Logic - Study Materials

Lesson 6: The Kinds of Statements and Relations of Opposition Between Them

In our last lecture we were introduced to the logic of the second operation. We noticed that, since the second operation is that by which we understand the true and the false, and truth and falsity always requires some combination or separation, then the second operation takes place through a combination and separation. The words which signify that operation, then, signify some kind of combination or separation. We called those words the statement. We defined the statement as a complex expression which is true or false and we said that it had two parts: the noun, which is the subject of the statement and which signifies something simple without indicating time; and the verb which is predicated of the noun and which signifies something simple implying time.

In this lesson we are going to look at the kinds of statements. What we mean by different kinds of statements are not statements with different parts, with a different noun and a different verb, but statements that are formed in different ways from each other, even though they have the same noun and verb. For example, the two statements "all men are mortal" and "no men are mortal" have the same noun and verb, but are different because they are formed in different ways and are different kinds of statement. After we have figured out the kinds of statements, we are going to look at the relations of opposition between them, how statements which have the same noun and verb can be opposed to each other. That opposition determines certain things about the truth and falsity of those statements.

Before we get to our two main topics, I would like to talk about the word "statement" itself. Other logicians often use the word "proposition" or "enunciation" instead of statement. I like the word statement because it is plain English. It clearly comes from the English word "to state" as in stating a fact and it connotes that we are talking about facts, things that are true or false. On the other hand, while the word "enunciation" is a Latin-based equivalent of statement, it suffers from having very different common meaning in English, a clear way of speaking, as in "his enunciation is very precise." The word "proposition" has a different problem. St. Thomas reserves the word "proposition" to refer to a statement insofar as it is part of a chain of reasoning. We are going to follow St. Thomas' practice and reserve the word "proposition" for the statements which is part of a chain of reasoning, and use the word "statement" for the statement taken by itself. Please keep in mind, however, the common equivalents to "statement."

The Kinds of Statements

Let us now determine the kinds of statements. Aristotle divides statements in three different ways. The first is not so much a distinction between kinds of statements as between the one and the many with regard to statements. Some statements are one, some look like one statement but are actually many. For example, the statement "Socrates is tan" is one, but "Socrates is wise and tan" is not one statement but many, although it might look like one. Aristotle explains it as follows:

We call those statements simple which indicate a single fact or the conjunction of the parts of which results in unity. Those statements on the other hand are separate and many in number which indicate many facts, and whose parts have no conjunction.

By Aristotle's account, "Socrates is tan" is a simple statement because it points to a single fact. Its parts, "Socrates" and "is tan," unite to make one whole. On the other hand, "Socrates is tan and wise" is two statements because it points to two different facts, Socrates' being tan and Socrates' being wise, one of which can be true without the other being true. In this case the verb is not one, and a noun and two verbs do not unite to form one whole. We can sum it up this way: if a statement has a single noun and a single verb, the statement is simple, otherwise it is complex.

To see if the noun and verb are simple is not always as easy as it seems: sometimes a noun or verb is simple, even though it is made of many words. For example, Aristotle considers "two-footed animal" to be a simple noun, even though it has many words, because the many make a unity. Why this is so, Aristotle points out, cannot be explained by the logician, but only by the metaphysician. Suffice it to say that a simple statement is made of one noun and one verb, however "one" is understood.

Since complex propositions can be reduced to simple propositions, Aristotle leaves them aside and focuses on the simple proposition. Then he proceeds to make the second division, a division of simple propositions into affirmations and denials, or negations. This division is very important to use because it is a division according to the form of the proposition.

What does it mean to say that this is a division according to the form of the proposition? The form of anything is what makes something be the thing it is. For example, the shape or form of the bronze makes the bronze actually be a statue. Now the fact that something is an affirmation or denial corresponds to what combines a noun and a verb into a statement. The noun and verb are a statement because they are joined. Thus the affirmation and negation relate to the form of the statement because they are two ways of joining a noun and a verb.

