15 Relational Arguments (1962)

Opponents of a predicative logic have commonly maintained that there are valid arguments of the form ArB, BrC ? ArC (arguments holding wherever the relation r is transitive; such validity, in fact, being what is meant by calling a relation transitive) which cannot be presented in ordinary predicative form, particularly in syllogistic form, but depend on principles other than the syllogistic. It may be admitted at once that there is no question of the putting forward of a “principle”, the transitiveness of r, as a major premise, the specific terms involved appearing in the minor premise and a syllogistic conclusion then being drawn; to which, however, it should be added that the “principle” of the syllogism (what is conveyed in the “dictum”) is equally not a premise but is the form of valid syllogism (or, putting it at its minimum, the form of the Barbara syllogism) and that in any argument of this form the conclusion follows from the premises (an X-Y conclusion from X-Z and Y-Z premises) without any superadded principle — it being just by the direct recognition of the implication of a given conclusion by given premises that we recognise the validity of a certain form of argument (recognise the “principle”). There is, however, no question of the validity of any argument of the form ArB, BrC ? ArC, and in order to distinguish valid from invalid “relational arguments”, or relations which are transitive from those which are not, we have to get a more detailed formulation of both. When we do this, I would argue, when we get a distinction between the valid and the invalid which can be “read off”, a distinction, i.e., which is determinable simply from the form and not at all from the material of the arguments, this can only be in terms of a predicative logic; and I shall endeavour to show that in a number of types of argument which have been taken to be striking exceptions to the syllogistic character of demonstrative reasoning, the formal distinction must actually be made in syllogistic terms — that the distinction is between valid and invalid syllogisms.

The standpoint of predicative logic involves an immediate questioning of the view that relational assertions (or relational propositions) are the constituents of relational arguments and are to be definitely distinguished from “subject-predicate” assertions. The line of criticism of this view is the same as I have followed in the article “Hypotheticals”. Any assertion whatever is an attempt to settle an issue, to give a decisive answer to the question “Is it so or not?” Whatever is called a relational assertion is open to formal contradiction and thus is properly stated, can only be unambiguously stated, in one of the four forms — in a

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proposition with a subject and a predicate and (besides the sign of quantity) the affirmative or negative copula — in exactly the same way as the assertion of the possession of a quality. We ask “Is A greater then B or is it not?” just as we ask “Is A red or is it not?”, and no use of symbols (A > B, A ? B; or, in general, ArB, ArB) can alter the fact that what is being asserted or denied is that something is so, and that only the is and is not (more properly, are and are not) really indicate the formal character of contradiction, whereas the > and the r run together or confuse between formal assertion and material features of what is being asserted, between the copula and something that belongs to the terms, between the notion of occurring or not occurring and the notion of what it is that occurs.

When these are clearly distinguished, we are faced, in the “relational” assertion just as much as in the “qualitative” assertion, with the independence of terms, the fact that subject and predicate are distinct and that the latter can be as significantly denied as it can be asserted of the former, and likewise with the “convertibility” of terms, the fact that any predicate can function as a subject and any subject as a predicate; and this, as before, leads on to quantification. Thus as the converse of “A is greater than B” we have to give “Some (things) greater than B are A”; similarly, the converse of “A is not to the left of B” will be “No (things) to the left of B are A”, and its converse will be “No A are (things) to the left of B”, which thus appears as a stricter form of the original assertion. In the same way “A is greater than B” will be replaced by “All A are (things) greater than B”, and, in general, assertion and denial of the occurrence of relations will take place in the four forms. But what must specially be emphasised, besides the need for quantification, is the fact that “a relation” is not a term, that there is no sense in any such formula as “A is greater than”; and since, as has been noted, the presentation of “is greater than” (perhaps in the form “exceeds”) as the relation simply amalgamates being so with some specific thing that is said to be so, the conclusion we are led to is that “the relation” falls not between the logical terms (subject and predicate) but within one (or both) of them — as is indicated in the formula “All A are things greater than B”, where the term “things greater than B” does not indicate any concrete quality of such things but indicates that they must have some concrete qualities if they are to have the relation; to which it may be added that some qualities must be recognised if any relation the things have is to be considered.

