The Doctrine of Signification, Propositions, and Future Contingents
The Subject and Purpose of the Work
Aristotle’s treatise Peri Hermeneias concerns the simple statement-making sentence, the proposition that is either true or false. It stands as a bridge between the Categories, which treats of simple terms signifying individual substances, qualities, quantities, and the other genera of being, and the Prior Analytics, which treats of syllogisms composed from propositions. The proper subject of this book is therefore the enuntiative speech, the declarative discourse that affirms or denies one thing of another. Other forms of speech, such as prayers, commands, questions, vocatives, and exclamations, belong to the orator and the poet rather than to the philosopher who pursues truth in the demonstrative sciences.
The title itself bears interpretation. The Greek term hermeneia signifies the act of expressing through articulate speech what the mind has conceived. To interpret is to bring forth the inward conception of the soul into outward voice for the hearer’s understanding. Hence this work treats of the way in which the mind’s apprehensions are made manifest by means of significant sound and assembled into the unit called the proposition, which is the seat of truth and falsity.
The order of the parts of speech follows the order of nature. The mind first conceives simple things, then joins them into compositions; the voice first utters simple sounds, then combines them into statements. Aristotle therefore begins with the simple parts, the name and the verb, then passes to the speech composed of them, and from speech to the enunciation, and from the enunciation to affirmation and negation, and finally to the various species of opposition between affirmations and negations, including the most difficult question of statements made about future contingent matters.
The Order of Spoken Sounds, Mental Conceptions, and Things
At the opening of the treatise Aristotle establishes the foundational doctrine that spoken sounds are signs of the passions of the soul, and written marks are signs of spoken sounds. The order is fourfold: things exist outside the soul; the soul receives the likenesses of things by way of conceptions; the voice utters sounds that signify these conceptions; and writing depicts the voice with visible characters.
This order is not arbitrary. Things themselves come first by nature, since nothing would be conceived if nothing existed. From things, the mind takes its likenesses or similitudes, which are the passions of the soul, the mental affections produced when intelligible or sensible objects act upon the apprehensive power. These conceptions in turn are signified by spoken sounds, and the sounds are recorded by letters. Thus writing is the sign of a sign, twice removed from the thing itself, while the voice is the immediate sign of the mind’s affection.
The names and verbs that men utter differ from nation to nation, since speech is established by convention. But the conceptions of the soul, which the voices signify, are the same for all men, because they arise from the same things which are the same for all. Likewise the things from which conceptions arise are the same in every place. From this twofold sameness it follows that men of every tongue think the same things, although they utter different sounds; the Greek who says anthropos, the Roman who says homo, the barbarian who utters yet another word, all alike conceive the same likeness of man in the mind, because all alike behold the same nature.
The likenesses are called passions because the mind is moved or affected by the things it apprehends, suffering an impression as wax suffers the impression of a seal. The mind is not the originator of these images by its own power but receives them through sensation and intellection of what exists outside it.
Concerning the relation of spoken sound to thought, sounds are not natural signs but conventional ones. They do not signify by nature, as smoke signifies fire or footprints signify the passage of an animal, but by institution and human agreement. This is shown by the diversity of languages and by the fact that the same articulate sound means different things in different tongues. A natural sign cannot be otherwise than it is, but a name might have been imposed upon any other thing, had men so chosen.
Yet though the imposition of names is voluntary, once a name has been instituted it functions stably within its community of speakers. Without such stability the use of speech for the transmission of thought would be impossible, and the offices of teaching, deliberating, and disputing would all fall into ruin.
The Composition of Truth and Falsity
Truth and falsity arise neither in things considered absolutely, nor in simple conceptions, nor in simple utterances. Things in themselves are neither true nor false but only exist or do not exist. A simple thought of man, of horse, of whiteness, considered by itself, is neither true nor false until something is joined to it. A simple name spoken alone, such as “man” or “white,” signifies something but neither affirms nor denies and therefore cannot be true or false.
Truth and falsity arise only in composition and division. When the mind joins one conception to another, saying that this is that or that this is not that, then for the first time arises an affirmation or denial that can be measured against the way things stand. If the joining or separating in the soul corresponds to the joining or separating in things, then the conception is true; if it does not so correspond, then it is false. The same holds for the spoken sentence, which is true or false insofar as it expresses a true or false composition of the soul.
The example often given is that of “goat-stag,” a name compounded from “goat” and “stag” to signify a creature that has both natures. The name signifies something to the mind, since it brings forth a conception of such a chimerical being, but until one says goat-stag is or goat-stag is not, no truth or falsity has been determined. The simple naming of the impossible signifies; only the assertion concerning the impossible falls into truth or falsity.
On the Name
A name is defined as a significant sound, established by convention, without time, no part of which signifies separately. Each part of this definition excludes something else from the genus of name.