A statement can join a noun and verb in such a way that it points to the things they signify being joined in reality, and such a statement is called an affirmation. For example, the statement "Socrates is tan" points to Socrates and tanness and says that the two are joined in reality. The denial also joins the noun and verb, otherwise it could not be a statement, but it joins them in such a way that the joining in words signifies a separation in reality. For example, the statement "Socrates is not tan" still joins the words "Socrates" and " is not tan," but using the term "not" points out that the two things, Socrates and tanness, are separated in reality. A joining of words that indicates a joining in reality is an affirmation, a joining in words that signifies a separation in reality is a denial or negation. That is the first division of statements into kinds.

The second division of statements into kinds looks to the matter, the noun, of the statement. Aristotle distinguishes nouns that are individual from nouns which are universal. A universal is a word that can be predicated of many things, while an individual is not predicated of many. "Man" is universal, "Socrates" is individual. In the statement "Socrates is mortal" the noun is individual, while in "men are mortal" the noun is universal. Of course, in the latter statement, "man" is the subject and so is not actually predicated of something else, but it could be predicated of something else, and that makes it universal. Thus some statements are about universals, some about individuals.

Notice that this distinction has nothing to do with how the noun and verb are joined. In our examples, the parts are joined in the same way because they are both affirmations. The distinction has to do with the noun itself, which is a kind of matter for the statement. Thus, we could say that the division according to universal and individual is division according to the matter of the statement.

A comparison might help to clarify the distinction between a division according to form and a division according to matter. The sports baseball and cricket both use bats to hit the ball. The difference is that the cricket bat is flat, it is shaped like a plank with a handle, while the baseball bat is round like a large stick. The division of bats into baseball bats and cricket bats is a division according to shape or form, and this is like the division of statements formally into affirmations and negations.

On the other hand, we could divide baseballs bats into wooden bats and aluminum bats. This does not have to do with the form, since both kinds have the same shape. It has to do with the matter of the bats, what they are made of. That corresponds to the division between the statement which has an individual noun, and that which has a universal noun. That is the difference between the two ways of dividing statements.

Aristotle makes a division of those statements which have universal nouns. Some statements take the universal noun in all of its universality, others do not. For example, the statement "man is mortal" can mean two things. If the noun is taken universally, it means that "all men are mortal." If the noun is not taken universally, it means that "some men are mortal." The division of statements with universal nouns into those which take the noun universally and those that do not is the essence of the division between the universal statement and the particular statement. "All men are mortal" is a universal statement, "some men are mortal" is a particular statement. "Men are mortal" is called an indefinite statement, since it is not clear whether it is meant to be universal or particular.

Once again, this division is different from the previous one. Statements about individuals are not particular statements, because particular statements have nouns which are universal. Thus, "Socrates is mortal" is an individual, not a particular, statement, while "some men are mortal" is a particular, but not an individual, statement.

St. Thomas will speak about the case this way: the universal statement takes the universal noun in virtue of its universality. The particular statements predicates something of the universal noun, not in virtue of its universality, but in virtue of a part of it. That is why it is called "particular." It would be fitting to take some time here to talk about how the notions of whole and part work in logic.

Digression on the Whole and Part in Logic

Aristotle and St. Thomas themselves compare the relation between the universal and the particulars to the relation between the whole and its parts in quantified things. The universal is a word that can be predicated of many. A species, for example, is universal because it can be predicated of many individuals: Socrates, Plato, and Aristotle are all men. The genus is universal because it can be predicated of many species, and the individuals under those species: man, dog, Socrates, and Fido are all animals. But the genus has a greater universality than the species, because the genus is said of more things. Animal is predicated of everything that man is, and more: not only are Socrates and Plato animals, but so are Fido and Rover. Thus the more universal is greater than the less universal.

Now we see a similar relation in wholes and parts. The famous axiom is that the whole is greater than any of its parts. Since the relation of whole to part is greater to less, and that of more universal to less universal is greater to less, the word "whole" may have its meaning extended to apply to the universal, and the word "part" is extended to the less universal. For example, the species man is called "part" of the "whole" genus animal. And the less universal is called "particular" in relation to the more universal.

We need to understand, however, that the terms "whole" and "part" are used in an extended, or analogous, sense when we say that the species is part of the whole genus. The first meaning of whole and part points to physical or quantitative wholes and parts. The child wants the whole candy bar, we only give him part of it. The semicircle is part of the whole circle. Universal terms are not quantitative wholes and parts. There are important differences between the two ways of talking about wholes and parts.

