Finite Alphabets

This is a question based on section 2.5.6 from the book Logic, laws of truth, by Nicholas J.J Smith, page 48.

What does Nicholas mean when he says:

Note that we can further specify exactly what we mean by "a finite string of the digits 0,...,9 that does not begin with 0" (henceforth fsd for short) using a recursive definition:

1. Each of the single digits 1,...,9 is an fsd
2. If x is an fsd and y is one of the digits 0,..,9 then xy is an fsd.
3. Nothing else is an fsd.

And why don't propositions have subscripts 0 and 1?

• Those that don't have access to the book have no idea what you mean by "propositions". The question should include this missing information. Sep 17 '19 at 12:57

This construct has a very useful property: there is a bijective mapping between fsd and natural numbers. In particular, it is designed to make sure it doesn't have to deal with mapping "123" and "000000123". There does exist a mapping that permits leading zeroes, but its a much more complicated one to define. By not permitting leading zeroes, the mapping is intuitive.

As for starting with 1 rather than 0, I see two reasons. The first is simply that natural numbers start with 1. The second is that "0" technically has a leading zero. Supporting it would require amending the rules, and there's just no benefit for doing so.

The author is explaining how the infinitely many basic propositions:

A, A2, A3, . . . , B, B2, B3, . . . , C, C2, C3, . . . , Z, Z2, Z3, . . .

can be generated starting from an alphabet of finite many symbols : the letters of the alphabet : A,B,C... and the ten digitis.

It seems to me that he had (arbitrarily) decided that A0 (and similar) is not used and that A1 (and similar) is abbreviated as A.

IMO, no specific reason for this; we may use different (infinite) list of basic propositions, like e.g.

p0, p1, p2, ...

A finite string of the digits

is exactly what has been described : a number in decimal base without leading zero, like e.g. 2, 234, 6666, etc.

Nicholas J. J. Smith is trying to show in section 2.5.6 on Finite Alphabets that one does not need to use infinitely many symbols: "five connectives, two parentheses, and infinitely many basic propositions." (page 48)

Instead one can replace the infinitely many symbols for basic propositions by recursive definitions. He replaces the definition of basic proposition as an infinite set of symbols with a recursive definition using only the letters of the alphabet and the digits from 0 to 9. This recursive definition uses the idea of a "finite string of digits" or "fsd" which he also recursively defines.

Consider the question, And why don't propositions have subscripts 0 and 1?

Basic propositions can have the symbols 0 and 1 along with the other digits. The only restriction is that 0 may not be used in the first position of an fsd. Smith is trying to avoid subscripts that start with 0.

Note the base clause:

Each of the single digits 1,..., 9 is an fsd.

This means that the first digit can be any one of 1, 2, 3, 4, 5, 6, 7, 8, or 9. However, it cannot be 0. Any following digit, however, may include 0 based on the recursive clause:

If x is an fsd and y is one of the digits 0,..., 9 then xy is an fsd.

Smith, N. J. (2012). Logic: The laws of truth. Princeton University Press.