There is a neat way of saying what we have just said, using the language of _mathematical logic_ (see Chapter 9 and Supplement 2). Suppose that we consider the statement “_a_ is greater than _b_” where _a_ and _b_ may be any of the numbers 0, 1, 2, 3. We can say that the _truth value p_ of a _statement P_ is 1 if the statement is true and that it is 0 if the statement is false: _p_ = 1 if _P_ is true, 0 if _P_ is false The truth value of a statement _P_ is conveniently denoted as _T_(_P_) (see Supplement 2): _p_ = _T_(_P_) Now we can say that the table for the operation _greater than_ shows the truth value of the statement “_a_ is greater than _b_”: _p_ = _T_(_a_ > _b_) Let us turn now to the operation _selection_. By _selecting_ we mean choosing one number _a_ if some statement _P_ is true and choosing another number _b_ if that statement is not true. As before, let _p_ be the truth value of that statement _P_, and let it be equal to 1 if _P_ is true and to 0 if _P_ is false. Then the operation of selection is fully expressed in the following table and logical formula (see Supplement 2): SELECTION _c_ = _a_·_p_ + _b_·(1 - _p_) _p_: 0 0 0 0 1 1 1 1 _b_: 0 1 2 3 0 1 2 3 _a_+—————————————————— 0 | 0 1 2 3 0 0 0 0 1 | 0 1 2 3 1 1 1 1 2 | 0 1 2 3 2 2 2 2 3 | 0 1 2 3 3 3 3 3 For example, suppose that _a_ is 2 and _b_ is 3 and the statement _P_ is the statement “2 is greater than 0.” Since this statement is true, _p_ is 1, and _a_·_p_ + _b_·(1 - _p_) = 2(1) + 3(0) = 2 This result is the same as selecting 2 if 2 is greater than 0 and selecting 3 if 2 is not greater than 0. Thus we have four operations for Simon that do not overstrain his mentality; that is, they do not require him to go to any numbers other than 0, 1, 2, and 3. These four operations are: addition, negation, greater than, selection. We label these operations also with the numbers 00 to 11 as follows: addition, 00; negation, 01; greater than, 10; selection, 11. SIMON’S MEMORY—STORING INFORMATION The _memory_ of a mechanical brain consists of physical equipment in which information can be stored. Usually, each section of the physical equipment which can store one piece of information is called a _register._ Each register in Simon will consist of 2 relays. Each register will hold any of 00, 01, 10, 11. The information stored in a register 00, 01, 10, 11 may express a number or may express an operation. [Illustration: S1-2 Relay energized] [Illustration: S1-1 Relay not energized FIG. 3. Register _S_1 storing 10.]