by Jonathan Dolhenty, Ph.D.
Ideas are the raw materials of knowledge but ideas are not in themselves true or false. There is no truth or falsity until we take two or more ideas, compare them, and express an agreement or disagreement between them. Only then can we speak of truth or error.
Truth and error lie in the judgment, not the idea. A judgment is an act of the mind pronouncing the agreement or disagreement of ideas among themselves. It is an act in which the intellect affirms or denys one idea of another.
The Nature of the Judgment
There are three things necessary for making a judgment. First, the mind must understand the two ideas about which it intends to make a judgment. Second, the mind must compare the two ideas under consideration. Third, the mind must express mentally the agreement or disagreement between two ideas. This latter act constitutes the essence of the judgment.
Put in the simplest terms, we take one idea, let’s call it the subject, and we say something about it (with another idea), let’s call this part the predicate, and we compare the two ideas. We then pronounce agreement or disagreement between the two ideas.
But how do we determine if a judgment is true or false? The discussion of this question does not, strictly speaking, belong to the science of logic. It belongs to a branch of philosophy called epistemology, which is the philosophic study of knowledge in its most general sense. Logic deals with the validity of an argument, not specifically with the truth of an argument.
Nevertheless, a brief discussion of truth and falsehood may be appropriate. We have said that an idea is fundamentally a representation of a thing as it is in itself, independent of the mind. Since judgments are constituted of ideas, the judgment is also a representation of things as they are in themselves, independent of the mind. When our minds compare two ideas with each other and pronounces an agreement or disagreement between them, it actually compares two things with each other and judges about their agreement or disagreement among themselves as they are in reality. If a judgment coincides with reality, it is true and, if not, it is false.
The “test” of truth is, therefore, agreement of the judgment with reality. We verify a judgment by comparing it with the reality it is supposed to represent. We refer to this as objective evidence and this is the criterion of truth for us.
The Nature of the Proposition
Ideas are expressed in words which we call “terms.” Judgments, the agreement or disagreement between ideas, are expressed in sentences we call “propositions.” All propositions are sentences but not all sentences are propositions.
There are different kinds of sentences in our language. We ask questions and these are expressed in interrogative sentences. We issue a command or make a request and this is expressed in an imperative sentence. We express joy, surprise, or some other emotion, and these may be expressed in exclamatory sentences. These types of sentences are of no concern to logic.
The Structure of the Proposition
Propositions are a special kind of sentence for they must contain a judgment. A proposition may be defined as a judgment expressed in a sentence. And three elements enter into the construction of a proposition: the subject, the predicate, and the copula.
The subject is the term designating the idea about which the pronouncement is made. The predicate is the term designating the idea which is affirmed or denied of the subject. The copula is the term expressing the mental act which pronounces the agreement or disagreement between subject and predicate. The copula is usually expressed with a term such as “is” or “is not.”
It should be noted that the copula always expresses the present act of the mind and will always be represented by the present tense of the verb “to be.” Every proposition can be reduced to this present tense even though the proposition may refer to some past or future event. Example: “The Republicans did not win the last election” can be restated as “The Republican party is not the party which won the last election.” The meaning of the proposition has not changed, merely the form has changed.
Sometimes the verb “to be” is hidden in a sentence. A sentence like “The cat bites,” which appears not to contain a form of “to be,” should be restated as “The cat is biting,” which does contain a form of “to be.” The meaning has not changed, merely the form has changed.
Many times in ordinary language, a judgment will be expressed in a form that is unsuitable for logic. We have the right to change the wording of a proposition to meet the needs of logic as long as the original meaning of the judgment remains the same. Sometimes the form of a proposition may appear clumsy or unusual when converted to a proposition useful in logic, but we are not concerned here with beautiful prose but with the substance of the thought expressed.
We are so use to excess verbiage and pompous speech, particularly in the political arena, that it may appear impossible to deal logically with complex judgments and complicated arguments. It doesn’t matter, however, how complex a sentence is; if it expresses a judgment it can be reduced to a simple proposition including a subject, a predicate, and a copula. Complicated arguments may have to be reduced to set of simple propositions in order to make sense of them logically. But it can done.
