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:

Also, later definitions will be shortened considerably if we can say briefly that a given individual is a characterof 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:

- D4.
- Char
x=. VeexV AcxV LParxV RParxV StrxV Epx.Convenience is similarly served by the definition of an inscriptionas an object composed of whole characters in normal orientation to one another. In view of the interpretation of "C" above, the definition is easy:

- D5.
- Insc
x=. CharxV (]y)(]z)Czyz.An inscription xis said to be aninitial segmentof another,y, ifxis identical withyor there is some inscriptionzsuch thatyconsists ofxfollowed byz.

- D6.
- InitSeg
xy=. Inscx&x=y.V (]z)Cyxz.The definition of

final segmentis strictly parallel:

- D7.
- FinSeg
xy=. Inscx&x=y.V (]z)Cyzx.

An inscription xis said to be asegmentofyifxis an initial segment of some final segment ofy.

- D8.
- Seg
xy= (]z)(InitSegxz& FinSegzy).A segment x-- whether initial, final, or interior -- of an inscriptionywill be continuous relative toy, in the sense that ifxcontains two characters ofythenxmust contain all the characters that occur inybetween those two. The characters of a segmentxofymay still be irregularly spaced, but only because of irregular spacing inyitself.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 thatxis just as big as every smallest character.

- D9.
- Bit
x=. (y)(Chary--> ~Bgrxy) & (]z)(Charz& ~Bgrzx) .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 xislonger thananother,y, ifxcontains more characters thany. Using the same method as for the example of cats and dogs in 4 above -- where a verbal explanation is given -- we define:

- D10.
- Lngr
xy=. Inscx& Inscy& (z){(w)[Charw& Segwx.--> (]u)(Bitu& Partuw& Partuz)] --> (]t)[(r)(Charr& Segry.--> (]s)(Bits& Partsr& Partst)) & Bgrzt]}.Two inscriptions are equally longif neither is longer than the other.

- D11.
- EqLng
xy=. Inscx& Inscy& ~Lngrxy& ~Lngryx.We can now define what we shall mean by saying that two inscriptions are likeone another. Two characters are alike if both are vees, or both are accents, etc. Two inscriptionsxandyare alike if they are equally long and if, for every two equally long inscriptionszandwsuch thatzis an initial segment ofxandwis an initial segment ofy, the segmentszandwend in like characters.

- D12.
- Like
xy=. EqLngxy& (z)(w){EqLngzw& InitSegzx& InitSegwy.--> (]s)(]t)(FinSegsz& FinSegtw: Vees& Veet.V. Acs& Act.V. LPars& LPart.V. RPart.V. Strs& Strt.V. Eps& Ept)}.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.

Notes ^{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 "Other definitions are to be construed analogously.x", "y", "z", and "w", provided that a variable distinct from them is used in place of "t".Contents -- Go to §7