UNITS OF THE MACHINE

The machine contains 16 units. These units are listed in [Table 11], in approximately the order in which they appear on the front panel of the machine—row by row from top to bottom, and from left to right in each row.

Table 11

UNITS, THEIR NAMES, AND SIGNIFICANCE
Unit	Row	Part	No.	Mark	Name	Significance
1	1	Small red lights	12	—	Statement truth- value lights	Output: glows if statement is assumed true in the case
2	1	2-position snap switches	12	~	Statement denial switches	Input: if up, statement is denied
3	2	14-position dial switches	12	V	Statement switches	Input of statements
4	3	4-position dial switches	11	k	Connective switches	Input of connectives: ∧ (and), ∨ (or), ▲ (if-then), ▼ (if and only if)
5	4	11-position dial switches	11	A	Antecedent switches	Input of antecedents
6	5	11-position dial switches	11	C	Consequent switches	Input of consequents
7	6	2-position snap switches	11	S	Stop switches	Input: if up, associates connective to main truth-value light
8	6	2-position snap switches	11	~	Connective denial switches	Input: if up, statement produced by connective is denied
9	7	Red light and large button	1	Start	Automatic start	Input: causes the calc. to start down a truth table automatically
10	7	Red light and 2 buttons	1	Start Stop	Power switch	Input: turns the power on or off
11	7	2-position snap switch and red button	1	Stop	“Stop-on-true-or- false” switch	Input: causes the calc. to stop either on true cases or on false cases
12	7	Yellow light	1	—	Main truth-value light	Output: glows if the statement produced by the main connective is true for the case
13	7	Large button	1	Man. Pulse	Manual pulse button	Input: causes the calc. to go to the next line of a truth table
14	7	11-position dial switch	1	kⱼ	Connective check switch and light	Output: glows when any specified connective is true
15	7	13-position dial switch	1	TT Row Stop	“Truth-table-row- stop” switch	Input: causes the calc. to stop on the last row of the truth table
16	Be- tween 6 & 7	Continuous dial knob and button	1	—	Timing control knob	Input: controls the speed at which the calculator scans rows of the truth table

Some of the words appearing in this table need to be defined. Connective here means “and,” “or,” “if ··· then,” “if and only if.” Only these four connectives appear on the machine; others when needed can be constructed from these. The symbols used for these connectives in mathematical logic are ∧, ∨, ▲, ▼. These signs serve as labels for the connective switch points. In this machine, when there is a connective between two statements, the statement that comes before is called the antecedent and the statement that comes after is called the consequent.

HOW INFORMATION GOES
INTO THE MACHINE

Of the 16 units 13 are input units. They control the setup of the machine so that it can solve a problem. Of the 13 input units, those that have the most to do with taking in the problem are shown in [Table 12].

Table 12

Unit	Name of Switches	Mark	Kind of Switch	Switch Settings
3	Statement	V₁ to V₁₂	Dial	Statements 1 to 12 or constant T or F
2	Statement denial	~	Snap	Affirmative (down) or negative (up)
4	Connective	k₁ to k₁₁	Dial	∧ (and), ∨ (or), ▲ (if-then), ▼ (if and only if)
8	Connective denial	~	Snap	Affirmative (down) or negative (up)
5	Antecedent	A₁ to A₁₁	Dial	V or various k’s
6	Consequent	C₁ to C₁₁	Dial	V or various k’s
7	Stop	S₁ to S₁₁	Snap	Not connected (down) or connected (up)

The first step in putting a problem on the machine is to express the whole problem as a single compound statement that we want to know the truth or falsity of. We express the single compound statement in a form such as the following:

V k V k V k V k V k V k V k V k V k V k V k V

where each V represents a statement, each k represents a connective, and we know the grouping, or in other words, we know the antecedent and consequent of each connective.

For example, let us choose a problem with an obvious answer:

Problem. Given: statement 1 is true; and if statement 1 is true, then statement 2 is true; and if statement 2 is true, then statement 3 is true; and if statement 3 is true, then statement 4 is true. Is statement 4 true?

How do we express this whole problem in a form that will go on the machine? We express the whole problem as a single compound statement that we want to know the truth or falsity of:

If [1 and (if 1 then 2) and (if 2 then 3) and (if 3 then 4)], then 4

The 8 statements occurring in this problem are, respectively: 1 1 2 2 3 3 4 4. These are the values at which the V switches (the statement dial switches, Unit 2) from V₁ to V₈ are set. The 7 connectives occurring in this problem are, respectively: and, if-then, and, if-then, and, if-then, if-then. These are the values at which the k switches (the connective dial switches, Unit 4) from k₁ to k₇ are set.

A grouping (one of several possible groupings) that specifies the antecedent and consequent of each connective is the following:

and

if-then

and

if-then

and

if-then

k₂

k₄

k₆

k₁

k₅

k₃

k₇

The grouping has here been expressed graphically with lines but may be expressed in the normal mathematical way with parentheses and brackets as follows:

{[ 1 and (1 if-then 2)] and [(2 if-then 3) and (3 if-then 4) ] } if-then 4.

So the values at which the antecedent and consequent dial switches are set are as shown in [Table 13].

Table 13

Connective	Antecedent Switch	Set at	Consequent Switch	Set at
k₁	A₁	V	C₁	k₂
k₂	A₂	V	C₂	V
k₃	A₃	k₁	C₃	k₅
k₄	A₄	V	C₄	V
k₅	A₅	k₄	C₅	k₆
k₆	A₆	V	C₆	V
k₇	A₇	k₃	C₇	V

In any problem, statements that are different are numbered one after another 1, 2, 3, 4 ···. A statement that is repeated bears always the same number. In nearly all cases that are interesting, there will be repetitions of the statements. If any statement appeared with a “not” in it, we would turn up the denial switch for that statement (Unit 2).

The different connectives available on the machine are “and,” “or,” “if ··· then,” “if and only if.” If a “not” affected the compound statement produced by any connective, we would turn up the denial switch for that connective (Unit 8).

The last step in putting the problem on the machine is to connect the main connective of the whole compound statement to the yellow light output (Unit 12). In this problem the last “if-then,” k₇, is the main connective, the one that produces the whole compound statement. So we turn Stop Switch 7 (in Unit 7) that belongs to k₇ into the up position. There are a few more things to do, naturally, but the essential part of putting the information of the problem into the machine has now been described.

HOW INFORMATION COMES OUT
OF THE MACHINE

Of the 16 units listed in [Table 11], 3 are output units, and only 2 of these are really important, as shown in [Table 14].

Table 14

Unit	Name of Light	Mark	Kind of Light
1	Statement truth value	V₁ to V₁₂	Small, red
13	Main truth value		Large, yellow

The answer to a problem is shown by a pattern of the lights of Units 1 and 13. The pattern of lights is equivalent to a row of the truth table. Each little red light (Unit 1) glows when its statement is assumed to be true, and it is dark when its statement is assumed to be false. The yellow light (Unit 13) glows when the whole compound statement is calculated to be logically true, and it is dark when the whole compound statement is calculated to be logically false.

The machine turns its “attention” automatically to each line of the truth table one after the other, and pulses are fed in according to the pattern of assumed true statements. We can set the machine to stop on true cases or on false cases or on every case, so as to give us time to copy down whichever kind of results we are interested in. When we have noted the case, we can press a button and the machine will then go ahead searching for more cases.