**608 Appendix**

11 Numbers

Numbers and arithmetic expressions are defined in the standard fashion.

12 Quantities, dimensions, and units

A quantity is just a dimensionalized number-a number of units along a given dimension.

13 Booleans and relations

Logical expressions involving and (L
), or (V), not (Ø
) implies (É
), equivalence **(º
), **and exclusive-or (Å
)** **are defined in standard fashion, as are expressions involving the six basic relations

1. Basic semantics

*1.1 *We will use the term "entity" to refer to all things designatable by expressions in the language.

*1.2 *An entity is assumed to be fully characterizable by a set of attributes and associated values, which are themselves entities.

COMMENT** **There will necessarily be entities with no further specification within the system-that, in effect, have only a name.

The semantics of the language consists in showing how expressions in the language determine the various attributes and values.

*1.3 *There are three types of expressions.

1 A definite expression designates an entity.

2 An indefinite expression defines a class of definite expressions; it designates one of the entities designated by members of this class.

3 A command designates the establishment of some purely linguistic convention.

EXAMPLES** **'IBM 7090 is a definite expression.

Mp is an indefinite expression (any primary memory).

SAM **= **Mp is a command to give the name SAM to an Mp.

*1.4 *There are also English language comments, which are connected with the language only in being associated with particular occurrences of expressions (on which they comment) and in having a punctuation convention that allows them to be unambiguously distinguished from expressions in the language.

1 In the book we use italics.

EXAMPLE** ***This is an example of a comment; *it *may appear anywhere.*

2. Metanotation

*2.1 *The language itself is described by giving various classes of expressions and assigning meanings to the members of these classes (i.e., telling what they designate). We will generally do this in English but with a few special notations.

*2.2 *Expression-variables

1 Let **a, b, . . . , A, B,...** be variables whose domain is a set of expressions.

2 Let **class(a)** be the set of definite expressions defined by the indefinite expression **a**. This is extended to definite expressions, x, by defining **class(x) = x.**

COMMENT** **Normally lowercase variables (e.g., **a**) stand for any legal expression, whereas uppercase variables (e.g., **A**) stand for any indefinite expression.

*2.3 *We will define the language by giving forms of expressions, that is, by writing down sequences of expressions and expression-variables. These forms are to be interpreted as permitting any expression that results from replacing the expression-variables with expressions from their respective domains.

EXAMPLE** **If the form x÷
y is legal, where x and y range over components, then the expression M ÷
P is legal.

*2.4 *The one special notation is the expression form

xox...

which is to be taken as permitting an indefinite sequence of **x**'s separated by **o**'s, terminating with an x, where each occurrence is to be viewed as an independent variable. That is, **x o x. . .** is equivalent to

x

or

xox

or

xoxox

or

xoxoxox

etc.

EXAMPLE** d a d. . .,** where **d** ranges over digits and **o** over arithmetic operations, could have as instances: 5, 6 + 6, 7 - 2 + 3, etc.