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Truth-based formal norms.

Two of the operational logical categories, the principles and the rules, can be --assimilated to norms. Yet, if rules, similarly to laws, are syntactically autonomous, being confirmed by and functioning on their terms, owing to correctness or vi formae validity, principles definitely involve semantic criteria, turning to norms in the distribution of truth or other subjacent values. From this point of view, the critique of the present-day picture of the logical principles and especially the rejection of their formal treatment which has blurred their specificity as against the laws or the theorems in various hypothetically-deductive constructions seems to be not only possible but also necessary steps. Taking these tasks upon ourselves, we shall mention the case of the deontic systems or algebras, as a starting point in finding out a solution which, we think, will throw light upon the status of principles in formal logic toto genere. Our attempt outlines a research program that the following considerations will but partly reveal. We have already presented some aspects of the "reform" of principles at various symposia or in papers (loan 1974: 69-76; loan 1975a: 79-81; loan 1975b: 7-8; loan 1990: 185-198) and, we believe, some others will be stimulated by the criticism of our endeavour. The topic has gained our attention thanks to the arguments it provides in discussing the problems of pluralism and of the unity of present-day logic.

Principles and values in deontic logic

Notwithstanding the multiplication of deontic formalisms, their afferent principles and values are in a very confused position. In the axiomatic reconstruction (Kalinowski 1972: 84-85) of the "old system" (the first version of deontic logic, dating from 1951), we will find four "principles" (von Wright 1951): extensionality (PE): PA [left and right arrow] PB if A and B are actions of the same agent and have the same value of achievement; (unilateral) permission distribution (PPD): P(A [disjunction] B) [left and right arrow] (PA [disjunction] PB); permission (PP): PA [disjunction] P([logical not] A); tautology (PT): -P(A & [logical not] A); we also find the principle of deontic contingence: "a tautological act is not necessarily obligatory and an obligatory act is not necessarily forbidden". (von Wright 1967: 310)

If (PP) and (PT) stand for the classic interdictions (tertium non datur and non contradictio), on the whole, the so-called principles inoperatively fill in the list of axioms [[logical not](OA & O([logical not]A)); O(A & B) [left and right arrow] (OA & OB)] and the rules of transformation (substitution, replacement and detachment). (OS), the old system of deontic logic, can be founded exclusively on some principles, (PPD) and (PP), taken as axioms and thereby as laws or logical theses (Prior 1954: 64-65; von Wright 1956: 507-509). In this case, a deduction rule would correspond to (PE); and (PT) asserts itself only morally; if we did not admit it, any act would be permitted (von Wright 1951: 38 sq; Kalinowski 1972: 86) (?!). Yet, for the logician (PT) must appear as a materially-decidable statement, true or false, according to circumstances. Whatever the case, the fact is that, in the economy of deontic formalisms, "principles" are either incorporated in the basic ingredients of the deductive system, operating as axioms (laws) or rules, or they decline their normative competence. We do not know this precarious situation, that von Wright illustrates, to have been solved by alternative versions of deontic logic but by simply ignoring the appeal to principles.

No less deficient is the condition of the basic values in deontic logic. If the first creator of such systems interprets the proposition on norms (id est, the propositions--and the truth functions between propositions--on coercitiveness, permission, interdiction and the other deontic--derived --characters of acts (and performance functions between acts--von Wright 1967: 306) in the field of the achievement or performance values of the normed acts restricted to the bi-valent "fulfilled" or "unfulfilled" register, Kalinowski (1972: 115) ensures entries for the deontic matrices in the trivalent value space: intrinsically good (action), [1.sup.*]; intrinsically bad (action), [0.sup.*]; intrinsically indifferent (action), [1/2.sup.*].

Taking into account, for example, the value itinerary of the negation and of the permission of an action (columns 1-2 of the matrix below), formula "CNPxN[alpha]Px[alpha]" (Kalinowski 1972: 113) will be taken as thesis of the logic of norms:

A             (1) N         (2) Px    (3) Px     (4) NPx    (5) CNPx
              [alpha]       [alpha]   N[alpha]   N[alpha]   N[alpha]
                                                            Px[alpha]

[0.sup.*]     [1.sup.*]     0         1          0          1
1/[2.sup.*]   1/[2.sup.*]   1         1          0          1
[1.sup.*]     [0.sup.*]     1         0          1          1


One could say that "the logical value of the normative relation is independent of choosing X (one of the terms of the relation!)" (Weinberger 1960: 16-7). Yet the respective change is not in the least less vulnerable. The deontic modalities themselves are proposed as values of the argument of action, thus "determining the meaning of the normative functors with respect to the other (normative) functors," i.e. establishing the itinerary of the functors "it is permitted," "it is compulsory" and "it is indifferent" with respect to the values (of the argument of action): "permitted," "compulsory" and "indifferent." The deontic functors "it is permitted," "it is obligatory," and "it is forbidden" could be defined in the same manner on the basis of the values of the action variable: "obligatory," "indifferent" and "forbidden" (Fisher 1961: 107-118). We shall well understand why, when no satisfactory choice is possible, many of the decision devices, operating in the field, ingeniously avoid resorting to deontic values--this being the case of the method of selective matrices, phrase normalizing, semantic table closing, etc. (Popa 1976a: 173-180; Popa 1976a: 267-286).

