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The Logician © Avi Sion All rights reserved
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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 19. MORE ABOUT QUANTITY.
In this chapter, we will look into various topics which involve
quantitative considerations.
3.
Collectives and Collectionals.. 4.
Quantification of the Predicate.
Substitution is a widely used, yet little noticed logical process, which
is open to formal treatment of sorts. It consists in replacing a term with
another which has the same units, but views them in a somewhat different
perspective. The entity referred to remains the same, only its label changes (qua
what it is referred to); the substitution is thus justifiable.
We may substitute a generic term for a species, if we keep the same
quantity, or a species for an individual. For instance: 'X has (some number of)
Y, All Y are Z, so X has (that many) Z', or 'X has this Y, this Y is Z, so X has
a Z'. Example: 'Man has a mind, a mind is an organ, so man has an organ (at
least one)'.
Note that this is not a normal, first-figure, classificatory syllogism.
Here, the major premise must be or be made affirmative and classificatory; but
the minor premise and conclusion are possessive (in this case). Needless to say,
verbs other than 'to have' are open to substitution, too. Example: 'Bill hit
Joan: Bill hit a woman'.
If the major premise is negative, it should be obverted before the
substitution. Thus, 'X has Y, No Y is Z' conclude 'X has nonZ', rather than 'X
doesn't have Z'. For example, 'Tom has a dog, a dog is not a cat'. Likewise, if
the major premise is not classificatory, it should first be permuted. Thus, in
'X has Y, all Y have Z', the term to substitute would be 'something which has
Z', rather than just 'Z'. Example: 'Tom has a dog, dogs have fleas'. We are said
to commit the 'fallacy of accident' when we make errors of this kind.
If the minor premise is negative, the conclusion must be formulated very
carefully, if at all. One is forced to keep the middle term in the conclusion,
only qualifying it with the major term, to ensure we do not change the implicit
quantity of units referred to. Examples: 'Tom doesn't have a dog: Tom doesn't
have an animal of the dog kind (though he may have some other animal)', 'Bill
did not hit Joan: Bill did not hit this woman (though perhaps another)'.
We often profit by substituting a subject. This process takes the third
figure form: 'Y is X, Y has Z, so an X has Z'. Here, the major premise may have
any copula or polarity (in this case we used 'has'), while the minor should be
affirmative and classificatory. Example: 'Joe is a man, Joe runs 40 miles a day,
so (there's) one man (which) runs 40 miles a day'. Substitution of a pronoun
would take this form.
Even the verb may be substituted, using an exact description of some
necessary aspect of it. Examples: 'the magnet was repelled 3 feet: the magnet
was caused to move 3 feet' or 'he sprinted to the finish line: he ran to the
finish line'.
What logicians call immediate inference by added determinants (e.g.
'horses are animals: therefore, the heads of horses are heads of animals') or
complex conception (e.g. 'Physics is a science: therefore, physical treatises
are scientific treatises'), involve substitutive syllogism, with a tacit minor
premise (e.g. 'horses have heads' or 'some treatises are about physics'), which
enables the conclusion to be drawn. These processes are illicit when the rules
of substitution are not properly obeyed (e.g. 'horses are animals: the majority
of horses are the majority of animals' or 'physics is fun, physical treatises
are funny').
Logic interfaces with mathematics, whenever we compare the number,
position, magnitude or degree of something, relative to something similar; or of
two things measured by reference to a third. We may call such propositions
'comparative'.
This concerns forms like 'X is more Y than Z', 'X is less Y than Z', 'X
is as Y as Z'. They affirm X to be greater, smaller or equal to Z, in some
respect Y (e.g. this metal is as strong as steel). It is implied that X and Z
are each Y, though to different extents.
Sub-categories of these measures may be defined by inserting more precise
quantities, like 'much more' or '30% more', say. Further complications are often
introduced, through the concepts of 'enough' and 'too much', which evaluate the
measurements in relationship to some goals.
Copulae other than 'is' may be involved, and the comparison may concern
the verb or an appendage (e.g. 'he ran faster than her' or 'they ordered more
food'). Often the comparative aspect is verbally concealed (e.g. 'she was
happier today' or 'that ball is the closest').
The corresponding negative forms are defined as follows. 'X is not more Y
than Z' means 'either X is not Y and/or Z is not Y or X is less or equally Y
compared to Z'. Similarly for 'not less' (= not at all, or more or as much), and
similarly for 'not as much'.
