Logical Ruler

(1) In the case of propositional logic, its use is rather straightforward. Given any premise in terms of (up to) 3 propositions (say "A or B"), you eliminate all logical alphabets incompatible with this premise (eliminate all cells with AB); "eliminate" means folding cells back behind the main body (out of sight), so the ruler now looks like this:

This operation is far easier than you may imagine, since in the case of the disjunction "A or B", each cell having either one of A or B remains, and all the rest should go out of sight.

If an additional premise is given (say "not-B and C"), perform a similar operation, and you can see only the following cell:

ABC.

Therefore, at this stage of reasoning, that's the conclusion, i.e., A and not-B and C! Thus you correctly did the following inference:

A or B; and not-B and C; therefore, A and not-B and C.

What you have done is basically an inference by elimination, one of the favorite patters of Sherlock Holmes! If two or more cells remain, what is common to all these cells is the conclusion from the premises. Suppose you now have

ABC

Then, the conclusion is "not-B". Try yourself.

(2) In order to apply the Logical Ruler to the Aristotelian syllogism, we have to adopt another convention (this is not the only way; Jevons chose another way, which was more in conformity with the traditional understanding of syllogism). We will adopt Venn's "existential" interpretation of categorical propositons (in the contemporary terminology, we may call it the "extensional interpretation"); that is, a universal proposition such as "All Apples are Beautiful" is interpreted as asserting non-existence of something ("A and not-B"), A and B (or, its negative term not-B) now referring to a class of objects, rather than a proposition. In contrast, a particular proposition such as "Some Apples are Cubic" is interpreted as asserting existence of something ("A and C"). As is obvious, non-existence is clearly expressed by eliminating appropriate cells from the Ruler.

But how should we represent existence? I will recommend raising cells halfway up; e.g., "Some Apples are Cubic" looks as follows on the Ruler（↑means up）:

ABC↑

ABC

ABC↑

ABC

But keep in mind that this means "ABC or ABC exists"; you cannot conclude a definite status of a single cell. But don't worry, since practice is easier than you may imagine.

You may recall the Aristotelian syllogism contains only four forms of (categorical) proposition, and any inference contains two premises and one conclusion, each in either of the four forms:

All X is Y, No X is Y, Some X is Y, and Some X is not Y.

Now, let me illustrate the process of the following inference, as an example.

This is valid, and you can see why, from the following table.

All A is B	Some C is not-B	∴ Some C is not-A
ABC ABC --- --- ABC ABC ABC ABC	ABC ABC --- --- ABC ABC ABC↑ ABC	Since the existence of ABC is definite, you can see that the conclusion is validly drawn.

Since ABC is already eliminated by the first premise, you don't have to worry about it when considering the second premise. Thus, by putting a universal proposition first into the Ruler, you can make the inference easier. Since the Aristotelian syllogism has only two premises (in either of the four categorical forms), all you have to do is to put the two premises into the Ruler, and see whether or not the proposed conclusion is definitely represented in the final state of the Ruler; if it is, the inference is valid, and if not invalid.

In short, you can do on the Ruler whatever you can do in terms of the Venn diagram. Thus, using the Logical Ruler, you can master the basics of propositional logic and the Aristotelian syllogism in an hour! If you meet a "logic teacher" who spends a half year for teaching these elementary subjects (except for the "history of logic"), you may safely conclude that he/she is a fake, knowing almost nothing about logic!