General Types of Propositions
Truth and falsity are found in the judgment and proposition. A knowledge of the various types of propositions is necessary and there are general types and special types.
The general types of propositions are based on the quality, quantity, and the relation of subject and predicate found in the proposition and it is to these general types we now turn our attention.
The Quality of Propositions
The quality of a proposition affects the copula, making the proposition either affirmative or negative. The predicate is either affirmed or denied of the subject.
Consider the following propositions:
- A dog is a mammal.
- Politicians are verbose.
Both of these propositions are affirmative. The copula affirms the predicate of the subject.
Consider these propositions:
- A dog is not an invertebrate.
- Criminals are not good members of a society.
Both of these propositions are negative. The copula denies the predicate of the subject.
Sometimes a sentence will have two copulas, one in the main proposition and the other in a qualifying clause. Here are two examples: [the clause is within brackets]
- A man [who is sick] is not healthy.
- A man [who is not sick] is healthy.
In both these sentences, the clause affects the subject “man.” Are these propositions affirmative or negative? If the copula of the main proposition is negative, it is a negative proposition. If the copula of the main proposition is affirmative, it is an affirmative proposition. It is clear that the first sentence is negative because of the copula “is not,” which is negative. The predicate “healthy” is being denied of the subject “man.” The second sentence is affirmative because of the copula “is,” which is affirmative. The predicate “healthy” is being affirmed of the subject “man.”
When we run across sentences such as the above, which contain qualifying clauses, we must look to the meaning of the sentence. The meaning can usually be discovered by some slight change of the words (but be careful not to destroy the original meaning).
For example, the first sentence could be restated, “A sick man is not healthy,” and the second could be restated, “A not-sick man is healthy.” This clears up the proposition without changing its meaning.
Since the predicate affirms or denies something of the subject, how does this affect the comprehension and extension of the predicate? Does the comprehension and extension remain the same or are they changed in any way?
This is an important question and has a vital bearing on the validity of an argument. The relation of predicate to subject from this viewpoint needs to be well understood. Here are the rules to follow:
If we affirm, for example, that “Dogs are mammals,” what do we mean to assert by applying the predicate “mammals” to the subject “dogs”? Of course, we assert an identity between the two ideas of “mammal” and “dog.” Therefore, the comprehension of the idea “mammal” must be found in the idea “dog.” And that, in fact, is the case. We are applying the whole of the comprehension of “mammal” to the subject “dog,” because the definition of “mammal” is contained in the definition of “dog.”
It’s different now when we consider the extension. We don’t mean to assert by the proposition that the whole of the extension of “mammal” applies to “dog,” since that would mean that “dog” would fill out the whole extension of “mammal.” There wouldn’t be any other things contained in the class of “mammal” except “dogs.” But we know this isn’t true since human beings are mammals, as are cats, mice, and raccoons.
In an affirmative proposition we intend to assert merely that the subject forms a part of the extension of the predicate. In an affirmative sentence the predicate is taken only as a particular term (a universal term taken partly and indeterminately with regard to its extension). Another way of saying this is: the predicate in an affirmative proposition is not distributed and therefore not used as a universal. Note the words “not distributed,” as these will become very important later.
If we state, for example, that “Dogs are not reptiles,” we deny the identity between the predicate “reptiles” and the subject “dogs.” The comprehension of “reptiles” contains something which is not found in the comprehension of “dogs.” By denying that the whole of the comprehension of “reptiles” is found in “dogs,” we realize that part of the comprehension may be found in the subject. For instance, the ideas “animal” and “vertebrate” are found in the comprehension of “reptile” and also of “dog.” In a negative sentence, therefore, the whole of the comprehension of the predicate never applies to the subject, but a part of the comprehension does.
Also, in a negative proposition, the predicate is always taken according to the whole of its extension and denied of the subject. When we state that “Dogs are not reptiles,” we intend to assert that “dogs” do not belong at all to the class of “reptiles.” They stand completely outside the class, because every one of them (all dogs) do not have all the characteristics that “reptiles” have.
