CHAPTER 8

LOGIC I:

PROPOSITIONS

[This subject is also treated in Modes of the Finite, Part 4, Section 2, Chapters 1-3.]

8.1. Science

The object of this chapter is to introduce the science of logic by first analyzing the structure of statements of fact.

We will discover in the next chapter how we can arrange them so as to get other statements which must be true if the original ones were; and then we will deal with logical operations involving multiple statements: the \syllogism."

But before we start, let us briefly pause to look at what a science is, and classify the various types of sciences.

DEFINITION: A SCIENCE is a set of factual statements on some subject together with the evidence for the statements' factuality, and the relation between the evidence and the statements' factuality.

DEFINITION: The SUBJECT-MATTER of the science is a set of objects related together in a certain way.

So every science deals with a definite subject-matter. Two sciences (like physics and chemistry) might deal with the same set of objects (bodies); but physics considers them as related by whatever produces \physical changes" and chemistry by whatever accounts for \chemical changes." The set of objects themselves are sometimes called the \material object" of the science, and the particular foundation of the relationship in question is called the \formal object" (because it is a kind of \form" that each object has).

This makes a science a definite \body" of knowledge. But in addition, the set of factual statements that constitute the \knowledge" of a science is not just a haphazard set of \known facts"; the facts are interrelated in that they depend on some kind of theory, which gives the evidence for their factuality.

Hence, a science is a \systematic" body of knowledge. If you have scientific knowledge, you know that X is true (e.g. that gases expand when heated), you know why you know that it is true (because in the one instance inductions have produced the \laws" of gases, and in the other, the Kinetic-molecular theory of heat explains why these inductions are based on the nature of a gas). Hence, you know what caused you to have this piece of knowledge, and why it is the cause, as well as how it fits into other related pieces of knowledge on the same subject.

Aristotle's definition of science as \knowledge through causes" amounts to the same thing; because the evidence for any factual knowledge is the cause of its being knowledge and not opinion.

8.1.1. Kinds of sciences

An analysis of science and scientific method is a whole science in itself, and it is not our purpose here to go into it.

I just want to give a brief classification of different types of sciences to show where logic fits in.

Theology differs from all other sciences in that the evidence it uses is the testimony of God. Its task is to understand this testimony, showing that the apparently contradictory statements aren't really contradictory, and that revelation does not contradict what is known from other sources of evidence (like philosophy or empirical science).

There are two sciences whose evidence is internal to the science itself: logic and mathematics. Logic uses nothing but statements, and does not investigate whether any of the statements are true or false; all it is interested in is whether the statement called the \conclusion" is one which cannot be denied without contradicting what was earlier said. (Of course, if what was said earlier is true, then the conclusion will then have to be true also, because you can't deny it without contradicting what is true. So logic has an indirect relation to the facts of the world.)

Mathematics deals with relationships and relata, and is basically the science of manipulating relationships so that new relationships are generated. Like logic, it is not interested in whether these relationships are facts or not, but in the characteristics of the type of relationship as such (e.g. \belonging to" \greater than") and what can be done with them based on these characteristics. Insofar these relationships express facts, then mathematics can be \applied" to the real world.

There are some philosophers who have tried to make logic a branch of mathematics, on the grounds that statements are ways of expressing relationships&&and others who have tried to make mathematics a branch of logic, on the grounds that manipulations of relationships is what logic does. But I don't think either of these has been successful, because statements are a peculiar kind of expression of a relationship (the indirect one which is the expression of the understanding of a fact); and because there are relationships which are not the same as what logic deals with, and whose manipulations are therefore different from those in logic.

Note that much of contemporary \symbolic logic" suffers from this mistake of confusing logic and mathematics.

The result is that, while with a certain forcing of statements, they can be made to fit what symbolic logic does, the logical sequence of contemporary symbolic logic is a different sequence from the logical sequences of statements, and it is not an expression of how we actually do logic. It turns language into a kind of mathematics, and is a very interesting, often fascinating, error.