The condition of principles in "theoretical" formal logic

Formulating out of complaisance, or even n the absence of some principles in eontic logic, meant to control the assumption of values in the domain and the co-domain of the functions of functions, set as theses or laws in various systems, reflect a reality in theoretical or alethic logic, with respect to "truth," "falsity" and intermediate values. In propositional, predicational and classial systems, the traditional principles of noncontradiction, of excluded middle, bivalence, identity and reasserting through double negation occur as theorems, at the best, whose demonstrations rely on axioms not necessarily more obvious or better-known than the theorems themselves. The natural reconstruction of propositional logic, by restoring the axiomatic dignity of the classical principles, failed (Dimitriu 1973: 73-80). The failure is illustrative of the illusions and confusions maintained by the usual translation of these principles with the help of the tautological formulae p / [logical not]p, p [disjunction] [logical not]p, p [disjunction][disjunction] [logical not]p, p [contains] [logical not] ([logical not]p), p [contains] p, and others. By virtue of the interdefinisability of truth functions, the number of equivalent formulae that claim to be principles is amazingly high: at least twenty-four for identity and reassertion through double negation, at least sixteen both for noncontradiction and middle term exclusion, at least eight for bivalence. In this context, the excessive use of synonymy is associated to the overbidding of homonymy. All the expressions of bivalence are to be found again in the identity register. Among the formulae of the identity principle there also appear the "most natural" translations of the contradiction exclusion [p / [logical not]p, [logical not]p / p, [logical not] (p & [logical not]p), [logical not] ([logical not]p & p)] and of the middle term exclusion [p [disjunction] [logical not]p, [logical not]p [disjunction] p, [logical not](p [perpendicular to] [logical not]p), [logical not]([logical not]p [perpendicular to] p)]; in addition, [logical not]p [contains] [logical not]p, [logical not]p [subset] [logical not]p, [logical not] ([logical not]p [not contains] [logical not]p) and [logical not]([logical not]p [not subset] [logical not]p) concurrently apply to the three principles of logical opposition.

And the more conventional by complication the beginning of the demonstration by such axioms taken as principles, the less acceptable, by simplification, is the understanding that all the propositional tautologies emerge from the formula of the excluded middle, as all antilogies proliferate the violation of the principle of non-contradiction --we are thus hinting at the decisional method by normalizing prepositional phrases, according to which the truth of the statement of the form "([p.sub.1] [disjunction] [logical not][p.sub.1] [disjunction] ...) & .... & ([p.sub.n] [disjunction] [logical not][p.sub.n] [disjunction]...)" sends us back to the truth of each sub-statement "[p.sub.i] [disjunction] [logical not][p.sub.i] [disjunction]....", guaranteed, in its turn, by the truth of the segment "[p.sub.i] [disjunction] [logical not] [p.sub.i]". In case the given statement is turned to a disjunction of conjunctions of the form "([p.sub.1] & [logical not] [p.sub.1] &...) [disjunction] ... [disjunction] ([p.sub.n] & [logical not] [p.sub.n] & ...)", the falsity of the whole formula results from the falsity of each conjunction term "[p.sub.i] & [logical not][p.sub.i] &...", a falsity which is determined, in its turn, by violating the principle of contradiction exclusion by the statement segments "[p.sub.i] & [logical not][p.sub.i]"--the statement being dictated by the presence of the excluded middle in formula p [disjunction] [logical not]p, and of non-contradiction in [logical not](p & [logical not]p). The same identification misguides logicians to the false problem of limiting and suspending the classical principles in the intuitionist restrictive logics as well as in the many-valued logics. To say nothing of the fact that, in the formal treatment under discussion, principles are not involved in the proper development of the logical problematique.