The name is a sound, which separates it from those things that signify without uttering sound, such as nods, gestures, and written marks taken in themselves. The name is a sound that signifies, which separates it from those sounds that mean nothing, such as inarticulate cries, the sounds of animals, and noises made by inanimate bodies. The name signifies by convention and not by nature, which separates it from those sounds that signify naturally, such as the groans of the sick that betoken pain or the cries of beasts that betoken their passions. The name is without time, which separates it from the verb, for the verb signifies time along with its principal signification, but the name does not. The name has no part that signifies separately, which separates it from speech, for in speech the parts also signify, as in “good man” both “good” and “man” signify separately.
When a name like “Philip” is uttered, the parts “phil” and “ip” do not signify anything when considered apart; the whole name signifies one thing, namely the person so called, by virtue of the imposition that established the whole. The same is true of names compounded from words that, were they uttered separately, would each have meaning. In “horseman” the parts “horse” and “man” each signify when standing alone, but in the compound name they no longer signify their proper objects separately; the whole signifies one thing, the rider, by a single imposition.
There are also names not strictly names, such as indefinite names, which signify nothing definite. The negation “not-man” is not properly a name, for it neither affirms nor denies, nor does it pick out any definite kind of being, but rather encompasses everything that is not man, whether existent or non-existent, whether animate or inanimate. It is called indefinite because what falls under it is not bounded but ranges across an indefinite multitude.
The cases or inflections of names, such as “of man,” “to man,” “by man,” are not names properly speaking but derivatives of names. The proper name is the nominative form; the oblique cases relate to the nominative as inflections, modifying it without being themselves the principal name. A statement made with an inflected case, joined to a verb, does not constitute a proposition in the strict sense, because the inflected name does not stand as the proper subject of predication.
On the Verb
A verb is a significant sound, established by convention, signifying with time, no part of which signifies separately, and which is always a sign of those things that are said of something else.
The verb shares with the name the conditions of being a significant sound, established by convention, and having no part that signifies separately. What distinguishes it is the signification of time and the function of being a sign of things said of something else.
Every verb consignifies time. When one says “runs,” the verb signifies not merely the action of running but also that this action is present; “ran” signifies the action together with its pastness; “will run” signifies the action together with its futurity. The principal signification of the verb is the action or passion; the consignification is the time in which that action or passion takes place.
The verb is a sign of what is said of another because every verb naturally functions as a predicate. To predicate is to attribute something to a subject, and the verb performs this attribution by signifying that the subject does or undergoes something. Even the verb of existence, the simplest of all verbs, when uttered with a subject signifies that the subject is, that something is predicated of it.
Verbs in their unmodified form are called verbs simply; their inflections, such as past and future forms, although they too signify time, are sometimes called by a different name to indicate their derivation from the principal verb. Yet for the purposes of logical analysis, all forms that signify time and predicate something of a subject function as verbs.
Indefinite verbs, such as “does not run,” parallel the indefinite names. They are not verbs in the strict sense, for they signify indefinitely, applying equally to one who walks, who sits, who stands, who sleeps, indeed to anyone whose state is not running. The indefinite verb does not pick out a determinate action but only the absence of a determinate action, and so applies to an indefinite multitude of things.
A verb uttered by itself, like a name uttered by itself, signifies something but neither affirms nor denies. “Runs” by itself signifies the action of running, but it does not yet say of whom this action is asserted nor whether it is asserted or denied. To produce a statement, the verb must be joined with a subject, and the affirmation or negation must be determined.
On Speech and the Statement
Speech is a significant sound, established by convention, of which some part signifies something separately, as a part not as an affirmation or negation.
Speech differs from the name and the verb because its parts signify separately. In the speech “good man,” both “good” and “man” signify; in “Socrates walks,” both “Socrates” and “walks” signify. The whole speech signifies one thing by virtue of the composition of its parts, but the parts themselves also have signification.
Yet the parts of speech, when taken separately, signify only as names or verbs, not as affirmations or denials. The single word “man,” extracted from “Socrates is a man,” signifies the nature of man but does not assert anything about Socrates or anyone else. To be an affirmation or negation, the speech must be a whole composed of subject and predicate joined into a unity.
Not all speech is statement-making. Prayer is speech, since it has parts that signify, but it is neither true nor false. The poet who composes a hymn or a lamentation, the orator who delivers a panegyric, the lover who utters a vow, all produce speech that is meaningful but not declarative. Such speeches belong to rhetoric and poetry; they pursue persuasion, beauty, or the moving of the affections. Only declarative speech, which affirms or denies one thing of another, can be true or false, and only such speech belongs to logic.
On the Statement and Its Unity
The declarative statement, the enuntiatio, is a speech in which truth or falsity inheres. It is constituted by joining a subject to a predicate by means of a verb, or by separating them through negation.
There are two species of statement: affirmation, which asserts something of something, and negation, which removes something from something. The affirmation declares that what is is, or that what is not is not; the negation declares that what is is not, or that what is not is. The two species exhaust the genus of declarative speech.