Sciences which use facts learned from our five senses (i.e. from observation) are called empirical sciences. The most general of these is philosophy, whose evidence is the effects contained within our experiencing&&which shows how we can know facts at all. Hence, philosophy deals with \ultimate causes."

Other empirical sciences suppose that our observation gets us factual knowledge; and the evidence in these sciences is various effects connected with our factual knowledge. The different empirical sciences differ not only in the different objects they deal with but in the different types of effects that form their starting-points.

8.2. Statements

Let that sketch suffice for an overview of the different sorts of sciences there are.

Now let us, as we said approach logic by analyzing statements of fact. Every statement of fact expresses a relationship understood among a set of objects; and because of this, every statement of fact has to have two basic parts.

DEFINITION: The SUBJECT of a statement is the object or object-class about which the relationship is to be understood.

DEFINITION: The PREDICATE of a statement expresses the relationship to be understood about the subject.

In English, the subject of the statement generally comes before the predicate; those statements that have the predicate first sound \poetic" to us: \Blessed are the poor in spirit," where the objects you are talking about are the people that are poor in spirit, and what you are to understand about them is that they are \blessed" (i.e. they are the same as people we call \lucky" or \privileged.").

The idea here is that the word-group that functions as the subject of the statement calls to mind the objects that you are supposed to understand something about; hence, the subject just points.

Notice that the words used as the subject may do more (in themselves) than point; for example, \poor in spirit" has a meaning as well as pointing to a class of objects. But the statement above is not interested in the characteristic quality of \poverty", (i.e. what poverty is or what it is to be poor) but in pointing to the people that have that quality. The idea is that if you know what it means to be poor, you will be able to bring to mind a \generalized image" of \a poor person." In that sense, the meaning is used as a means for the pointing-function.

The predicate, on the other hand, is the word-group that expresses the meaning of the statement: what is to be understood about the subject, or what you are trying to say about the subject. Predicates include the verb of the sentence, which, according to grammarians, expresses the \action or state of being" of the subject: in other words, what the subject is doing (because if it just is or is like something, this turns out to be some activity). But what something does relates it to itself or other objects in an intelligible way.

Another way you can look on the predicate is that it is a kind of \adjective" describing the subject. For instance, \John is running" could be expressed in a peculiar but understandable way by \John, runner (now)," where you can see the relationship John has in a little better way than the verbal way of putting it.

Remember, actual statements in a given language can use words any way they please to get across these two basic functions. For instance, in English, we say, \I run" or \I am running," and Latin would express this same statement by the single word \Curro," where the \-o" ending of the word tells you that the speaker is talking about himself (as opposed to \Curris"&&\you run"&&or \Currit"&&\he/she/it runs"[no sexist problem here]).

8.3. Logic

That is, I think, enough about statements as they actually appear in language.

Let us now try to move into the science of logic.

DEFINITION: LOGIC is the science which arranges statements in such a way that the final statement cannot be denied without contradicting what was already stated.

The first thing to note here is the following:

Logic is not concerned with the truth or falseness of the statements it uses.

That is, logic is simply interested in generating a statement whose denial involves a contradiction of previously made statements; whether those previously made statements are true or are mistakes or lies is irrelevant. Now of course, as I said earlier, when we use logic (when we reason), we understand the truth of the conclusion based on the fact that it has to be true because we know the previous statements to be true (from extra-logical evidence); but the logic itself is self-contained and is not concerned with this.

In one sense, you could say that logic \assumes" that the original statements are true \for the sake of argument," and then concludes that the statement it generates is true (on the supposition of their truth); but this rigmarole obscures the fact that logic is the pure manipulation of statements and \assumptions of truth" and so on have no real part in it.

There are logicians who don't subscribe to this; for instance, some who say the statement \The present King of France is bald" is a meaningless statement because nothing is referred to by the subject (there is no King of France at the moment, so how could it be true or false that he's bald?). But I think this misses the point. The statement is no more meaningless than, \Frodo Baggins was a very brave hobbit," which refers to an imaginary character. There is no Frodo Baggins.

