Punch-card machines do many operations of reasoning or logic that do not involve addition, subtraction, multiplication, or division. Just as we can write equations for arithmetical operations, so we can write equations for these logical operations using mathematical logic (see Chapter 9 and Supplement 2). If any reader, however, is not interested in these logical equations, he should skip each paragraph that begins with “in the language of logic,” or a similar phrase. Translating Reading and writing are operations perhaps not strictly of reasoning but of _translating_ from one language to another. Basically these operations take in a mark in one language and give out a mark with the same meaning in another language. For example, the interpreter takes in punched holes and gives out printed marks, but the holes and the marks have the same meaning. The major part of sorting is done by a punch-card sorting machine and can be considered an operation of translating. In sorting a card, the machine takes in a mark in the form of a punched hole on a punch card and specifies a place bearing the same mark where the card is put. The remaining part of sorting is done by human beings. This part consists of picking up blocks of cards from the pockets of the sorter and putting the blocks together in the right sequence. Comparing [Illustration: FIG. 13. Comparer.] The first operation of reasoning done by punch-card machines is _comparing_. For an example of comparing in the operation of the tabulator, let us take instructing the machine when to pick up a total and print it. As an illustration, suppose that we are making a table by state, county, and township of the number of persons counted in a census. Suppose that for each township we have one punch card telling the total number of persons. If all the cards are in sequence, then, whenever the county changes, we want a minor total, and, whenever the state changes, we want a major total. What does the machine do? The tabulator has a mechanism that we shall call a _comparer_ (Fig. 13). A comparer has 2 inputs that may be called _Previous_ and _Current_ and one output that may be called _Unequal_. The comparer has the property of giving out an impulse if and only if there is a difference between the 2 inputs. In the language of the algebra of logic (see Supplement 2 and Chapter 9), let the pieces of information coming into the comparer be _a_ and _b_, and let the information coming out of the comparer be _p_. Then the equation of the comparer is: _p_ = _T_(_a_ ≠ _b_) where “_T_ (···)” is “the truth value of ···” and “···” is a statement, and where the truth value is 1 if true and 0 if false.