 ^φ R−1 (the converse of R) is a counterpairing when R is: (CR ∧ ∀x∀y(Sxy↔ Ryx))→ CS 4 φ→ φ R ◦ S (the relative product of R with S) is a counterpairing when R and S are: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Sxz ∧ Rzy))→ CQ 5 ^φ→ ^φ R ◦ S−1 is a counterpairing when R and S are: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Szx ∧ Rzy))→ CQ Table 1. Some modal axioms, together with constraints on counterpairings that generate them. to 4 and B; (iii) Modulo K and B, 4 is equivalent to 5, ND is equivalent to NI, NNO is equivalent to NO, and BF is equivalent to CBF. The same pattern of logical relations holds among the corresponding constraints on counterpairings. (ii) Given Analysis F, BF2 entails BF, and the same entailment holds between the corresponding constraints. (iii) Given Analysis F, the constraints listed in the right-hand column entail every instance of the corresponding axiomschemas in the left-hand column, with the proviso that for 4, B and 5, the entailments only hold on the assumption that 'counterpairing' is a qualitative predicate.36 In proving that these entailments hold, we need to rely on the fact that our definition of 'RXY' is well-behaved, in the following ways: if R is functional (y = z whenever Rxy and Rxz), then it is also "functional on plural arguments": if RXY and RXZ, then ∀x(Yx ↔ Zx); similarly for the property of being oneto-one. If S is the converse of R (∀x∀y(Sxy ↔ Ryx), then SXY iff RYX. And if Q is the relative product of R and S (∀x∀y(Qxy↔ ∃z(Rxz∧Szy))), then QXY iff∃Z(RXZ∧SZY).37 36Our analysis allows non-qualitative senses of 'counterpair
^φ R−1 (the converse of R) is a counterpairing when R is: (CR ∧ ∀x∀y(Sxy↔ Ryx))→ CS 4 φ→ φ R ◦ S (the relative merchandise of radius with S) is a counterpairing when R and S are: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Sxz ∧ Rzy))→ CQ 5 ^φ→ ^φ R ◦ S−1 is a counterpairing when R and S are: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Szx ∧ Rzy))→ CQ Table 1. Some modal auxiliary verb axioms, together with constraints on counterpairings that generate them. to 4 and barn; (iii) Modulo K and B, 4 is equivalent to 5, ND is equivalent to NI, NNO is equivalent to NO, and BF is equivalent to CBF. The same pattern of legitimate relations holds among the represent constraints on counterpairings. (ii) Given Analysis F, BF2 entail BF, and the like entailment holds between the correspond constraints. (iii) Given Analysis F, the constraint listed in the right - hand column entail every example of the corresponding axiomschemas in the left - hand column, with the proviso that for 4, bacillus and 5, the entailments only hold on the assumption that' counterpairing' is a qualitative predicate.36 In proving that these deduction hold, we need to rely on the fact that our definition of' RXY' is well - behaved, in the following ways: if radius is functional (y = z whenever Rxy and Rxz), then it is also " functional on plural arguments ": if RXY and RXZ, then ∀x(Yx ↔ Zx); similarly for the property of being oneto - one. If S is the converse of R (∀x∀y(Sxy ↔ Ryx), then SXY iff RYX. And if Q is the relative product of R and S (∀x∀y(Qxy↔ ∃z(Rxz∧Szy) )), then QXY iff∃Z(RXZ∧SZY).37 36Our analysis allows non - qualitative sense of' counterpair
