-nJHERE is a contrast between the traditional theory of positive and negative propositions and what occurs in the conduct of inquiry. The contrast invites examination. In scientific inquiry there is scrupulous attention to exceptions and whatever appear to be exceptions. The technique of inquiry is concerned as much with effective eliminations as with noting agreements. No amount of agreement among the traits of phenomena investi- gated suffices of itself to establish a conclusion; agreements have to be safeguarded at every point by observation of differences.

Experimental operations are undertaken with the express object of instituting deliberate variations of conditions in order to bring out negative traits which serve to test currently accepted con- clusions. Should logical theory take its cue for interpretation of affirmative and negative propositions from what happens in the conduct of inquiry, it would be evident that (1) such propo- sitions are functional in resolution of a problematic situation, and are (2) conjugate or functionally correspondent in relation to each other.

Traditional theory, however, takes the propositions as given ready- made and hence as independent and complete in themselves. They are just there to be noticed, with description of whatever proper- ties they present. This mode of treatment becomes intelligible when it is viewed in conjunction with its derivation from the onto- logical logic of Aristotle, whence it ultimately derives. In the latter logic, species or kinds are the ultimate qualitative wholes or real individuals. Some of these species are by nature, or by in- herent essence, exclusive of others. The negative proposition was 181 182 THE STRUCTURE OF INQUIRY thus a cognitive actualization of a fundamental ontological form.

Species also are ordered hierarchically. Hence affirmation of in- clusion of some species in species that are more comprehensive was also a cognitive actualization of the ontological.

Positive and negative propositions are on this basis immediate apprehensions or "notations" of what exists in and by nature.

What has just been said applies also to universal propositions i. e., to those about wholes. Similar considerations apply to particular propositions, and hence to the so-called square of opposition with its relations of contrariety, sub contrariety, contradiction and sub- alternation. Things that change are by inherent nature incom- plete and partial. Hence they are apprehended in particular propositions. The connection between partial and particular was more than etymological. In traditional formal theory "some," the verbal mark of the particular, has come to mean "some, per- haps all." But in the Aristotelian theory, some meant some only.

By the ontological nature of the case, wherever the affirmative "some are" applies, the negative proposition "some are not" holds also. The relation of subcontrariety was as ontological as was that of the contrariety of mutually exclusive universals. Particu- lars, or that which by nature is incomplete, because changing, can be known only through fixed limits imposed by the essence that defines a universal. Consequently, subalternation is in so far ontologically grounded. As for contradiction, it is evident that a proposition which is restricted by its ontological subject- matter to some only contradicts a proposition which by nature is about a whole.

The development of modern science destroyed the conceptions of fixed species, defined by fixed essences, upon which the Aris- totelian logic rested. This destruction affected, therefore, the classic conceptions of universal and particular, whole and part, and the scheme of their relationships with one another. Modern logic, however, attempted to retain the scheme but with the understand- ing that it is purely formal, devoid of ontological import. The inevitable consequence is the mechanical way in which affirm- ative and negative propositions and their relationships are con- ceived in both traditional and modern formalistic logic. They AFFIRMATION AND NEGATION 183 have lost their ontological basis without gaining a functional re- lation to the conduct of inquiry.

The old designation, quality of propositions, is retained in con- nection with affirmative and negative propositions but is hardly- more than a mechanical label. From the standpoint of the func- tional connection of positing and negating with determination of unsettled or indeterminate situations, they are means, through the operations of selection and elimination they respectively prescribe, of ^qualifying the original indeterminate situation. Affirmative propositions represent the agreement of different subject-matters in their evidential capacity; they agree in that they support or are taken to support one another cumulatively in pointing in the same direction, in spite of the fact that existentially the subject- matters involved occur at different times and places. Negative propositions, on the other hand, represent subject-matters to be eliminated because of their irrelevancy or indifference to the evidential function of material in solution of a given problem. Ul- timately, the fact that certain facts or ideas are excluded means that the original indeterminate situation can be transformed or re- qualified into a determinate one only through existential experi- mental operative elimination of some of its constituents; affirma- tion of certain data or ideas means that they are operatively selected to reinforce one another in institution of a unified situa- tion. If these statements sound odd in contrast with the traditional interpretation of affirmation and denial, one has only to think of what happens in the conduct of scientific inquiry to see that they have a solid base and a pertinent meaning.