Every statement, whether affirmation or negation, must possess unity. A statement is one when it signifies one thing about one thing. If many things are predicated of one subject, or one thing of many subjects, the statement may be one in form but many in content; it is one statement only equivocally if the predicates do not coalesce into a single signification, or only by accidental aggregation.
A statement may also be one because of conjunction or because of nature. By conjunction, when the parts are joined by connectives or by the verb of existence into a continuous expression, the speech appears one externally. By nature, the statement is truly one when its predicate signifies one thing said of one subject, with no equivocation or accidental multiplication.
The simple statement is the most basic kind, in which a single predicate is attributed to a single subject. From simple statements compound statements arise through conjunction, conditional joining, or other modes of combination. The treatise here treats principally of the simple statement and the relations of opposition that arise from it.
Affirmation and Negation: The Foundation of Opposition
Every affirmation has its corresponding negation, and every negation has its corresponding affirmation. The negation that opposes an affirmation is that negation which denies the very thing the affirmation asserts of the very thing of which it asserts it. The affirmation “Socrates is well” is opposed by the negation “Socrates is not well.” These two cannot both be true at the same time concerning the same Socrates in the same respect, nor can they both be false; one or the other must hold.
Such opposition is called contradiction. In contradictory opposition the same predicate is affirmed of the same subject by the affirmation and denied of the same subject by the negation, with no shift of time, of respect, of meaning. If any such shift occurs, the opposition is not genuine contradiction but only apparent contradiction.
The principle that of every contradictory pair one is true and the other false governs all logical analysis. Whoever affirms what is must speak truly; whoever affirms what is not must speak falsely; whoever denies what is must speak falsely; whoever denies what is not must speak truly. By this measure of correspondence between speech and being, truth and falsity are determined.
Yet not every pair of opposed statements forms a true contradiction. Two affirmations may be opposed, as “Every man is just” and “Every man is unjust”; but these are not contradictories, for both may be false if some men are just and others unjust. They are contraries, opposed in the strongest way among affirmations, but not standing in the precise opposition that requires one to be true whenever the other is false. The contradictory of “Every man is just” is rather “Not every man is just,” which denies the same predicate of the same subject taken in the same universal way.
Universal and Particular: The Square of Opposition
Some things are universal and some are particular. The universal is what is naturally predicable of many; the particular is what is not so predicable. Man, animal, white, just—these are universals, since they may be said of many singulars. Socrates, this man, this whiteness here—these are particulars, since they cannot be said of more than one in the manner of an essential or accidental predicate.
When statements are made about universals, the predication may be universal or particular. The universal affirmation says that the predicate belongs to every instance of the subject: “Every man is an animal.” The universal negation says that the predicate belongs to no instance of the subject: “No man is a stone.” The particular affirmation says that the predicate belongs to some instance: “Some man is just.” The particular negation says that the predicate does not belong to some instance: “Some man is not just.”
The four statements stand in a network of oppositions. The universal affirmation and the universal negation are contraries: both cannot be true together, but both may be false together, as when the subject is found in some instances but not in others. The particular affirmation and the particular negation are subcontraries: both may be true together but both cannot be false together. The universal affirmation and the particular negation are contradictories, as are the universal negation and the particular affirmation: in each contradictory pair, one must be true and the other false, with no remainder.
There is also the relation of subalternation: the particular affirmation follows from the universal affirmation, since if every man is an animal then some man is an animal; the particular negation follows from the universal negation, since if no man is a stone then some man is not a stone. The converse does not hold, since the truth of the particular does not establish the truth of the universal.
When statements are made about singulars, such as “Socrates is just” and “Socrates is not just,” only a single pair of contradictories arises, since the singular subject does not admit of being divided into universal and particular instances. Of every such contradictory pair concerning a singular and a present or past state, one must be true and the other false.
“It is necessary that of every contradiction one part be true and the other false.”
This principle governs all contradictories regarding things that are or have been. The question whether it also governs contradictories regarding future contingents constitutes the most difficult inquiry of the treatise.
Indefinite Statements
Between the universal and the particular falls the indefinite statement, in which the subject is named without being marked as universal or particular. “Man is just” does not say whether every man is just, or some man, or no man. The indefinite statement is logically less determinate than either the universal or the particular and may be interpreted as either, depending on context and intention.
For purposes of strict analysis, indefinite statements are reduced to particular ones, since the bare assertion that man is just is verified if any man is just. Yet in ordinary speech the indefinite often functions as a universal, as when one says “Man is mortal” intending that all men are mortal. The logician must be careful to determine whether the indefinite is to be taken universally or particularly, and to fix the analysis accordingly.
The statements that contain indefinite names or indefinite verbs, such as “Not-man is just” or “Man does not run,” generate further complications. The indefinite name, prefixed with a negative particle, applies to all things that are not man, both existent non-men and non-existent non-men. The indefinite verb similarly applies to all states that are not the determinate state signified by the verb’s affirmative form. From such indefinite terms arise statements whose truth conditions extend over wider domains than the corresponding definite statements.