On this understanding, on this doctrine of the four forms and the unambiguous copula, the strict form of “relational statements” is the predicative form and the strict form of “relational arguments” may quite well be syllogistic form and will at any rate be one whose validity can be determined quite independently of the matter or terms. It is still to be understood that syllogism is not the only form of demonstrative reasoning, though it is by far the most important; we can have, without any departure from admittedly qualitative assertions, the non-syllogistic forms of conjunctive argument (combinations of predicates) and disjunctive argument (combination of subjects) — e.g., (1) AaX, AaY ?

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AaXY (All A are both X and Y); (2) AaX, BaX ? (A or B) aX (What ever is either A or B is X). Both these arguments differ from syllogism in the particular point that the two premises can be inferred in turn from the conclusion, and hence it might be argued that no real inference is involved; but this is not uniformly the case with such arguments (e.g., the premises of the conjunctive argument AaX, AiY ? AiXY cannot be inferred from the conclusion, though AiX, AiY can). It should be noted, moreover, that even in the conjunctive argument first given neither of the premises by itself implies that the conjunctive term XY is a real term (that XiY), so that a certain inferential discovery has been made; and, equally, neither premise in the given disjunctive argument implies that (A or B) is a real term, since it does not imply that the opposite, (non-A)(non-B), is a real term, and, unless it is (or if “everything” is A or B), saying that any given thing is A or B is not settling any real issue, is not distinguishing that thing from anything else.

The special importance of disjunctive argument, or of the conclusion to which it leads, is that it presents us with the notion of a plurality of things (terms) that have a common character and thus with the notion of “class”, or, more broadly, that the disjunctive term, A or B, adds to the notion of quality, of a predicate which A singly or B singly may possess, the notion of juxtaposition, of the possession of a quality by A and B even if neither qualifies the other and thus of their falling within a common expanse or range in which they can still be distinguished. These notions make possible the recognition, within a predicative logic, of distinctions of extent or amount and of the distinctions among class-relations, particularly the relations of co-extension and inclusion, in terms of which arguments involving quantitative relations can be presented. (It might be observed that disjunction points directly to these two class-relations, since A or B includes B, except where AaB, and includes A, except where BaA, and in the exceptional cases the relation is co-extension.) Intersection and exclusion, in which there is no question of transitiveness, do not enter into these arguments; but if quantitative equality can be presented in the form of co-extension of classes, and quantitative inequality in the form of the inclusion of one class in another, then quantitative arguments can proceed by means of ordinary predicative assertions and, in particular, syllogistically.

The initial step is the representation of A = B by AaB, BaA (A and B are co-extensive classes), even if A in these two propositions had to be understood as “measurable by the quantity A” and similarly with B — the meaning of “measurable by” is a point I shall take up later — and the representation of A > B by AoB, BaA (the class A includes the class B); it will hardly be denied on any view that the co-extension of classes is at least comparable to equality in amount, and class-inclusion to excess or deficiency in amount. The predicative form which can thus be given to quantitative arguments can be carried over to a considerable number of relational arguments (including, as observed above, many which have been taken as beyond the scope of predicative and syllogistic logic, as

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demonstrating its inadequacy) and indeed, I should further hold, to all in which formal criteria, irrespective of any material consideration, can be found for validity — this being, of course, the condition under which objective validity can be intelligibly spoken of.

The position is, then, that two quantitative or extended things, two things of some amount, are equal when and only when anything measurable by the one is measurable by the other. Thus, when A and B are equal quantities, what I am also calling A and B are equal classes, in that anything measurable by A is measurable by B and vice versa; and thus A = B can be represented by the two propositions AaB, BaA. The transitiveness of equality is then exhibited in a pair of syllogisms. Replacing A = B and B = C by AaB, BaA and BaC, CaB, we have the two Barbara syllogisms AaB, BaC ? AaC and BaA, CaB ? CaA, the conclusions AaC, CaA being, on the view presented, the logical form of A = C. We thus see that the so-called axiom “Things which are equal to the same thing are equal to one another” is just an assertion of the validity of the Barbara syllogism and is not an additional premise; or, more exactly, the “axiom” is the form of the two AAA arguments, and it is from A = B and B = C, and not from any “axiom”, that A = C is proved. We may note also that the treatment of equational reasoning as syllogistic reasoning gets rid of any doctrine of an “equational logic” in contrast to the ordinary predicative logic — an outstanding defect of such a supposedly special logic (and likewise of the view of logic as essentially equational) being that it cannot cope with ordinary logical relations, notably contradiction, since the contradictory of an assertion of equality is not an assertion of equality (“‘A = B’ is false” is not an equation), whereas the contradictory of a predicative assertion is a predicative assertion and any particular case of co-extension that happens to arise can be presented in a pair of predicative assertions and suggests no logical “realm” distinct from that of the four forms.

