Propositions and their parts
The whole discussion about the truth and meaningfulness of statements was necessary, because even though logic doesn't care whether a proposition is true or not, it must be a statement, and so it must be either true or false. And meaningless "statements" are just not statements, because they can't express a judgment.
But because propositions are statements to be arranged in such a way that they generate other statements, then it would not be surprising to find that propositions were placed in a stylized form that made them easy to manipulate.
Logical form is the form into which a statement is cast to make it a proposition easily operated on in logic.
In logical form, a proposition has three parts:
The subject of the proposition is the term that refers to a class of objects.
The predicate of the proposition is the term that expresses the proposition's meaning.
The copula is the present indicative active of "to be" used as a "link" between the subject and the predicate.
A term is a word or group of words which functions grammatically as a noun.
Note carefully that though what I am calling the "subject" and the "predicate" of a proposition are terms, they have different definitions from what I will later call the "subject-term" and the "predicate-term" of a categorical syllogism (which are the terms that form the subject and predicate of its conclusion, though they may not be so in the premises). I want to mention this early to minimize confusion as much as possible.
Let us look, then, at the difference between a proposition and a statement. A statement, first of all, has only two parts, not three; if there is a verb "to be" in it, this is part of the predicate, not a "link" between subject and predicate. In a statement, as in a proposition, the subject is the word-group that calls to the mind of the hearer the object(s) he is supposed to be seeing a relationship among or within; but in the predicate of a statement, a verb is always included, indicating the act of this subject--in its relation to itself or to other acts. Thus, "John talked for two hours" refers to John and means the act of talking (i.e. he did what other people do when they make articulate sounds, and he did it for this length of time). So the hearer understands the fact that John performed this act, which he understands as in the past, as the certain kind of act, as lasting that long.
Statements often cannot be taken over into logic as they stand, because stylistic considerations often cast them in a form which is clear enough, but which does not easily reveal how they go together to generate conclusions. Even when the progression of statements is obviously logical, the word-groups get transformed in the process (repeating phrases exactly is generally bad English style), and the logic is rather connected with the progressions of meanings rather than staring at you out of the words.
For instance, if a person said, "I can't stand John; he talked for two hours, and I hate people who do things like that," it is perfectly clear from the meaning that the first statement is a conclusion based on the combination of the last two. But "John" gets changed to "he" and "talked for two hours" to "do things like that." On the other hand, saying, "John is a long-winded person, and a long-winded person is a person I can't stand; and so John is a person I can't stand," it becomes perfectly clear that it would be self-contradictory for the speaker to say that he "stands" John.
Logic, then, has two functions: (a) to allow the easy manipulation of statements so that new statements can be generated, and (b) to reveal how the statements are being manipulated so as to test whether the logic is valid or not.
Notice that if the propositions above were arranged, "John is a long-winded person, and a person I can't stand is a long-winded person," the proposition, "John is a person I can't stand" doesn't follow, any more than it follows from "Horses have four legs and dogs have four legs" that horses are dogs. We will see why this is so later; the point I am making here is that the statements in their normal form can mask fallacies like this; and that is why propositions have a special form.
As can be seen from the valid inference above, the term, "a person I can't stand" is used as the predicate of one proposition and the subject of another; and it is this manipulation of subjects and predicates of propositions that is what Aristotle gave to the world as the "categorical syllogism." But we will, as I said, see this later. Our focus right now is on what the parts of a proposition are.
First of all, a term can be a single word, as long as it is a noun or pronoun (which functions grammatically as a noun); or it can be any group of words that performs the same function: a phrase, a clause, a complex of linked clauses, or what have you. Something like "he" would be a term, as would "John"; and so would, "The Queen of England" or "The red-haired man who is stooping down to pick up the package of Tums he just dropped out of his shopping bag."
One thing to note carefully, especially if the term is a single word.
Rule: The same word can be different terms, depending on the class of objects it refers to in the context of its use.