There are thus statements in which both subject and predicate are definite, both indefinite, or one of each. From these combinations arise eight species of simple statement when considered together with their negations. The logical analysis of these species, their truth conditions, and their oppositions occupies considerable attention, since the proper handling of indefinite terms is essential to the construction of valid syllogisms.
Statements with the Verb of Existence as Third Element
Sometimes the verb of existence joins the predicate to the subject, as in “Socrates is just,” where “is” stands as a copula between “Socrates” and “just.” In such statements the verb “is” functions as a third element, neither the subject nor the predicate proper but the bond between them. The principal verb in such a statement, when one analyzes its structure, may be considered to be the predicate together with the copula, or the predicate alone with the copula serving merely to mark the assertion.
In other statements, the verb of existence stands alone as predicate, as in “Socrates is,” where “is” signifies existence simply. Here the verb is not a mere copula but a principal predicate, asserting that the subject exists. The two uses of the verb of existence, as copula and as principal predicate, must be carefully distinguished, for the rules of contradiction and conversion vary somewhat between them.
When negation is applied to a statement with the copula, the negative particle may attach to the copula or to the predicate. “Socrates is not just” denies that justice belongs to Socrates and is the contradictory of “Socrates is just.” “Socrates is not-just” affirms an indefinite predicate, namely the predicate “not-just,” of Socrates and is not strictly the contradictory but rather affirms something concerning Socrates with respect to an indefinite predicate.
The distinction between these two kinds of negation has consequences. “Socrates is not just” may be true if Socrates does not exist, for what does not exist cannot be just. “Socrates is not-just” requires that Socrates be something, and that the something he is be other than just. The former is a simple denial; the latter is an affirmation with an indefinite predicate.
Multiple Predicates and Subjects
A statement may have many predicates joined to one subject, or one predicate joined to many subjects, or many to many. “Man is a rational mortal animal” predicates several things of one subject; “Cicero and Demosthenes are orators” predicates one thing of two subjects.
When the predicates joined to the subject coalesce into a single signification, the statement is one in nature as well as in expression. Man is one being, and rational mortal animal is one description of his nature, so that the statement composing these predicates is one statement signifying one thing. But when the predicates do not coalesce, as in “Socrates is musical and walking,” the statement is one in form but many in content, since musical and walking do not unite into a single signification.
For the purposes of contradiction, the unity of the statement must be precisely determined. The contradictory of a statement that combines two predicates into a single signification denies that single signification; the contradictory of a statement combining two distinct predicates by accidental conjunction may have to deny each predicate separately to constitute a full contradiction. The logician must therefore analyze the unity of the proposed statement before applying the rules of opposition.
The same considerations apply to subjects. When many subjects are joined in one statement, the statement may signify a single subject taken collectively or many subjects taken distributively. “The Greeks defeated the Persians” may signify the collective body of Greeks acting as one or the individual Greeks acting in concert. The truth conditions vary accordingly, and so do the rules of contradiction.
The Problem of Future Contingents
The most weighty inquiry of the entire treatise concerns statements about future contingent matters. Of every contradiction concerning what is or has been, one part must be determinately true and the other determinately false. Concerning future contingents, however, a great difficulty arises.
If it is now true that there will be a sea-battle tomorrow, then the sea-battle must occur tomorrow, for what is true cannot be otherwise than as it is said to be. If it is now false that there will be a sea-battle tomorrow, then the sea-battle cannot occur, for what is false cannot turn out to be the case. But if the truth of every future statement is fixed from eternity, then every future event is fixed; nothing happens by chance or by deliberation; the very possibility of action in council, of choice, of effort, is destroyed, since whatever will happen will happen by necessity, and whatever will not happen could not have happened.
The chain of reasoning may be made still tighter. If at any present moment it is true that something will be tomorrow, then ten thousand years ago it was likewise true that this same thing would be at the appointed time. And what was true ten thousand years ago has been settled in its truth from then until now; the eternity of past truth secures the necessity of the present and future. By this argument, all things would come to pass by an inflexible fate, and deliberation would be vain.
“If everything that is, is by necessity, and everything that will be, will be by necessity, then there is no need to deliberate or to take pains; for whatever we do, that will be which is determined, and whatever we omit, that will not be which is not determined.”
Yet experience and reason both teach that many things come to pass through human deliberation, that many things are open to be or not to be, that the future is not in all respects settled. Men take counsel about what to do; they invent arts to bring about what nature would not provide; they restrain themselves from what they might otherwise do; they alter their actions according to circumstance. If all things were fixed by an antecedent necessity, all such deliberation would be empty, all moral effort would be misdirected, and the praise or blame of human conduct would have no foundation.