First, the quantitative whole is made by putting its parts together, but the universal whole is made by taking something away from the parts. Thus, I can make a whole circle by putting two semicircles together, but I make the whole "animal" by taking something away from its parts, man and beast. That is, both man and beast are sensitive living things, but man is rational and the beasts are irrational. The genus "animal" cannot be made by putting man and beast together, otherwise the genus would have conflicting parts; that is, "animal" would be at the same time both rational and irrational. Rather, the genus comes to be when the differences between the species under it are taken away or put aside. Thus, when we say that man and beast are the species of the genus "animal," in the genus we take away from the species what makes them different, rational and irrational. "Animal" just means "sensitive living thing" and it leaves aside the question of rational or irrational altogether.

Second, the quantitative whole is never predicated of its parts, while the universal whole is always predicated of its parts. For example, I never say that a semicircle is a circle, but I do say that man is an animal. I must say that man is an animal, because being predicated of its parts is of the very definition of the universal: the universal is what is predicated of many. Thus while the analogy between the universal and quantitative whole helps us to understand the universal better, we must be careful not to confuse the two.

Return to Kinds of Statements

Now we can see what Aristotle means when he talks about statements which take the subject either universally or particularly. When the subject is universal, it can be taken as a whole, and we indicate that by the terms "all" or "every" or even "no." Thus, every man is mortal and no man is mortal both take the subject man universally. We can also take the subject particularly, that is, we can predicate something of the subject because some particular underneath the universal has that predicate, and we indicate this by words like "some" or "many" or "a few." For example, "some man is tan" is true, not because man as a whole is tan, but in virtue of that fact that some particular individual, let us say Socrates, is tan. Similarly, we can say that some animals are rational, not because rational belongs to the whole genus, but because part of the genus, the species man, has it. That is what makes statements universal or particular.

In philosophy we are for the most part concerned with statements that have universal nouns for their subjects. The philosopher is not so much interested in Socrates as in the universal "man," which signifies human nature itself. We are going to focus, then, on just two divisions of statements, that between affirmations and negations and that between universal and particular statements. The division between affirmations and negations is a division according to the quality of the statement, while that between universal and particular statements, since it is analogous to quantitative whole and part, is a division according to the quantity of the statement. When combine, or cross, the two divisions, we get four basic kinds of statements, as is shown in Chart #1.

The first kind is in the upper left, the universal affirmation. It has a universal quantity and an affirmative quality. An example is "all men are mortal." It is symbolized (we will see why it is symbolized this way later) by the letter A. The second kind is the universal denial. It has a universal quantity and a negative quality. An example is "no men are mortal" and it is symbolized by the letter E. The third kind is the particular affirmation. It has a particular quantity, but an affirmative quality. An example is "some men are mortal" and it is symbolized by the letter I. Finally, there is the particular denial. It is particular in quantity, negative in quality, and is symbolized by the letter O. An example is "some men are not mortal." Thus, there are four kinds of simple statement with a universal noun, the universal affirmation and denial, and the particular affirmation and denial.

One thing to note about the two particular statements is that they do not imply each other. The statement "some men are mortal" points only to a group of men and says that they are mortal. We cannot infer from it that "some men are not mortal." The first statement implies nothing about whether the other men are mortal or not mortal. The rest of the men might also be mortal, or they might not be. Thus, it could be the case that some men are mortal and others not, or it could be the case that all men are mortal. The particular statement, both affirmative and negative, speaks only about part of the whole, and implies nothing at all about the rest of the whole.

These are the basic, though not the only, kinds of statements which Aristotle covers in Peri Hermeneias. Later in the book he covers statements which have negative nouns and verb, such as non-Catholic, and also statements which talk about the possible and necessary, such as "it is possible that all men are tan." We are going to skip those discussions and focus on the four basic kinds of statements. Our next task is to talk about the relations of opposition between these statements.

Oppositions Among Statements

Remember, statements are concerned with the true and the false. The relations of opposition, then, are specified by how statements are related looking to their truth and falsity. What I am saying will become clearer when we look at examples of these relations.