In a negative sentence, therefore, the predicate is always taken according to its whole extension as a universal and then denied of the subject. Both subject and predicate belong to totally different classes and neither one belongs to the class of the other.
The Quantity of Propositions
The quantity of a proposition affects the whole judgment as a judgment and it expresses the number of individuals to whom the judgment or proposition applies.
Since the predicate is referred to the subject, the proposition will be true of all the individuals contained in the extension of the subject. From the viewpoint of quantity, propositions will be universal, particular, singular, or collective, depending on the way the subject is taken.
- All male mosquitoes bite.
- Every cat purrs.
There can be no doubt about the term “every.” But the term “all” may be ambiguous. Does “all” mean “all taken collectively,” and apply to each and every member of the class?
If we say “All members of the club were present at the meeting,” we are using the term “all” distributively. We mean that “Every member was present.” But if we say “All members of the club filled the room,” we are using the term “all” collectively. We don’t mean that “Every member filled the room.” We have to look to the meaning.
The universal negative proposition is expressed by putting “no” in front of the subject, as in:
- No dogs are green.
- No man is an angel.
- No rectangle is round.
The following examples are particular propositions:
- Some dogs are fairly intelligent.
- Not all males are brash.
- Some politicians are verbose.
- Not all reptiles are poisonous.
Be cautious, however, about some sentences. Words can be deceiving. The sentence “All men are not drunkards” seems at first to be universal (because of the term “all” in front of the subject). But if such a sentence was universal, it would mean “No men are drunkards” and this is clearly not intended. What is meant is probably “Not all men are drunkards,” which is the same as saying “Some men are not drunkards,” and which makes this a particular (not a universal) proposition.
Consider the following propositions:
- That dog is huge.
- This man is our general manager.
- The president of the United States is not young.
Singular propositions have the same value as universal propositions and are treated the same way. The subject is taken according to the whole of the extension, which in this case is one.
Consider the following propositions:
- The Germans were defeated in World War I.
- The flock is flying south together.
- All his books filled his briefcase.
In these propositions we mean the Germans as a nation, the flock as a group, and all his books as a set of books. A collective term represents many considered as one. It is taken according to the whole of its extension and it, too, is treated as a universal.
Consider the following propositions:
- Children are tiresome.
- Men are verbose.
- Athletes are strong.
- Cops are well-trained.
These propositions indicate no definite quantity. They evidently mean “some” or “all” and are either particular or universal propositions. To determine the exact quantity, the propositions must be evaluated from the sense of the statement or the context in which they appear.
Since singular and collective propositions are equivalent to universal propositions, all judgments have the value of either universal or particular propositions. And as all propositions will be either affirmative or negative, we arrive at the following results:
- The universal affirmative
- The universal negative
- The particular affirmative
- The particular negative
|Every man is mortal.|
|No man is an angel.|
|Some men are kind.|
|Some men are not content.|
The Relations of Propositions
Another general division of propositions is based on the relation between subject and predicate.
The subject and predicate of every proposition have the relation of agreement or disagreement among themselves. This relation, however, may be either necessary or contingent. This means that the connection between the subject and predicate is either absolutely necessary and unchangeable or it is contingent and changeable.
- The whole is greater than any of its parts.
- Man is an animal.
- A square is a quadrangle.
We can tell just by looking at these propositions that the connection between the subject and the predicate is absolutely necessary and unchangeable.
The subject “whole” is related to the predicate “greater than any of its parts” by a necessary and unchangeable relation. We cannot say that the “whole” is “equal to” or “smaller than” any of the parts which makes up the “whole.” We know this simply by analyzing the meanings involved. The predicate must belong to the subject. The same holds true for “man is an animal” and “a square is a quadrangle.”
On the other hand, it is possible for the subject-predicate relation of propositions to be contingent and changeable.
- Salt is an inexpensive mineral.
- Alaska is the largest state in the United States.
- Cats are playful all their life.
