Chapter 9
REASONING:
THE KALIN-BURKHART
LOGICAL-TRUTH CALCULATOR
So far we have talked about mechanical brains that are mathematicians. They are fond of numbers; their main work is with numbers; and the other kinds of thinking they do are secondary. We now come to a mechanical brain that is a logician. It is fond of reasoning—logic; its main work is with what is logically true and what is logically false; and it does not handle numbers. This mechanical brain was finished in June 1947. It is called the Kalin-Burkhart Logical-Truth Calculator. As its name tells, it calculates logical truth. Now what do we mean by that?
TRUTH
To be true or false is a property of a statement. Usually we say that a statement is true when it expresses a fact. For example, take the statement “Salt dissolves in water.” We consider this statement to be true because it expresses a fact. Actually, in this case we can roughly prove the fact ourselves. We take a bowl, put some water in it, and put in a little salt. After a while we look into the water and notice that no salt whatever is to be seen.
Of course, this statement, like many another, occurs in a context where certain things are understood. One of the understandings here, for example, is “a small amount of salt in a much larger amount of water.” For if we put a whole bag full of salt in just a little water, not all the salt will dissolve. Nearly every statement occurs in a context that we must know if we are to decide whether the statement is true or false.
LOGICAL TRUTH
Logical truth is different from ordinary truth. With logical truth we appeal not to facts but to suppositions. Usually we say that a statement is logically true when it follows logically from certain suppositions. In other words, we play a game that has useful, even wonderful, results. The game starts with “if” or “suppose” or “let us assume.” While the game lasts, any statement is logically true if it follows logically from the suppositions.
For example, let us take five statements:
- 1. “The earth is flat like a sheet of paper.”
- 2. “The earth is round like a ball.”
- 3. “John Doe travels as fast as he can, without turning
- to left or to right, for many days.”
- 4. “John Doe will fall off the earth.”
- 5. “John Doe will arrive back at his starting point.”
Let us also take a certain context in which: We know what we mean by such words as “earth,” “flat,” “falling,” etc.; we have other statements and understandings such as “if John Doe walks off the edge of a cliff, he will fall,” “a flat sheet of paper has an edge,” etc. In this context, if statements 1 and 3 are supposed, then statement 4 is logically true. On the other hand, if statements 2 and 3 are supposed, then statement 5 is logically true. Of course, for many centuries, nearly all men believed statement 1; and the importance of the years 1492 to 1521 (Columbus to Magellan) is linked with the final proof that statement 2 expresses a fact. So, depending on the game, or the context, whichever we wish to call it, almost any statement can be logically true. What we become interested in, therefore, is the connections between statements which make them follow logically.
LOGICAL PATTERNS
Perhaps the most familiar example of “following logically” is a pattern of words like the following:
- 1. All igs are ows.
- 2. All ows are umphs.
- 3. Therefore, all igs are umphs.
If statements 1 and 2 are supposed, then statement 3 is logically true. In other words, statement 3 logically follows from statements 1 and 2. This word pattern is logically true, no matter what substitutions we make for igs, ows, and umphs. For example, we can replace igs by men, ows by animals, and umphs by mortals, and obtain:
- 4. All men are animals.
- 5. All animals are mortals.
- 6. Therefore, all men are mortals.
The invented words “igs,” “ows,” “umphs” mark places in the logical pattern where we can insert any names we are interested in. The words “all,” “are,” “therefore” and the ending s mark the logical pattern. Of course, instead of using invented words like “igs,” “ows,” “umphs” we would usually put A’s, B’s, C’s. This logical pattern is called a syllogism and is one of the most familiar. But there are even simpler logical patterns that are also familiar.
THE SIMPLEST LOGICAL PATTERNS
Many simple logical patterns are so familiar that we often use them without being conscious of doing so. The simple logical patterns are marked by words like “and,” “or,” “else,” “not,” “if,” “then,” “only.” In the same way, simple arithmetical patterns are marked by words like “plus,” “minus,” “times,” “divided by.”