Aristotle himself gives names for two of these relations, so we will cover those first. Aristotle writes:

If a man makes a positive and negative statement of a universal character with regard to a universal, these two statements are contrary. For example, Every man is white, no man is white.

When I have two statements which have the same noun and the same verb, and both are universal, but one is an affirmation and the other a denial, the statements are called contrary statements. We need to keep in mind that this sense of the term "contrary" is different from the one used by Aristotle in the Categories. There, contraries referred to simple things which were most different from each other in the same genus. White and black are contraries in the sense of that term used in the Categories. Here we are talking about something complex, the statement. Statements are not contrary in the sense in which that term is used in the categories, but because certain statements are opposed to each other in a similar way to how certain simple things are opposed to each other, by an analogy these statements are called contraries.

What is the similarity between simple contraries and contrary statements? Simple contraries are opposed so that one excludes the other, and yet there is a middle ground possible, so that at one time one contrary is present, at another time the other contrary, but at other times neither contrary is present. For example, sometimes a colored object is white, at other times it is black, and at still other times it is neither, but some intermediate color, such as blue. Contrary statements are like this. The truth of one of the statements excludes the other, yet there is a middle ground in which neither statement is true. For example, the statements "every man is tan" and "no man is tan" are contrary statements. The truth of one excludes the truth of the other. If the first is true, the second must be false, while if the second is true, the first must be false. There is, however, a middle ground because it is possible for both to be false: instead, it might be true that some men are tan, and some are not tan. Since statements opposed in this way exclude each other, and yet admit of a middle ground, they are like simple contraries and are called contrary statements.

We can sum up contrary statements as follows. Statements are contrary to each other when, having the same noun and verb, and both being universal, one is affirmative and the other negative. It is impossible for both contrary statements to be true, but it is possible for both to be false. Chart #2 is a variation on Chart #1 and is called the square of opposition. The contrary relation is that which goes across the top of the chart between the two universal propositions.

The next relation is called "contradiction." Aristotle writes:

An affirmation is opposed to a denial in the sense which I denote by the term "contradictory" when, though the subject remains the same, the affirmation is of a universal character and the denial is not.

Then Aristotle gives an example:

The affirmation "every man is white" is the contradictory of the denial "not every man is white." Or again, the statement "no man is white" is the contradictory of the statement "some men are white."

Two statements are contradictory when, while they have the same noun and verb, they differ from each other both in quantity and quality. This is different from contraries, which differ in quality but have the same quantity, universal. In the square of opposition, the contradictory relations are signified by the two diagonal lines. The universal affirmation and the particular denial are one set of contradictories, while the universal denial and the particular affirmation are the other set. Aristotle gives simple examples of contradictory statements: every man is white and some men are not white are contradictory to each other, and no man is white and some man is white are also contradictory to each other.

Contradictories have a similarity to the simple contradictories we talked about in the Categories, although I should note that the order of analogy is reversed from that of the contraries: the simple contradictories were called contradictory because they are like contradictory statements. The likeness is the following: with simple contradictories, it was always the case that either one or the other belonged to every subject. There was no middle ground. In the same way with contradictory statements, there is no middle ground. It is always the case that one is true and the other is false, even though we might not know which. For example, if "every man is white" is true, then "some man is not white" must be false. On the other hand, if "every man is white" is false, then "some man is white" must be true. We may not know which one is true, but we know that one always is.

We can sum up contradictories in this way: two statements are contradictory when, having the same noun and verb, they differ both in quality and in quantity. The rule that fits this relation is that, of every pair of contradictories, one is true and the other is false.

The later scholastics pointed out two other relations, which are shown in the chart. First, there is the relation of subcontraries. Statements which have the same noun and verb, and both are particular in quantity, but differ in quality, are called subcontraries. For example, "some men are white" and "some men are not white" are subcontraries. The rule for this opposition is the reverse of that for contraries: it is possible that both are true, but not that both are false. This rule follows from the two rules stated by Aristotle. Remember, as we said before, that one particular statement does not imply the other. Thus subcontraries do not imply each other.