For instance a "pen" that you write with would be a different term from a "pen" that you keep pigs in. These are traditionally called "equivocal terms," but in my terminology, they are equivocal words, and are simply not the same term in any sense. The case is similar with analogous words; they are not analogous terms, but different terms, even though the analogous words have a common core of meaning. Thus, it is illogical to say, "My complexion is healthy, and what is healthy eats well, and so my complexion eats well" because "healthy" in the first sense means "a sign of health" and so refers implicitly to a different set of objects from healthy living bodies. Similarly, if words are taken in two different suppositions, as we saw in Chapter 4 of Section 3 of the third part 3.3.4, they are different terms; and this is why it is illogical to say, "Clint Eastwood is a star and star is a four-letter word; and so Clint Eastwood is a four-letter word."
Now I said that a term refers to a class of objects, which seems to imply that the word-groups above would not be terms, because they only refer to one object. But first of all, individual references aren't terribly productive in logic, and so aren't much used; but since they can be, the convention is to regard them as classes that have only one member in them--and so when you talk about John, you are talking about the whole class of John--that is, not all the people named "John," but the "class" which consists of this individual John; because it turns out that individual objects function logically in the same way as classes taken as a whole, as we will see shortly.
Terms, as nouns, have two possible functions: (a) they can refer to a class of objects, or (b) they can express what the relationship is among the objects in the class. The first I will call the reference-function of the term (its traditional name is the term's "denotation"); and the second the meaning-function (traditionally designated as its "connotation"). For instance, in Every human being is mortal" the term "human being" is being used in its reference function, to call up the class into your consciousness, and what is being said about it is the fact that it eventually dies. But in "Every being that can laugh is a human being," the term is being used in its meaning-function so that you understand that the beings that can laugh have also the relationship that defines what human beings are. In other words, as I said in Chapter 5 of Section 3 of the third part 3.3.5, the subject uses the word to bring up an image, and the predicate to bring up a concept to the hearer's mind. This is also--not surprisingly--true in logic; but there are oddities about logical form, because we want to be able to use the term (in different propositions) in its two different functions.
Many contemporary logicians think that the meaning-function of a term is adjectival, because it expresses a "quality" or "property" the subject has; and, of course, this is very often the case (the common quality whose possession relates by similarity all the objects in the class). But first of all, there are other relationships besides similarity, and secondly, predicate adjectives only "mean" and can't grammatically be used in a reference-function (You can't say, "Every blue is ..." unless you are thinking of it as a noun: "every blue thing is ..."). Hence, to exploit the double functions of terms, we want them always to be nouns.
And precisely because the term, even when used in its meaning-function, also has (in itself) a reference-function, which we might want to exploit, logic tends to concentrate on the reference-function. This is the reason why I said that terms are different depending on the class of objects they refer to, rather than that they are different depending on whether they have the same meaning. But because terms are sometimes used in propositions in their meaning-function, then (as we will see) it becomes tricky to say what objects they would be referring to at that moment if they were referring to objects rather than expressing meaning. But to discuss this, we have to discuss subjects and predicates.
Now as I said, the subject is the term used in its reference-function, to point to the class of objects in question.
But in this pointing, you may or may not be pointing to the whole class, and so we have to make a distinction here. Strictly speaking, when you are referring to the "whole class," you are not referring to the class as such, but to every member of the class; and when you are referring to "part of it," you are not referring to a sector of it as if it were a pie you cut up, but to a number of the members of it. Logicians talk about "extension" of the term, (or its "quantity") and refer to the "distributed" or "undistributed" use of the terms; and call the "distributed" use (dealing with every member) the "universal term" (or the term used "universally") and the "undistributed" use the "particular" term (or the term used "particularly").
But it turns out that this only leads to confusion, because ordinary language uses "particular" to mean "individual" (as in "that particular man,") and the logical meaning of "particular" is that the reference is to some indefinite number of members of the class. "That particular man" meaning "that individual" would then be a universal term, as would "All of the students except these twenty-five."