The argument must therefore be answered. The answer lies in a careful distinction between two kinds of necessity. There is the necessity that belongs to a thing absolutely, considered in itself, and there is the necessity that belongs to a thing conditionally, considered in relation to another. The proposition “What is, while it is, is necessary” expresses a conditional necessity, since it says that when something is, it cannot at that very moment also not be; but it does not say that the thing must be at all times and in all circumstances. The proposition “What is, is necessary absolutely” would express an absolute necessity, and this does not follow from the conditional necessity.
Now of every contradictory pair concerning the future, it is true to say that one will be true and the other false, but it is not true to say that determinately this one will be true and determinately that one will be false. The disjunction “Either there will be a sea-battle tomorrow or there will not be a sea-battle tomorrow” is necessarily true, since the disjunction encompasses all possibilities. But the individual disjuncts are not yet determinately true or false. The truth or falsity of each member of the contradiction is not fixed in advance but waits upon the outcome of contingent causes.
The matter is illustrated by considering what kind of necessity holds for things that exist now. If Socrates sits, it is necessary, while he sits, that he sit; he cannot at the same time both sit and not sit. But it is not absolutely necessary that Socrates sit, for he might have stood, or walked, or done many other things, and tomorrow he may rise. The necessity of his sitting is the necessity of the present moment, not the necessity of the unalterable nature. So too with future contingents: it is necessary that one or the other member of the contradiction be true, but it is not necessary that this particular member rather than that one be true.
Boethius extends this analysis by distinguishing the modes of being among things. Some things are by necessity, such as the regular motions of the heavens and the essential properties of natural kinds. Some things are by impossibility, such as the union of contradictories. Many things are contingent, neither necessary nor impossible but admitting of being and not being. Of these contingents, some happen most often, some happen rarely, and some happen indifferently. The course of nature admits of all these modes, and the structure of speech corresponds to the structure of being.
For things that are by necessity, the affirmation and the negation are determinately true or false from eternity. For things that are by impossibility, similarly. But for contingent things, the determination of truth waits upon the actual unfolding of the contingent cause. The contradictory pair is determinately one true and one false at the time of the event, but not before; and the principle that of every contradiction one is true and the other false must be understood in this way for future contingents.
“It is necessary that there will be or will not be a sea-battle tomorrow, but it is not necessary that there will be a sea-battle tomorrow, nor is it necessary that there will not be.”
The disjunction as a whole bears necessity, but the disjuncts taken individually do not. From this it follows that deliberation has a real object, that human choice is genuinely open, that the future is not in all respects fixed by an antecedent inflexible fate. There is room for chance, for fortune, for free choice, and for the operation of human prudence within the order of contingent causes.
This solution preserves both the universal principle of contradiction, which is the foundation of all logical reasoning, and the freedom of human action, which is the foundation of moral life. The principle of contradiction governs being absolutely, holding for what is and what is not; but its application to the future is qualified by the contingency of the events spoken of. The contradiction is not destroyed, but its members do not yet stand in determinate truth and falsity until the time appointed for the event.
The defense of contingency also bears upon the providence of God. If everything happened by absolute necessity, then divine providence would be a kind of fatal mechanism, and the moral order would be a deception. But because contingents are genuinely contingent, providence ordains them according to their nature, working through free causes when the cause is free and through necessary causes when the cause is necessary. The divine foreknowledge does not impose necessity upon contingents but knows them as they are: necessary, contingent, possible, or impossible according to the proper mode of each.
The Possible, the Necessary, and the Impossible
After treating of statements about future contingents, the inquiry turns to statements that include the modes: possible, contingent, necessary, impossible. These modes qualify the relation between subject and predicate, indicating not only what is the case but in what manner it is or is not the case.
A statement is necessary when its predicate belongs to its subject in such a way that it cannot fail to belong; impossible when the predicate cannot belong; possible when the predicate can belong or fail to belong; contingent when the predicate may or may not belong indifferently. The modes are reduced to one another by various relations: what is necessary is possible to be, but not possible not to be; what is impossible is not possible to be; what is contingent is possible both to be and not to be.
The contradictories of modal statements require careful analysis, since the placement of the negation alters the meaning. “It is possible to be” is opposed by “It is not possible to be,” not by “It is possible not to be.” For “It is possible not to be” affirms a possibility, namely the possibility of not being, and is compatible with “It is possible to be,” whereas “It is not possible to be” denies the possibility outright. Similarly, “It is necessary to be” is opposed by “It is not necessary to be,” not by “It is necessary not to be,” for the latter affirms a different necessity.
The placement of the negation is therefore critical. To negate a modal statement properly, the negation must attach to the mode, not to the predicate. The statement “It is possible for Socrates to be just” is contradicted by “It is not possible for Socrates to be just,” meaning that the possibility itself is denied. The statement “It is possible for Socrates not to be just,” by contrast, affirms a different possibility, the possibility of not being just, and is compatible with the first.