That inquiry selects appropriate evidential data by means of comparison of what is found to exist or occur in different existen- tial cases is a commonplace. Without collection of phenomena observed at different times and places under different conditions, grounded inquiry, whether of common sense or science, can make no headway. Deliberate experimentation is resorted to for the express purpose of varying conditions, or so that observed conse- quences will so vary that comparison may have more extensive and more definite subject-matter to operate with. Collection of many cases with a view to institution of differences and agreements (in evidential force) is a kind of relatively uncontrolled experi- 184 THE STRUCTURE OF INQUIRY ___ mentation. Comparison is so involved in all inquiries that reach grounded conclusions it is usually taken for granted. 1 Now it is impossible to define comparison except operationally.

It is a name for all operations in which identities and incompati- bilities in evidential force are determined. It is a name for any and all of the operations by means of which alleged or provisional data are determined to be data with respect to the problem set by a given indeterminate situation; by which some facts are determined to be the "facts of the case" in hand and other facts not to be.

It is impossible to give an independent definition of comparison and then derive the operations of establishing agreements and differences in evidential capacity from that definition. It is a blanket term for the entire complex of operations by which some existences are selectively instituted as data and other existential materials are eliminated as having nothing to do with the case; as, in fact, obstructive in the required work of requalification of the existential situation.

Mr. Bosanquet, one of the idealistic logicians referred to above, says "Comparison in the ordinary sense is a name applied to the intentional cross-reference of two or more given contents, in order to establish between these contents as given, a general or special identity, or partial identity (likeness)." 2 The view ex- pressed in this passage serves to bring out, by contrast, the meaning of the position here held. The italicized words of the text cited, as given, involve, positively, an affirmation of the antecedent on- tological basis of comparison and, negatively, a denial of the func- tional or operative force of propositions of identity agreement and difference contrariety, subcontrariety, and contradictoriness.

In contrast, the position of the text is that what is meant by com- parison is institution of selected facts on the basis of equivalent (similar) evidential force in a variety of cases which are existen- tially different, this determination being grounded only as the operations of observation involved in the selection eliminate, pari 1 Examination of logical texts will show that the word rarely appears. The exception to this statement is found in the case of the writings of logicians of the rational idealistic school. They are interested in it as a somewhat elementary exemplification of their ontological proposition that "reality" as such is always a system of differences-in-identity or identity-in-diiferences, or what is called the "concrete universal." 2 Logic, Vol. II, p. 21, italics in original text.

AFFIRMATION AND NEGATION 185 passu, other existential constituents as irrelevant to the problem in hand; as non-evidential, and indeed misleading unless eliminated.

The view of Mr. Bosanquet reduces comparison to an act that can be and is performed within the "mind." The view here taken is that it is operational in the existential sense of effecting modifica- tions in what antecedently existed as does controlled experimenta- tion. "Similarity" is the product of assimilating different things with respect to their functional value in inference and reasoning.

There is much common sense inference in which similarity is im- plicitly postulated. When the assumption is stated in a proposition (as it needs to be if the conclusion of inquiry is to be grounded) a proposition of similarity is, in effect, an affirmation that there is sufficient probability of equal values to serve as ground for tenta- tive assimilation.

The foregoing discussion has contrasted a theory of affirmation and negation based upon the practice of present scientific inquiry with the Aristotelian doctrine and with that later formalization of his doctrine which emptied it of all content. The connection of our view with the general theory of judgment will now be con- sidered. Indeterminate situations are marked by confusion, ob- scurity and conflict. They require clarification. An unsettled situation needs clarification because as it stands it gives no lead or cue to the way in which it may be resolved. We do not know, as we say, where to turn; we grope and fumble. We escape from this muddled condition only by turning to other situations and searching them for a cue. What is borrowed pro- vides a new attitude as the means for directing observational operations performed on the common sense level through sensori- motor organs. These operations make some aspects of the given situation stand out. The attitude, when made explicit, is an idea or conceptual meaning.