So I think that the traditional terminology is misleading. If you know logic, what you realize is that references to definite individuals behave differently from indefinite references to individuals; and so let us call them by the names that show what we're talking about.
A definite term is a term in which the objects referred to can in principle be designated.
The word every is the primary sign of a definite term.
An indefinite term is a term in which the objects are known only in relation to the class they belong to.
The phrase at least one is the primary sign of an indefinite term.
I chose "every" instead of the traditional "all" for the definite reference (the "universal" one), because "all" can refer to the class as a whole (collectively) and not each member of it (distributively), as in "All the students weighed exactly one ton," which certainly means something different from "Every student weighed exactly one ton." Further, there is an ambiguity in definite negative statements using "all." For instance, does "All children are not mature" mean "Not all children are mature" (i.e. some are not mature) or "No children are mature" (not one is mature)? Grammatically, it should mean the former, since the proper form to deny a definite negative proposition is "No X's are Y's," while "Not all X's are Y's" simply denies the "universality" of the subject's reference, and "All X's are not Y's" is ambiguous. But "Every X is not a Y" is clear, as is "Not every X is a Y"; and these are the only forms possible using "every."
I chose "at least one" here instead of the traditional "some" because in ordinary language, "some" means "some are and some aren't," whereas the indefinite reference doesn't necessarily exclude the possibility that you might be referring to every member of the class; it's just that you don't know whether you are or not; and "at least one" has this connotation of not excluding "every." Hence, "At least one palomino is a pony" is obviously true, while it is not intuitively obvious that "Some palominos are ponies" is true if you happen to know that there is no instance of a palomino that isn't a pony.
Further, if you say, "At least one X is a Y," then you're grammatically in the same form as "Every X is a Y"; whereas if you say "Some X's are Y's," you're using the plural, and it's not as easy to leave the terms unchanged as you manipulate the propositions'--and logic becomes clearer the less you modify the terms themselves.
Rule: The subject of a proposition must be preceded by "every" or "at least one" to indicate whether it is a definite or indefinite term.
The proposition is called definite or indefinite depending on whether its subject is definite or indefinite.
So the statement "John spoke for two hours" becomes "Every John is something that spoke for two hours." Since John is one definite person, the reference is definite (and so the proposition is a definite proposition); and this means that the definite reference "every" must precede it. It makes it clear that John is now a class, and that every member of the class (the one member, John) is what spoke for two hours.
You are going to lose some information in this transformation of statements into propositions; the point is that you won't lose any logically relevant information; and once you have done your logic, then you can always "substitute back" if you do it carefully enough and get back a statement that looks more like standard English. For instance "All of the students but one" translates into logical form as "At least one student," because the words don't indicate which student was left out, and so you can't "point to" the ones that are left. "All the students but this one," however, is definite, and would probably translate simply into "Every student" if it were clear that what "student" now referred to was the subgroup.
Logic's purpose, as I said, is to make manipulation easy; and so the transformation process should not obfuscate more than is necessary. If it is clear what is being referred to, then let it stand; if not, add as few words as possible to the term to make it clear. For instance, if there might be a confusion between the students in the class above, then you could say, "Every sub-student" or something, defining "sub-students" as "The students in the class minus John."
In order to do the transforming, you have to know what sorts of words in ordinary English indicate definite and which indefinite references. Since this chapter is about logic rather than being a textbook on logic, I will simply mention that definite references are words like the following: this, that, these, those, the, all, any, every, each'--and a, when it means "any example of," as in "A horse is an animal." Indefinite references are indicated by the following: at least one, some, one, ten (or any number without "these") many, part of, a few, all but one (or any number without "these")'--and a, when it means "some unspecified one of" as in "A man spoke to me." Note also that "not every" as in "Not every dog has fleas" is an indefinite term; it is not the logical equivalent of "Every dog does not have fleas."