The relations among the modes generate a square of opposition parallel to that of the assertoric statements. The necessary affirms the strongest connection; the impossible affirms the strongest disconnection; the possible to be and the possible not to be stand between these extremes, both affirming weaker connections that admit of the contrary. The square of modal opposition exhibits contraries, subcontraries, contradictories, and subalternates analogous to the square of assertoric opposition.
There is also the question of whether possibility in the strict sense includes necessity or excludes it. In one sense, what is necessary is also possible, since whatever is the case is possible to be the case. In another sense, possibility is restricted to what may either be or not be, and in this restricted sense the necessary is not possible, because the necessary cannot fail to be. The two senses of possibility correspond to two different ways of conceiving the relation between the predicate and the subject: as merely consistent, in the broad sense; or as genuinely open to either outcome, in the narrow sense.
The treatment of modal statements is essential for the construction of demonstrative science and for the analysis of practical reasoning. Demonstrations proceed from necessary premises to necessary conclusions; deliberations and counsels concern what is possible and contingent; the impossible is excluded from both. Hence the proper logic of the modes is indispensable for both theoretical and practical philosophy.
The Convertibility and Equipollence of Modal Statements
Modal statements, like assertoric statements, admit of various transformations that preserve their truth. Equipollent statements are those that, though differing in expression, signify the same truth conditions. “It is necessary that something be” and “It is impossible that this not be” are equipollent, since both assert the same relation between predicate and subject under different modal expressions.
The chart of equipollences may be set out by considering all the combinations of mode and negation. To say that it is necessary that something be is equivalent to saying that it is not possible that it not be, and to saying that it is impossible that it not be. To say that it is possible that something be is equivalent to saying that it is not necessary that it not be, and to saying that it is not impossible that it be. Similar equivalences hold for the negative modes.
Through these equipollences, every modal statement may be expressed in alternative forms without change of meaning. This permits the logician to choose, in any given analysis, the form most useful for showing the logical relations under examination. In demonstrative reasoning, the form expressing necessity is often most convenient; in deliberative reasoning, the form expressing possibility is more useful.
The Order of Modal Negation and Affirmation
A particular difficulty arises concerning the proper placement of the negative particle in modal statements. In the assertoric statement, the negative particle attaches to the verb and produces a contradictory whose truth conditions are clear. In the modal statement, the negative particle may attach to the mode or to the verb expressing the action or state, and these two placements produce statements with different truth conditions.
If one wishes to contradict the statement “It is possible for Socrates to walk,” the proper contradiction is “It is not possible for Socrates to walk,” in which the negation attaches to the mode of possibility. The statement “It is possible for Socrates not to walk” does not contradict the original but rather affirms a different possibility, namely the possibility of not walking.
This distinction must be carefully maintained, for failure to observe it leads to fallacies. One who replies to “It is possible that you do this” with “It is possible that you not do this” has not contradicted the first speaker but has affirmed a compatible possibility. The genuine contradiction would be “It is not possible that you do this,” which denies the possibility outright.
The same considerations apply to statements concerning the necessary. “It is necessary that Socrates walk” is contradicted by “It is not necessary that Socrates walk,” which denies the necessity. The statement “It is necessary that Socrates not walk” affirms a necessity of not walking, which is a stronger claim than the simple denial of necessity. The latter merely denies that Socrates must walk; the former asserts that Socrates must not walk. The two are by no means equivalent.
Statements That Cannot Both Be True nor Both False
Some pairs of opposed statements are constructed in such a way that they cannot both be true together and cannot both be false together. These are the strict contradictories, which exhaust the possibilities. Other pairs may not exhaust the possibilities, leaving room for a third option, or they may overlap, allowing both to be true.
The strict contradictory of “Every man is just” is “Not every man is just,” and not “No man is just.” The first denial states only that the universal does not hold, leaving open whether some men are just and others not. The second denial states the universal opposite, that justice belongs to no man. Between “Every man is just” and “No man is just,” there is the intermediate possibility that some men are just and others not, and this intermediate falsifies both universal statements without falsifying the genuine contradictory.
The proper construction of contradiction must therefore consider the quantity of the subject term. For a universal subject, the contradictory of the universal affirmation is the particular negation, and the contradictory of the universal negation is the particular affirmation. For a singular subject, the contradictory of the affirmation is the simple negation. For an indefinite subject, the contradictory is determined by interpretation; the indefinite is usually treated as the particular for purposes of strict contradiction.
The square of opposition exhibits all these relations together. At the corners stand the universal affirmation, the universal negation, the particular affirmation, and the particular negation. Across the diagonals stand the contradictories: universal affirmation against particular negation, universal negation against particular affirmation. Along the top stands the contrary pair of universals; along the bottom stands the subcontrary pair of particulars. Along the sides stand the subalternate pairs of universal and particular of the same quality.
The Application of the Doctrine to Singulars and to Future Singulars
The doctrine of contradiction applies most clearly to statements about singulars in the present or past. “Socrates is walking” and “Socrates is not walking” are strict contradictories: one is true, the other false, with no remainder. The same holds for statements about the past: “Socrates walked yesterday” and “Socrates did not walk yesterday” exhaust the possibilities.