The very operations that select certain conditions, taken as potential clews to the problem to be dealt with, also rule out other conditions and qualities of the total given situation. Selection involves rejection and the latter act is rudimentary negation. The unsettled situation is also usually such as to evoke contrary modes of response. Attitudes and habitual modes of treating situations clash. This conflict is involved in confused and blind situations.

186 THE STRUCTURE OF INQUIRY But sometimes the conflict is so uppermost that the main problem is that of reduction to unified significance rather than clarification.

Some constituents stand out but point in opposed directions. To solve the problem resort must be had to other experienced situa- tions. These suggest additions and eliminations which, when ef- fected, will bring together the materials that first evoked conflicting responses.

The process of eliminating materials that are irrelevant and obstructive goes hand in hand with that of rendering other ma- terials definite in their indicative force. Negation is thus the restrictive side of the selection involved in all determination of material as data. What is selected is provisionally positive. This positive phase is at first identical with taking and using the ma- terial in order to try it out. But control of this taking and using demands that the material be formulated. The propositions (that are the formulation) thus differ from the final assertion which is characteristic of judgment. Dependence of the rejection-selection operation upon suggestions supplied by other situations explains the emphasis put, in the traditional theory, upon "common" factors and upon agreement. Comparison is at the same time contrast, expressed in the rejection and the elimination of those elements and qualities in the situation which other situations indicate are irrelevant.

It is sometimes said that affirmation and negation cannot be made coordinate with each other because there then arises a re- gressus ad infinitum. Such would be the result if they trod upon each other's heels. But in fact they are strictly conjugate. Not only is all determination negation but all negation is (or moves in the direction of) positive determination. The relation of affirmation-negation is no more successive than the taking of food by an animal is prior to or after rejection of other materials as non- food. Acts "which at one and the same time accept for use and that shut out are not sequential.

The connection between organic selection-rejection and logical affirmation-negation is, moreover, a special case of a general prin- ciple already laid down. The organic function provides the existential basis of the logical. Transition from one to the other occurs when the direct existential commitment involved in organic AFFIRMATION AND NEGATION 187 acceptance-rejection is deferred till the functional capacity of materials has been determined in inquiry. This postponed de- termination is made possible by language, by propositions about final decisive action. There are, for example, historical reasons for believing that processes of blame and accusation in connection with attempts to support and refute allegations were a main factor in developing the positive-negative aspect of inquiry. Then came argumentation pro and con in relation to some proposal advanced for social adoption. Argument still means reasoning. Crimen means judgment in the Latin tongue, and its root is found in our words discrimination and crime. The Greek aitia, usually trans- lated cause, had a definitely legal origin. Transition from the cultural to the logical status is manifest in the change from assent and dissent to affirmation and denial on specified grounds. To admit and to refuse to admit may be acts performed either for social reasons or because of reference to demands that are imposed by grounded inquiry. In the latter, they have explicit logical status.

Affirmation is unambiguously a logical term. We affirm only that which we take to be capable of confirmation.

There is another objection to the idea that affirmation is logically coordinate with denial. When the functional nature of affirma- tive propositions is overlooked that is, their office in institution of data and meanings to be operatively employe they are given direct existential reference. They are taken to be declarative of what is existentially there. The same thing cannot be said of negative propositions. Hence it is denied by some writers that negative propositions have any logical import at all. They are at most, according to them, rejections of suggestions that have arisen in our own minds and hence they have only a personal or psychological standing. In the words of one writer on logic "There is no such thing as a negative copula but only a negated copula." 3 Mere negation, however, reminds one unpleasantly of the dis- putes of children, consisting of reiterations of "Tis, Tisn't." The important point is that the view in question follows from the postulate that all propositions about fact are complete and final because enunciative of antecedent existence. The doctrine which 3 Sigwart, Logic, Vol. I, p. 122.