The last opposition on our chart is between subalternates. Statements which have the same noun and verb, and the same quality, but differ in quantity, are related as subalternates. More strictly, the particular is the subalternate of the universal. For example, "some men are white" is the subalternate of "all men are white." The rule for this relation also follows from the rules for the first two relations: if the universal is true, the particular is also true, while if the particular is false, the universal is also false. It does not work the other way around: if the universal is false, that tells us nothing about the particular, and if the particular is true, that tells us nothing about the universal.

Since the last two rules follow from the first two, it is not necessary to memorize them because one can always figure them out when he needs to. For example, if "no man is tan" is true, then I can figure out its subalternate, "some men are not tan" is also true. For if "no man is tan" is true, then "every man is tan" is false by the rule of contraries, and then by the rule of contradictories "some man is not tan" must be true. I can do likewise with the subcontraries. That is why Aristotle himself lays out only the two basic kinds of opposition, and leaves the rest to be inferred by us.

The last thing I want to speak about is the necessity of logic for the second operation of the intellect. Of course, we do not need logic to teach us to make statements because we do that naturally. We even argue about things and oppose the opinions of others naturally. It might seem, then, that there is no real need for a logic in this operation of the intellect.

It is true that we naturally make statements, but the statements we make naturally are imperfect and imprecise. They do not fully convey the truths we wish to affirm. We might mean to affirm that every human soul is immortal, but our statement "the soul is immortal" does not necessarily convey that. The logic of the second operation teaches us how to make precise statements which convey exactly what we mean. Moreover, when we oppose what we consider falsity in the opinions of others, we initially do it badly because we do not understand how their statements should be opposed. When another says that no soul is immortal, we do not initially know whether we should oppose it by saying that all souls are immortal, or just that some are. Logic helps us to craft statements that precisely oppose the false opinions of others. The kinds of statements and the square of opposition are tools which we need to affirm clearly what is true and to deny precisely what is false.

Conclusion

In this lesson we have covered the fundamental kinds of statements and the relations of opposition that occur between them. We have distinguished them into four kinds according to quantity and quality: namely, the universal affirmation and denial, and the particular affirmation and denial. We have seen that the two universals are contrary to each other, but that statements which differ in both quantity and quality strictly contradict each other. We have then looked at why we need these tools for the second operation of the intellect.

For the rest of the course we will be dealing with the third operation of the intellect, discursive reasoning. In our next lessons we will begin considering the fundamental tool of the third part of logic, the syllogism. We will look at the definition and parts of the syllogism. Then we will look at the principles of the syllogism. And we will end by considering the fundamental rules of syllogistic reasoning.

Exercises

1. Every grain is nutritious.
2. Some heavenly bodies are unchanging.
3. Some heavenly bodies are not unchanging.
4. All pearls are valuable.
5. No gift is unwelcome. 
6. The arches of ancient Roman buildings are never pointed.
7. Every proof for the existence of God begins with creatures.
8. Metaphysicians are always also logicians.
9. Sometimes logicians are not metaphysicians.
10. A few scientists are also logicians.

Identify the kind of opposition between the propositions. If for some reason they are not strictly opposed, explain why.

1. Every grain is nutritious. - No grain is nutritious.
2. Some pearls are valuable. - No pearls are valuable.
3. No gift is unwelcome. - Some gifts are not unwelcome.
4. Metaphysicians are also always logicians. - Scientists are not always logicians.
5. Some churches are Gothic. - Some churches are not Gothic.

Mark the later statements as either TRUE, FALSE, or UNKNOWN, given the truth or falsity of the first statement. For example, if "Every grain is nutritious" is TRUE, then "No grain is nutritious" is FALSE.

1. If "Every philosopher loves wisdom" is TRUE, then:

Some philosophers love wisdom.
No philosophers love wisdom.
Some philosophers do not love wisdom.

2. If "Every philosopher loves wisdom" is FALSE, then:

Some philosophers love wisdom.
No philosophers love wisdom.
Some philosophers do not love wisdom.

3. If "Some forms are material" is TRUE, then:

Every form is material.
No form is material.
Some forms are not material.

4. If "Some forms are material" is FALSE, then:

Every form is material.
No form is material.
Some forms are not material.

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