I suppose I have to say a few words here about contemporary logic and references. Bertrand Russell says, I think, that a statement like "The present king of France is bald" is meaningless, because if "the present king of France" is to be taken as a substitute for a proper name (such as Louis XVIII), then it is a term that refers but has no referent (because there is no king of France at present).
What I gather Russell's position is based on is a kind of naive realism where an "object" referred to is not a being but something like a "shaped patch of color" or in other words whatever "out there" (the set of energies) produced the percept.
Subjects of statements, therefore, can only be proper names, because they merely point and so there is no understanding connected with them at all. And since, I think he is saying, they point, then obviously (according to Russell) they have to have something to point to, or something that directly affects some sense organ. Everything else is actually a predicate. Hence, all objects allegedly pointed to by common nouns, such as "the human being over there" are actually implicit statements, of the form "There is an X such that X is a human being and X is over there," where X is simply a "place-holder" and "human being over there" are predicates describing X.
For him, references to every member of a class of objects have to have this form, because obviously you can't "point" in any meaningful sense to every single human being, say, or every dog. Hence, for contemporary logic, the proposition "Every human being is something mortal" becomes a hypothetical inference: "(For any X) if X is a human being, then X is a mortal thing."
Presumably, you are simply declaring this to be true, because I don't see how you could predict it unless you had actually checked out, not only every single human being, but absolutely every object in the universe, to see that all of them that were human beings also had the property of mortality. If this is true, then the little "place holder" (for any X) is just terminologically different from (for every X), and this has to be the equivalent of "every single thing there is or could be."
The interesting thing here is that contemporary logic's view of the inference in question allows it to be valid when the "if" sub-proposition is false; and so it is true (they say) that every unicorn has no more than three legs because there aren't any unicorns. That is "(for any X), if X is a unicorn, then X has three legs" is true for these people because There aren't any X's that are unicorns, which means that "X is a unicorn" is false, making the inference valid (and so the statement true).
Note that if you accept this way of looking at things, then it doesn't follow that if every human being is mortal, at least one human being is mortal. Why? Because the indefinite proposition can't "point to" an empty set of objects (since it says "at least one" is something or other), while the definite proposition can, on this theory (because as hypothetical it doesn't point, but only links two predicates). For example, it doesn't follow from "Every unicorn has three legs" on this view that "At least one unicorn has three legs," because "every unicorn has three legs" is true given that there aren't any unicorns at all, while "at least one unicorn has three legs" isn't true because there aren't any unicorns.
But that's silly.
I will get to the business of the validity of a hypothetical inference later; but the reason why definite propositions ("universal" ones) are turned into hypothetical inferences is basically the same as Russell's claim above, that any reference can't do anything but point and therefore must be a pure demonstrative (like "this" or "that" or some proper name) without any content that could be understood, since that would make it a predicate.
But this would make it impossible to say what you are talking about when using a common noun like "human being," because as having content it merely means the "quality" of humanity, and doesn't point to a class of objects. If you are pointing, you'd say, "That's a human being, and that's a human being, and that's a human being..."; but then if you say "the human beings" you're not re-pointing at the objects you pointed out, you're saying "These exist, and they are human beings."
Now granted, you can transform any subject of a statement (if it isn't a pure demonstrative) into the predicate of a different statement that has a pure demonstrative for its subject (or into a hypothetical inference); but that's a far cry from saying that that's what subjects of statements "really are," and that common nouns merely express concepts and don't also link up to a generalized image and so point to a class of objects.
That is, I when you hear, "Every human being is mortal," you don't go through a little process in your brain searching through its files and first picking out all the X's that you understand as human beings and giving each of them the additional quality of being mortal. When you hear the statement, the word "human being" simultaneously links up to the generalized image you have of a human being (that set of nerves that is activated when you see human beings, which could be conjured up sensitively as a generalized image), and recalls the concept. Since in this case, the concept is not relevant, you ignore it and understand that everything referred to by this generalized image has also the relationship of mortality (which is the concept you understand in this judgment). To put it another way, you don't understand anything at all about the meaning of humanity in this proposition; the only meaning it has is that of mortality.