The logicians would say that I have made a false parallel here, because my present King of France contradicts the fact that France is a real country which has no King. I am aware of this retort, which I think misses the point; but I do not want to argue the matter in a book like this. The difficulty comes, I think, from trying to give some connection with facts to logic, and not leave it just a connection of statements. So let us say that the more reasonable theory of logic is that it doesn't concern itself with the truth of its statements, but simply their form and the fact that the conclusion can't be denied without contradicting the premises (the other statements).

8.3.1. The proposition

Now then, it turns out to be confusing to do logic with the actual languages that exist.

The reason is that logic uses its own rules for manipulating statements, and these rules are not quite the way we actually do things as we use real languages. That is, logic manipulates the subject and predicate functions to make new statements out of old ones; and so no matter what the language you use for logic, logic is always doing basically the same thing&&and yet it might appear very different because of the different grammatical structures of the different languages.

Hence, logic creates an artificial sort of language which can be \translated" easily into the particular language of the culture; but it is a language which makes it easy to see what the logical operations are.

Some logicians get away from words altogether, replacing them with quasi-mathematical symbols, which then can translate into any language at all. I think the extremes of this generate more confusion than they eliminate, because they make it almost impossible to do the reasoning which is symbolized. As I said at the beginning of the book, reasoning is doing logic and knowing what you are doing; but if all you have is a bunch of p's and q's and squiggles, you can do the logic all right by mechanically following the rules; but it takes a superhuman effort to understand what is being done to the statements you are supposed to be transforming.

We will be doing some symbolizing; but I hope to make it a clarification and simplification rather than an obfuscation.

Then let us begin building our artificial way of making statements.

DEFINITION: A PROPOSITION is a factual statement expressed in logical form.

Two things to note: First, it is a \factual statement" in the sense that it is that kind of sentence. We are not interested, as I have said so often, in whether it is true or false. It is called a \proposition" and not a statement precisely because it is \proposed" or \put forward" rather than actually asserted as factual or claimed to be factual.

Secondly, whether the proposition looks like a statement in any real language is irrelevant. \Logical form" means that there is a special grammar for the language of logic&&though in general, it borrows words from existing languages (except in symbolic logic, where the symbols are the logical words).

To create a proposition, we have to know something about the grammar of ordinary languages:

NOUNS in existing languages have in themselves either or both of the functions of factual statements: they can (1) point to an object or object-class, and they can (2) express the relationship among the objects that they point to (or the foundation of that relationship).

Thus, the word \horse" points in the statement, \a horse is an animal"; and it means (relates) in the statement, \a maverick is a horse." Depending on whether the noun is the subject or the main part of the predicate of the statement, it is used in its referent (pointing) or its meaning function.

Now then:

DEFINITION: A TERM in logic is a word or word-group that is used as a noun.

Terms in logic, then, either point to objects or sets of objects, or express relationships among objects (or foundations of a relationship).

A term may be a single word, a phrase, a clause, or even a set of interconnected clauses. They may be one or ten or fifty or any number of words long. The point is that the word-group is unified and has one of the two functions above in a proposition.

In the proposition, \Every person who is reading this book during the fall of 1989 is something that is human," the first term is \person who is reading this book during the fall of 1989," and the other term is \something that is human."

Notice that the language called \logic" sounds a lot like English; but it looks a little peculiar.

Two things to note:

1. A term always stays the same through logical transformations of propositions. This is true except for the tag called its \quantity" which we will discuss below (the \every" or \some" prefix that logic attaches to it. If you consider the quantity as not part of the term, then the term remains the same.

That is, terms are deliberately constructed in such a way that the term itself does not change when the propositions containing it change; in this way it is clear what the new propositions refers to and means.

2. A term is defined by the objects it refers to and the relationship among them.

That is, terms in logic have one and only one referent (object or class), and one and only one meaning.

The same word, therefore, can be more than one term.