Since only one of the three affirmative measures may be true, and,
granting that Y is applicable to X and Z, one of them must be true, they are
contrary to each other. It follows that the negatives which contradict them are
subcontrary to each other.
Comparative propositions can be commuted. If X is (or is not) more/as
much/less Y in comparison to Z, then Z is (or is not) less/as much/more Y in
comparison to X, respectively. Example: 'he left just before sunrise' to 'the
sun rose soon after he left'.
Syllogistic style arguments can be constructed. For instances, using the
symbols of mathematics (>, =, <; and / for their negations), to signify
the possession or lack, of some common character, we can predict that 'If A>B
and B>C, then A>C' or that 'If A¹B
and B=C, then A¹C'.
In some cases, no deduction is possible; as for instances in 'A>B and B<C'
or 'A¹B
and B¹C'.
These relations are generally well known, and need not be pursued further here.
Comparative propositions are significant in so-called a-fortiori
arguments. These arguments are quite important and very commonly used in
everyday reasoning, but apart from a brief mention of them in a later chapter,
they will not be analyzed in detail in this volume. I hope to deal with them in
a later work. 3.
Collectives and Collectionals.
A proposition is 'dispensive', or 'collective', or 'collectional',
according to the way its subject subsumes its units for the predication.
a.
Most propositions are dispensive (many authors prefer the name 'distributive' instead), and this is the way of subsumption we have
dealt with so far, in detail. A plural dispensive refers us to its class members
severally, each one singly; it is simply a conjunction of a number of mutually
independent singulars. Thus, 'all or some S are (or are not) P', means 'this S
is P; that S is P;… and so on', until all the S intended to be included under
the all or some quantifier have been enumerated.
b.
In contrast, some propositions are collective,
applying a predicate to the units included by their subject only if they are
taken together, and not separately. Such propositions are effectively singular;
they conjoin the units into a group, rather than a class. A collective has the
form 'these S together are P', meaning 'this S and that S and…so on, taken as
one, are P'. Note that, unlike with dispensives, 'all S together are P' does not
imply that 'some S together are P'.
The group may have a summational property, which sums up the lesser
measures or degrees of the same property displayed by the parts (e.g. we each
have ten dollars, but both of us together have twenty). Or the group may have a
composite property, due to the causal interaction of the parts, which is not
found in any measure or degree in the parts themselves (e.g. as individual cells
together make up a human being, the whole having various powers the parts lack).
In the latter case, a certain arrangement of the parts may be tacitly required
for the predication to work, so that a statement more descriptive than mere
conjunction may be needed for accuracy.
The logical subject here is 'these S together'; we may, if we so wish,
form a new collective term from it (like 'crowd' or 'society'). Note that, in
some propositions, the intention is a dispensive summary of collectives (e.g.
'fifty books form a big pile' means any set of fifty books). This may be
formalized as follows (where 'n'
signifies some number): 'Any nS
together are P' or 'Certain nS
together are P'; here, 'nS together'
refers to a class of collectives.
c.
Some propositions are collectional. These differ from dispensives and collectives, in
that, although they refer to events each one singly, they also tell us whether
these can or cannot, are bound to or may not, be actual jointly —
simultaneously, at the same definite or indefinite instant or period of time.
This is usually signified by stressing the quantity (by the tone of voice or
italics).
Thus, here, 'All S can be P' means 'the conjunction of this S as P and that S as
P and… so on — is potential': this does imply the dispensive 'all S can be
P', but further reveals that the actualization of these potentials can take
place all at once. We would use 'All S
can be nonP', if we want to say 'it can happen that all S are simultaneously
nonP'.
Accordingly, 'All S cannot be P' denies the potential for simultaneous
actualization, the 'not' being directed at the 'all' (rather than at the 'can
be'): it is formally compatible with 'all S can be P' in a dispensive sense;
though usually used in such context, it does not imply it. We would use 'All
S cannot be nonP', if we want to say 'it cannot happen that all S are nonP at
once'.
(In contrast, the form 'All S
must be P' would be interpreted as 'it cannot happen that some S are not P: if
any S are P, then all are P'; similarly with 'All
S must not-be P'; to deny these statements, we would say 'it can happen that
some S… etc'.)