We can tell that the predicate “inexpensive mineral” is related to the subject “salt,” but it is not necessarily related to it. Under certain circumstances, salt could be or become expensive. We only know the truth of the proposition from experience. A mere analysis of the subject and predicate terms is not sufficient.
The same is true of the other two propositions. There is no absolutely necessary relation between “Alaska” and “largest state.” Another state may be admitted to the United States and be larger in area. “Cats” are not absolutely necessarily playful all their life.
The difference between the two types of propositions, absolutely necessary (unchangeable) and contingent (changeable), is easy to see.
The first set of propositions involves something essential. By essential we mean the whole or part of the essence (species, genus, differentia) or something necessarily resulting from the essence (property). The relation between the subject and the predicate is such that the one is the species or genus or differentia or property of the other. One of the terms is contained in the comprehension of the other.
For example, a “quadrangle” is a plane figure with four sides and a “square” is a plane figure having four equal sides with four right angles. A “quadrangle” is the genus of the “square” and is contained in its comprehension. An analysis of “square” reveals the predicate “quadrangle” as part of the comprehension and and essence of the subject “square.”
The second set of propositions, in which the relation between subject and predicate is contingent and changeable, presents us with something different. This set contains only accidental attributes. The subject is not contained in the comprehension of the predicate nor is the predicate contained in the comprehension of the subject. The relation between the two is one of contingent fact only and while it may be actually so, it could be otherwise.
Salt may be an inexpensive mineral but it is not necessarily so. It could be otherwise. Cats may be playful all their life but not necessarily. It could be otherwise. These attributes of salt and cats are merely accidental and not part of the comprehension or essence of salt and cats.
We now come to some new words which will be used to designate these different relations between the subject and the predicate of a proposition.
You may recall that the relation between the subject and the predicate in the first set of propositions could be seen to be absolutely necessary and unchangeable. We could actually determine this relationship by an analysis of the terms. By analyzing the subject “square” and the predicate “quadrangle,” we could determine that there was an absolutely necessary relation between them. A square will always be a quadrangle. It cannot be otherwise.
If the relation of subject and predicate is necessary and unchangeable, we say the proposition is analytic (from, of course, the word analysis). Another term you may hear is a priori. This means the same thing. Analytic propositions are necessary, essential, and a priori. Knowledge is said to be a priori when it is obtained by reasoning from the whole to the parts.
We may, therefore, define an analytic proposition (or a priori proposition) as one in which either the predicate is contained in the comprehension of the subject, or the subject is contained in the comprehension of the predicate.
Now let’s consider the second set of propositions. These, as you recall, contained a relation of the subject and predicate which was contingent and changeable. The predicate was only accidentally (not essentially) related to the subject. We cannot determine this relation by analysis. We can do so only from experience. There was no absolutely necessary relation between the subject and the predicate; the relation was merely contingent and changeable. When this is the case, we say the proposition is synthetic. Another term you may hear is a posteriori. This means the same thing. Synthetic propositions are contingent, accidental, and a posteriori. Knowledge is said to be a posteriori when it is obtained by reasoning from the parts to the whole.
We may, therefore, define a synthetic proposition (or a posteriori proposition) as one in which neither the subject nor the predicate is contained in the comprehension of the other.
The Main Divisions of Propositions
The function of language is to convey thought and truth from one mind to another. The complexity of language, however, tends to cover up the truth of a judgment with words. Language is only an imperfect medium of expression. It is the imperfection of language which forces the mind to weave it into intricate textures of words and many times the truth is almost more hidden than revealed.
Since we tend to express ourselves in complicated sentences in a variety of ways using words which may be subtle and involve nuances of meaning, the task of the person who wants to think logically can become difficult. It is our job, then, if we want to be good logicians, to resolve these complicated sentences into simpler forms, so we can uncover the hidden truth of their meaning.
Truth, as we’ve already learned, lies ultimately in the judgment and the proposition. It becomes necessary, therefore, for us to learn to classify and analyze the various types of propositions. The two main divisions of propositions we will be concerned with are the single and the multiple, the categorical and the hypothetical.