If Russell's view is true, then when a woman says, "Those three people stole my purse!" she is really making the compound statement, "Those are three and they are people and they stole my purse." Even if she might be trying to convey that there are three of them, clearly she has no interest whatever in making the policeman understand that they are people, and anyone who claims that this is what she is really saying simply doesn't know how we use language. For a woman in this situation, "Those three people stole my purse" is exactly equivalent to "Those three stole my purse," or even "Them! They stole my purse!" The only function of "three people" here is to make the pointing more accurate--to single out the objects pointed to verbally from the background information that is also coming into the policeman's eye.
I rest my case.
Let me therefore make the following rule:
Rule: For logical purposes, it is to be assumed that classes referred to are not empty.
That is, if you are using a term in its reference-function, then for logical purposes it refers to something; and when it is used in its meaning-function it potentially refers to something for logical purposes. The only way you could know whether a term (like "unicorn") actually referred to something or not would be to check your experience; and this is extra-logical verification, something that we are precisely trying to avoid by using propositions instead of statements. Propositions are only "proposed as true for the argument," not stated as true in fact. The only verification we are interested in in logic is that connected with the "verification" of the conclusion based on the validity of the logic; and even here, whether the conclusion is factually true (i.e. true as a statement) is irrelevant, but only whether it follows from the premises (i.e. whether it must be true if they are statements of fact). Remember, in logic, we affirm and deny, we don't "recognize the truth" of something (even though, in using logic, we recognize the truth of the conclusion based on the knowledge that the logic is valid and the premises are true statements).
Hence, the proposition, "Every unicorn is something that has four legs" is not "taken to be true" (if you affirm it) because it's a fancy hypothetical inference, but because if you're going to be talking about unicorns, the rule above says that you're talking about something. But by the same token, the proposition "At least one unicorn has four legs" need not be denied (and in fact cannot, as we will see, be denied if the definite proposition is affirmed).
So much, then, for the subject of the proposition. The logical form of the copula is simple: it can be either affirmative or negative. Hence, there are only a few forms that the copula can take: am (am not) are (are not) is (is not).
The proposition is affirmative or negative depending on whether the copula is affirmative or negative.
That should be obvious. But note that a definite negative statement can look like an affirmative proposition with a negative subject: "No horse is a dog" is a statement that translates into the proposition "Every horse is not a dog." I mentioned this, if you will recall, when I was justifying my choice of "every" and "at least one" as the reference-indicators (the "quantifiers," in traditional terminology).
This points up the fact that negatives can appear all through the proposition--in the subject, in the copula, and in the predicate; but the proposition is negative only if the copula is negative, irrespective of the negativeness of either the subject or predicate or both. For example, "At least one non-horse is a non-dog" is an affirmative proposition, as would be its expansion, "At least one thing that is not a horse is something that is not a dog." Even though "not" appears here, the "not's" are in clauses forming parts of the subject and predicate respectively, and do not affect the copula.
Tense is not included in the copula; the present is actually an "aorist," or timeless use of the verb. Any tense in the statement has to be translated into a clause or phrase in the predicate.
The predicate of the proposition must be recast as a noun so that it can be used as the subject of a different proposition. Traditionally, the predicate could be a noun or an adjective; but adjectives cannot be used as they stand as subjects of other propositions, and so they should be ruled out as predicates. Thus, "the cars are all red" translates into "every car is something red." For instance, as we will see shortly, from "every car is something red" you can get "At least one red thing is a car"; but "Red is a car" obviously doesn't follow. (Note, by the way, that "every car" in the proposition refers to the definite class of cars referred to in the "the cars" of the sentence.)