Let us see what some of these simple logical patterns are. Suppose that we take two statements about which we have no factual information that might interfere with logical supposing:
- 1. John Doe is eligible for insurance.
- 2. John Doe requires a medical examination.
In practice, we might be concerned with such statements when writing the rules governing a plan of insurance for a group of employees. Here, we shall play a game:
(1) We shall make up some new statements from statements 1 and 2, using the words “and,” “or,” “else,” “not,” “if,” “then,” “only.”
(2) We shall examine the logical patterns that we can make.
(3) We shall see what we can find out about their logical truth.
Suppose we make up the following statements:
3. John Doe is not eligible for insurance.
4. John Doe does not require a medical examination.
5. John Doe is eligible for insurance and requires a medical examination.
6. John Doe is eligible for insurance, and John Doe is eligible for insurance.
7. John Doe is eligible for insurance, or John Doe requires a medical examination.
8. If John Doe is eligible for insurance, then he requires a medical examination.
9. John Doe requires a medical examination if and only if he is eligible for insurance.
10. John Doe is eligible for insurance or else he requires a medical examination.
Now clearly it is troublesome to repeat quantities of words when we are interested only in the way that “and,” “or,” “else,” “not,” “if,” “then,” “only” occur. So, let us use just 1 and 2 for the two original statements, remembering that “1 and 2” means here “statement 1 AND statement 2” and does not mean 1 plus 2. Then we have:
| 3: | not-1 |
| 4: | not-2 |
| 5: | 1 and 2 |
| 6: | 1 and 1 |
| 7: | 1 or 2 |
| 8: | if 1, then 2 |
| 9: | 1 if and only if 2 |
| 10: | 1 or else 2 |
Here then are some simple logical patterns that we can make.
CALCULATION OF LOGICAL TRUTH
Now what can we find out about the logical truth of statements 3 to 10? If we know something about the truth or falsity of statements 1 and 2, what will logically follow about the truth or falsity of statements 3 to 10? In other words, how can we calculate the logical truth of statements 3 to 10, given the truth or falsity of statements 1 and 2?
For example, 3 is not-1; that is, statement 3 is the negative or the denial of statement 1. It follows logically that, if 1 is true, 3 is false; if 1 is false, 3 is true. Suppose that we use T for logically true and F for logically false. Then we can show our calculation of the logical truth of statement 3 in Table 1.
| Table 1 | Table 2 | |||
| 1 | not-1 = 3 | 2 | not-2 = 4 | |
| T | F | T | F | |
| F | T | F | T | |
Our rule for calculation is: For T put F; for F put T. Of course, exactly the same rule applies to statements 2 and 4 (see Table 2). The T and F are called truth values. Any meaningful statement can have truth values. This type of table is called a truth table. For any logical pattern, we can make up a truth table.
Let us take another example, “and.” Statement 5 is the same as statement 1 and statement 2. How can we calculate the logical truth of statement 5? We can make up the same sort of a table as before. On the left-hand side of this table, there will be 4 cases:
- 1. Statement 1 true, statement 2 true.
- 2. Statement 1 false, statement 2 true.
- 3. Statement 1 true, statement 2 false.
- 4. Statement 1 false, statement 2 false.
On the right-hand side of this table, we shall put down the truth value of statement 5. Statement 5 is true if both statements 1 and 2 are true; statement 5 is false in the other cases. We know this from our common everyday experience with the meaning of “and” between statements. So we can set up the truth table, and our rule for calculation of logical truth, in the case of and, is shown on Table 3.
Table 3
| 1 | 2 | 1 and 2 = 5 |
| T | T | T |
| F | T | F |
| T | F | F |
| F | F | F |
“and” and the other words and phrases joining together the original two statements to make new statements are called connectives, or logical connectives. The connectives that we have illustrated in statements 7 to 10 are: or, if ··· then, if and only if, or else.