188 THE STRUCTURE OF INQUIRY denies logical standing to denials thus gives indirect support to the position they are instrumental and functional. Existences and meanings are referred to, both in affirmation and negation, not just for the sake of mentioning them, but with respect to their function in requalification of an indeterminate situation; for this requalification can be effected (with respect to negation) only by elimination of obstructive materials and of suggestions that lead nowhere. If negative propositions are ruled out of the logical domain, comparison must go too.

In short, negation is other than mere omission or dropping out of certain considerations, factual and ideational. Some facts and some meanings have to be actively eliminated because they are obstacles that stand in the way of resolution of an unsettled situa- tion. The idea that negation is connected with change, with be- coming other or different, is at least as old as Plato. But in Plato change, altering or othering, has a direct ontological status. It is a sign of the defective ontological character of that which changes, its lack of full Being. The negative proposition, which dealt with change, was thus the counterpart in knowledge of the ontological inferiority of one kind of existential material. But in modern science, correlations or correspondences of change are the chief object of determination. It is no longer impossible to treat the relation of the negative proposition to change and alteration as declarative of defective being. On the contrary, the negative proposition as such formulates a change to be effected in existing conditions by operations which the negative proposition sets forth.

It is an indication of an experimental operation to be performed such that conditions will be so varied that the consequences of the operation will have an evidential significance lacking in the condi- tions as they existed at first.

The affirmative proposition also has intrinsic connection with change. Take the proposition "This is red." On its face, it is purely affirmative; it carries with it no suggestion of negation or elimination. But the bare existence of a red thing is not a suffi- cient ground for the affirmation that "It is red." To be grounded, alternative possibilities must be ruled out. There is no logical necessity why this should be red; it may have been some other color a moment ago and become another color a moment from AFFIRMATION AND NEGATION 189 now. The proposition is "synthetic" in the Kantian sense; it can- not be grounded in a mere intellectual analysis of this. Valid determination that "this is red" depends upon (1) exhaustive dis- junction of alternative possibilities of color and (2) upon elimina- tion of all other possibilities than the one affirmed, the elimination resulting (3) from a series of hypothetical propositions such as "If blue, then such and such consequences," etc., in contrast with the proposition "If red, then such and such other and differential consequences." I do not mean of course that as a matter of fact such an elaborate process of determination is often gone through.

I do mean, however, that for complete logical validity there is required a proposition like the following: "Only if this is red, will observed phenomena be what they are." "Ofily" in this prop- osition depends upon a series of eliminations expressed in negative propositions. Whenever a scientific determination of color quality is required in solution of a scientific problem, inquiry proceeds in the direction of just such an exhaustive disjunctive system and of systematic elimination of all alternatives save one for which positive grounds are found.

The connection of this determination with deliberate institution of change should be obvious. A series of experimental operations has to be performed with and upon the existential material indi- cated by the demonstrative this. The changes which follow as consequences of the execution of these experimental operations provide grounds for denying that it is blue, yellow, purple, green, etc., and for affirming that it is red. If one is inclined to doubt this account, especially on the ground that the proposition in ques- tion, if not "self-evident," is at least not nearly as highly mediated as the account assumes, let him recall that scientifically color is determined only by operations which identify colors with certain rates of vibration, and red with one particular exclusive number.

In other words, the proposition "This is red" means, logically, that a certain differential change has occurred or may be predicted to occur when certain operations are undertaken. In the latter case, the logical meaning is "This will become red or will redden something else," certain conditions being postulated. If the propo- sition be interpreted to mean "This has been red for a long time," even more extensive mediation is required to warrant a con- 190 THE STRUCTURE OF INQUIRY elusion about the added trait of temporal duration. If it is taken to mean "It is red by nature or necessity," reference to change is excluded but this is the only case in which the proposition is not about a change.