Fundamental types of logical relations

While the principles found out by Aristotle hinted only at the logical opposition (non-contradiction and the exclusion of the middle between opposites), Leibniz was to explain the principle of identity (foreseen by Aristotle) and to introduce the principle of the sufficient reason, yet, without pointing to their relationship and to their relevance for a fundamental type of logical relationships that should accompany the spectrum of oppositions. The peremptory distinction between truths of reason and truths of fact has influenced the homogeneity of principles with respect to field jurisdiction, as well as to the value of functionality. But the principle of sufficient reason might not be "a principle proper, nor a law but a presupposition"; it "can fail as principle, since it expresses a tautology" (Noica 1972: 66-67). And this is not an isolated view, because the same principle might well be a postulate of rationalism, "which must be abandoned simultaneously with the latter." (Verneaux 1960: 39)

Though supporting the remark that this principle could often connote the ontological principles of finalism and causalism--Schopenhauer (1864) having shown that the principle of sufficient reason, unique in its essence, varies according to the nature of the objects to which it applies, becoming (principium rationis sufficientis fiendi): cognition (p.r.s. cognoscendi); existence (p. r. s. essendi); action (p. r. s. agendi) (Botezatu 1969: 18)--we consider possible the dissociation between the philosophical prescription of "sufficient reason" and its logical conversion in a series of principles of conditioning. In this respect Constantin Noica's suggestion is quite adequate. According to him, the place of the principle of sufficient reason, meant to fill in a blank space in between the principles of identity and contradiction, which refer to extreme situations, "rather belongs to another principle, i.e. of the necessary connection, that is sure to come out in the logical process and that waits, as it seems [....], but for Leibniz' authority to get its proper formula and be concurrently admitted to citizenship in the table of principles." (Noica 1972: 7)

Subscribing to this verdict, found out by an author who does not make a point of calling himself a logician, we will express the reserve that identity may well be considered the maximal form of connection. Instead of the proposed identity-connection-contradiction triad we would rather consider the identity and contradiction marginal cases of the fundamental relationships: conditioning and opposition. So, we basically agree that the main logical relationships are exclusion and consequence (Jorgensen 1962: 33-35), that logic is "whatever bears the mark of affirmation and negation, identity and non-identity" (Lupasco 1951: 6), that, in the wave of psychology like Stefan Odobleja, logic could be founded on the ideas of consonance and dissonance. Considering the logical principles in the context of opposition and conditioning, we shall approach, from within the "orthodox" formal logic, the tendencies of connoting and multiplying the acts of negating and affirming in the logics of change or in "productive" logic. (cf. Sesic 1972: 15 sq; Spisani 1970, 1971, 1972a, 1972b, 1973, 1974a, 1974b, 1975, 1976)

The parallelism between principles and relationships

In our systematization, the logical principles norm the degrees or the hypostases of the fundamental logical relationships: the proposition opposition and conditioning, concept exclusion (opposition) and ordering. The logical-formal aspect of each principle is marked by the game field of a dyadic function of truth, of an interclassial function, respectively. Two opposed propositions observe the principle of the exclusion of the co-truth of the contraries if they cannot be simultaneously true but they can be simultaneously false, thus satisfying the function of anticonjunction: D[T.sub.p][T.sub.q] = [F.sub.Dpq]; the principle of the exclusion of the subcontrary co-falsity if they cannot be false simultaneously and from the same point of view, but they can be simultaneously true, thus satisfying the disjunction matrix: A[F.sub.p] [F.sub.q] = [F.sub.Apq]; the principle of the exclusion of the co-valency of the contradictories if they can simultaneously be neither true, nor false, and therefore they meet the requirements of the strong disjunction: J[T.sub.p][T.sub.q] = J[F.sub.p][F.sub.q] = [F.sub.Jpq].

On the contrary, two connected propositions can be: co-true, but not necessarily, also co-false (observing the principle of the sufficiently-necessary conditioning or of co-ascertaining, i.e. the value itinerary of implication: C[T.sub.p][F.sub.q] = [F.sub.Cpq]); co-false but not necessarily co-true (when they are influenced by the principle of necessarily sufficient conditioning or of co-falsifying, that the replication function reveals: H[F.sub.p][T.sub.q] = [F.sub.Hpq]); co-valent (in concordance with the principle of necessary and sufficient conditioning, of co-valency or of identity set by the value itinerary of equivalence: E[T.sub.p][F.sub.q] = E[F.sub.p][T.sub.q] = [F.sub.Epq]).