The two syllogisms which indicate the transitiveness of equality also bring out the fact that it is a symmetrical or reversible relation (as co-extension is). In the case of unequal quantities we recognise that one quantity is greater than the other, and this is presented in argument by means of the class-relation of inclusion, which is a transitive but asymmetrical relation (as “greater than” is). Using, as before, the same symbol for the term (whose extension is the “class”) and the quantity, we represent A > B by BaA, AoB (which, as before, might be presented in the alternative formulation as “All things measurable by B are measurable by A” and “Some things measurable by A are not measurable by B” — assertions capable of precisely the same syllogistic use as BaA, AoB). And, with this representation, we can exhibit the transitiveness of the asymmetrical relation “greater than” (and similarly of “less than”) as definitely as that of the symmetrical relation “equal to” was exhibited above; though there are complications in the case of inequality or class-inclusion which do not appear in the case of equality or co-extension.

Thus we can argue syllogistically from the premises BaA, AoB and

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CaB, BoC to the conclusions CaA, AoC (from A > B and B > C to A > C) with the same validity as we can argue from BaA, AaB and CaB, BaC to the conclusions CaA, AaC. In the latter case, however, we use all the premises in arriving at the required conclusion, whereas in the case of inclusion, while we have to use both of the A premises, we need use only one of the O premises. We argue to the conclusion CaA from the premises CaB, BaA, but to prove AoC we may use either BaA, BoC (OAO, third figure) or AoB, CaB (AOO, second figure). In the former case we do not use AoB as a premise and, in fact, we could use the OAO argument even if AaB — i.e., since BaA, even if A and B were equal. Similarly in the second case, where BoC is not used, we could get the required conclusion even if B and C were equal. The common way of expressing this in mathematical reasonings is to say (first case) that if A is greater than or equal to B (A = B) and B is greater than C, then A is greater than C; similarly (second case) if A > B and B = C, A > C. An alternative way of expressing “greater than or equal to” is “not less than”. Thus when we prove CaA, AoC from BaA, CaB, BoC, we are treating BaA as simply asserting that A is not less than B, and when we get the same conclusions from BaA, AoB, CaB, the last of these is expressible as “B is not less than C”. Now it appears strange that “A is less than B” (AaB, BoA) should be contradicted simply by the denial of the O proposition; but the point is that terms quantitatively comparable belong to a single scale or range, so that, if any two do not coincide, one must include the other (one quantity must be higher in the scale than the other) and thus there will always be one true A proposition having them as terms and such a position as AoB, BoA (or again AeB) could not arise. I do not here argue the question whether there must be some scale on which any two quantities or measurable things can be compared; but, in any argument concerning things all of which are quantitatively comparable, the relation “not less than” (and similarly “not greater than”) is conveyed by a single proposition.

Thus, in the argument BaA, CaB, BoC ? CaA, AoC, the premises leave it an open question whether AaB or AoB; indeed, the single proposition BaA does this — there is no question of this proposition's sometimes meaning co-extension and sometimes meaning inclusion, no question of qualifying the copula in the manner suggested by such confused and confusing conceptions as “the is of identity” or “the is of inclusion in a class”. The making of such distinctions is an error of the same type as that of importing some of the material of an assertion into the formal sign of something's being asserted; and their pointlessness becomes clear when it is noted that co-extension, and similarly inclusion, can be quite directly and unambiguously presented by the assertion of a number of propositions, each with the unambiguous copula of occurrence or “being so”. Thus, while BaA never means BaA and AaB and never means BaA and AoB, the possibility of presenting either situation in two propositions leaves BaA itself quite unambiguous. The point is reinforced by the consideration that “not less than” leaves “not greater than” an open question, and that to take each