Impersonal propositions such as "It is raining" have been the theme of more or less discussion. The natural interpretation of such propositions is that in them a total prior qualitative situation is negated-affirmed by specification of a change. "It" refers to the gross environing perceptual field; "rain" to the gross alteration it is undergoing. If the proposition is "It is only sprinkling" or "It is raining hard," the qualification is more differentiated because more specific negations have been introduced. Propositions of gross qualitative change are the starting points for a set of dis- junctive propositions in which a single continuous change is formulated in terms of a scale or spectrum of degrees. It is not sufficient to take a gross change in its given discreteness. It has to be resolved into a series of changes, each of which is determined by reference to its position in a continuum of changes. Such determination involves a disjunctive set of propositions. In each determination of position in the scale a negation of all disjunctive possibilities except one is involved. 4 After these general remarks, I come to the specific forms of the relations of affirmative and negative propositions designated as contrariety, subcontrariety and contradiction. From what has been said, it follows (1) that these relations have to be understood in the functional office they exercise in inquiry, (2) as correlative or conjugate determinations, not as independent sets of propo- sitions which happen to sustain to each other the relation des- ignated. (The ordinary square of opposition is likely to be interpreted in the latter sense, since failure to connect the proposi- tions which are contrary, etc., with the process of inquiry has the effect of setting-up a purely mechanical scheme of propositions each logically independent of the other.) I. Contrariety or logical opposition obtains between affirmative and negative propositions when both are general. The relation is such that only one can be valid and both may be invalid. The relation between "All marine vertebrates are cold-blooded" and 4 The topic of scales receives further attention in the next chapter.

AFFIRMATION AND NEGATION 191 "No marine vertebrates are cold-blooded" exemplifies the relation of contrariety. Contrariety of propositions sets the limits within which specific determinations must fall In themselves, they are indeterminate; that is, if they are taken as final and complete, rather than as expressing a certain necessary stage in the progress of controlled inquiry, they are logically defective. This logical defect is apparent in the fact that both may be invalid. Contraries are a stage in institution of the set of exhaustive disjunctives which, as we have seen, are required for adequate affirmative-negative determinations. They do not of themselves constitute the re- quired disjunctives, for (as is evident in the illustration just given) they permit of alternatives, such as "Some are and some are not." But they set the limiting termini for intermediate alternatives.

They serve to delimit the field of inquiry, and thereby to give direction to subsequent observational and ideational operations.

The traditional A and E propositions represent the limits within which alternatives fall, but the fact that both may be invalid proves that they do not do more than that. As contraries, they represent, then, not conclusions but the results of a preliminary survey of the total problematic field, the survey being made to circumscribe the field within which further determinations must occur. A process of groping reaches its initial termination when we can state the extreme boundaries within which a solution must be sought.

We have, then, the following logical situation. (1) On the one hand, the field of possible propositions must be bounded or else inquiry will roam all over the lot. This delimitation is effected by means of contrary general propositions. (2) On the other hand, when the strictly functional nature of the propositions having the relation of contrariety to each other is overlooked, these delimiting propositions are supposed to exhaust possible alternatives. Then the rigid type of Either-Or reasoning results, a type which is common in thought about social and moral issues. Either "The Individual" or "Society" as a fixed entity; either freedom from all restraint or coercion from without; either the bourgoisie or the proletariat; either change or the unchanging; either the continuous or the discrete, and so on. Only when the strictly functional nature of contrary propositions is seen do we 192 THE STRUCTURE OF INQUIRY escape from the unending and inherently endless round of con- troversies generated by this mode of thought. When their func- tional and instrumental nature is perceived, they are seen to be necessary, but necessary only because they set the boundaries within which a set of more determinate disjunctive alternatives are to be sought for. They are functional directives for further, more discriminating, determinations. 5 In logical theory, the rigidity and hence apparent finality of contrary propositions is often enforced by use of symbols that have no meaning or content of their own. A and Not~A are, for example, such symbols. These purely formalistic contraries can- not possibly have directive force. For if, say, "virtue" be assigned to A as its meaning, then Not-A includes not only vice but trian- gles, horse races, symphonies and the precession of the equinoxes.