The decided sentences, for "ever" set as true or false, will get involved in well-determined relationships, as instances of contradiction or of neutral conditioning. Two opposed enunciations with such an alethic regime cannot be co-valent with respect to the principle of the contrarily-subcontrary-proposition exclusion (on the logical path of the difference or non-implication function, S[T.sub.p][F.sub.q] = [T.sub.Spq]) or to the principle of the subcontrarily-contrary proposition exclusion (in the game field of converse difference or non-replication, B[F.sub.p][T.sub.q] = [T.sub.Bpq]). For statements of the same kind, in a connection relationship, there will operate the principle of sufficiently-sufficient conditioning (in the sense of co-ascertaining revealed by the function of conjunction: K[T.sub.p][T.sub.q] = T[K.sub.pq]) and that of necessarily-necessary conditioning or of co-falsifying on the logical itinerary of the function of anti-disjunction or rejection: X[F.sub.p][F.sub.q] = [T.sub.Xpq]).

The formalization we are supporting is marked

We shall formulate each logical principle in accordance with the exigencies of an interpropositional operation and we shall locate the corresponding relationship when the operation with truth values leads to the value "true," i.e. the principle is observed. Since, for example, the opposed statements "the Romanians are more numerous than the Hungarians" and "the Hungarians are more numerous than the Romanians" satisfy the matrix of the incompatibility function (and, as such, observe the principle of co-truth exclusion), we shall declare them to be contrary. Or, as the statements "Caesar did not cross the Rubicon" and "the Roman republic did not fall" are under the incidence of the rejection function (and the co-falsifying principle), we shall declare them to be in a necessarily-necessary conditioning relationship.

The same correspondence between operations and relations is to be found with notions.

In general, the relationships are reverse-operations (Tutugan 1960: 179). As prescriptions, logical principles determine the value or the result of an operation; the reunion of two contradictory notions is exhaustive (it coincides with the universe of discourse); the conjunction of contraries cannot be a true sentence, the intersection of two subcontrary notions cannot be an empty set; the conjunction of two contrarily-subcontrary propositions is (a) false (sentence); etc. As reverse operations, the same relationships allow us to establish the value of a sentence with respect to the other sentence and the result of the operation: if a disjunctive sentence is true, its terms are subcontrary sentences, one being false, the other one is true; similarly, if a replicative sentence is true, the relationship between its terms is subject to necessarily-sufficient conditioning, the antecedent being false, the sequent cannot be true. Also, we shall say that if the intersection of two notions is empty, their relationship is one of contrariety, i.e. while one is admitted as predicate by a given subject, the other is not. Or, if the intersection between a notion and the complementary of another one is an empty notion, the relation of the given positive notions is one of subordination, the former being the predicate of a certain subject, the more so will the other be.

Distinct epistemic levels of logical prescriptions

By taking into account the ten normal, or non-deteriorated, dyadic propositional functions, and their corresponding functions in the logic of classes, we have succeeded in getting two categories of logical relationships controlled by principles of opposition and conditioning (ordering). Six distinct relations can be established between propositions and concepts with an intermittent character, i.e. between propositions that are now true, now false, respectively between concepts which are neither empty, nor total.

Other four relationships, as hypostases of contradiction and equivalence (identity), are established for decided propositions, with non-fluctuating truth--which, in von Wright's terminology are individual statements with uniquely determinate truth values, unlike the generic statements whose truth-value is determinate when they are associated with a circumstance, a spatio-temporal or just a temporal location (Stoianovici 1972: 511), such a distinction being thoroughly convenient from a logical point of view, unlike the usual alternative "analytic" and "synthetic"--and for concepts with marginal, empty or total, extension. The contradiction between the statements "the soul is mortal" and "the soul outlives the body" will be read as a contrary opposition in the stated order, and as subcontrary opposition in the reversed order. We shall call it contrarily-subcontrary opposition, but it must be understood as a contradiction between propositions with a constant logical truth, the former being definitely established a true proposition, while the latter is definitely established a false proposition. And so is the contradiction between the statements 'the Earth is the hub of the solar system" and "the Earth is not the hub of the solar system" that represents the subcontrarily-contrary opposition. Both of them being hypostases of contradiction (S[T.sub.p][F.sub.q] = [T.sub.Spq]; B[F.sub.p][T.sub.q] = [T.sub.Bpq]; Jpq = Spq [disjunction] Bpq), difference will be given only by the order in which we consider the opposed propositions. We shall call sufficiently-sufficient the conditioning between such statements as "the soul is mortal" and "man must achieve his personality in his earthly life" while we shall consider necessarily-necessary conditioning the one between such statements as "the soul is immortal" and "the soul's reincarnation is possible," which are scientifically false. In a formal context, Epq = Kpq [disjunction] Xpq formally points to the position of these relationships with respect to equivalence (K[T.sub.p][T.sub.q] = [T.sub.Kpq]; X[F.sub.p][F.sub.q] = [T.sub.Xpq]).