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as conveyed by an A proposition leaves it possible that both propositions, BaA and AaB, are true — so that the truth of both, the co-extension or equality of A and B, is equally expressible by “A is not less than and not greater than B”. When we understand the A proposition in this way, we get a further type of quantitative argument involving A propositions alone; A = B, B = C ? A = C (or A is not less than B and B is not less than C, therefore A is not less than C) takes the form of the Barbara syllogism BaA, CaB ? CaA, where no decision is involved as to the truth of falsity of the propositions AaB, BaC, AaC, and it is perfectly possible that the argument B = A, C = B ? C = A has true premises and thus (being valid) establishes its conclusion, C is not less than A.

One important point in connection with all this is the clear understanding of what a class-relation (or a relation between the extensions of two terms, an “extensive relation”) actually is. There is no question of the enumeration of “individual instances”; for, if we say that any subject is a predicate (or has subjects), there will be no limit to the list of instances, no question of “setting it out in full”, any more than there will be a question of setting out in full a term's intension, of enumerating its properties, if every predicate is a subject or has further predicates. This is what gives the extension-intension rule, similarly to quantitative arguments, no application to the cases of intersection and exclusion where any quantitative comparison would have to rely on the (impossible) full count. It applies only to co-extension, where it can be said that two terms having the same subjects have the same predicates, and inclusion where the term that has more subjects has fewer predicates (general properties) — A's “having more subjects” than B being conveyed in the two propositions BaA and AoB (there are no B that are not A, there are A that are not B); while, strictly, “having the same subjects” must be brought down to what is conveyed by AaB, BaA, since a term is not one of its own subjects. Arguments involving quantitative comparison, then, will take one of the syllogistic forms presented above.

What I should further maintain is that any type of relational argument in which formal validity (the only kind of validity) can be distinguished from formal invalidity can be set out in the same way as quantitative arguments; and that it is not merely a question of one form of argument being “akin” or analogous to another, but that to see that some relation is transitive and symmetrical, to see that some relation is transitive and asymmetrical, to see that some relation is transitive and “non-symmetrical” (that B may or may not have it to A when A has it to B), is to recognise the existence of some class-relation of co-extension or of inclusion or (in the “non-symmetrical” case) of either co-extension or inclusion. I select for special consideration the transitive and asymmetrical relation “to the right of” which, for some reason, has been taken as an outstanding example of non-syllogistic arguments that have as great and as obvious cogency as syllogism; and my contention here will be that the relations “to the right of”, “to the left of” and “level with” not only can be treated in the same way as “greater than”, “less than” and “equal to”, that is, as

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indicating relations of inclusion and co-extension so that arguments involving them will appear in syllogistic form, but that only in this treatment is their cogency, their formal character, definitely brought out — in other words, that syllogistic argument is their strict logical form.

The first point to be noted with regard to the relation “to the right of” (and similarly with any other relation of spatial direction) is that it is meaningless except with reference to (a) some field which the things related occupy (most commonly, some field of vision) and (b) some point from which the field is to be taken or before which the field is spread out, so that rotations or turnings round the point in one or other direction can be considered. If that point is not kept constant, if some other point before which the field is spread out could be taken instead, then A, which was first of all said to be to the right of B, could just as easily be said to be to the left of B. The question, then, when the relation is spoken of with any settled meaning, is of comparisons between turnings round a point O (“origin” or observer), within a field “in front of” it, in a rightward (or, similarly, in a leftward) direction; and A is to the right of B when and only when the turning to the direction of A, perhaps from a postulated “extreme left” position or simply “from the left” if it were maintained that the extremities of such a field could never be precisely marked, is greater than the turning to the direction of B. Thus we are back at the question of unequal quantities and hence at the question of class-inclusion, and we can have exactly the same sorts of syllogisms conveying the transitiveness of “to the right of” as we have in the case of the transitiveness of “greater than”, with the very same complications, or special arguments, involving “not to the left of” as we have in the case of “not less than”. We say that A is further to the right than B (that, as it might be put, A has “greater rightwardness”) in the given field and from the given origin or point of view in the same way as we say that A is of greater quantity than B, and we can have arguments concerning the interrelation of several things in different directions from O in the same forms as the arguments concerning several things of different quantities or amounts. The position is roughly illustrated in the following diagram —


where the upward direction from the line LOR represents the outward direction (towards the field) from the origin.

