XXXXXXX Generalizations and compositions of modal squares of oppositions XXXXXXX

Abstract. The paper aims at showing that the notion of an Aristotelian square may be seen as a specialcase of a variety of different more general notions:1) the one of a subAristotelian square, 2) the one of asemiAristotelian square 3)the one of an Aristotelian cube, which is a construction made up of sixsemiAristotelian squares, two of which are Aristotelian. Furthermore, if an Aristotelian square is seenas a special ordered 4-tuple of formulas, there are four 4-tuples with the same properties describingdifferent rotations of the same square, i.e. of the standard Aristotelian square. The second part of thepaper focuses the notion of a composition of squares. After discussing some possible alternativedefinitions, a privileged notion of composition is identified, thus opening the road to introducing anddiscussing the wider notion of composition of cubes.

KEYWORDS: Square of oppositions; Modal logic; Contingency; Hexagon; Bimodality

§1. This paper contains a development of some ideas introduced in XXXXX (2014). An obvious but frequently neglected proviso concerning the modal squares of oppositions is that therelations which are claimed to hold between the formulas of the square only subsist with reference tosome given modal system. Look for instance at the following square:

Fig.1

This square is Aristotelian with reference to system S4, but is not such with reference, say, to KT. Onthe other hand the standard modal square (which is as in Fig.1 with □A replacing evereywhere □□A) isvalid with respect to any modal system at least as strong as the deontic system KD, but invalid in anynormal system strictly weaker than KD. So the existence of a modal square of oppositions depends on some sets of given axioms and on theinfinite substitution-instances of them. But this obvious fact suggests that any Aristotelian square mayhave also a different kind of dependence, i.e. a dependence on one or more specific hypotheses. Forinstance, on the basis of the supposition ◊p É □p the relations between the following fourformulas may be graphically represented as a square having ◊p É □p at its center:

1

Fig.2

We have in fact, with respect to KD, ├ (◊p É □p) É (□p q º Ø( Ø□ p ˄ Øq)) but ┤ (□p q º Ø( Ø□ p ˄ Øq))and obviously├ (□p ˄ q) É (□p q) Without the premise ◊p É □p the square would then not be an Aristotelian square. The above square is then a conditioned or subordinated Aristotelian square: we shall call it asubAristotelian square with respect to the given supposition. The standard modal square ofoppositions might then be thought as a subAristotelian square subordinated to some logical truth of thereference system. A set-theoretical representation of a modal square of oppositions is provided by identifying it as a 4-tuple of formulas Q= <W, X, Y, Z> called vertices and ordered from left to right by the well-knownrelations governing the square. The pairs of wffs <W, X> <X,Y>, <Y, Z> ,<Z,W> will be called corneredges or simply edges of Q, while the proposition W which is the first member of the 4-tuple will becalled the origin of Q. In XXXX (2008) the notion of an Aristotelian cube was introduced as a generalization of thenotion of an Aristotelian square. Intuitively, an Aristotelian cube is a six-sided structure based on twoAristotelian squares such that the vertices of the former have with the vertices of the latter someof the relations subsisting inside an Aristotelian square: contrariety, subcontrariety, subalternance(contradiction being a special case of contrariety). The graphical representation of the two Aristoteliansquares related in the above sense (see Fig. 3) provides a picture in which the two basic Aristoteliansquares of the cubes are combined with four other squares located on the sides of the cube. Such newsquares are of a kind which will be called here semiAristotelian, in the sense that each of the their edgeshave some of the properties required for the edges of an Aristotelian squares. A rigorous definition of Aristotelian and semiAristotelian squares is as follows. A semiAristotelian square with respect to a given normal modal system S 1 is an ordered 4-tuple Γ= <A,B, C, D>, such that each one of the edges of Γas above defined consists in a pair of formulas which areeither contraries or subcontrares or connected (i.e.such that either A entails B or B entails A) with respect toS. An Aristotelian square with respect to a given normal modal system S is a semiAristotelian square Γ=<A, B, C, D>, with respect to a given system S such that the edges in Γ have the following propertieswith respect to S: in <A,B> A and B are contraries; in <B, C> B entails C; in <C,D > C and D are subcontrares; in <D,A> Aentails D. An Aristotelian cube is a set C of six squares in which each edge of the squares is in common with anedge of some other square in C anda) all squares in C are semiAristotelianb) two squares in C are Aristotelian. More rigorously, an Aristotelian cube in S is a set C = {Q*1 ... Q*6} where Q*1 ...Q*6 are 4-tuples