Similarly, the contrarily-subcontrary opposition is to be found between a total notion and an empty one ("animal" and "immortal" in the universe of discourse ensured by the notion "man"). The subcontrarily-contrary opposition, between an empty notion and a total one ("pegas" and "quadruped" in the universe of discourse set up by the notion "horse"). Sub-superordination: between two total notions ("quadrangle" and "quadrilateral" in the universe of discourse expressed by the notion of "rhombus"). Super-subordination: between two empty notions: "tricephalic" and "door keeper of the Inferno" in the universe of discourse set up by the notion "dog."

Yet, what part will be played by the dyadic degenerated functions, i.e. those which iterate the monadic functions? Since Ipq = p (prependency), Fpq = q (postpendency), Gpq = Np (prenonpendency) and Lpq = Nq (post-nonpendency), we will be tempted to thus ensure the game space of the relationships of independence or, at best, of quasidependence. Yet, we can find out that Ipq = Hpq & Apq, and we shall realize that the prependency function is the matrix of a transition relationship between conditioning and opposition, marking the concomitance of superalternation and subcontrariety. In such a relationship are, for example, the statements "some people are poets" and "some people are mortal." Also, the function of postpendency can promote a subalternation-subcontrariety relationship (Fpq = Cpq & Apq), like the one between the statement "some rhombs are square" and "some rhombs are parallelograms." In both cases we have to deal with dyads of propositions where the truth of one of them bears the stamp of necessity. In the context of inter-classial relationships, the same situation is revealed by orderings in which one of the partners is a total notion (the subordination of the notion "philosopher" with respect to the notion "man," in the universe of discourse ensured by the notion "European"; the superordination of the notion "parallelogram" with respect to the notion "square" in the universe of discourse set up the notion "rhombus"). Going further with the translation, by normal functions, of the degenerated dyadic functions, we shall realize that prenonpendency can render an intermediary relationship of subalternation-contrariety (Gpq = Cpq & Dpq), while postnonpendency expresses a similar relationship, of superalternation-contrariety (Lpq = Hpq & Dpq). This is the case for the couples of the opposed propositions out of which one is necessarily false, and for the dyads of notions out of which one is empty ("sportsman"-"immortal," "centaurs"-"friendly beings," respectively the propositions built up with such predicates).

After having turned to good account fourteen interpropositional functions and their similar functions on the interclassial level, it is left for us to make inquiries into the uses--in the table of the logical relationship and of the principles that norm them--of the dyadic functions which iterate the constant or the medadic functions. The intersection of notions is an example of constant function, valid in the whole game space. At the level of interpropositional functions this intersection corresponds to tautology, the identically true function. In both cases there is a perfect equilibrium between opposition and connection (conditioning, ordering respectively). In the Wittgensteinian-Carnapian terminology, the game space of the function is maximal and the logic content is null. On the contrary, the extralogical content, the cognitive information included in the matrix of such a relationship, meets the requirements of a positive determination. Not of the least concern is the fact that the intersection of notions and what we prefer to call the interference of statements are the measure of progress, through synthesis in cognition, be it a matter of successive specification of the object (swans can be black, too), phenomenon intensification (rain may be accompanied by lightning), discovering compatibilities (prime numbers can also be parse) or the extension of cases (propositions can differ according to quantify, to quality, etc). It is surprising how logicians have stopped half-way homologating the relationship of intersection between notions, but making indistinct the frequency and the importance of the interference relationship or of minimal conditioning pointed to only as a marginal case in the formal combinatorial game of truth functions. Rejecting the label under which it is vehicled in the symbolic logic texts, that of tautology, we shall say that it is interference itself which is the fullest interpropositional relationship, vi forma as well as vi materia (Freudenthal 1958: 47). On the contrary, the identically false function, respectively the corresponding interclassial function, identical with the empty class, are the most entitled to express logical independence. We can grasp the "nonrelationality" between empty and total terms, such as 'winged man" and "celestial being" in the universe of discourse ensured by the notion "angel," respectively between the statements built up with the two predicates. As we cannot aspire to a total overlapping between the criterion of the extensionality of propositions and the criterion of the extensionality of terms or names, we shall consider that any couple of factual propositions, or propositions with a fluctuating alethic nature, but conflicting with respect to meaning and informational content (and the class of such associations is practically infinite) reveal the matrix of the inter-propositional function Opq = F, improperly called "contradiction."

Ontologies and value criteria

Owing to the above remarks, the logical relationships and their corresponding prescriptions, taking the form of principles of opposition and conditioning, make up a complete and thoroughly symmetrical table subdivided into epistemical levels, according to the "contingent" or "necessary" character of the statement truth or, in von Wright's terminology, according to the "individual" or "generic" nature of the respective relata.