Table 4 shows the truth table that applies to statements 7, 8, 9, and 10. This truth table expresses the calculation of the logical truth or falsity of these statements.
Table 4
| 1 or 2 | if 1, then 2 | 1 if and only if 2 | 1 or else 2 | ||
| 1 | 2 | = 7 | = 8 | = 9 | = 10 |
| T | T | T | T | T | F |
| F | T | T | T | F | T |
| T | F | T | F | F | T |
| F | F | F | T | T | F |
The “or” (as in statement 7) that is defined in the truth table is often called the inclusive “or” and means “and/or.” Statement 7, “1 or 2,” is considered to be the same as “1 or 2 or both.” There is another “or” in common use, often called the exclusive “or,” meaning “or else” (as in statement 10). Statement 10, “1 or else 2,” is the same as “1 or 2 but not both” or “either 1 or 2.” In ordinary English, there is some confusion over these two “or’s.” Usually we rely on the context to tell which one is intended. Of course, such reliance is not safe. Sometimes we rely on a necessary conflict between the two statements connected by “or” which prevents the “both” case from being possible. In Latin the two kinds of “or” were distinguished by different words, vel meaning “and/or,” and aut meaning “or else.”
The “if ··· then” that is defined in the truth table agrees with our usual understanding that (1) when the “if clause” is true, the “then clause” must be true; and (2) when the “if clause” is false, the “then clause” may be either true or false. The “if and only if” that is defined in the truth table agrees with our usual understanding that (1) if either clause is true, the other is true; and (2) if either clause is false, the other is false.
In statement 6, there are only two possible cases, and the truth table is shown in Table 5.
Table 5
| 1 | 1 and 1 = 6 |
| T | T |
| F | F |
We know that 6 is true if and only if 1 is true. In other words, the statement “1 and 1 if and only if 1” is true, no matter what statement 1 may refer to. It is because of this fact that we never use a statement in the form “1 and 1”: it can always be replaced by the plain statement “1.”
LOGICAL-TRUTH CALCULATION BY
EXAMINING CASES AND REASONING
Now you may say that this is all very well, but what good is it? Almost anybody can use these connectives correctly and certainly has had a great deal of practice using them. Why do we need to go into truth values and truth tables?
When we draft a contract or a set of rules, we often have to consider several conditions that give rise to a number of cases. We must avoid:
1. All conflicts, in which two statements that disagree apply to the same case.
2. All loopholes, in which there is a case not covered by any statement.
If we have one statement or condition only, we have to consider 2 possible cases: the condition satisfied or the statement true; the condition not satisfied or the statement false. If we have 2 conditions, we have to consider 4 possible cases: true, true; false, true; true, false; false, false. If we have 3 conditions, we have to consider 8 possible cases one after the other (see Table 6).
Table 6
| Case | 1st Condition |
2nd Condition |
3rd Condition |
| 1 | T | T | T |
| 2 | F | T | T |
| 3 | T | F | T |
| 4 | F | F | T |
| 5 | T | T | F |
| 6 | F | T | F |
| 7 | T | F | F |
| 8 | F | F | F |
Instead of T’s and F’s, we would ordinarily use check-marks (✓) and crosses (✕), which, of course, have the same meaning. We may consider and study each case individually. In any event, we must make sure that the proposed contract or set of rules covers all the cases without conflicts or loopholes.
The number of possible cases that we have to consider doubles whenever one more condition is added. Clearly, it soon becomes too much work to consider each case individually, and so we must turn to a second method, thoughtful classifying and reasoning about classes of cases.
Now suppose that the number of conditions increases: 4 conditions give rise to 16 possible cases; 5, 6, 7, 8, 9, 10, ··· conditions give rise to 32, 64, 128, 256, 512, 1024, ··· cases respectively. Because of the large number of cases, we soon begin to make mistakes while reasoning about classes of cases. We need a more efficient way of knowing whether all cases are covered properly.