Thus, \pen" can refer to the things you write with or the things you keep pigs in. But the term \pen" has only one of those referents when you use it in a logical process; and if you introduce the other use of the word, you have introduced another term (and often ruined the logic), as in the following fallacy (called the \4-term syllogism") \A Bic is a pen, and every pen is something that can hold animals, and so a Bic is something that can hold animals."

Terms exist in propositions in the following way:

Every proposition has two and only two terms: a subject-term and a predicate-term.

DEFINITION: The SUBJECT-TERM of a proposition is the word-group that is used in its reference-function in the proposition. This is the term that points.

DEFINITION: The PREDICATE-TERM of a proposition is the word-group that is used in its meaning-function in the proposition. This is the term that relates.

Every proposition has three and only three parts, arranged in the following order: First, the subject-term; second, the copula; third, the predicate-term.

DEFINITION: The COPULA of a proposition is the appropriate form of the PRESENT TENSE of the verb \TO BE."

That is, the copula (the \link") can be nothing other than \am," \are," or \is," depending, of course, on whether the subject-term is singular or plural and first, second, or third person.

Now the idea here is that, since nouns tend to have either of two functions, logical form enables the predicate of one proposition to become the subject of another proposition. In grammar, the verb is part of the predicate; in logic, the grammatical predicate is the copula-predicate complex.

An example of what is going on can be seen from this: Every maverick is a horse; and every horse is an animal; and therefore every maverick is an animal. In the first of these propositions, the term \maverick" is the subject and \horse" is the predicate; but in the second, \horse" is used as the subject&&which allows us (for reasons we will see) to attach the predicate \animal" now to the subject \maverick." Or we can say that since every maverick is a horse, then some horses are mavericks. Here the subject and predicate-terms are interchanged.

If you were using ordinary language, you couldn't do this. \Horses run in races" can't use the predicate as the subject of anything. \Horses run in races, and run in races is exciting, and so horses are exciting" is nonsense. But \horses are things that run in races, things that run in races are things that cause excitement, and so horses are things that cause excitement." That isn't profound, but at least it makes sense.

8.3.2. The "quantity" of terms

But what about those \every's" and \some's" that appeared and disappeared in the propositions?

These are not part of the term itself, but logical modifiers of the term. If they were part of the term itself, they would have to stay the same in the transformations.

DEFINITION: Terms are said to have QUANTITY in their reference-functions; the quantity of a term is of two types:

DEFINITION: The reference is DEFINITE if the exact objects referred to can be known from the use of the term.

DEFINITION: The reference is INDEFINITE if the exact objects referred to cannot be known from the use of the term.

Traditionally, \definite" reference is called \universal" (or individual) quantity; and indefinite is called \particular." I have several problems with these designations, however, and they are serious enough to make me abandon them. First of all, when we say \particular" in ordinary language, we mean \definite," and this is exactly opposite to the meaning in logic. Secondly, \universal" seems to refer to the class as a whole, and not to each member of it. I think the designations I have made are less confusing.

Now in both cases, the term itself will refer to an object or class of objects. But it can refer to the whole class or a definite (point-outable) subset of designable members of that set, or it can refer to a part of the set without specifying which individuals make up that part. In the last case, the reference is indefinite, even though the reference might be to some defined fraction of the class (e.g. \half of the students in this class" doesn't tell you who these are, in spite of the fact that you may know that there are ten of them).

\Ten students" is an indefinite reference; \These ten students" is definite. \Students" is an indefinite reference, while \all students" is definite. \Every student" is definite, as is \each student" or \any student." \Some students" is indefinite, as is \many students" or even \all but one student" (because you don't know which one is left out, and so you don't know exactly which ones are in). \All but this student" is definite.

Let me list some words in English that indicate definite or indefinite references:

DEFINITE

This, that, these, those, the, all, any, every, each. Also \a," when it means \any example of" as in \A horse is an animal."

INDEFINITE

Some, one, ten (or any number without \this"), many, part of. Also \a" when it means some unspecified one of" as in \A man spoke to me."