Since the time of Aristotle, the nugatory nature of "infinitation of the negative" has been generally recognized. "What has not been so generally recognized is ( 1 ) that failure to recognize the inter- mediary function of contrary proposition tends in the direction of infinitation, and (2) that any purely formalistic either-or formu- lation of contraries (such as A and Not-A) eliminates reference to any universe of discourse and, hence, when any value is assigned to the positive expression, renders the negative wholly inde- terminate. Nevertheless, the institution of opposites in hypo- thetical form, when interpreted as a means of fixing the limits within which determinate disjunctive alternatives fall, is a necessary preparatory logical procedure.

II. Subcontrary propositions of the form "Some are . . ." and "Some are not" may both be valid while one must be valid, when they are determinate. "Some marine vertebrates are cold-blooded" and "Some are not" are subcontraries both of which are now known to be valid. The phrase "are now known" is related to 5 The dialectic of thesis, antithesis and synthesis recognizes that the initial con- traries are not final. But it suffers from the logical vice of supposing that the "synthesis" grows directly out of the contraries, instead of from determinate in- quiries which the contraries indicate. In scientific inquiry, thesis and antithesis are never treated as generating a synthesis. For example, the relationship between "heredity" and "environment" as contraries sets an important problem, as at one time in physics a problem was set by the relation of centrifugal and centripetal "forces." But the scientific problem is handled by means of analysis of the subject-matter of these highly general terms into specific conditions, not by manipulation of the concepts.

AFFIRMATION AND NEGATION 193 the clause in the previous sentence "when they are determinate." The formal logical relation involved is, in other words, a form of existential contents determined by observation. However, like any form, it may be abstracted, while the abstract form has logical meaning only in its possible application to material contents. As far as mere form is concerned, both of the propositions cited might be invalid. For apart from what is existentially determined, the valid proposition might be "No marine vertebrates have blood." Only because a conjunction of the traits of possessing a spinal column and possessing blood has already been established, are the propositions subcontraries.

Subcontraries are more determinate than contraries but are still indeterminate as compared with final judgment. For fully de- terminate propositions regarding the subject-matter in question would be u All marine vertebrates marked by -such-and-such traits (say, bringing forth young alive and breathing with lungs) are warm-blooded" and u All marine vertebrates having such-and-such other differential traits are cold-blooded." If they were final and complete, subcontrary propositions as logical forms would be even more slovenly than contraries. As a matter of fact, however, they record the results of observation in such a way as to provide factual data that set a definite problem. The subcontrary proposi- tions cited represented the state of zoology at a given date when the discovery of two kinds of marine vertebrates, marked off by differences in quality of blood, definitely set a problem; namely, the problem of discovering the conditions in which some marine animals are of one kind and others of another kind. It did so because of a material postulate, namely the postulate that blood plays such an important role in animal life that a difference with respect to it is, to a high degree of probability, bound up with other important characteristics. Propositions marked by "some," affirmative and negative, thus present the results of a relatively in- complete empirical state of inquiry, where "empirical" means a valid statement of results of actual observation without insight into the conditions upon which observed traits depend. The de- pendence of valid conclusions in existential matters upon factual observation shows that such propositions, while not final, represent 194 THE STRUCTURE OF INQUIRY a definite stage in the conduct of inquiry and perform a necessary office in carrying it forward to a conclusion.

Inquiry with respect to light is at the present time in this stage. There are grounds for holding that "Light in some respects is a radiant phenomenon and in some respects is not, being cor- puscular." Granting the adequacy of the observations upon which these propositions rest, no one will deny that they mark a scientific advance. On the other hand, few would contend that scientific inquiry can be content with these propositions as final.

They institute a definite problem for further investigation: Under what conditions is light vibratory and under what conditions is it discrete?

III. The discussion of subcontrariety leads up to the conception of subalternation. If it has once been determined that all marine vertebrates marked by a specified conjunction of traits are warm- blooded, the so-called subaltern that some such animals are warm- blooded is trivial. Reference to the general proposition may serve upon occasion as a reminder to some person who is temporarily forgetful, but it has no logical force. Suppose that inquiry at a certain stage has determined only that in the case of a shipwreck some passengers have been saved and some lost. Suppose further inquiry determines specifically the names of all who are saved and all who are lost. In the latter case, it is silly to recur to the weakened form "some" when the tabulated list of all of each kind is at hand. The name of any given person must appear in one list or the other. About a specified person there are no alternatives.