1 1 As a matter of fact, the properties of an Aristotelian square are independent from the □-distributivity law □(A ˄ B) º □A ˄ □B and from the Necessitation rule, which are characteristic features of normal modal systems. So in principle modal squares could be defined also in non- normal modal system. However, the relation between different squares which is analyzed in the second part of the paper is more easily treated in the framework of normal modal systems.

such that (i) each Q*i (1 < i < 6) in C is a semiAristotelian square in S and(ii) at least two of the Q*i in C are Aristotelian squares in S.(iii) each edge of each square in C is coincident with aN edge of some other square in C. It is proved in XXXX (2008) that an Aristotelian cube cannot contain more than two distinctAristotelian squares. Such Aristotelian squares belonging to the cube are called the basic squares of thecube. Every cube is univocally determined by its basic squares. If the basic squares are <A, B, C, D> and<E, F, G , H>, the cube is conventionally designed as <A, B, C, D> + <E, F,G, H>. If the two basic squares are identical square the cube <A,B,C,D> + <A,B,C,D> is a degenerated cube,and may identified with the square which is duplicated in it. Thanks to such identification, any Aristoteliansquare may be seen as a degenerate Aristotelian cube. The relations of connection inside a cube deserve attention. If two formulas A and B are such that Abelongs to Q1 and B belongs to Q2 - Q1 and Q2 being the basic Aristotelian squares of the same cube -they also belong to some semiAristotelian square that has some edge in common with one of them. Given the definition of an Aristotelian square there are then three possible relations:1) A and B are contraries (so possibily contradictories), i.e. ├ A É ØB. But ØB belongs to Q2, being thecontradictory of B. So A entails an element of Q22) A and B are subcontraries, i.e. ├ A v B and ├ ØA É B. But ØA belongs to Q1, being the contradictoryof A, so an element of Q2 is logically implied by an element of Q1.3) A and B are connected. So there is some entailment relation between elements of the two squares. We can then say that, given the definition of a cube, there is always some entailment relation which, givenan element of Q1, links it to some element of Q2. However, we may ask some stronger condition about entailment, requiring that the entailment relationssubsist among formulas having an homologous position inside the squares. We introduce then thestronger notion of a connected Aristotelian cube. An Aristotelian cube <A,B,C,D> + <E,F,G,H> isconnected iff A and E, B and F, C and G, D and H are connected formulas, i.e. each one of the wffs insuch couples entails or is entailed by the other one. A noteworthy fact concerning cubes is that the composing the squares may be written in languagesbased on two different primitive operators, and that their relations may be described in a language ofsome normal bimodal system. Let us consider, for example, the formulas □A, OA, OØA, Ø□ A, A , ◊ PA,PØA , ØA◊ where of course PA = Df ØOØA. In any bimodal system written in □ and O (let us say, analethic-deontic system) in which □A entails OA, the basic relations among the eight formulas may berepresented by the following connected cube (the arrows symbolyzing entailment relations): Fig.3

This does not exclude, however, that the two basic squares in the cube may be written in one and thesame language. In every normal modal system which contains the deontic system KD, for instance, thecube denoted as < (□ A ˄ B), □(ØA ˄ ØB), ( ◊ ØA v ØB), ( A◊ v B)> + < A□ , □ØA, ◊ØA, A)> ◊ isaconnected cube. To conclude this first approach to cubes we have to remark, however, that there are also squareswhich we shall call “section squares”: they result from sectioning the cube along the diagonals. Look forinstance, in the following figure, at the square drawn in red and at the other drawn in blue:

Fig.4 For every cube there are four section squares, and one can check that none of them is Aristotelian (PA ,for instance, is not the negation of Ø□ A), but all of them are semiAristotelian.