LOGICAL-TRUTH CALCULATION
BY ALGEBRA
One of the more efficient ways of reasoning is often called the algebra of logic. This algebra is a part of a new science called mathematical logic. Mathematical logic is a science that has the following characteristics:
- It studies chiefly nonnumerical reasoning.
- It seeks accurate meanings and necessary consequences.
- Its chief instruments are efficient symbols.
Mathematical logic studies especially the logical relations expressed in such words as “or,” “and,” “not,” “else,” “if,” “then,” “only,” “the,” “of,” “is,” “every,” “all,” “none,” “some,” “same,” “different,” etc. The algebra of logic studies especially only the first seven of these words.
The great thinkers of ancient Greece first studied the problems of logical reasoning as these problems turned up in philosophy, psychology, and debate. Aristotle originated what was called formal logic. This was devoted mainly to variations of the logical pattern shown above called the syllogism. In the last 150 years, the fine symbolic techniques developed by mathematicians were applied to the problems of the calculation of logical truth, and the result was mathematical logic, much broader and much more powerful than formal logic. A milestone in the development of mathematical logic was The Laws of Thought, written by George Boole, a great English mathematician, and published in 1854. Boole introduced the branch of mathematical logic called the algebra of logic, also called Boolean algebra. In late years, all the branches of mathematical logic have been improved and made easier to use.
We can give a simple numerical example of Boolean algebra and how it can calculate logical truth. Suppose that we take the truth value of a statement as 1 if it is true and 0 if it is false. Now we have numbers 1 and 0 instead of letters T and F. Since they are numbers, we can add them, subtract them, and multiply them. We can also make up simple numerical formulas that will let us calculate logical truth. If P and Q are statements, and if p and q are their truth values, respectively, we have Table 7.
Table 7
| Statement | Truth Value |
|---|---|
| not-P | 1 - p |
| P and Q | pq |
| P or Q | p + q - pq |
| if P, then Q | 1 - p + pq |
| P if and only if Q | 1 - p - q + 2pq |
| P or else Q | p + q - 2pq |
For example, suppose that we have two statements P and Q:
- P: John Doe is eligible for insurance.
- Q: John Doe requires a medical examination.
To test that the truth value of “P or Q” is p + q-pq, let us put down the four cases, and calculate the result (see Table 8).
Table 8
| p | q | p + q - pq |
| 1 | 1 | 1 + 1 - 1 = 1 |
| 0 | 1 | 0 + 1 - 0 = 1 |
| 1 | 0 | 1 + 0 - 0 = 1 |
| 0 | 0 | 0 + 0 - 0 = 0 |
Now we know that P or Q is true if and only if either one or both of P and Q are true, and thus we see that the calculation is correct.
The algebra of logic (see also Supplement 2) is a more efficient way of calculating logical truth. But it is still a good deal of work to use the algebra. For example, if we have 10 conditions, we shall have 10 letters like p, q to handle in calculations. Thus we need a still more efficient way.
CALCULATION OF CIRCUITS BY
THE ALGEBRA OF LOGIC
In 1937 a research assistant at Massachusetts Institute of Technology, Claude E. Shannon, was studying for his degree of master of science. He was enrolled in the Department of Electrical Engineering. He was interested in automatic switching circuits and wondered why an algebra should not apply to them. He wrote his thesis on the answer to this question and showed that:
- (1) There is an algebra that applies to switching circuits.
- (2) It is the algebra of logic.
A paper, based on his thesis, was published in 1938 in the Transactions of the American Institute of Electrical Engineers with the title “A Symbolic Analysis of Relay and Switching Circuits.”
Fig. 1. Switches in series.
For a simple example of what Shannon found out, suppose that we have two switches, 1, 2, in series (see Fig. 1). When do we get current flowing from the source to the sink? There are 4 possible cases and results (see Table 9).