For singulars in the future, the doctrine is qualified, as has already been explained. “There will be a sea-battle tomorrow” and “There will not be a sea-battle tomorrow” exhaust the possibilities in the sense that one will be true and the other false at the time of the event, but neither is determinately true now in such a way as to impose necessity upon the event. The disjunction holds with necessity; the disjuncts hold contingently.
This qualification preserves the universality of the principle of contradiction while accommodating the contingency of future events. The principle is not abolished but is applied according to the nature of the things spoken of: determinately for things determinately existing or determined in the past, indeterminately for things whose causes have not yet produced their effects in the contingent order.
Statements About Non-Existent and Universal Subjects
Statements may be made about subjects that do not exist, such as “The chimera is winged,” and about universal subjects taken as such, such as “Man is mortal.” In the first case, the question arises whether such statements can be true at all, since the subject is nothing. In the second case, the question arises how the predicate is attributed: to the universal nature itself, to all individuals falling under it, or in some other way.
Concerning non-existent subjects, the affirmation requires that the subject exist in some manner, at least conceptually, for the predication to take effect. The chimera, considered as a mental construct, may be said to have the property of being winged in the construction; but in the order of being it is nothing and cannot truly be said to be anything. The proper handling of such statements requires distinguishing between predication in thought and predication in being, and between universal predication of a kind and singular predication of an individual.
Concerning universal subjects, the predication may be essential when the predicate belongs to the subject by its nature, as “Man is rational”; or accidental when the predicate belongs to the subject only contingently, as “Man is musical.” The essential predication is true of every instance and at every time; the accidental predication is true only of those instances at those times when the accident is present. The logician must determine which mode of predication is in play before applying the rules of contradiction and conversion.
Conclusion of the Treatment of Opposition
The full doctrine of opposition among statements may now be summarized. Statements differ in quantity (universal, particular, indefinite, singular) and in quality (affirmative, negative). Universal affirmative and universal negative are contraries, both false together but not both true. Particular affirmative and particular negative are subcontraries, both true together but not both false. Universal affirmative and particular negative, universal negative and particular affirmative, are contradictories: of each pair, one is true and the other false in determinate cases, and the truth value waits upon outcome in cases of future contingents. Universal and particular of the same quality are subalternates, with the particular following from the universal but not vice versa.
The same doctrine applies, with appropriate qualification, to modal statements. The necessary affirmative and the impossible (necessary not to be) are contraries; the possible to be and the possible not to be are subcontraries; the necessary affirmative and the possible not to be, the necessary not to be and the possible to be, are contradictories. The full square of modal opposition exhibits these relations and provides the foundation for syllogistic reasoning involving modes.
The Goal of the Doctrine
The doctrine of the proposition, of opposition, of modes, of contradiction, leads toward the science of the syllogism. In the syllogism, propositions are joined as premises from which conclusions follow by logical necessity. The propositional analysis here developed is the foundation of that further science.
The categorical syllogism requires that its premises be propositions of definite quantity and quality, that they share a common term, and that they be related in one of the figures and modes that permit a valid conclusion. The modal syllogism requires further attention to the modes of its premises and the rules governing their combination. The hypothetical syllogism, which involves conditional and disjunctive premises, depends upon the proper analysis of compound statements built from simple ones.
Beyond the syllogism lies the demonstrative science, in which conclusions are drawn from necessary premises through valid syllogisms to produce certain knowledge of the causes of things. Beyond demonstration lies the dialectical art, in which probable premises are used to test opinions and to discover truth through orderly disputation. Beyond dialectic lies the rhetorical and the poetic, which use probable and apparent premises for persuasion and for moving the affections.
All these arts depend upon the proper handling of the simple statement, which is the seat of truth and falsity, the basic unit of logical analysis, and the foundation of the entire structure of rational discourse.
The Order of Inquiry Within Logic
The arrangement of the works of logic follows the order from the simple to the complex. The Categories treats of simple terms, which are the parts of the proposition. The present treatise treats of the proposition itself, composed of simple terms joined by the verb of existence or by other verbs. The Prior Analytics treats of the syllogism, composed of propositions. The Posterior Analytics treats of demonstration, the highest form of syllogism. The Topics treats of dialectical syllogism, working from probable premises. The Sophistical Refutations treats of apparent syllogisms that fail to demonstrate, exposing the fallacies that imitate true reasoning.
The order is necessary because each successive work presupposes the materials treated in the preceding. One cannot understand the syllogism without understanding the proposition; one cannot understand the proposition without understanding its terms. The student of logic must therefore proceed in order, mastering each part before advancing to the next.
The present treatise occupies a central place in this order. It transmits from the doctrine of terms to the doctrine of syllogisms. It treats of the basic unit in which truth and falsity arise and in which the relations of opposition are first manifested. It distinguishes the modes of being and the corresponding modes of statement, providing the materials from which modal syllogisms will be constructed. It addresses the difficult question of future contingents, preserving both the principle of contradiction and the freedom of human action.