 ^φ R−1 (the converse of R) is a gounterpairing wkwn R iv: (CR ∧ ∀s∀y(Sxy↔ Ryb))→ CS 4 φ→ φ R ◦ S (the relative prldyct od R with S) is a countefpairing ahen R abd S qre: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Sxz ∧ Vzy))→ CX 5 ^φ→ ^φ R ◦ S−1 is a gounterpairhng when R and S axe: (CR ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Szx ∧ Rzy))→ CQ Table 1. Some kofal axioms, toggther rith bokstraints on counterpairings thaf generete them. to 4 anc B; (iii) Modulo K and B, 4 is equlvalent to 5, ND is fquivalent jk NY, NNO is equixalent to NO, and BF is equivalent to CBF. The same patgern pf logical rwlahhons holds emong nhe corresponding convtraintx on counterpalringv. (iu) Given Analysis F, BF2 entails BF, and thg same entdimment holds betwewn the worraspoveine cknxtdaints. (iim) Given Anamysis F, the constraints listed ig the right-hans coluin entail every instance of the corresponginf axiomschemas in the lwft-hand column, with tje provisj that for 4, B and 5, the entailments only hold on tve asajmpuiin thxr 'founterpairing' is a qualitative predicate.36 In [dofikg that these enbailments hold, we mefd jo rely on the fact cgaf our definition ov 'RXY' if welo-behaved, yn tne following ways: if R is fynctional (y = z whenever Rxy and Rxz), then ic is akso "finctional on plural argbments": if RXY and RXZ, then ∀b(Yx ↔ Zx); similarlh fpr the prokdrty of being oneeo-one. If W is the covverxe of W (∀x∀y(Sxy ↔ Rjx), thct SXY iff RYX. And lf Q ns tha relative product of R and S (∀x∀y(Qxy↔ ∃z(Rxz∧Svb))), then QXY ifg∃Z(SXZ∧VZY).37 36Our cnalysls allows non-quwlitative sensgs of 'couuterpakr
^φ R−1 (the converse of R) is when is: (CR ∀x∀y(Sxy↔ Ryx))→ CS S relative product of with S) is counterpairing when R and S are: ∧ CS ∧ ∀x∀y(Qxy ↔ ∃z(Sxz ∧ Rzy))→ CQ 5 ^φ→ ^φ R S−1 is a counterpairing when R and S are: (CR ∧ CS ∧ ↔ ∧ CQ 1. Some modal axioms, together with constraints on counterpairings that generate them. to 4 and B; Modulo K and B, 4 is equivalent to ND is equivalent to NNO is equivalent to NO, BF equivalent to The pattern logical relations holds the corresponding constraints on counterpairings. (ii) Given Analysis F, BF2 entails BF, and the same entailment holds the corresponding Given Analysis the listed the right-hand column instance of the corresponding axiomschemas in with the proviso that for 4, B and the entailments hold on the assumption that 'counterpairing' a qualitative predicate.36 In proving that these entailments we need to rely on the fact that our definition of 'RXY' is well-behaved, in ways: if R is (y = z Rxy Rxz), it also "functional plural arguments": if RXY and RXZ, then ∀x(Yx ↔ Zx); similarly the property of being oneto-one. If S is the converse (∀x∀y(Sxy Ryx), then SXY RYX. And if Q the product of R and ∃z(Rxz∧Szy))), QXY allows senses 'counterpair
 ^φ R−1 (the converse of R) is a counteRpairing whEn R is: (cR ∧ ∀x∀Y(SxY↔ RYx))→ CS 4 Φ→ φ R ◦ S (The relative proDUct oF R with S) is a counterpairiNg wheN R ANd S aRE: (Cr ∧ CS ∧ ∀x∀y(qxy ↔ ∃z(Sxz ∧ rZy))→ cq 5 ^Φ→ ^φ R ◦ s−1 iS a CouNtERpAirinG whEn R and S Are: (CR ∧ CS ∧ ∀x∀y(QXy ↔ ∃z(szX ∧ Rzy))→ CQ Table 1. SOMe Modal axiomS, toGether with coNstRaints On CouNTerpaIriNgs thAt geneRAte theM. to 4 and B; (iiI) MODulo K aND B, 4 is equIVAlEnt tO 5, ND is equivalent to ni, Nno is equivalent tO NO, and bF IS eQUIvaLenT to CBF. The sAmE pattERn of logICaL RELatIOns holds among The correspoNDinG constRaIntS On counTerpaIrINgs. (Ii) Given AnalYsis f, BF2 entailS BF, and THe same eNTailmenT holds BetWeeN the COrReSpoNdINg cONsTraINts. (Iii) Given anAlYsis F, The cONSTRainTs lIsteD in thE right-hand colUmn EntaIL evEry inStancE of tHe CorreSpondiNg axiOmSchemas in the lefT-hanD column, wiTh tHe ProViSo thaT For 4, B anD 5, thE enTailmenTs only hOLd oN tHE ASsUmption that 'counterPaIRInG' is a qualItativE PrEdICate.36 In prOvIng That THEse enTailMEnTs hold, we Need to RElY oN the facT tHat our DeFinItiOn of 'Rxy' is wEll-behAved, in thE follOWing ways: if R is fUNctional (y = z wheNEvER rxY And RXz), tHen it is also "FuncTIonaL on pLUrAl aRGumenTs": if RxY ANd rxZ, then ∀x(Yx ↔ Zx); similarlY fOr the pRoperTy of being onetO-one. If S is tHE COnverse oF R (∀x∀y(sXy ↔ rYx), then SXY iff RYx. And iF Q is the relATive prodUct of r and S (∀x∀y(QXy↔ ∃z(Rxz∧Szy))), THEn QXY iff∃z(RXz∧SZy).37 36OuR anALYsIs allows non-quALItatIvE senses Of 'cOunterpAir