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Leaving aside for the moment the question of the precise terms to be used in the relevant syllogisms, we see that the relation “to the right of” involves a relation of greater and less, and that in this special case we shall be able as before (a) to exhibit the transitiveness of the relation (to bring out the validity of the argument “A is to the right of B, B is to the right of C ? A is to the right of C”) in the form of two valid syllogisms, one in the AAA form and one with an O premise and an O conclusion, (b) to distinguish the cases in which the two different O premises are used and thus to distinguish the arguments (equally valid) “A is not to the left of B (it is either to the right of it or level with it, i.e., in the same direction from the origin), B is to the right of C ? A is to the right of C” and “A is to the right of B, B is not to the left of C ? A is to the right of C”, (c) to recognise the valid AAA syllogism expressed more loosely in the argument “A is not to the left of B, B is not to the left of C ? A is not to the left of C”, (d) to recognise that the truth of the premises here is quite compatible with the truth of “A is not to the right of B, B is not to the right of C”, and thus recognise a possible argument of the same form as that of “A = B, B = C ? A = C” — an argument roughly expressible as “A is level with (or in the same direction as) B, B is level with C ? A is level with C”; where “in the same direction as” is a particular example of being of the same quantity as, the quantities in this case being turnings. Thus we find, in addition to the transitiveness of “to the right of” (and “to the left of”), that of “not to the left of” (and “not to the right of”) and that of “neither to the right of nor to the left of” (or “level with”); so that the argument involving the transitiveness of “to the right of” is only one example of a whole class of “angular” arguments, the distinction and the interconnection of which are brought out by their presentation in syllogistic form.

To see in what terms these arguments may be set out syllogistically (or in strict form) we can consider swings or turnings from left and observe that any such swings which pass through A must pass through B, whereas those that pass through B need not pass through A; or we can say, in terms of positions and directions, that to be beyond A in a rightward swing is to be beyond B, while to be beyond B is not necessarily to be beyond A. Putting this more concretely in terms of things beyond or not beyond a given thing in a rightward swing, we come to the pair of assertions “All things to the right of A are (things) to the right of B” and “Some things to the right of B are not (things) to the right of A”; that is to say, to a relation of inclusion. Adopting for brevity the notion RA for “things to the right of A”, etc., we can present the types of argument above referred to in the form of syllogisms with the following premises and conclusions:-

(a) RAaRB, RBoRA; RBaRC, RCoRB ? RAaRC, RCoRA (the transitiveness of “to the right of”);

(b) RAaRB, RBaRC RCoRB, ? RAaRC, RCoRA (the case where A is given as “not to the left of” B) and RAaRB, RBoRA, RBaRC ?

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RAaRC, RCoRA (the case where B is given as “not to the left of” C);

(c) RAaRB, RBaRC ? RAaRC (the transitiveness of “not to the left of”);

(d) RAaRB, RBaRC ? RAaRC and RBaRA, RCaRB ? RCaRA (the transitiveness of “neither to the right of nor to the left of”).

In cases (a), (b) and (c) we can get similar syllogistic arguments for leftward as contrasted with rightward turnings by simply substituting left for right and right for left; but it should be noticed that there is no need to introduce an LA notation, since the formal statement of “A is to the right of B” (RAaRB, RBoRA) will also serve for “B is to the left of A”, and the same form serves for “B is not to the right of A” as for “A is not to the left of B”.

