class X, it necessarily has members. Without this assumption, even the very first
example he gives is not a valid inference. He gives the following table by way of
introduction to the symbolic logic that he is about to introduce:

Instead of:

All men will die

All men are rational beings

Therefore some rational beings will die

Write:

Every Y is X

Every Y is Æ

Therefore some Zs are X's.