Two generic, or truth-fluctuating, propositions, now true, now false, seem to us compatible with one of the following relationships: interference (the weakest conditioning, neither necessary nor sufficient); subalternation (the sufficiently-necessary conditioning); superalternation (the necessarily-sufficient conditioning); equivalence (the necessary and sufficient conditioning); nonrelationality (the weakest opposition, neither contrary nor subcontrary); contrariety (the contrarily-contrary opposition); subcontrariety (the subcontrarily-subcontrary opposition); contradiction (the contrary and subcontrary opposition). The individual propositions, definitely decided, can enter, in their turn, in equivalence relationships (the sufficiently-sufficient conditioning or the necessarily-necessary conditioning) and in contradiction relationships (contrarily-subcontrary opposition or subcontrarily-contrary opposition). Joint associations--a generic proposition and an individual preposition--reveal intermediate situations, between conditioning and opposition: sub(super)alternation-subcontrariety, sub(super)alternation-contrariety.

Yet, the reference to the value course of the dyadic functions allows us not only to find out the epistemic levels (contingency--certainty; individual-context--generic-context ; etc.) afferent to their correlated proposition but also the very protection of the logical principles in the "non-standard contexts mentioned by Aristotle (the sentences about the contingent future), by Leibniz (the sentences about continuum and its limit), and systematically by the intuitionist mathematicians (sentences about the infinite).

In our formalization, logical principles express interdictions or obligations of connecting the sentences with one another vis-a-vis truth values. We can show that they substantiate scientifically affirmation, respectively the negation of a statement with respect to other statements. Each type of conditioning supports a course of relative affirmation, such as each opposition appears as a support of a relative negation:

1) Sufficiently      Kpq
  sufficient         p(q) = T
  affirmation        [therefore] q(p) = +p
                       (q) = T
2) Necessarily       Xpq
  necessary          p(q) = F
  affirmation        [therefore] q(p) = +p
                       (q) = F
3) Sufficiently      Cpq                      Cpq
  necessary          P = T                    q = F
  affirmation        [therefore]q = +p = T    [therefore] p =
                                                +q = F
4) Necessarily       Hpq                      Hpq
  sufficlent         p = F                    q=T
  affirmation        [therefore] q = +p = F   [therefore] p =
                                                +q = T
5) Necessarily       Epq
  and sufficient     p(q) = T(F)
  affirmation        [therefore] q(p) = +p
                       (q)=T(F)
6) Contrarily        Dpq
  contrary           p(q)=T
  negation           [therefore] q(p) = -p
                       (q) =F
7) Subcontrarily     Apq
  subcontrary        p(q)=F
  negation           [therefore] q(p) = -p
                       (q)=T
8) Contrarily        Spq                      Spq
  subcontrary        p=T                      q = F
  negation           [therefore] q = -p = F   [therefore] p =
                                                -q=t
9) SulKOiilrarlry-   Bpq                      Bpq
  contrary           p = F                    q=T
  negation           q = -p=T                 [therefore] p =
                                                -q = F
10) Contrarily and   Jpq
  subcontrarily      p(q) = T(F)
  (contradictory)    [therefore] q(p) =-p
                       (q) =F


Since the other six dyadic functions are degenerated into monadic and medadic functions, they do not admit of any determined inferential situations and they do not express determined courses of the relative affirmation and negation. On the contrary, they express the affirmation and negation of a proposition irrespective of the truth-value of its correlative, or leave undetermined the value of each of the correlated statements.

The terms contrary, subcontrary, necessary, sufficient qualify the alethic circuit between the correlated sentences, respectively the transition from true to false, from false to true, from false to false and from true to true. Affirmation and negation as principle-normed acts, acquire an interpropositional meaning in our approach, coming back to the correlation of two co-valent sentences, respectively alethically hetero-valent. Thus are set the limits between the suspension of the excluded middle and of the reaffirmation by double negation. We shall say that, with sentences referring to infinite, to potentialities, to continuum, etc. the operations validated by principles, respectively relative affirmation and negation are not in force. In our case we cannot pass from the falsity of a sentence to the truth of its subcontrary, for the simple reason that in the universe of discourse we are referring to there are no decided sentences: Apq; p(q) = ? [therefore] q(p) = ?

Such situations can hinder any principle (not only the excluded middle, as the intuitionists put it!) with logical indeterminate sentences, multiply decidable or undecided, pure and simple. Unappliable or inoperative in the above contexts, the logical principles maintain their validity because they only prescribe that we should not simultaneously falsify two opposed sentences; that we should simultaneously validate two sentences with respect to the sufficiently-sufficient conditioning, etc. As Aristotle specified, "it is necessary that a sea battle should take place tomorrow or should not take place tomorrow, but it is not necessary that it should take place tomorrow and nor is it necessary that it should not take place" (De Interpretatione 9: 19a). None of the alternatives being decided, the principle of contradiction (the co-valent exclusion) can be put in the service of indirect cognition ensured by the relative negation.