It will not necessarily be obvious what words in a given sentence are to be translated as the subject of the proposition and what is to be included as the predicate; this will depend on what you think the sentence means.
For instance, "Fourscore and seven years ago, our fathers brought fourth upon this continent a new nation" can be translated variously depending on whether you think Lincoln was talking about the fathers (and meaning what they did) or what the fathers did, or what they "brought forth" or when they did it. Thus, depending on your interpretation of the statement, the proposition might be "At least one of our fathers is something that brought forth...," or "Every thing [i.e. that definite thing] our fathers did fourscore... is an act of bringing forth...," or "Every thing our fathers brought forth fourscore and seven years ago upon...is a new nation," or "Every fourscore and seven years ago [that definite time] is the time when our fathers..." By my reading of the speech, what Lincoln was driving at was the third meaning; but in different contexts, the others might also be legitimate renditions of the statement. The point is that it is not cut-and-dried.
Obviously also, once you translate a statement into a proposition, it is apt to look funny.
But since the translation's function is to make it easy to do logic, then a complex statement like Lincoln's, with all of its qualifying phrases and clauses, should be reduced to the simplest form possible consistent with not losing anything that is logically relevant. Hence, "Every brought forth thing is a new nation" would probably serve as a reminder of what the statement is; and once the logical manipulation is over, the reverse substitution could be made to make a statement out of the conclusion, referring back to Lincoln's statement rather than the original proposition.
To take one more example, the statement of Jesus is traditionally translated into English as "Blessed are the poor in spirit," where the Greek word-order is used, and the subject of the statement comes last. As a proposition it would read "Every poor in spirit thing is something blessed."
But how do you know what reference to give the predicate term? It doesn't really seem to have one in the propositions I have stated so far.
Ah, here is one of the secrets of logic. Textbooks will give you rules indicating the "extension" of the predicate. I will also give them, and reveal the mystery of why the rules are what they are.
Rule: If the copula is affirmative, the predicate is indefinite; if the copula is negative, the predicate is definite.
So you don't need a word to indicate the reference of the predicate, because it is determined by the "quality" (affirmativeness or negativeness) of the copula.
Now why is this? Because, as I said earlier, the predicate doesn't actually refer to a class of objects, but to the relationship among the members of the class; and so it expresses the meaning of the proposition, not its referent. But since we want to be able to use the predicate as the subject of a different proposition, then we have to know what it would refer to if in fact it were pointing out a bunch of objects: is its pseudo-reference definite or indefinite?
In the proposition, "Every horse is a four-legged thing" what you are saying is that if you take any horse out of the class of horses, you will find that it is similar to anything that is four-legged. Then what does this say about horses and the class of four-legged objects? It should be obvious that it doesn't say that horses are the only four-legged objects there are; and so horses form an indefinite subset of the class of four-legged things.
Note, by the way, that an indefinite subset could be the whole set; it's just that you don't know this by what is actually said in the proposition. For instance, "Every proposition is a statement in logical form" has an indefinite predicate, even though there are no statements in logical form that aren't propositions (since the definition of "proposition" is "a statement in logical form"). But you don't know from the proposition that it is a definition.
Negative propositions, even indefinite ones, as I said, have definite predicates, because if the subject does not have the relation in question, then it belongs outside the whole class of objects the predicate would be referring to (it wouldn't be like any member of the class that has the relationship). Thus, "Every horse is not a dog" indicates that every single member of the class of horses is outside the class of dogs (since no horses have "dogginess" and every dog does). But even "At least one horse is not a palomino" means that at least one horse is totally outside the class of palominos (though there may be some other horses that are not).
The easiest way to remember this rule is that it is the opposite of what you would superficially expect. Since we tend to think of definite references as in some sense the "good" ones and affirmative copulas as the "good" ones, we would tend to infer that affirmative copulas "ought to have" definite predicates. But, as I said, the exact opposite is the case.Next