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FUTURE LOGIC

© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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CHAPTER 5. PROPOSITIONS.

1. Terms and Copula.

2. Polarity and Quantity.

3. Distribution.

4. Permutation.

1. Terms and Copula.

Logic looks upon sentences as attempts to record or predict reality, which may or may not be correct. For this reason, it calls them propositions, to stress their fallibility. Logic develops by scrutiny of ordinary thought and language, but also sets especially rigid structural standards in order to be able to develop systematically.

Looking at many propositions, we see that irrespective of their particular contents, they appear to share certain ‘forms’. Our job is to analyze each form, how it is structured, what it means and implies, what are its interrelationships with other propositions, and how it can be known to be true.

Our study begins with one form shared by many propositions, ‘S is P’. Propositions of this sort are characterized as categorical, meaning that they are unconditional. We call ‘S’ the subject; ‘is’, the copula; and ‘P’ the predicate. The subject and predicate are both called terms. The copula relates the terms together in a certain way. We may view the subject as our center of interest, while the predication (copula and predicate) provides us with additional information concerning it.

Note well how the terms are treated as ‘variables’, while other features such as the copula (so far) are kept ‘constant’, like in algebra. In this way, we can theoretically concentrate on the properties of a kind of proposition, without regard to the specific ‘values’ which might take the place of the symbols S and P. Form is released from content.

We owe this artifice to Aristotle’s genius. In one stroke, it made possible the development of a science of logic, because the study of relations and processes was thereby greatly facilitated, as we shall see.

We will concentrate mainly on categoricals called classificatory. Here, the subject and predicate are classes, and their copula informs us that they contain members in common. Typically, in a general proposition, the subject is a species and the predicate a genus; for example, ‘trees are plants’. Other forms will be dealt with eventually.

2. Polarity and Quantity.

Propositions may be distinguished by the polarity of their copula. Thus, ‘S is P’ is said to have a positive copula; ‘S is not P’, a negative one. (Polarity is traditionally also known as ‘quality’, note, but since this word has other meanings it will be avoided here.)

We could view ‘is’ and ‘is not’ as two distinct relations (which happen to be contradictory), or as respectively the presence and absence of the same relation of ‘being’ (so that ‘is-not’ means ‘not-is’); logically, these viewpoints are equivalent.

The characterization of propositions as affirmations or denials has accordingly two senses, one absolute and the other relative. Normally, an assertion with a positive copula is called affirmative, and that with a negative copula is called denying; but also, we say of either polarity that it affirms itself and denies the other.

Another relevant distinction between propositions refers to their quantity. This primarily concerns the subject, clarifying how much of it we intend by our statement. The quantity is often left tacit in everyday discourse, but for the purposes of science, we have to be more explicit.

If S is a specific, recognizable individual, we use the designation ‘this S’, and the proposition is said to be singular (and indicative). Any proposition which is not singular may be called plural. If S refers to the whole class, we say ‘all S’, and the proposition is called general or universal. If S is a loose reference to some unspecified member(s) of the class, we say ‘some S’, and the proposition is called particular.

Other quantifiers define ‘some’ more precisely. Thus, ‘a few’ or ‘many’ mean, a small or large number; ‘few’ or ‘most’ mean, a minority or majority, a small or large proportion. These for most purposes have the same logical properties as particulars, though the latter two sometimes require special treatment.

By combining these different features, the various polarities and quantities, we obtain the following list of classificatory propositions. These are traditionally assigned symbols as shown to facilitate treatment (from the Latin words AffIRmo and nEGO, which serve as mnemonics).

A

All S are P

E

No S is P

R

This S is P

G

This S is not P

I

Some S are P

O

Some S are not P



The other quantities are also applicable to the two polarities, of course, as in ‘Few or Most S are or are not P’, but have not been traditionally symbolized.

All such propositions are called actual, because they suggest the relation they describe as taking place in the present. In that case, they imply that the units which their terms referred to do exist, i.e. that there are S’s and P’s in the world at the time concerned. This claim is open to debate, but will be taken for granted for now — later, we will clarify the issues involved, and look into the implications of not making such an assumption.

3. Distribution.

Plural propositions normally refer us to their class members each one singly; the plural is simply a shorthand statement of a number of independent singulars. Each individual, subsumed by the subject, and included in the all or some enumeration, is separately and equally related to the predicate. The predication is intended to be ‘dispensively’ applied; meaning severally, not jointly or collectively.