Formulated in terms of co-truth, co-falsity or co-valency exclusion, respectively in terms of claiming these value combinations, the principles of opposition and conditioning apply, with no alteration, in polyvalent logics in which prescriptions [D.sub.11] = 0, [A.sub.00] = 0, [J.sub.10] = [J.sub.01] = 0, etc. are preserved. Worth mentioning is the fact that the acceptance of some intermediate values between true (1) and false (0) does not substantially modify but completes the operational steps supported by principles. In trivalent logic, for example, contrary negation will indicate the same transition to falsity, but starting from possible too, not only from truth. The subcontrary negation will allow for the advance to truth as well, starting, however, from the possible too, not only from the false. In the negation schemes belonging to bivalent logic, the third value is now ascribed to truth (contrary negation), now to falsity (subcontrary negation), now both to truth and falsity (contradictory negation):

Even in these few suggestions we shall realize that polyvalent logic regulates by its truth functions, set as principles, as well as by the reverse operations that express steps of relative affirmation and negation, the formal framework of the transition from not yet decided sentences to decided sentences, correlative to the former ones. The sui generis bivalence, "decided" and "undecided," is also revealed, all the principles being apt to be rephrased in the context of this value spectrum.

The question is whether we can also insert other semiotical criteria besides the alethological or the apophantic one that such a long tradition in the science of logic has identified with. (Botezatu 1974: 61)

The vague idea that "on a logical level, if accepting as logical forms of connection: identity, causality, substantiality, systematicity (the connection between part and whole) and contradiction--obviously so, with no clear reason for this enumeration, we shall not have only one type of relationship" (Noica 1972: 72) is made clear by the above-mentioned method, in a theory of adapting value criteria to the nature of logical objects, reinforcing the presentiment of principle universality and of logical relationships controlled by the former. Concurrently with the truth and falsity of declarative or constative statements, more exactly in the context of last moment operance of the respective criterion, we could take into account--for the logical norming of language performances--the probability that the question is asked or not, that the order is given or not, that the wish is expressed or not, that the norm is ratified or not, etc., etc. In the corresponding atheoretical logics (erothetic, of imperatives, proieretic, deontic, etc.) should be included principles according to which two contrary questions must not accept the same answer; it is not permitted that none of the subcontrary given orders be carried out by the agent under the circumstances that have been set for it; two contradictory wishes must neither be preferred nor abandoned; if a norm has been ratified, its subordinate must be automatically assumed; etc.

The relativity of principles as to the domain and the construction level of the logico-formal theory is expected to infringe upon the interpretation of the propositions which embody the theses and the laws of the theory. Yet, not even the simplest propositional tautologies adapt themselves too easily to such a "principial" reading. We shall accept with no reserve the translation of p [disjunction] p by the statement that "two contradictory propositions are (also) subcontrary"; of p/[bar.p] by the statement that "two contradictory propositions are (also) contrary" or of p [contains] P by the statement that "any proposition is its self-sufficient condition." On the contrary, we shall express our reserve in interpreting the valid formula p&q [contains] j p by the phrase "any proposition (p) is sufficiently-necessarily conditioned by its sufficiently-sufficient interconditioning with another proposition (q)," and we shall have reserves in interpreting p [contains] p v q by the phrase "any proposition (p) sufficiently-necessarily conditions its subcontrariety with another proposition (q)." And the suggested reading gets more and more complicated while the logical analysis goes ever deeper into the "atomic" layer. In deontic logic, for example--among whose versions mention must be made of von Wright's on the basis of the logic of action relying, in its turn, on change--one could find practically all the categories of functors: connectors (making up statements out of statements), predicators (making up statements out of names), operators (making up names from other names) and subnectors (making up names from statements) (Ziembinski 1976: 93). Such a relatively simple valid formula as P(A [disjunction] B) [contains] (PA) [disjunction] (PB), will be read as: "the permission of the subcontrariety of two actions sufficiently and necessarily interconditions with the subcontrariety of the permissions of the two actions." Another simply realizable formula in its symbolic structure, (PA)&0(A [contains] B) [contains] (PB), requires a sufficiently laborious principial reading: "the sufficiently-sufficient interconditioning between the permission of an action and the obligation that this action should sufficiently-necessarily condition another action sufficiently-necessarily conditions the latter action." With respect to formula "P(pT(qIr)) [equivalent to] P(pT(tIr)&P(tTq/pT(tIr))" the interpretation we suggest is already rhebarbative: "the permission that up to the moment of the given time p should occur, and from this moment onwards the world should be in state q, implying the (active) presence of the given agent, or it should be in state r if this agent were still passive, this permission sufficiently and necessarily interconditions with the sufficiently-sufficient interconditioning between the permission that: up to the moment of the given time p should occur and from that moment onwards the world should be in state t, implying the (active) presence of the given agent, or the world should be in state r if this agent were still passive, respectively the permission that: up to the given moment of time the world should be in state t, implying the (active) presence of the given agent, or the world should be in state q if this agent were still passive, on condition that: p should occur up to the given moment of time and from that moment the world should be in state t, implying the active presence of the given agent and the world should be in state r if this agent were still passive."

