PROPOSITION

A proposition is often said to be a verbal expression that can be true or false. This definition is imprecise for two reasons. First, a proposition is the sense of the sentence that expresses it; one proposition can be expressed by several sentences in different languages or in the same language. Secondly, truth and falsity belong neither to sentences nor to propositions, but to judgments; only the intellectual act of judgment assenting to the proposition expressed by the sentence is in the proper sense true or false. To say that a proposition is true or that it is false is to say that anyone asserting it would be judging truly or falsely. Nevertheless, with the understanding that the usage is elliptical, sentences are here called propositions, and they are declared to be true or to be false.

Kinds of Proposition. Propositions are either assertoric or modal. The former merely declare that a given state of affairs obtains; the latter qualify such declarations as necessary, contingent, possible, or impossible. Hence there are four kinds of modals. Obviously, every assertoric proposition can be qualified by each mode, so that there are four modals for every assertoric. Assertoric propositions are of two kinds: categorical and compound. A categorical proposition is one in which an attribute (signified by the predicate) is said to characterize or not to characterize something (signified by the subject). These terms, subject and predicate, constitute the matter of the categorical. Its form, making it a proposition of a specific kind, is the copula, which affirms or denies the predicate of the subject. There are thus two species of categorical, affirmative and negative. Based upon the matter is the division into singular propositions, whose subjects are proper names or demonstrative words or phrases, and general propositions, whose subjects are common nouns or descriptive phrases. Also based upon the matter is the division into universal and particular propositions. universals are those in which the predicate is affirmed or denied of everything denoted by the subject; they are identified by some such modifier of the subject as "all," "every," "none." The subjects of universals are said to be distributed. Particular propositions are those in which the affirmation or denial is extended only to some of the things the subject signifies; such restriction of predication is indicated by the use of "some," "not every," or the like to modify the subject. The restriction does not positively exclude from the predication anything the subject denotes; the particular proposition is noncommittal as to whether the predication could be truly made of everything the subject denotes. Subjects of particulars are said to be undistributed.

Reflection on the nature of affirmation and denial discloses that the distinction between distributed and undistributed terms applies to predicates also; for to deny a predicate of a subject is to state that nothing the predicate denotes is referred to by the subject, or that everything the predicate denotes is excluded from the subject's referents. Affirmation has no such exclusive force, but leaves unanswered the question whether the attribute the predicate signifies belongs to other things besides the subject's referents. Hence, predicates of negative propositions are distributed, while those of affirmatives are undistributed. Since both of these can be either universal or particular, there are four kinds of categorical propositions: A. Universal affirmative, with subject distributed, predicate undistributed (Every S is P ). I. Particular affirmative, with subject undistributed, predicate undistributed (Some S is P ). E. Universal negative, with subject distributed, predicate distributed (No S is P ). O. Particular negative, with subject undistributed, predicate distributed (Not every S is P ). When these propositions have the same subjects and predicates, they constitute two pairs of contradictories: A and O are mutually exclusive and also exclusive of a middle, as are I and E.

Existential Import and Equivalence. With what has been said up to this point, logicians are in general agreement; concerning another aspect of categoricals, their existential import, there is no such consensus. Concerning this aspect, there are three schools of interpretation: the first understands particulars as stating that what their subjects signify exists in some realm of discourse, and understands universals as leaving the question of existence open; the second understands affirmatives as requiring for their truth the existence of whatever their subjects signify, and understands negatives as true if what their subjects signify does not exist; the third holds all categoricals to be alike in existential import, either all existential or all nonexistential.

Conversion and Obversion. Modification of a proposition's matter or form produces a new proposition, whose relation to the original interests the logician. Conversion affects the matter; it is the interchange of subject and predicate, the copula remaining unaffected. There are two kinds of conversion: simple conversion, in which the propositions are the same in quantity; and conversion by limitation, in which the new proposition is particular, the original being universal. Obversion has to do with the proposition's form; it is the change of quality from affirmative to negative or vice versa, with the compensating substitution for the predicate of its contradictory term. The older logicians discussed another process, called equipollence, by some mistakenly identified with obversion, which it superficially resembles. It is effected by the insertion of non at appropriate places in Latin sentences, expressing propositions to produce other propositions equivalent to the contradictories, contraries, subalterns, and superalterns of the original (see opposition). The resulting equivalents are not obverses, however, for they are of like quality, both affirmative or both negative. They are not different propositions at all, but different sentences expressing one proposition; they have identical matter and form, and differ only in syntax.

