Among the great pioneers of logic, Gottlob Frege (1848-1925) should
certainly be mentioned. His magnum opus, **Begriffschrift** (in English,
"Concept Writing") introduced many basic ideas of modern logic that are also
very important for the study of programming languages, including the idea of
applying a function to an argument (as opposed to the classical Greek way of
doing it with subject and object), propositional logic, quantification, and
the modern idea of formal proof (i.e., proofs based on the form of what is
to be proved, as opposed to the classical Greek proofs by syllogism).

Well founded induction for the Peano signature does require proving P(0): if we let x=0, then we get the implication true => P(0), which is equivalent to P(0), as our proof obligation for this case.

It is also worth mentioning that all of the rules of deduction for first order logic can be seen as part of a Post system.

Two assertions about initial algebras not proved in the book have short
proofs, which are, however, quite abstract and algebraic. They rely on the
characterizization of an **initial algebra** as a Sigma-algebra having
exactly one Sigma-homomorphism to any other Sigma-algebra; this is called
its **universal property**. Here are the two properties to be proved:

- Any two initial Sigma-algebras are isomorphic.

- An initial Sigma-algebras has no proper sub-Sigma-algebras.

If ` f: I -> I `

is a homomorphism from an initial algebra to
itself, then ` f `

is the identity map on ` I `

.

This lemma follows from noticing that the identity map on ```
I
```

is a homomorphism on ` I `

, and that the universal
property then implies it is the only one.

For the first assertion, let ` I, I' `

be initial algebras.
Then there are homomorphisms ` f: I -> I' `

and ```
g: I' -> I
```

, and hence ` f;g: I -> I `

and ```
g;f: I' -> I'
```

are also homomorphisms. By the lemma, ` f;g `

and ```
g;f
```

must be identity maps, which implies that they are inverses,
which means that ` I `

and ` I' `

are isomorphic.

For the second assertion, let ` I' `

be a subalgebra of an
initial algebra ` I `

, and let ` f: I' -> I `

be the
inclusion. The universal property implies there is also a homomorphism
` g: I -> I' `

, and the lemma implies ` g;f: I -> I `

is the identity, which in turn implies that ` f `

is surjective.
Now since ` f `

is both surjective and injective (i.e. onto and
one-to-one, in the older terminology), it is an isomorphism.

Everything above about initial algebras generalizes to the case where we consider not just Sigma-algebras, but Sigma-algebras that satisfy a given set E of equations. Here the pair (Sigma, E) defines an abstract data type, and its initial algebra (up to isomorphism) is a canonical model for the abstract type. (See the bottom of page 252 of Sethi.)

To CSE 230 homepage

To CSE 230 notes page

Maintained by Joseph Goguen

© 2000, 2001, 2002, 2003 Joseph Goguen

Last modified: Wed Mar 5 20:21:50 PST 2003