§2. Let us now introduce two new notions: the one of a degenerate square and the one of a rotation ofa square. An Aristotelian square is a degenerate one when subalternance holds in the two directions, i.e. whenthe two subalternant formulas are logically equivalent with respect to the reference system. A good example of degenerate squares is provided by the logic of contingency (see XXXX (2014)).As is well-known, contingency and absoluteness are defined as follows in any normal modalsystem: (Def Ñ) ÑA = Df ◊A ˄ ◊ØA (bilaterality principle)(Def D) DA = Df ØÑA Proper contingency is often confused with the notion of non-necessity, or weak contingency, which maybe defined as follows:( Def Ñw) ÑwA = Df Ø□A So DA º □A v Ø□ A is a theorem in every normal modal system extended with (Def D) and (Def Ñ),according to the bilaterality principle, the following equivalences are also theorems: ÑA º ÑØA,DA º DØA. The representation of a square of contingency notions is then as follows: Fig.5

In the degenerate square of Fig.5 the contrary formulas are also contradictory, so there is no distinctionbetween the diagonal and the upper line of the square. Given the logical equivalence subsisting betweenthe upper formulas and the lower formulas, we could then flatten the square into a simple line, redrawing itas follows:

Fig.6

A special case of equivalence is provided by identity. The most simple degenerate square is then constitutedof the couple [ A, ØA]. Let us go now to the notion of a rotation of a square. Such a notion is straightforwardly suggested by theset-theoretical representation of an Aristotelian square as an ordered 4-tuple. In fact, according to such adefinition, the squares <A, B, C, D>, <B, C, D, A>, <C, D, A, B>, <D, A, B, C> are different squares evenif they have common internal relations and a common origin. If Q1=<A, B, C, D> is a standardAristotelian square having A has its origin, the other squares may be intended as rotations of Q1 (intending with this terms the result of a process of rotation) . In Q1 the starting point of the square iscoincident with the origin, while in other rotations such coincidence does not subsists. The distinctionbetween different rotations of the same square is irrelevant when we take into consideration a singularsquare, but may make a difference if we consider the relations between different squares inside a cube. Speaking generally, the rotation of a square is always performed on some pre-established axis passingyhrough the center of the square, as for instance in the following picure:

The rotation axis may be also vertical and diagonal. In all such cases the processes of rotation share theproperty of yielding a maximum of two distinct outputs, since a double application of the rotation yieldsagain the initial square. But things are more interesting if we choose an axis crossing the center of thesquare in a three-dimensional space: the result is a clockwise process of rotation . For instance,the process of rotation beginning from the first of the following squares goes on as shown in the nextfigure –where the star * points out the moving origin. The first rotation is coincident with the standardAristotelian square. Fig.10

R1 R2 R3 R4

In principle a cube may be composed of rotations of one and the same square. Any couple of rotationsof one and the same standard Aristotelian square as in Fig.10 always yields some Aristotelian cube, but theonly connected cube resulting from such process of rotation is some cube described by the schema <x, y, z,d > + <x, y, z, d>, so a cube having the property of being a degenerate cube. A connected cube may be a subordinated cube in the sense akin to the sense in which we spoke before ofsubordinated squares. It may happen, in fact, that the connections inside the basic squares of thecube depend essentially on some specific hypothesis . Look at for instance at the following picture, where Øq is essential to grant the connection betweensquares according to the laws of system KD. The remarkable fact is that the square located in the anteriorface of the cube is a rotation of a standard square of modalities, whose origin is located at the bottomcorner at the left.Fig.11

In KD one is able to prove:

1) Øq É ((□p ˄ q) É (◊p v q))2) Øq É (◊p É (Ø□p v Øq)) (= Øq É Ø(□p ˄ q))3) Øq É ((□p ˄ q) É Ø□ p) 4) Øq É (( Ø□ p ˄ q) É ◊Øp) (= Øq É (□p É (◊p v q))

The formulas 2) and 3) cannot be proved in KD if the antecedent Øq is omitted, so Øq is essential to grantthe properties of an Aristotelian cube. We spoke before of a degenerate cube intending that the two Aristotelian squares composing the cubeconsist of identical formulas (so the cube is simply a reduplication of the same square). But there is asecond sense in which a cube may be said to be a degenerate one: it may happen, in fact , that some ofthe squares composing it are themselves degenerate squares. It may be of some interest to note that the well-kown Blanché’s hexagon (see Blanché (1953))may be seen as a degenerate cube in the latter sense. Let us look in fact to a cube based on twosquares, one written in □ and ◊, and another written in Ñ and D, having the following shape:

Fig.12

The cube is Aristotelian and connected since, of course, □A implies DA and ÑØA implies ◊A. The star nearÑA indicates the origin of the second basic square, giving evidence that such square is a rotation of thestandard square of contingency. Since DA and DØA on one side and ÑA and ÑØA on the other areequivalent in any normal system extended with the standard definition of such operators, they can berepresented by a straight line segment (see Fig. 6), so that the above cube could be redrawn in thefollowing way:

Fig.13

Blanché’s hexagon is then a bidimensional graphical representation of this degenerate cube.

§3. The notion of a composition of Aristotelian squares has been introduced in XXXX [2014]. What isthere defined as °-composition of two squares Q1 and Q2 consists in the conjunction of the upper verticesof Q1 and Q2 and in the disjunction of the lower vertices of Q1 and Q2: <A, B, C, D> ° <E, F, G, H> =<A ˄ E, B ˄ F, G v C, D v H>Fig.14

As already remarked, an Aristotelian square need not be necessarily a standard Aristotelian square but canbe a rotation of it. In XXXX (2014) the following results receive a proof.1) The °-composition of the first rotation of two Aristotelian squares is still an Aristotelian square.2) Let Q be the first rotation of an Aristotelian square and Q* an arbitrary rotation of an Aristoteliansquare. Then the °-composition of Q and Q* is the first rotation of an Aristotelian square. Prof. Lloyd Humberstone of Monash University correctly remarked (in correspondence) that theproposed method of composition is not the only possible one. Other two possibilities he suggestedare the following: <A, B, C, D> °' <E, F, G, H> = <A ˄ E, B v F, C v G, D ˄ H> <A, B, C, D> °'' <E, F, G, H> = <A v E, B ˄ F, C ˄ G, D v H> The reader can easily check that such operations of composition preserve the required properties

of Aristotelian squares, as it should be evident from the two following pictures:

There is, however, at least one intuitive property which the °-composition has but the other have not.To clear the point, let us consider the example provided by the simple bimodal cube of Fig.3 we remark (i) that the stronger formulas in the “boxed” square, □A and Ø□ A, entail to the strongerformulas of the second square, OA and OØ A respectively bu are not equivalent to them (ii) the weakerformulas in the former square, ◊A and ◊ØA, are weaker than the weaker formulas in the latter, PA and PØArespectively, and are not equivalent to them. When an Aristotelian square Q1 has with anotherAristotelian square Q2 the same relation which the above □◊-square has with the above OP-ssquare we stipulate to say that the Q1 is stronger than Q2. Recalling that ├ A É B, where A is stronger than B, is equivalent to ├ A ˄ B º A, by analogy weexpect that when in a connected cube one of the two basic squares, say Q1, is stronger that the other, sayQ2 , the composition of Q1 and Q2 yields again Q1 and not some different square2. Now, according to the definition of °-composition , with reference to Fig. 3 the square resulting fromthe °-composition of the two basic squares is the following: <□A ˄ OA, Ø□ A ˄ O ØA, ◊A v PA, ◊ØA vPØA>. But, given the equivalences├ □A ˄ OA º □A, ├ Ø□ A ˄ O ØA º Ø□ A, ├ ◊A v PAº ◊A, ├◊ØA v PØAº ◊ØA, it turns out that the square which is the output of the °-composition is equivalent(i.e. it has vertices which are equivalent) to the “stronger” of the two input squares. Let us look, by contrast, at the result which one finds by applying the alternative composition operationsprovided by the operators °’ and °’’. Recalling that <A, B, C, D> °' <E, F, G, H> = <A ˄ E, B v F, C v G,D ˄ H>, the resulting °'-composition yields <□A ˄ OA, Ø□ A v O ØA, ◊A v PA, ◊ØA ˄ PØA>. The originof the output square □A ˄ OA is indeed equivalent to □A, but the formula occupying the second vertex,

Ø□ A v OØA, is not equivalent to Ø□ A. The resulting square boils down to < □A, OØA, ◊A, PØA >, whichis neither a standard Aristotelian square nor a rotation of it. Let us look at the third composition operator °'' which, as we recall, is defined as <A, B, C, > °” <E, F,G, H> = <A v E, B ˄ F, C ˄ G, D v H>. The result of the °'' -composition in the given example is thesquare <□A v OA, Ø□ A ˄ OØA, ◊A ˄ PA, ◊ØA v PØA> which reduces to <OA, Ø□ A, PA, ◊ØA >, which is

2 This property is obviously satisfied by squares Q1 and Q2 ina connected cube in which the enatilment relations are equivalence relations.

neither a standard Aristotelian square nor a rotation of it. Of course we can extend the notion of °-composition of squares to the more general notion of °-composition of cubes. Intuitively, the composition of cubes consists in the °-composition of its basic squares, as it isexemplified in the next figure. Fig.16

We will say that a °-composition of two input cubes C' and C” is regular iff the stronger square of C’is composed with the stronger square of C”, provided a comparison of strength is possible. Regularityimplies that the weaker square in C' is composed with the weaker square in C”. An open problem is to determine the conditions at which a regular composition of cubes preserves the properties of being a connected Aristotelian cube. A simple result in this concern is that the compositionhas this welcome result when the two input cubes are both Aristotelian and connected. It may be shown infact in such simple case the composition of the two stronger squares in the input cubes yield thestronger square in the output cube, while the composition of the weaker squares in the inputcubes yields the weakest square in the output cube. The theorem which may be proved is as follows:T. Let C1= <ABCD> + <ILMN> and C2=<EFGH> + <OPRQ> two connected Aristotelian cubes. LetABCD and EFGH the stronger squares in C1 and C2 respectively. Then the cube C3= C1 ° C2= <A˄ E, B ˄ F, C v G, D v H,> + <I ˄O, L˄P, M v Q, N v R> is also a connected Aristotelian cube in whichA ˄ E, B ˄ F, C v G, D v H is the stronger Aristotelian square.Proof. The main problem is to prove that among formulas of the squares belonging to C3 a relation ofconnection exists.For the proof one may use Leibniz’s Theorema PraeclarumLTP: ((X É Y) ˄ (W É Z)) É ((X ˄ W) É (Y ˄ Z) and its variantLTPv: ((X É Y) ˄ (W É Z)) É ((X v W) É (Y v Z)) Let us suppose that A entails I and E entails O, in the sense that in some background system X which isan extension of KD ├ X A É I and ├ X E É O. Then by LTP and standard logic ├ X (A ˄ E) É (I ˄ O). The same result obviously holds for the wffs B, L, F, P. B entails L and F entails P: in this case B ˄ Fentails L ˄ P. By the definition of connection and of strength of squares we have that D entails N and H entails R. Soby LTPv D v H entails N v R. Furthermore, M entails C and Q entails G; so by parity of argument M v Qentails G v Q. Thus the cube < A ˄ E, B ˄ F, C v G, D v H,> + <I ˄O, L ˄ P, M v Q, N v R> has the required properties:1) it is connected since its vertices are pairwise connected, and 2) the square < A ˄ E, B ˄ F, C v G, D vH,> is stronger than the square <I ˄O, L˄P, M v Q, N v R>. (Q.E.D.)

§4. The basic idea of seeing Aristotelian squares and hexagons as special cases of more abstract structureshas been recently at the origin of a lively trend of studies in logics and mathematics. In the realm of whatis called now “n-oppositional geometry” see for instance Moretti (2009), Moretti (2012) and Pellissier(2008). The present paper has been worked out indipendently from such a direction of inquiry and fromothers which have been recently developed in the analysis of the square of opposition. The notion of andAristotelian cube employed here has no apparent relation, for instance, with the notions of a “cube ofopposition” introduced by Dubois and Prade in (2012). However, there are many reasons to expect thatsuch different approaches could be unified or synthesied in some more general theory which, to beginwith, should be expressed in a language richer than the one of propositional modal logic.

Acknowledgment. Part of the content of §3 develops a suggestion received by Lloyd Humberstone alongwith other comments to a preceding draft of the present paper. The author wishes to thank him and in themeantime to release him from every responsibility about the use which is made here of his remarks.

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Blanché, R.(1953), Sur l’opposition des concepts, Theoria, 19, pp. 89-130

Dubois,D. And Prade H. (2012), From Blanché's Hexagonal Organization of Concepts to Formal ConceptAnalysis and Possibility Theory, Logica Universalis, 6, pp.149-169

Moretti, A. (2009) The Geometry of Logical Opposition, PhD Th., Neuchâtel, Switzerland, 2009

Moretti, A. (2012) Why the Logical Hexagon?, Logica Universalis, 6(1-2), pp.69-107

Pellissier, R. (2008) Setting n-opposition, Logica Universalis, 2(2), pp.235-263 XXXX (2008) XXXXXXXXXXXX (2014) XXXXXXXX