By mastering this treatise, the student gains the tools necessary for the further study of logic and for the application of logic to the demonstrative sciences, to dialectical inquiry, and to the right ordering of human discourse generally. The principles here established underlie all subsequent treatment of inference, of argumentation, and of the relation between language and reality.
Statements About What Is Said of Several Things at Once
A final difficulty concerning the unity of the statement arises when several predicates are joined to a single subject or when a single predicate is attributed to several subjects. The unity of the statement requires that the predication signify one thing of one thing. When the predicates joined to the subject do not coalesce into one signification, the statement may appear to be one but is in reality many.
“Man is a rational mortal animal” predicates several attributes, but these attributes together constitute the definition of man and signify one nature. The statement is therefore one in nature as well as in expression. By contrast, “Man is white and musical” predicates two accidents that do not coalesce into one nature. The statement is one in expression but two in content: it asserts that man is white and that man is musical, and its contradictory must deny each separately, or at least one of them.
The same considerations apply to subjects. “Animal is rational” predicates rationality of the universal animal, but only of those species of animal that are rational. The statement is true if taken to mean that some animals are rational and false if taken to mean that every animal is rational. The proper interpretation depends upon whether the predication is essential or accidental, universal or particular, and the unity of the statement varies accordingly.
When subjects are joined by conjunctions, the predication may be collective or distributive. “Cicero and Demosthenes are orators” may signify that each is an orator individually or that the two together are orators in some collective sense. “The Greeks defeated the Persians” usually signifies collective action by the body of Greeks, not the individual action of each Greek. The truth conditions vary accordingly.
For the purpose of constructing valid contradictions, the unity of the statement must be precisely determined. A contradiction must deny exactly what the original asserts; otherwise it is not a true contradiction but only a partial denial or a denial of something other than what was asserted. The fallacies of equivocation, of composition and division, and of accident, all arise from failure to observe the proper unity of the statement.
The Final Application to Singular Statements About the Past
Statements about the past, when the subject is singular and the predicate is definite, exhibit the principle of contradiction in its most exact form. “Caesar crossed the Rubicon” and “Caesar did not cross the Rubicon” exhaust the possibilities. One is true, the other false, with no remainder, no qualification, no intermediate state. The past is fixed; what has been cannot be otherwise; the truth value of the contradictory pair is determinately distributed.
The same exactness obtains for statements about the present, with the qualification that “the present” is a moving point and that statements about the present must specify the moment to which they refer. “Socrates is sitting now” is determinately true or false at the moment of utterance; one moment later, the truth value may shift if Socrates changes his posture.
For statements about the future, the qualification developed earlier obtains. The contradictory pair exhausts the possibilities in the sense that one will be true and the other false, but the present truth value of each is not determinately fixed in advance for contingent matters. The principle of contradiction is preserved at the level of the disjunction; the determinate truth value waits upon the contingent outcome.
By this careful analysis, the universal principle is preserved without prejudice to the contingency of future events, the freedom of human action, and the order of providence that governs the world through both necessary and contingent causes.
The Place of This Treatise in the Course of Studies
The student who has completed this treatise has acquired the doctrine of the proposition in its various species and the doctrine of opposition in its various modes. The student now possesses the tools necessary for the analysis of arguments, for the detection of fallacies, and for the construction of valid inferences. The student is prepared to proceed to the Prior Analytics, where the syllogism is treated; to the Posterior Analytics, where demonstration is treated; to the Topics, where dialectic is treated; and to the Sophistical Refutations, where fallacies are treated.
Beyond logic lies the application of logic to the various sciences. In natural philosophy, demonstrations proceed from the causes and principles of natural things; in mathematics, demonstrations proceed from the principles of quantity and number; in metaphysics, demonstrations proceed from the principles of being as being. In each of these sciences, the propositional doctrine here established underlies the construction of demonstrative arguments. The properly trained logician advances to these sciences equipped to analyze their reasonings, to verify their conclusions, and to detect any failures of inference.
The treatise thus serves not as an end in itself but as an instrument for the further pursuit of wisdom. Its proper use lies in the unfolding of philosophical knowledge, in the right ordering of discourse, in the preservation of truth, and in the avoidance of error. By the careful study of the simple statement, of its parts, of its species, of its oppositions, of its modes, of its application to contingent matters, the mind is prepared to approach the higher questions of being, of knowledge, and of the order of the universe with the discipline required for sound conclusions.
The principle that of every contradictory pair one is true and the other false, preserved in its proper application, stands as the foundation of all rational discourse. The freedom of the future and the contingency of human action stand as the foundation of moral life. The doctrine of modes stands as the foundation of demonstrative knowledge and of practical deliberation. Together these doctrines constitute the heart of the treatise and the bond between the analysis of language and the contemplation of being.