The real function of the proposition of the form of some is in the opposite direction from that of the traditional table. Instead of movement from "all" to "some," there is a reaction from some into all. At an early stage of inquiry that "some" are saved indi- cates that perhaps "all" on board have been saved. At the stage when the inquiry is completed, the transition is from the indefinite "some" of "all" who were on board, to all of a specified group.

In a strictly empirical proposition (in the sense of "empirical" defined above), there is no difference in logical form between the proposition "All cases so far observed are such and such" and the proposition "Some cases out of all existential cases, past, present and future, are such and such." The logical sense of both lin- AFFIRMATION AND NEGATION 195 guistic forms is "Perhaps all cases are such and such." When the conditions under which phenomena are such and such have been exclusively determined (through a set of affirmative-negative prop- ositions) then a general proposition in the form of a law is possible:

Whenever conditions are such and such, consequences are such and such. 6 IV. The foregoing analysis has one main purpose, namely to indicate, on one side, that when affirmative and negative proposi- tions are taken to be final and complete (as they must be when their operative connection with the progressive conduct of inquiry is ignored) the forms in question are mechanical and arbitrary; and, on the other side, to indicate that when their functional ca- pacity is taken into account the relations of contrariety, sub- contrariety and subalternation mark definite stages in the advance of inquiry toward a final warranted judgment. These considera- tions come, as it were, to a head in the case of contradictory propo- sitions, those that are such that if one is valid, the other is invalid, and if one is invalid the other is valid. In the traditional square of opposition this relation of contradiction is symbolized by the diagonal lines from the general affirmative to the negative particular (some, meaning one or more) and from the general negative to the affirmative particular. Formally speaking, it is certainly true that the proposition "all men are white" is contra- dicted if a single case of a colored person is observed, while the proposition "No men are red" was negated as soon as the first North American Indian was encountered.

But the essential logical point here is that the general (affirmative or negative) is negated not by the indeterminate "some" but by the determinate singular. "Some" is logically either excessive or de- ficient. It is excessive, if a singular case has been determined (not in fact an easy matter) ; it is defective, if "some" is understood in its strict logical force, namely, as an indication of a possibility, of the form "may be" or "perhaps." The fact that a given Z or O proposition may be invalid is enough of itself to prove that it can- not contradict in any strict logical sense a general proposition of the opposite quality. The proposition "Some men are not white" 6 The difference between the two kinds of general propositions in both of which the word all may appear is discussed in Chap. XIII.

196 THE STRUCTURE OF INQUIRY indicates that an object may be colored and yet be a human being, or may be a human being and yet not be white. We are familiar with the warning against vague generalities. The warning is de- cidedly relevant at this point. "Some," if unspecified with refer- ence to singulars, is of the nature of a vague generality. If it is specified, then it assumes one of two forms: "Such and such determinate singulars are of a given kind," which negates a general proposition that "all are of some other kind"; or, still more deter- minately, "All singulars marked by specified traits are of a certain kind." In either case, it is not an indeterminate "some" which contradicts the general A or E proposition.

As has already been indicated, the proposition about a number of (or some) singulars being such and such sets a problem. It suffices to negate a general proposition of the opposite quality.

But the negation is in so far incomplete or indeterminate. It does not of itself establish a valid, universal proposition. It warrants a contradictory universal only when two logical conditions are satisfied: (1) Determination of a set of alternative disjunctives as exhaustively as possible, and (2) determination of the differen- tial traits which are evidential signs of one and not another kind.

At a given stage of scientific inquiry, an exception is discovered to some previously accepted generalization. If careful inquiry substantiates the authenticity of the exceptional singular, then the generalization in its previous form is certainly negated. But no scientific inquirer would suppose for a moment that this negation was equivalent to establishment of a valid universal proposition.