Table 9
| Switch 1 is closed |
Switch 2 is closed |
Current flows |
|---|---|---|
| Yes | Yes | Yes |
| No | Yes | No |
| Yes | No | No |
| No | No | No |
Now what does this table remind us of? It is precisely the truth table for “and.” It is just what we would have if we wrote down the truth table of the statement “Switch 1 is closed and switch 2 is closed.”
Fig. 2. Switches in parallel.
Fig. 3. Switch open—current flowing.
Suppose that we have two switches 1, 2 in parallel (see Fig. 2). When do we get current flowing from the source to the sink? Answer: when either one or both of the switches are closed. Therefore, this circuit is an exact representation of the statement “Switch 1 is closed or switch 2 is closed.”
Suppose that we have a switch that has two positions, and at any time must be at one and only one of these two positions (see Fig. 3). Suppose that current flows only when the switch is open. There are two possible cases and results (see Table 10).
Table 10
| Switch 1 is closed |
Current flows |
|---|---|
| Yes | No |
| No | Yes |
This is like the truth table for “not”; and this circuit is an exact representation of the statement “Switch 1 is not closed.” (Note: These examples are in substantial agreement with Shannon’s paper, although Shannon uses different conventions.)
We see, therefore, that there is a very neat correspondence between the algebra of logic and automatic switching circuits. Thus it happens that:
1. The algebra of logic can be used in the calculation of some electrical circuits.
2. Some electrical circuits can be used in the calculations of the algebra of logic.
This fact is what led to the next step.
LOGICAL-TRUTH CALCULATION
BY MACHINE
In 1946 two undergraduates at Harvard University, Theodore A. Kalin and William Burkhart, were taking a course in mathematical logic. They noticed that there were a large number of truth tables to be worked out. To work them out took time and effort and yet was a rather tiresome automatic process not requiring much thinking. They had had some experience with electrical circuits. Knowing of Shannon’s work, they said to each other, “Why not build an electrical machine to calculate truth tables?”
They took about two months to decide on the essential design of the machine:
1. The machine would have dial switches in which logical connectives would be entered.
2. It would have dial switches in which the numbers of statements like 1, 2, 3 ··· would be entered.
3. It would scan the proper truth table line by line by sending electrical pulses through the dial switches.
4. It would compute the truth or falsehood of the whole expression.
CONSTRUCTION AND COMPLETION OF THE
KALIN-BURKHART LOGICAL-TRUTH
CALCULATOR
With the designs in mind, Kalin and Burkhart bought some war surplus materials, including relays, switches, wires, lights, and a metal box about 30 inches long by 16 inches tall, and 13 inches deep. From March to June, 1947, they constructed a machine in their spare time, assembling and mounting the parts inside the box. The total cost of materials was about $150. In June the machine was demonstrated in Cambridge, Mass., before several logicians and engineers, and in August it was moved for some months to the office of a life insurance company. There some study was made of the possible application of the machine in drafting contracts and rules.
GENERAL ORGANIZATION
OF THE MACHINE
The logical-truth calculator built by Kalin and Burkhart is not giant in size, although giant in capacity. Like other mechanical brains, the machine is made up of many pieces of a rather small number of different kinds of parts. The machine contains about 45 dial switches, 23 snap switches (or two-position switches), 85 relays, 6 push buttons, less than a mile of wire, etc. The lid of the metal box is the front, vertical panel of the machine.
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:
| 1 | and | 1 | if-then | 2 | and | 2 | if-then | 3 | and | 3 | if-then | 4 | if-then | 4 |
| | | | | | | | | | | | | |||||||||
| 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.
A COMPLETE AND CONCRETE EXAMPLE
The reader may still be wondering when he will see a complete and concrete example of the application of the logical-truth calculator. So far we have given only pieces of examples in order to illustrate some explanation. Therefore, let us consider now the following problem: