N. Goodman and W. V. Quine, "Steps Toward a Constructive Nominalism", Journal of Symbolic Logic, 12 (1947).

6. Some Auxiliary Definitions

We now proceed to define certain useful auxiliary predicates. First, it is convenient to have four-, five-, and six-place predicates of concatenation. The definitions are obvious:

Cxyzw = (]t)(Cxyt & Ctzw),13

Cxyzwu = (]t)(Cxyt & Ctzwu),

Cxyzwus = (]t)(Cxyt & Ctzwus).

Also, later definitions will be shortened considerably if we can say briefly that a given individual is a character of our object language. Since a character is any concrete object that is either a vee or an accent or a left parenthesis, etc., the definition runs:

Char x =. Vee x V Ac x V LPar x V RPar x V Str x V Ep x.

Convenience is similarly served by the definition of an inscription as an object composed of whole characters in normal orientation to one another. In view of the interpretation of "C" above, the definition is easy:

Insc x =. Char x V (]y)(]z)Czyz.

An inscription x is said to be an initial segment of another, y, if x is identical with y or there is some inscription z such that y consists of x followed by z.

InitSeg xy =. Insc x & x = y .V (]z)Cyxz.

The definition of final segment is strictly parallel:

FinSeg xy =. Insc x & x = y .V (]z)Cyzx.

An inscription x is said to be a segment of y if x is an initial segment of some final segment of y.

Seg xy = (]z)(InitSeg xz & FinSeg zy).

A segment x -- whether initial, final, or interior -- of an inscription y will be continuous relative to y, in the sense that if x contains two characters of y then x must contain all the characters that occur in y between those two. The characters of a segment x of y may still be irregularly spaced, but only because of irregular spacing in y itself.

We shall later want to be able to say that two inscriptions are equally long, not in the sense that their ends are equally far apart but in the sense that each inscription has as many characters as the other. Since the characters comprising any inscription are discrete from one another, this numerical comparison can be handled in a way explained in 4 above. We begin by so defining "Bit" for our present purposes that "Bit x" means that x is just as big as every smallest character.

Bit x =. (y)(Char y --> ~Bgr xy) & (]z)(Char z & ~Bgr zx) .

It must not be supposed that, because accents are in general the smallest characters of our object language, every accent will be a bit. For accents may vary in size, and only the smallest characters, along with everything that is just as big, will be bits.

An inscription x is longer than another, y, if x contains more characters than y. Using the same method as for the example of cats and dogs in 4 above -- where a verbal explanation is given -- we define:

Lngr xy =. Insc x & Insc y & (z){(w)[Char w & Seg wx .--> (]u)(Bit u & Part uw & Part uz)] --> (]t)[(r)(Char r & Seg ry .--> (]s)(Bit s & Part sr & Part st)) & Bgr zt]}.

Two inscriptions are equally long if neither is longer than the other.

EqLng xy =. Insc x & Insc y & ~Lngr xy & ~Lngr yx.

We can now define what we shall mean by saying that two inscriptions are like one another. Two characters are alike if both are vees, or both are accents, etc. Two inscriptions x and y are alike if they are equally long and if, for every two equally long inscriptions z and w such that z is an initial segment of x and w is an initial segment of y, the segments z and w end in like characters.

Like xy =. EqLng xy & (z)(w){EqLng zw & InitSeg zx & InitSeg wy .--> (]s)(]t)(FinSeg sz & FinSeg tw : Vee s & Vee t .V. Ac s & Ac t .V. LPar s & LPar t .V. RPar t .V. Str s & Str t .V. Ep s & Ep t)}.

Note that only inscriptions can be 'alike', in the sense here defined, since only inscriptions can be equally long; and further, that likeness depends solely upon the component characters and their order of occurrence, not upon identical spacing.


13 The sign "=", when it occurs as the main connective in definitions in this paper, is not to be thought of as expressing identity. It is to be regarded rather as constituting, in combination with the "D" which precedes each definition-number, a mark of definitional abbreviation; and it may occur between name-matrices and statement-matrices indifferently. The definition D1 is to be understood as a convention to this effect: "Cxyzw" is to be understood as an abbreviation of "(]t)(Cxyt & Ctzw)" and a similar understanding is to obtain when any other variables are used in place of "x", "y", "z", and "w", provided that a variable distinct from them is used in place of "t". Other definitions are to be construed analogously.

Contents -- Go to §7