CSE 230: Principles of Programming Languages
Notes on Post Systems, Chapter 2 of Stansifer

Emil Post is one of two great early American logicians (Charles Sanders Peirce is the other). Post systems provide an alternative way of defining general recursive functions, but they are little used for that today, the main use being to give a precise meaning to formal systems of deduction. For example, it is no coincidence that the rules of inference in Hoare logic use the same notation as Post systems, since these rules can be formulated as the "productions" of a Post system.

The first two occurrences of the word "list" in section 2.4.1 (page 62) should be replaced by "set". Stansifer uses the phrase "tally notation" the notation that represents 0, 1, 2, 3, ... by the empty string, |, ||, |||, ...; this is a variant of what we call Peano notion. There is a typo on page 64, where it says that || + || = |||, i.e., 2 + 2 = 3! There is an OBJ version of the Post system for the propositional calculus. Also, the production

```     xax
---
xxx  ```
on page 65 is called an "axiom" but it isn't.

The "proof" on page 68 does not really deserve to be called a proof, because it only sketches one direction and it completely omits the other direction, which turns out to be much harder than what is sketched. It is remarkable that the term "theorem" appears at different three levels: (1) a theorem of the predicate calculus, i.e., some `x` for which `Th x` is provable; (2) a theorem of the Post system for the predicate calculus, which means a derivable term of that system, which includes some terms of the form `Th x`, others of the form ```P x```, etc.; and (3) a theorem of mathematics, the proof of which is discussed in the previous sentence.

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