But since English often uses words capriciously, it isn't a good idea to rely slavishly on the lists above; the criterion is whether the words tell you you could point to every one of the objects referred to.

All those \quantity-words" are used for translating English sentences into Logic, so that the term in Logic will refer to what the word as used in the English sentence does.

But

Blairian logic always uses \every" for every definite reference, and \some" for every indefinite reference.

Traditionally, \all" and \some" are the quantity-signals. But the negative \all" becomes \none"; and this causes confusion, for various reasons. \Every horse is not a cow" makes sense and is clear, and it means the same as \no horse is a cow." But \No horse is a cow" doesn't show clearly that the copula is negative, and some other formulations are even more confusing: \Not every horse is a cow" has an indefinite subject (it means that there is at least one horse that is not a cow&&but it doesn't tell you which one or ones).

My terminology circumvents this problem&&once you have translated your English (or French or whatever) statement into Logic.

Note that even proper names will have a quantifier. Since a proper name refers to a definite person (e.g. George Blair), then the reference will be definite, and in Logic it will read, \Every George Blair," as in \Every George Blair is a teacher of philosophy." (Note that this proposition is not the same as the statement "Every person named George Blair is a teacher of philosophy." "Every" is tacked onto a proper name simply for logical purposes, so that the term can be manipulated properly.)

Note that, since \every" and \some" are the only allowable quantifiers in Blairian logic, you are going to lose some information (sometimes) in translating from English to Logic. For instance, \All the students in this room but one wear ties" translates into \Some students in this room are things that wear ties."

It also means that in order to make the sentence in Logic readable, you might have to do some recasting of it: \All our ancestors" would translated into \Every one of our ancestors," for instance, or \Every ancestor of ours."

NOTE WELL: The Logical quantifier \some" means \AT LEAST ONE"; it does NOT imply \some are and some aren't." \Some horses are animals" is a true statement. Every horse is an animal, and if every one is, then at least one is. (Oddly enough, there is a controversy here, which I will discuss in a later chapter.)

Every subject of a proposition will have to have a quantifier in Logic

That is, every logical proposition will begin with \Every" or \Some."

8.3.3. The pseudo-quantity of the predicate

I said that the predicate-term (the noun-like term after the copula) is the word-group that is used in its meaning-function in the proposition.

If so, of course, it does not refer to a class of objects, but to a relationship they have or to the foundation of that relationship. Hence, the predicate of a proposition does not have a quantity, actually. This is something that some theoreticians of logic have not noticed.

Nevertheless, it is often the case that the logical manipulations of propositions moves a term from the predicate of a proposition to the subject of another one (when it is now used in its reference-function). In order to avoid saying more than you have a right to, you have to know what quantity the predicate would have had if it had one. This I call the \pseudo-quantity" of the predicate term.

DEFINITION: The PSEUDO-QUANTITY of the predicate term is the objects the predicate would be referring to if it actually were referring to objects.

Again, this reference might be definite and it might be indefinite. In the proposition, \Every maverick is a horse," you are clearly not referring to the whole set of horses, but only some indefinite part of them (the ones that are mavericks). But in the proposition \Every maverick is not a pig," you know that the way pigs are related does not apply to mavericks&&and if this is true of pigs as such, then it applies to every pig; and so the pseudo-quantity is definite. You can see this if you say, \If every maverick is a horse, it does not follow that every horse is a maverick," but \If every maverick is not a pig, then every pig is not a maverick."

Oddly enough, the definiteness or indefiniteness of the predicate's pseudo-quantity does not depend on the definiteness or indefiniteness of the subject, but on the affirmativeness or negativeness of the copula.

RULE: If the COPULA is AFFIRMATIVE, the PREDICATE is ALWAYS INDEFINITE.

If the COPULA is NEGATIVE, the PREDICATE is ALWAYS DEFINITE.

This is not actually as arbitrary as it seems. A statement, as I mentioned, in fact expresses a relationship among the members of some class of objects; and the predicate indicates either the relationship itself or (more often) the aspect that all the members have in common.

