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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 5.
PROPOSITIONS.
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.
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).
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.
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.
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. |
