In this chapter, we will look into various topics which involve quantitative considerations.

1.Substitution.

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 (quawhat 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’).

2.Comparatives.

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-calleda-fortioriarguments. 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 aredispensive(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 arecollective, 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): ‘AnynS together are P’ or ‘CertainnS together are P’; here, ‘nS together’ refers to a class of collectives.

c.Some propositions arecollectional. 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, ‘AllS 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 ‘AllS can be nonP’, if we want to say ‘it can happen that all S are simultaneously nonP’.

Accordingly, ‘AllS 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 ‘AllS cannot be nonP’, if we want to say ‘it cannot happen that all S are nonP at once’.

(In contrast, the form ‘AllS 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 ‘AllS must not-be P’; to deny these statements, we would say ‘it can happen that some S… etc’.)

The particular versions of such statements, ‘SomeS… 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 allotherS 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: ‘AllS are sometimes P’, ‘AllS are never P’, and so forth. Note that collectionality is used in a modal context; the actual proposition ‘AllS 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, ‘whenallthe cog-wheels are aligned, the key is able to turn’ or ‘when the button is pressed,allthe 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 establishedA,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.