But this aspect is very often an aspect of what in in fact a larger class of objects than the one that you picked out to point to for the subject. Hence, implicitly, in an affirmative statement, the predicate brings in a larger class of objects, of which the subject-class pointed to forms an indefinite part.

On the other hand, a negative statement says that the objects referred to by the subject are not related in the way mentioned in the predicate, or do not have the aspect which is the foundation of that relationship; hence, no one of the subject-things belongs to the class of objects implicitly referred to by the predicate.

This is why many logicians think of logic in terms of the mathematics of class-inclusion. There is a kind of implied class-inclusion or class-exclusion in what we do in statements; and for logical purposes this is useful. But it must be remembered that logic is supposed to reflect what happens with statements, and so it is not simply class inclusion.

That is, the predicate does not really refer to a class of objects; but the fact that it does so allows us to exploit the class that it implicitly brings in in order for us to be able to use predicate-terms (in our peculiar logical form) as subject-terms of new propositions, and so to draw logical conclusions.

In any case, you must keep in mind that the \quantity" of the predicate-term depends solely on the affirmativeness or negativeness of the copula, and has nothing to do with the quantity of the subject.

Thus: Every maverick is (some) horse. Some horses are (some)brown things. Some horses are (some) animals. Every horse is not (every) pig. Some horses are not (every) pig.

8.3.4. Some notes on translation

Once a statement has been transformed into a proposition and is in the language I am calling Logic, things should be simple and clear.

The problem is often getting the English statement into Logic without changing what the statement says.

If you think, you can generally do the translating without too much trouble. Basically, you have to ask yourself two questions: \What is the statement talking about?" (Alternatively, What objects is the statement referring to?") Put brackets around all of that; it is the subject of the proposition in Logic. \What is it saying about the subject?" This will be all the rest of the sentence; put brackets around it to make it the predicate-term.

Introductory adverbs and adverbial phrases and clauses belong to the predicate. E.g. \In the beginning, God created the heavens and the earth." What is the sentence talking about? \Creation," you say. No. It is talking about God. So the subject is just God and the predicate is \in the beginning created the heavens and the earth." Creation is the general subject-matter the statement is dealing with; but it is talking about what God did, and hence about God.

RULE: The subject of a statement will always be a noun or pronoun with its accompanying adjectives (which may be phrases or clauses).

To translate into Logic, remove any quantifying modifiers in the statement, and put either \every" or \some" in their place, depending on whether the reference in the statement is definite or not.

RULE: Choose the appropriate copula (is or are, is not or are not) depending on the quantity of the subject and the affirmativeness or negativeness of the statement to be translated.

Beware of making negatives in modifying phrases into negative copulas. \Every non-student is ignorant" is an affirmative proposition.

RULE: Form the predicate in the following way:

If the predicate of the statement was \is (are)+a noun" then it can simply be copied.

If the predicate of the statement was complex, then after the copula put \a thing that" or \things that" (depending on what makes sense) and add the rest of the predicate.

Don't panic.

It's simpler than it sounds, actually. \Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal."

In Logic:

\Some of our fathers are things that fourscore and seven years ago brought forth...etc." Or, if you think that Lincoln was referring to all of \our fathers," it would be \every father of ours is a thing that fourscore and seven years ago brought forth..."

\Now we are engaged in a great Civil War, testing whether that nation, or any nation..."

In Logic:

\Every one of us is a thing that is now engaged in a great..."

To avoid writing that whole mess every time you transform such enormous propositions, you can resort to symbolization, in the following way:

RULE: Use (e) [parenthesis, lower-case \e", parenthesis] to symbolize \every." Use (s) in the same way to stand for \some".

Reduce the subject-term to a single word that more or less is the same as the whole complex subject (you will be de-symbolizing at the end of the logical operations; this is just to remind you). Add this word to the quantifying symbol.

Symbolize an affirmative copula by - [space, hyphen, space]; symbolize a negative copula by n [space,lower-case \n," space].