Thus, ‘All S are P’ or ‘Some S are P’, here means ‘S1 is P’, ‘S2 is P’, ‘S3 is P’, … and so on; ‘No S is P’ or ‘Some S are not P’ here means ‘S1 is not P’, ‘S2 is not P’, …etc. — until every S, this one, that one, and the others, which are included by the quantity have been listed.

The doctrine of distribution is that if all the members of a class are covered, the term is called ‘distributive’; otherwise it is not.

This means that the subjects of universals, A and E, are distributive; whereas those of particulars, I and O, are not, since the instances involved are not fully enumerated. With regard to singulars, R and G, they are effectively distributive, insofar as they point to unique subjects.

What of the distribution of predicates? The predicates of negatives, E, G, and O, are distributive, because P is altogether absent from the cases of S concerned ; while in affirmatives, A, R, and I, the predicates are undistributive, since things other than the cases of S concerned might be P.

These properties can be illustrated by means of Euler diagrams, named after the Swiss logician who invented them. In these, S and P are represented by the areas of circles, which overlap or fail to overlap to varying degrees. The reader is invited to explore these analogies. (Very similar are Venn diagrams, named after another logician; the latter differ in that they stress the areas outside the circles, the areas of nonS or nonP.)

Diagram 5.1 Euler Circles.

In A propositions, the S circle is wholly within the P circle, and smaller or equal in size to it. In E, the circles are apart, whatever their relative sizes. In I propositions, the two circles at least partly intersect, whether each covers only a part of the other’s area, or S is wholly embraced by P, or P by S, or they both cover one and the same area. In O, the two circles at least partly do not overlap, whether each only covers only a part of the other’s area, or neither covers any part of the other’s area.

The forms in current use, listed above, are so designed that we can specify alternate quantities for the predicate, if necessary, simply by making an additional statement, in which the original predicate is subject and the original subject is predicate, with the appropriate distributions.

As a result of the distribution doctrine, there have been attempts to invent forms which quantify the predicate, but they have not aroused much interest, being artificial to our normal ways of thinking.

4. Permutation.

Classification is a special outlook, but one we can use to develop Logic with efficiently, because it allows us to standardize statements. Classification is more mathematical in nature, and so easier of treatment, than other relations. The process of rewording a proposition, so that its terms are overlapping classes, is called ‘permutation’.

Note that, in formal logic, the word ‘universal’ is used in a quantitative sense, to apply to general propositions, which address the totality of a class. But in philosophy, a ‘universal’ is understood as the common factor, resemblance, similarity, which led us or allowed us to group certain units into a class; in this sense every term is a universal for its members, and even a particular proposition contains universals, except that they happen to be only partially addressed.

Likewise, the word ‘particular’ refers to less than general propositions, in formal logic; whereas, in philosophy, it is understood to mean concrete individuals, as distinct from abstract essences. Normally, the context makes clear what sense of each word we intend.

a. The equivocation of the word ‘universal’ is not entirely an historical accident. A proposition may have a ‘quality as such’ as its subject, and only incidentally imply a quantifiable subject-class. Thus, for example, ‘greenness is a (kind of) color’ and ‘all green things are colored’ do not mean quite the same, though their truths are related.

Propositions which have as their subject a quality as such, a universal in the philosophical sense, are virtually singular in format. To be quantified, their subject must be reworded somewhat. This is called permutation of the subject.

b. As Logic has developed, it has come to focus especially on the classificatory sense of ‘is’, because attribution, and other relations, can be reduced to it. Colloquially, the ‘is’ copula first suggests that the subject has a certain attribute, viz. the predicate, as in ‘trees are green’. But attribution is a more complex and qualitative relational format than classification, requiring more philosophical analysis.

Many propositions which normally are thought without the classifying ‘is’ copula, can be restructured to fit into it, while more or less retaining the same meaning. Thus, in our example, we would shift from the sense ‘trees have greenness’ to the sense ‘trees are greenness-having-things’. This is called permutation of the predicate.

Most logical processing of categoricals assumes that the statements involved have been permuted into classificatory form. Note well that permutation merely conceals the previously intended relationship in a new term, it does not annul or replace it. The difficult relation is once-removed, put out of the way; it is not defined.

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