The fact that we can argue on leftward turnings by using “rightward” symbols raises the question whether we can treat the two assertions, “A is to the right of B” and “B is to the left of A”, as saying exactly the same thing — as asserting the very same proposition in different words. We can certainly treat them as equivalent (so that whatever, other than the two themselves, is a consequence of one is a consequence of the other, and either is a consequence of whatever, with the same proviso, the other is a consequence of) and thus substituting either for the other in an argument will not affect its validity though it might affect the rapidity with which the implication could be grasped. The point of this equivalence is that the relation in question is a relation of “right and left” (comparable to “north and south”, “above and below”, “before and after”), that it is impossible to recognise either “rightward” or “leftward” without recognising both. But if these considerations justified the denial of any difference between the two assertions, the same would apply to any pair of “equivalent” assertions; and this, in the case of XeY and YeX (which, on certain views, would each be rendered by “XY do not exist”), would lead to the rejection of the distinction between subject and predicate (between location and description) which is essential to logical form, even with the recognition of the “convertibility” of terms, and which would still be required by the “existence, non-existence” theorists if there was to be any sense in asserting or denying “existent” of this or that subject, or, for that matter, if there was to be any meaning in XY (things which are X and are Y — a conjunctive term giving two descriptions of something, or being a combination of predicates, even when it is used as a subject, just as a disjunctive term gives two locations or is a combination of subjects). Thus the necessary distinction of subject and predicate shows that “No cows are horses” and “No horses are cows” are different assertions even though they imply one another; the assertion that cows lack the equine character can be put in conjunction with the recognition of the characters cows do possess, as well as raising the question of how they are connected with and distinguished from other non-equine things, and thus has a quite different place in discourse and investigation from that of the assertion that horses lack the bovine character.

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Equivalence, then, does not mean identity; and the necessity of thinking right and left together, of seeing their interconnection, is nothing against seeing their opposition but indeed depends on this. The irreversibility of each of the relations “to the right of” and “to the left of” reinforces the point that, in spite of the equivalence referred to, quite different questions arise concerning things to the left of A from those concerning things to the right of B. Alexander (Space, Time and Deity, Vol. I; first edition, p. 243), discussing, with special reference to north and south, such differences in the “sense” or direction of relations, says that when “the same situation is expressed in two different senses by interchanging the terms (Edinburgh is north of London, London is south of Edinburgh), the difference is not indeed a merely verbal one, though perilously near to it, but a difference of aspect or description, what Aristotle expressed by saying that the two things are the same but not in their being” (italics in text). It is not at all clear what would be meant by being “perilously near to” a merely verbal difference without being precisely there, nor what, in the Aristotelian reference, is being attributed to the two things other than their being and not being the same. The point is that, taking it that we have two different assertions, we can indicate the difference quite clearly on the predicative view (any assertion having subject and predicate with different “functions”); in the one case, the subject is Edinburgh, and the predicate applied to it, in spite of being only a “relational description”, enables us to classify it with other things north of London, to say “All (E. or A or B…) are north of L.”, while in the other case the subject is London and we can similarly proceed to the assertion that “All (L. or X or Y…) are south of E.”; and here it might incidentally be noted that some things north of London are not south of Edinburgh and some things south of Edinburgh are not north of London. Thus the two assertions Alexander has difficulty in differentiating not only have different subjects (in fact they have no common terms) but raise the quite distinct questions of what is north of London and what is south of Edinburgh. Understanding of the north-south range still shows the two assertions to be equivalent; but, to see how this is so, we have to see that, if a “relational assertion” has a subject, it must have a predicate — that, unless this is so, we can give no account of the possibility of contradicting such an assertion or of the form of the arguments (implications) in which it is involved.

It might be contended that if we identify “A is to the right of B” with “All things to the right of A are to the right of B” and “Some things to the right of B are not to the right of A”, and particularly if we take the former of these propositions to be part of what it signifies, we are assuming, not demonstrating, the transitiveness of “to the right of”, and that it is pointless to go on to any syllogistic argument in which this is brought out. And a similar point might be made about the interpretation of A > B as the inclusion of the class of things “measurable by” B in the class of things “measurable by” A: viz., that “measurable by” has to be understood as “not greater than”, in that

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anything not greater than A can be marked off on A, if it is not precisely equal to A which would thus still give its measure; so that the assertions in which A > B was presented in syllogistic argument would have to be understood as “All things not-greater-than B are not-greater-than A” and “Some things not-greater-than A are not not-greater-than B” or, taking the equivalent contrapositives, “All things greater than A are greater than B” and “Some things greater than B are not greater than A” — where, as before, it could be contended that the former of these propositions conveyed the transitiveness of “greater than” and that nothing was contributed to the understanding of the relation by the syllogistic arguments in which this proposition was a premise. (The same types of relation would hold and the same problems would arise if “Measurable by” were taken as meaning “not less than” — i.e., if a thing were taken to be “measured” as so many unit measures, together with a marked off part of a unit if the units did not fit exactly. The question would still be one of class-inclusion and its transitiveness, etc.)