Insert the appropriate quantifying symbol for the predicate term [(e) if the copula is n, (s) if the copula is - ].

Reduce the predicate-term to a word that means more or less what the statement's predicate means, and put that after the quantifying symbol.

That's it.

In Blairian symbolism, \Fourscore and seven..." becomes \(s)fathers - (s)bringers-forth"; and \Now we are engaged..." becomes \(e)we - (s)engagers."

It's a lot easier to work with these things than with the propositions containing all the words of the statement; and you can always substitute the whole subject and predicate for the words after you get through, and then of course translate back into English so that you get a result that sounds as if you didn't just move here from Afghanistan.

We are now in a position actually to do something with logic&&or should I say, \Every one of us is a thing that is now in a position actually to do something with logic"? [(e)we - (s)eager people].

Let's go. (Can't translate that into Logic, because it's an exhortation, not a statement.)

Summary of Chapter 8

A science is a set of factual statements on some subject along with the evidence for the statements' factuality; its subject-matter is the set of objects the science considers.

Theology investigates testimony by God. Logic investigates how statements link to force other statements; mathematics investigates relations as such and what is related. Logic is not really a branch of mathematics, nor is mathematics a branch of logic. Philosophy is an empirical science whose evidence is effects contained within the act of experiencing, and so it deals with ultimate causes. Other empirical sciences differ in the objects they deal with and in the particular types of effects they focus on.

Statements have two parts: a subject, which points to an object or class of objects, and a predicate, which expresses the relationship to be understood. The subject may or may not come first in an actual language.

Logic is the science of arranging statements so that new ones are generated which cannot be denied without contradicting what was already said. Logic is not concerned with the truth or falsity of the statements.

Logic transforms statements into propositions so that the manipulation of them will be easier; a proposition is a statement in logical form. Propositions have terms, which are words or word-groups which function like nouns (point to objects or express relationships); terms are so constructed that they remain the same through logical manipulation. Terms are defined by the objects they point to and the relationship they express, and have one and only one referent. The same word can therefore be more than one term, depending on how it is used.

Propositions have two and only two terms: a subject-term (which points) and a predicate-term (which expresses meaning). These two terms are linked by a copula, which is the present tense of "to be."

The quantity of a term is what it refers to. Its quantity is definite if the term is used so that you can point exactly to the objects it refers to; it is indefinite if it points, but does not point out each object it refers to. "Every" is used for definite references; "some" for indefinite ones. "Some" is taken to mean "at least one, but maybe all." Every subject of a proposition must have its quantity named.

Predicates have a pseudo-quantity, the quantity they would have if they actually referred to objects. The "quantity" of a predicate is indefinite if the copula is affirmative, and definite if it is negative.

To translate statements into propositions, find the objects referred to, bracket the words that do this function, put the appropriate quantifier before it; then find the predicate (the rest of the sentence), make a noun out of it, and separate it from the subject by the appropriate copula.

For convenience, propositions can be symbolized by reducing each term to a single word, using (e) or (s) for the quantifiers and - and n for the affirmative or negative copulas. The pseudo-quantity of the predicate should also have its symbol.

Exercises

A. Translate the following into propositions:

1. In the beginning was the Word.

2. In the beginning, God created the heavens and the earth.

3. I saw John running down the street yesterday.

4. Those of us who care about the meanings of words will not misuse the gift of language we have had bestowed on us by society.

5. Three of the basketball players, it is alleged, have shaved points because of drugs and money.

6. All the perfumes of Arabia cannot sweeten this little hand.

7. But in the sleep of death, what dreams may come when we have shuffled off this mortal coil must give us pause.

8. I regret that I have but one life to give for my country.

9. Never have so many owed so much to so few.

10. Even though, my noble Theophilus, there have been many attempts to give a description of the events that have taken place among us--apparently based on what we have been told from the original eye-witnesses who dedicated themselves to the service of what they were affirming--I still thought it would be useful to research the whole matter from the beginning and write you the results of a careful study, so that you would know what would be safe to consider factual in what you have been told.

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