Contraposition and Inversion. The product of obversion is a proposition having the same subject as the original; that of conversion is a proposition whose subject is the predicate of the original. Two other processes, contra-position and inversion, consist of alternating obversions and conversions. The contrapositive, product of the former, is a proposition whose subject is the contradictory of the original predicate. The full contrapositive is of the same quality as the original and has for its predicate the contradictory of the original subject; the partial contrapositive is of opposite quality and has the original subject as its predicate. The inverse has for its subject the contradictory of the original subject. The distinction between full and partial inverses is analogous to the distinction between full and partial contrapositives.

Implications. A proposition implies another derived from it in any of these ways, if and only if (1) any term distributed in the derived proposition is also distributed in the original, and (2) either the original is existential or the derived proposition is nonexistential. It follows that when distribution and existential import are the same for both propositions they imply each other, that is, they are equivalent. Whichever interpretation be accepted, these conditions are fulfilled in the simple conversion of I and But since subject and predicate differ as to distribution in A and O, neither of these implies its converse; A and O are independent of their converses. For the second school of interpretation, however, a universal implies its converse by limitation: "Every man is just" implies "Some just being is a man"; and "No man is just" implies "Not every just being is a man." The first school, of course, rejects this implication. For this school, on the other hand, every proposition implies its obverse, and A and O imply their contrapositives. Contrariwise, for the second school, only affirmatives imply their obverses. It holds, therefore, that A propositions imply their partial contrapositives, but that full contrapositives are independent. For instance, "Every man is just" implies "No not-just being is a man," but neither implies nor is implied by "Every not-just being is a not-man."

The first and second schools hold that a proposition and its inverse are independent. Since for the second, the full contrapositive is not implied by the original, neither is the inverse, which is derived from it. The first accepts the equivalence of contrapositives, but rejects conversion by limitation, whereby one derives the inverse from the contrapositive. However, the third school holds that A and E imply their inverses. It teaches that "Every man is just" implies "Some not-man is not-just" and "Not every not-man is just." The term "just," however, which is undistributed in the original, is distributed in the partial inverse. For this reason the first and second schools hold that contradictories differ not only in quality and quantity, but in existential import also.

Compound Proposition. A compound proposition has for its matter two propositions, categorical or themselves compound; its form is a conjunction affirming or denying a given relation between these two. Instead of the one relation of characterization they may hold between terms, four basic relations are possible between pairs of propositions, namely: 1. Implication, the first implying the second; 2. Subimplication, the first being implied by the second; 3. Contrariety, the two excluding each other; 4. Subcontrariety, the two excluding any other that denies both. Letting p and q represent propositions, and ∼p and ∼q, their contradictories, one may affirm and deny these relations by means of the following compound propositions: Of the propositions affirming relations, the first two are conditional, the third is disjunctive, the fourth is alternative; all those denying relations are of the same kind, and are called conjunctive. These then are the four species of compound propositions.

Every conjunctive denies each of these relations between one or another pair of propositions, and consequently contradicts four other compound propositions. Thus "Both p and q " contradicts these: 1. If p, then ∼q 2. If q, then ∼p 3. Not both p and q 4. Either ∼p or ∼q Consequently, these four are equivalent. Similarly, each of the other three conjunctives composed of p, q, ∼p, and ∼q contradicts two conditionals, a disjunctive, and an alternative, which are therefore equivalent. The equivalent conditionals are said to be contrapositives, the relation among their parts being analogous to the relation among the parts of categorical contrapositives. The contradictories of converse conditionals are contraries to each other: "If p, then q " and "If q, then p " contradict respectively "Both p and ∼q " and "Both q and ∼p, " which are contraries. Since contradictories of contraries are subcontraries, such is the relation of these conditionals: one of the converse conditionals must be true.

Modal Proposition. Concerning the modal propositions corresponding to categoricals little need be said. To determine whether a modal implies another derived from it by obversion or conversion, one must add to the conditions mentioned above a third: that either the original proposition is in the mode of necessity or of impossibility or the derived one is in the mode of possibility. Between modal and assertoric compounds, however, there is an important difference. The necessary conditional, like the assertoric, is equivalent to its contrapositive; unlike the assertoric conditional, it is not subcontrary to its converse. For the contradictory of the necessary conditional is a conjunctive in the possible mode, and a possible proposition does not imply a necessary one. Thus the contradictory of "If p, then necessarily q " is "Possibly both p and not ∼q, " which does not imply "If q, then necessarily p. " A necessary conditional and its converse are therefore independent of each other.

See Also: logic, symbolic; term (logic).

Bibliography: v. miano, Encilopedia filosophica (Venice-Rome 1957) 3:1662–63. j. a. oesterle, Logic: The Art of Defining and Reasoning (2d ed. Englewood Cliffs, N.J. 1963). v. e. smith, The Elements of Logic (Milwaukee 1957). e. d. simmons, The Scientific Art of Logic (Milwaukee 1961). o. bird, Syllogistic and Its Extensions (Englewood Cliffs, N.J. 1964).

[j. j. doyle]