The particular versions of such statements, 'Some
S… etc'. may be similarly analyzed. There are also singular versions, like
'this S can be P, alone', which tells us about the potential for actualization
of 'this S is P' when all other S are
nonP. More broadly, any quantity 'n'
may be specified: thus, for instance, 'nS
can be P' informs us that this number of S can be P at the same time; in some
cases, we additionally specify 'at least' or 'no more than' to open or limit the
statement.
The above concerns natural modality, but equivalent statements involving
temporal modality are conceivable: 'All
S are sometimes P', 'All S are never
P', and so forth. Note that collectionality is used in a modal context; the
actual proposition 'All S are not P'
(meaning that not-all S are P, meaning that some are not, though some are), is
not really collectional.
Collectional intent is often encountered in the antecedent or consequent
of conditional propositions (for examples, 'when all
the cog-wheels are aligned, the key is able to turn' or 'when the button is
pressed, all the lights come on').
I will not here work out the logics of collective and collectional forms
in detail. Each form needs to be analyzed for its exact implications, then the
interactions of all the forms with each other and with dispensives (including
all immediate and mediate inferences) must be looked into. 4.
Quantification of Predicate.
The forms people currently use, and accordingly adopted by the science of
Logic, are so designed that we can specify alternate quantities for the
predicate, if necessary, simply by making another, additional statement in which
the original predicate is subject and the original subject is predicate, with
the appropriate distributions.
However, as an offshoot of the distribution doctrine, there have been
attempts to invent forms which explicitly 'quantify the predicate' of
classificatory propositions. Let us look into them briefly.
a.
On a singular level, the basic form would be 'this S is this P'. The
contradictory 'this S is not this P' would be compatible with 'this S is that
(meaning, some other) P'.
Normally, we need to know, say, 'whether the girl is or is not (at all)
pretty', rather than 'whether she is or not that pretty thing'. We may of course
say 'her dress was this shade of brown'; but here the indicative only specifies
a kind of color, not an individual qualitative phenomenon. Someone may tell me
'the girl I mean is the one we met last week'; but here the predicate is
intrinsically a one-member class.
Normally, we use indicatives in the subject, rather than the predicate.
The indicative is used to 'hold down' a first appearance, as our initial
designation of the object: once, that is settled, we are only interested in
discovering its further attributes as such.
Suppose I see a green and blue object, I may say 'this green thing is
blue' (or vice versa), but I would have no need to establish class
correspondence, since the object is already one and the same right before my
eyes. It is not inconceivable that I perceive a green object and later a blue
object, and then equate (or distinguish) them, saying 'this green thing is (or
is not) the same as that blue thing'; but this is a rare exception, and is it
really classification?
When we say 'this S is P' we first intend to qualify the subject by the
predicate (e.g. that baby was rather cute). We cannot transfer the designation
'this' from the subject to the predicate without missing the point, which is
attribution. Also, we normally use 'this' to refer to entities, rather than
qualities (though we can say 'this green is rather dark').
Still, theoretically, 'this thing' under discussion is indeed
theoretically an instance of P as well as S, so that permutation to
classificatory form is feasible. We have to remain formally open in this issue,
since we do regard 'all S' as implying 'this S'.
b.
With regard to plurals, 'quantification of the predicate' would give rise
to the following forms: 'all (or some) S are all P' (both implying that all P
are S); 'No S are certain P' (implying some P are not S) and 'some S are not
certain P' (the latter two not excluding that all S be P — i.e. other
instances of P).
The forms: 'all S are some P', 'some S are some P', 'no S are any P' and
'some S are not any P', would be equivalent to the established A,
I, E,
O. The rest would be relatively new.
Only the form 'some S are not certain P' contains information we cannot
express in natural language: but that may be simply because we never need to
make such a statement of partial exclusion in practise.
These forms have not aroused much interest, because they are artificial
to our normal ways of thinking. If we have so far managed very well without
them, why complicate things and try to introduce something no one will ever use?
However, to be fair, such statements are indeed used by logicians, if not
by laymen, to clarify the distributions of terms. We would speak in that way to
explain Euler diagrams, mentioning the one-one correspondence of individual
members of distinct classes, or the overlap or separation of segments of
classes. Thus, we may view them as specialized, rarely used — but still
legitimate.
Quantification of the predicate could also be viewed as a special case of
substitution.
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