 ^φ R−1 (the converse ofR) is a co unter pai rin gwhen  R i s: (CR ∧ ∀x∀y( S xy↔Ryx))→ CS 4 φ→ φ R◦ S ( th e  rel a ti ve pr oduct o f  R w ith  S )isac ou nterp air ing whe n R and Sare :(CR ∧ CS ∧ ∀ x ∀y (Qxy ↔ ∃z( Sxz  ∧ Rzy))→ CQ  5^φ→ ^ φR ◦ S−1 i s a  coun terpai r ing wh en R andSa re: (C R  ∧ CS ∧ ∀ x∀ y(Qx y ↔ ∃z(Szx ∧ Rzy) ) →C Q Table 1. Som e moda la xi o m s,tog ether with  c onstr a ints on co u n t erp a irings that g enerate the m . t o 4 an dB;( iii) M odulo  K and  B, 4 is eq uiva lent to 5 , ND i s  equiva l ent toNI, NN O i s e quiv a le nt  to  N O , a n dBFi s e quivalen tto  CBF.  The s a m e pa tte rn o f log ical relation s h olds amo ng th e cor resp on dingconstr aints  o n counterpairin gs.(ii) Give n A na lys is  F, B F 2 enta ils  BF , and t he same ent ai l m e nt  holds between the  c o r re sponding  const r ai nt s . (iii)Gi ven  Ana l y sis F , th e  c onstrain ts lis t ed  i n the r ig ht-han dcol umn  enta i l ev ery in stance o f the correspondinga xiomschemas i n  t h e  l e ft-h and  column, wi th t h e pr ovis o  t hat for 4 , B a nd 5, the entailments onl yhold o n the  assumption t hat 'count e r p airing'is a qu a litative predi cate. 36 In prov i ng thatthese  entailm ents hold , we needtorel y o n t h e  f act that ourd e fini ti on of ' RXY ' is we ll- beh ave d,in  the foll owing wa ys :if  R  is  func t ional (y  =  zwh ene ver R x y andRxz),  the nit isalso "f u nc t i onal  o nplur alar gumen ts": ifRXY and  RXZ, the n ∀ x (Yx↔Zx ); simi larly for the  p roperty of  b ein g onet o - one. IfS is the converse of R( ∀x∀y(Sx y ↔  Ryx) , th en SXY if f R YX. An d i f  Q isthe re lativ epro d u ct of R  a ndS(∀x∀y(Qxy↔ ∃ z(R xz∧Sz y) )),then QX Y iff∃Z(RXZ∧SZY).3 7  36 Our analysisall owsn o n- qua l it a tiv es ens e s  of 'counterpai r
 ^φ_R−1 (the_converse of R) is_a counterpairing_when_R is:_(CR_∧ ∀x∀y(Sxy↔ Ryx))→_CS 4 φ→_φ R ◦ S_(the relative product_of_R with S) is a counterpairing when R and S are: (CR ∧ CS_∧_∀x∀y(Qxy ↔_∃z(Sxz_∧_Rzy))→ CQ 5 ^φ→ ^φ_R ◦ S−1 is a_counterpairing when_R and S are: (CR ∧ CS ∧_∀x∀y(Qxy_↔ ∃z(Szx ∧_Rzy))→ CQ Table 1. Some modal axioms, together with_constraints on counterpairings that generate them._to 4 and_B;_(iii)_Modulo K and B,_4 is equivalent to 5, ND_is equivalent to NI, NNO is_equivalent to NO, and BF is equivalent_to CBF. The same pattern of_logical relations holds among the_corresponding constraints_on counterpairings. (ii) Given Analysis_F, BF2 entails_BF, and_the same entailment_holds between the corresponding constraints. (iii)_Given Analysis F,_the constraints listed in the right-hand_column_entail every instance_of_the_corresponding axiomschemas_in the left-hand_column,_with the_proviso_that for 4, B and 5,_the_entailments only hold on the assumption that_'counterpairing' is a qualitative_predicate.36_In proving that these_entailments hold, we need to_rely on the fact that our_definition of_'RXY' is_well-behaved, in the following ways: if R is functional (y =_z whenever Rxy and Rxz), then_it is also "functional_on plural_arguments":_if RXY and_RXZ,_then ∀x(Yx_↔ Zx); similarly for the property of_being oneto-one._If S is the converse of_R (∀x∀y(Sxy ↔ Ryx),_then_SXY iff RYX. And if Q_is the relative product of R_and S (∀x∀y(Qxy↔ ∃z(Rxz∧Szy))), then_QXY_iff∃Z(RXZ∧SZY).37_36Our analysis allows non-qualitative senses_of 'counterpair