The question at once arises as to the exact conditions under which the exceptional and negative case occurs. As soon as this is done, we have another generalization: "All cases marked by certain traits are such and such." In short, the discovery of singulars or a singular that negates a generalization is but the antecedently conditioning means to further inquiries. The proposition in which it is embodied is not final or complete, for it functions as occasion and stimulus of further inquiries with view to determin- ing honv and ivhy the exception occurs. When these inquiries are satisfactorily concluded, then and only then do we have a final proposition, which takes the form of a new general proposition.

In no case of controlled inquiry is a flat negation of a generaliza- AFFIRMATION AND NEGATION 197 tion taken to be final If it were so taken, a former generalization would simply be abandoned and that would be the end of the matter. What actually happens is that the prior generalization is modified and revised by discovery of the contradictory instance.

Certain data discovered by using the Einsteinian theory of rela- tivity contradicted the Newtonian formula of gravitation. If such negations had the independent and final logical status attributed to them by traditional formalistic logic, either the Newtonian for- mula would have been declared invalid and the matter would have ended there, or else the observational data would have been de- clared false and impossible because they contradicted the general proposition. Even in the cases in which an exception turns out to be apparent rather than actual, the older generalization is not simply confirmed, but gains a new shade of meaning because of its capacity to apply to the unusual and seemingly negative in- stance. It is in this sense that "the exception proves the rule." The logic of the contradictory relation of propositions thus af- fords a crowning proof of the functional and operative import of affirmative-negative propositions. Nothing is more important in inquiry than institution of contradictory propositions. Since one must be valid and the other invalid, they are determinate in a way in which contraries and subcontraries are not. But if the tradi- tional theory were sound, inquiry would have to stop right there.

There would be no ground upon which to decide which one of the two is valid and which is invalid. Those who prefer to trust to the "evidence of the senses" would hold that the generalization had been proved false. Those who distrust sense and exalt "rea- son" would be inclined to reverse the conclusion and hold the singulars are not "really" what they seem to be. Institution of contradictories in the actual procedure of scientific inquiry is crucially important just because it does not adopt the canons of any theory that makes contradictories final and complete. In the conduct of inquiry, institution of a contradictory negation is treated as a step in the continuation of inquiry towards final judgment. The final effect is to revise the generalization reached in earlier inquiries. Through this modification a generalization becomes applicable to both the old evidential material which sup- 198 THE STRUCTURE OF INQUIRY ported it and to the new evidential material which contradicts the earlier generalizations.

The original Aristotelian conception of affirmation and nega- tion at least corresponded to what was supposed to be the onto- logical nature of the objects to which affirmative and negative propositions apply. The functional conception here advanced denies that affirmative and negative propositions have a one-to-one correspondence with objects as they are. But it gives them the operative and instrumental force of means of transforming an unsettled and doubtful existential situation into a resolved de- terminate one. The modern theory, derived, as has been said, from the attempt to retain forms after their material or existential content had been abandoned, is grounded in nothing and leads nowhere. It is formal only in the sense of being empty and mechanical. In neither reflects existence already known nor for- wards inquiry into what may and should be known. It is a logical vermiform appendix.

In view of the fact that the metaphysical problem of the One and the Many has at various times had a very considerable influ- ence upon logical theory, it may be appropriate, in concluding this chapter, to say a few words on that topic as it affects logical theory. Unity, or what is termed The One, is the existential counterpart of the product of operations which, by institution of agreement of different contents in evidential force, establish war- ranted identities. Negation, on the other hand, discriminates and produces differences. The latter when hypostatized constitute the Many. The problem when approached from the logical side is one of operations of unifying and discriminating. These oper- ations have of course an existential basis and matrix. Integration and differentiation are biological processes foreshadowing the logi- cal operations just mentioned. They are themselves prepared for and foreshadowed in physical processes of conjunction and sepa- ration. The insoluble problems which have led to speculative metaphysical constructions about the One and the Many arise from making entities, expressed in nouns, out of processes and operations properly designated by active verbs and adverbs.