But such considerations would be an objection to the view here taken of the way in which the arguments can be precisely formulated only if, e.g., the two propositions which, in such arguments, represent “A is to the right of B” were taken as explaining what is meant by “to the right of” — which, since “to the right of” is contained in them, would be obviously circular. What can be said, in the first place, is that the two propositions (RAaRB, RBoRA) are true when and only when A is to the right of B, so that, in using them as syllogistic premises, we can formally exhibit what does follow from “A is to the right of B”. But we are aware of this “when and only when” relationship, we can carry out the “transformation” in question, we can understand and recognise its justification, only in being aware of a continuous range, a field of things in various “angular” relations, apart from which “to the right of” would have no meaning and, incidentally, in recognising which we recognise what extends beyond as well as what falls between the things compared; in order to speak of A's being to the right of B, we have to be aware of a greater stretch to A than to B from the left and a smaller stretch from A than from B to the right, and thus there is no impropriety, no deviating from or passing beyond what we are observing from the outset, in speaking of “things to the right of A”.

Being acquainted, then, with such a range (field of vision or whatever it may be) we can see how to present “angular relations” (involving greater and smaller turnings) as examples of inclusion and co-extension, and we can make clear in syllogistic argument the inter-connection of a set of such relations. As we saw, the transitiveness of “to the right of” is only one member of a whole class of connections which are formally presented in syllogisms and which are all involved in the recognition of a range of directions from an origin. It should still be noted that what we are doing in such arguments is not to establish transitiveness (even though in establishing the particular conclusion we come to — say, A is to the right of C — we exhibit the transitiveness of “to the right of”; and

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similarly with the rest of the class referred to) and that, unless we could observe this transitiveness as a feature of the situation in which A, B C, etc., are (of the whole field or range), we could never argue in the way we do; our distinction between transitive and non-transitive relations, between valid and invalid arguments, would be quite arbitrary unless we were able to see, to take in a single view, the validity of the whole process and unless we had some way of setting it out so that formal criteria for determining the distinction were apparent. The syllogistic form shows us more clearly not only how a given conclusion is established but what relations are transitive and what are not.

These considerations, of course, apply to syllogism in general and not just to the cases (“angular arguments”) we have been specially considering; unless we can observe in one situation the whole “syllogistic principle”, unless we can see as a fact the force of the syllogistic argument, i.e., the implication of the conclusion by the two premises, any assertion we make of the validity of some forms and the invalidity of others will be quite arbitrary. If we obtained our belief in the validity of the Barbara syllogism, for example, from something other than observation, we could never apply this belief to what we observe; we could never say “This is an example of that sort of interrelation of propositions (situations)” unless we could observe that interrelation as a single situation, observe, that is to say, the implication which holds in any Barbara syllogism. We could still say that the syllogism BaC, AaB ? AaC assumes and does not demonstrate what we may roughly call “the transitiveness of the relation a” (having as a property); what it demonstrates is AaC. But we could not see that it does, unless we could see BaC and AaB implying AaC, just as much as we can see the truth of any of these three propositions. The position is similar, then, with any argument, like the “angular” arguments, which involves a continuous range; it is no objection to a particular way of setting out such an argument that it implies that we can see transitiveness, that we can observe it within the observed complex situation, since, if we could not, we could not see the force of the argument or set it out in any way.

Since the treatment I have given of “angular” arguments depends simply on the recognition of a continuous range, it would appear that the same treatment would be applicable to any transitive relation, with the same complications when it is asymmetrical — that arguments involving it would start from class-relations of co-extension and inclusion and would take the same syllogistic forms, supplemented in some cases perhaps by conjunctive and disjunctive arguments. In the simplest case of transitive asymmetrical relation, that of before and after — simplest because the question is just of continuity, because there is no range more special than Time itself and no peculiar “origin” or point of view — the main type of relation, “A is before B”, will again be presented formally as inclusion, i.e., by the two propositions “All things before A are before B” and “Some things before B are not before A”, while “B is not after C” (B is either before or simultaneous with C) will take the form of a single A

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proposition, “All things before B are before C”, which can be taken along with the previous two propositions or with other “not after” assertions, just as in the “not to the left of” examples.