The truth is that not only the principial reading but also the functional one is made difficult by the formulae that get to have a certain degree of complexity. It is the very essence of formal language that it should encode the entries of a relationship and offer for decoding only the result of afferent calculations. Or, the formulae that are taken as laws most often concentrate inferential situations: if certain relationships are given or assumed, others will be derived from the former. Our point of view on logical principles makes us identify the opposition or/and conditioning/ ordering relationships given in premises and point to the ones deriving from the former, in conclusions. When considering the value of principles in the input and the output points on the calculus itinerary of formal language, we actually insist on the duality between form and content in its employment. Adapting a conclusion of Reymond's (1936: 67), we shall say that the formal principles of thought escape any demonstration because they are the condition of any reasoning activity as formal logic has it: they help mould the content of thinking into adequate forms and allow us to homologate the results (we get in the syntactic approach thus programmed) into new contents of thinking.

So we mark the role of the meaning or of the relata intension in penetrating the formal universe of the "any object" as well as in transcending this universe, simultaneously with interpreting the "schematized reality" and "the abstract object" (Gonseth 1936: 4, 18) by determined objects of thinking. We thus avoid the redundancy of the usual formalization of the excluded middle, non-contradiction, identity, bivalence and reaffirming by double negation. On the other side, we shall thus ensure an adequate translation of the degrees of opposition and connection, revealed by the natural language, and we enrich their register by recuperating and revaluating some intermediate forms. But, first of all, the prospect we have in view for logical principles and for the relationships they norm reveals a deep communion between logical theories.

Stimulated by Wittgenstein's second philosophy on meaning as usage and the multitude of language games, as well as by the conception of the Oxford School, influenced by Wittgenstein (especially by the theory of language forces, developed by Austin), the distinction between theoretical logic and atheoretical logics (Aqvist 1964: 246), interfering with the distinction between pure logic and applied logics (Rescher 1966: 35-6; Rescher 1968: 1) relies on the distinction between the descriptive discourse (informative, factual, cognitive) and the prescriptive discourse (emotional, normative, expressive, etc.). Von Wright goes even further separating the logic of the practical discourse as "theory of the conceptual frame of the dynamic world," "of changes and processes," from the "theory of the conceptual framework of the static world" (von Wright 1968: 150 sqq). The former refers to states or state descriptions, while the latter deals with changes or change descriptions, actions and processes (von Wright 1968: 40-41). Most varied arguments can be brought against the clear-cut distinction between the two types of discourse, from the dialectic principle of the unity between theory and practice and from the circulation of the syntagm "theoretical practice" to the reconsideration of the meanings of the concept of application itself (Botezatu 1973: 21 sqq). There are many reductions of the (modal) deontic logic to the (theoretical) modal alethic logic (cf. Prior and Anderson), to say nothing of reducing the erothetic logic (of questions) to the logic of imperatives (Kubinski 1968: 187). The possibility of creating a generic deontic logic has also been considered. This logic should be apt to take the form of the logic of orders, the logic of wishes, the logic of promises, the logic of decisions (or resolutive logic), the logic of intentions, etc. (Aqvist 1964: S3). Deontic logic could be as theoretical as classical logic and it is but reasonable that we should associate the distinction between theoretical and practical reasoning with the one between the logic of purely formal demonstration and the logic of the argumentation techniques. (Perelman 1968: 170)

In the context of a variety of opinions which we have partially presented, we shall mention the generous idea of the forerunner of logic, concerning the bivalence of structures that this science has to reveal, their relevance both for thoughts and things. We can identify such an aspiration in the sketch of a general theory of alternatives as an equivalent of the usual considerations on "technical logic" (Touchais 1956: XIX). This conception which we agree to as far as logical principles are concerned brings together the domains and the constructs of formal logic in terms of a general theory of logical relationships.

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Petra Ioan

Alexandru Ioan Cuza University

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