Chapter 4

Operations using a single proposition

That, then, is what the proposition looks like. Now is there anything we can infer from a proposition as it stands? It turns out that there are a couple of things.

First, there is the operation called conversion, in which you interchange the subject and the predicate, drawing, in other words, an inference about the class of objects implicitly referred to by the predicate, based on the meaning implicit in the subject.

Let me define some terms, and then give the rules for this operation and say a little about them.

Conversion is the logical inference involved in interchanging the subject and predicate of a proposition.

The converse of a proposition is the conclusion that results from conversion.

Rules for conversion

1. Leave the copula alone.

2. Interchange the subject and the predicate.

3. Check to see that the new subject has the same reference (definite or indefinite) as was implicit in the old predicate.

4. Check the implicit reference of the predicate against the reference it had as subject. If it is not the same:

a. If the term became indefinite from being definite, this is permitted.

b. If the term became definite from being indefinite the inference is not valid.

Since you can control the reference of the new subject, then you just explicitly give it the reference it implicitly had as the old predicate, without changing it. This is the point of Rule 3.

Rule 4 is based on the fact that you can't control the reference of the new predicate, because it doesn't depend on what it was before, but on whether the copula is affirmative or negative. Hence, this might necessitate a change in its implicit reference. And you can't conclude to a definite reference (knowing what the objects referred to actually are--being able to point to them individually) from an indefinite one (which supposes that you don't know which ones they are).

That's why you can't "conclude to the universal from the particular"; it's not that you're concluding to a larger class from a smaller one, it's that the "particular" is known not as an object, but only indefinitely, in its relation to the class (i.e. as belonging to it, not as "this thing"); and you can't point to something if you couldn't point to it before.

To take a couple of examples, "Every horse is an animal" converts into "At least one animal is a horse." "Every man is not an island" converts into "Every island is not a man." "At least one human being is a typist" converts into "At least one typist is a human being." But note that "At least one human being is not a typist" can't be converted, because "human being" would become the predicate of a negative proposition, and so definite; but it was indefinite before.

So essentially, what Rule 4 says is that indefinite negative propositions can't be converted.

There is one other thing to beware of in converting propositions: you must not base your conclusion on what you happen to know is true of the proposition as a statement, but only on the proposition as it stands. That is, you might know that only human beings can laugh (because spirits have no bodies, and non-human animals can't understand, and so their laughing sounds aren't real laughter). Hence, you might be tempted to convert the proposition, "Every human being is a laughing thing" into "Every laughing thing is a human being." But this doesn't follow, even though it happens to be true, because "human being" is now definite, whereas before it was indefinite. You can see that it doesn't follow from substituting "mortal thing" for "laughing thing."

The other operation with a single proposition changes its "quality," or the affirmativeness or negativeness of the copula.

Obversion is the logical inference involved in changing the copula from affirmative to negative or vice versa.

The obverse is the conclusion of an obversion.

Rules for obversion:

1. Leave the subject alone.

2. Change the copula from affirmative to negative or vice versa.

3. Add a negative to the predicate term.

4. Cancel pairs of negatives.

In Rule 3, the negative added to the predicate (a "non-" if it is a single word, or a "not" in some clause within it) is to make it "refer" to the contradictory class of objects from the preceding predicate (i.e. to the class of "everything else but" that one). Here, there is no need to worry about the predicate's changing from indefinite to definite (as it will if the original proposition was affirmative), because it is a different predicate, and the new predicate doesn't "refer" to the original class at all, but to an entirely different set of objects.

For example, the obverse of "Every human being is a mortal thing" is "Every human being is not a non-mortal thing." Here, the class of non-mortal things is definite, while that of mortal things is indefinite; but as you can see intuitively, if every human is within the class of mortal things, this will put every human outside the class of "everything but mortal things." Similarly, the obverse of "Every human being is not an island" is "Every human being is not not a non-island," which, by Rule 4 becomes "Every human being is a non-island." In this case, every human's being outside the class of islands automatically puts every one within the class of non-islands.

The indefinite propositions are the same: The obverse of "At least one human being is a typist" is "At least one human being is not a non-typist," and similarly, "At least one human being is not a typist" becomes "At least one human being is not not a non-typist, and, canceling the double negative, "At least one human being is a non-typist."

The only fallacy to watch out for in obversion is using the contrary class instead of the contradictory. The contrary is the opposite class on the scale, as in white is the contrary of black; and the assumption in contraries is that there are things in between. Contradictories exhaust the whole universe, as non-black is the contradictory of black, and involve all the other objects there are, whether they are in the category in question or not. For instance gray things, red things, things weighing two pounds, dogs, and even nothingness are all included in the class of "non-black."

Note that two obversions in a row get you back where you started, while this is not the case with conversions, since the references change because of the shift from subject to predicate.

Logicians talk about other operations such as "contraposition," but these are just alternate conversions and obversions, and have nothing special about them. They do, however, show how many different propositions you can generate just from one original. For instance:

"Every human thing is a mortal thing" obverts to

"Every human thing is not a non-mortal thing," converts to

"Every non-mortal thing is not a human thing," obverts to

"Every non-mortal thing is a non-human thing," converts to

"At least one non-human thing is a non-mortal thing," obverts to

"At least one non-human thing is not a (non-non) mortal thing," which cannot be converted, because "non-human thing" would become definite from being indefinite.

This is perfectly straightforward, following the rules; Note that we said "non-mortal" and not "immortal" (which is the contrary of "mortal"), because stones are not mortal, since, not being alive, they can't die, but by the same token, they're not mortal either. Note also that it takes a good deal of puzzling to think out whether "At least one non-human thing is not a mortal thing" actually follows from "Every human thing is a mortal thing" or not. That is, is it actually the case that if in fact every human being is mortal, it can't be false that there's a non-human that isn't mortal? (Always supposing, as we said earlier, that there are humans, non-humans, mortals, and non-mortals. The rule is that all classes have at least one member. It does not follow, of course, from the mere fact that every human being is a mortal thing that there are any non-mortal things.)

Logicians talk about the "Square of Opposition," which consists of the four possible propositions with the same subject and predicate: that is, the definite affirmative, definite negative, indefinite affirmative and indefinite negative. These propositions are related among each other in interesting ways; but I will discuss them after we discuss the various ways of joining propositions into a compound proposition (because the "square" happens to embody all the ways you can join two propositions into a compound).