The case of north and south, previously referred to, is one in which the range has definite extremities, so that, if we were to say “The North Pole is north of Greenland”, we could not present this as “All things north of the North Pole are…”, etc. It might be suggested, however, that, when we are comparing the latitudes of two things, we are concerned with which of them is further from the North Pole (or, similarly, from the South Pole), and that we treat the extremes not as things within the range but, like an “origin”, as part of the background or the “terms of reference” of the relation. At the very least, we have to distinguish, where there are specific limits, the recognition of differences within a continuous range from the recognition of such limits, and what confronts us, in our consideration of transitiveness or continuous transition in a range, is the shading off of a specific segment into an indefinitely extended background. This shading off, with non-specification of extremes, is also to be found, as suggested earlier, in right and left judgments, so that, even though we always recognise a “beyond” and could not make a specific comparison without doing so, our definite reference is always to things (or positions) within the range. The distinction between the question of limits and that of internal comparisons is emphasised by the fact that where, as with before and after, there is no question of such limits, the types of “transitive” argument are the same as where there are limits; and it is of some relevance to observe, in the case of north and south, that distinctions and relations of this sort were recognised before there was any question of extreme points.

Such special considerations are secondary to the general contention that recognition of implication requires recognition of the formal validity which may be seen in syllogism and in other arguments (conjunctive and disjunctive) set out in strictly predicative form. But it is interesting to note that the question of ranges and of “shading off” also has a certain application to syllogism in general. I have spoken of “the transitiveness of the relation a”, i.e., of the A proposition — though a proposition is not strictly a relation; and the point to be observed here is that, in asserting XaY (as, indeed, in using the terms X and Y at all), we always have the sense of a background — of further subjects of which X is a predicate or locations of which it is a description, of further predicates of which Y is a subject or descriptions of which it is a location. We have, in other words, a sense of an indefinitely continued range of extensions and intensions; we have, in knowing any proposition, a background of possible syllogisms, of lines of further inquiry and discovery. This is nothing against the definiteness of any given proposition or of any given syllogism; though recognition of further possibilities may at times make us uncertain of our “grasp” of some sort of thing with which we had supposed ourselves well-acquainted, it is only in so far as we have definite knowledge that we can take any step in inquiry or distinguish one line of inquiry from another.

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But the recognition of such “lines” is recognition of some range or ranges, exhibiting what may be roughly called the transitiveness of “qualifying” and of “being qualified by”; though this does not mark off “qualitative syllogisms” from “relational syllogisms”, since it is only on the predicative view of the proposition, on the three-term view of the syllogism, that validity in syllogistic reasoning can be recognised.

This assimilation of relational and qualitative arguments does not mean a breaking down of the distinction between relations and qualities. It is, I have suggested, from a qualitative starting-point, specifically from seeing that a given predicate has many subjects, that we pass to the question of relations, by way of the observation of juxtapositions or differences of place within a certain region or range. But it should be emphasised on the other side that even the distinguishing of various predicates of a given subject proceeds by finding the place of each quality in a set of “exchanges” or interactions between it and its surroundings (other subjects). Thus the question of ranges or lines of connection arises in regard to any issue whatever, whether it is primarily a qualitative or a relational one. The point is that we are always confronted simultaneously with questions of relations and questions of qualities, that relations and qualities are linked in the recognition, as in the existence, of any situation, any complex state of affairs, and that there is nothing less, and nothing more, than a complex (spatio-temporal) situation that we can be confronted with in dealing with any material, i.e., in any recognition of or search for connections and distinctions. The attempt to have separate relational and qualitative logics can only lead to confusion and insoluble problems; what this attempt misses is the fact that any object (any known thing and any existing thing) is a complex situation involving both relations and qualities, so that there will always be connections to be found between any object and any other object, between any and any other problem or line of investigation.