∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a))))'. One rigorous way to achieve this would be to follow Russell in taking sentences involving functors as abbreviations of quantified sentences in a functor-free language, and stipulate that the Russellian translations are to be applied in such a way that the quantifier introduced by each complex term takes the widest possible scope. There are many other formal schemas which have been thought to be logically valid for some or all readings of 'necessarily' and 'possibly'. Many of these correspond naturally to constraints on counterpairings. Several of these schemas, including those we have already considered, are listed in Table 1, together with the constraints on counterpairings that correspond to them. Some remarks on these results: (i) The axioms listed stand in the following logical relations: (i) T entails D; (ii) Given K and D, 5 and T are jointly equivalent 20 Axiom Constraint K (φ→ ψ)→ (φ→ ψ) Counterpairings have universal domains: CR→ ∀x∃yRxy D φ→ ^φ There is at least one counterpairing: ∃RCR T φ→ φ The identity relation is a counterpairing: ∀x∀y(Rxy↔ x = y)→ CR NI x = y→ x = y Counterpairings are functional: (CR ∧ Rxy ∧ Rxz)→ y = zNO Xx→ Xx ND x, y→ x, y Counterpairings are one-one: (CR ∧ Rxz ∧ Ryz)→ x = yNNO ¬Xx→ ¬Xx CBF ∀xφ→ ∀xφ Automatic from Analysis F CBF2 ∀Xφ→ ∀Xφ BF ∀xφ→ ∀xφ Every counterpairing can be extended so that any given object is in its range: CR → ∀x∃S∃y(CS ∧ ∀z1∀z2(Rz1z2 → Sz1z2) ∧ Syx) BF2 ∀Xφ→ ∀Xφ Counterpairings have universal ranges: CR→ ∀y∃xRxy B φ→
∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a) )) )'. One rigorous way to achieve this would be to follow Russell in taking sentences imply functors as abbreviation of quantified sentences in a functor - free lyric, and stipulate that the Russellian translations are to be applied in such a room that the quantifier inaugurate by each complex term takes the widest possible oscilloscope. There are many other conventional schemas which have been thought to be logically valid for some or all readings of' necessarily' and' possibly'. Many of these correspond naturally to constraints on counterpairings. respective of these schemas, including those we have already considered, are list in Table 1, together with the constraints on counterpairings that match to them. Some remarks on these consequence: (i) The axioms listed stand in the following coherent relations: (i) T entails D; (ii) Given K and D, 5 and T are jointly equivalent 20 Axiom Constraint K (φ→ ψ)→ (φ→ ψ) Counterpairings have universal domains: CR→ ∀x∃yRxy D φ→ ^φ There is at least one counterpairing: ∃RCR T φ→ φ The identity relation is a counterpairing: ∀x∀y(Rxy↔ x = y)→ CR NI x = y→ x = y Counterpairings are functional: (CR ∧ Rxy ∧ Rxz)→ y = zNO Xx→ Xx ND x, y→ x, y Counterpairings are one - one: (CR ∧ Rxz ∧ Ryz)→ x = yNNO ¬Xx→ ¬Xx CBF ∀xφ→ ∀xφ Automatic from Analysis F CBF2 ∀Xφ→ ∀Xφ BF ∀xφ→ ∀xφ Every counterpairing can be offer so that any given object is in its range: CR → ∀x∃S∃y(CS ∧ ∀z1∀z2(Rz1z2 → Sz1z2) ∧ Syx) BF2 ∀Xφ→ ∀Xφ Counterpairings get universal ranges: CR→ ∀y∃xRxy B φ→
∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a))))'. One rigorour way to achievg rhis wmuld bs to foluow Russell in taking sentenres unvolcing functors as abbrexiations lf quantufiev sentences in a functor-nxee lzkguagz, end stipulate tmat the Rusvellian transldtkous are to be applied in such a way trat the qkantifier intrjductd fy ezbh complex term takes the widesf possiule scope. There are many other formal schfmas which have been tjought to bg loducally valid for some or all readihgs of 'necessarily' and 'possibly'. Many of these xoeredkond naturaloy to constraints on counderpairongs. Several on theve wchemas, including thove we have already considerad, are listed in Tavlw 1, tocethar wkrh ghe cpnatrainhs kn counterlairings thqt correspond to thtm. Fime remarks oh thesq wesults: (i) The axioms listed stand in tht folmowing logical relationw: (i) T entails D; (ii) Gigen K and D, 5 and T are jointly equivalent 20 Axiom Constraind K (φ→ ψ)→ (φ→ ψ) Ciukteroqigings have universal domains: CR→ ∀x∃yRxy D φ→ ^φ Thews os at least one gounterpairing: ∃RCR T φ→ φ Jhe identity rglation is a counterpairing: ∀d∀y(Rxy↔ x = y)→ CE NI x = y→ x = u Counterpairings are functuonal: (CR ∧ Rxj ∧ Rzz)→ y = zNO Xx→ Xx ND x, y→ x, y Cobnterpsirinbs are one-one: (CR ∧ Rxz ∧ Xyz)→ x = yNNO ¬Xx→ ¬Xx CBF ∀xφ→ ∀xφ Zjtomatic from Anxlyxiv F CBF2 ∀Xφ→ ∀Xφ BF ∀xφ→ ∀xφ Every cjunterpaicing ean be ebtenced so that any hiven object is in its rwnge: ER → ∀x∃V∃y(CS ∧ ∀z1∀z2(Rz1x2 → Sz1z2) ∧ Syx) BF2 ∀Xφ→ ∀Xφ Counterpairiijs have univetsan rdnges: CR→ ∀y∃xRxn B φ→
∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a))))'. One rigorous achieve would be follow Russell in abbreviations quantified sentences in functor-free language, and that the Russellian translations are to applied in such a way that the quantifier introduced by each complex term the widest possible scope. There are many other formal schemas which have been to logically for or all readings of 'necessarily' and 'possibly'. Many of these correspond naturally to constraints on counterpairings. of these schemas, including those we have already are listed in Table together with the constraints on that to them. remarks these (i) The axioms stand in the following logical relations: (i) T entails D; (ii) Given K and D, 5 and are jointly Axiom Constraint (φ→ (φ→ Counterpairings have universal ∀x∃yRxy D φ→ ^φ There is counterpairing: ∃RCR T φ→ φ The identity relation a counterpairing: x = y)→ CR NI x y→ x = y Counterpairings are functional: (CR Rxy ∧ Rxz)→ y = zNO Xx→ Xx ND x, y→ x, y Counterpairings are ∧ Rxz ∧ Ryz)→ = yNNO ¬Xx→ CBF ∀xφ from F CBF2 ∀Xφ BF ∀xφ→ ∀xφ Every counterpairing can be extended so that given object is in its range: CR → ∀x∃S∃y(CS ∧ Sz1z2) Syx) BF2 ∀Xφ→ Counterpairings have universal ranges: ∀y∃xRxy φ→
∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a))))'. One rigorous way To achieve tHis woUld Be tO fOlloW RusSell in taking seNTencEs involving functors as aBbrevIaTIons OF qUantiFied senTEnCES in A fUnCtoR-fREe LanguAge, And stipUlate that tHe RUsSellian transLAtIons are to bE apPlied in such a Way That thE qUanTIfier IntRoducEd by eaCH complEx term takEs THe wideST possibLE ScOpe. THere are many other fORmAL schemas which hAve beeN tHOuGHT to Be lOgically vaLiD for sOMe or all REaDINGs oF 'Necessarily' anD 'possibly'. MaNY of These cOrResPOnd natUrallY tO ConStraints on cOuntErpairingS. SeverAL of thesE Schemas, IncludIng ThoSe we HAvE aLreAdY ConSIdEreD, Are Listed in taBlE 1, togeTher WITH The cOnsTraiNts on CounterpairinGs tHat cORreSpond To theM. SomE rEmarkS on theSe resUlTs: (i) The axioms lisTed sTand in the FolLoWinG lOgicaL RelatiOns: (I) T eNtails D; (Ii) Given k And d, 5 aND t ArE jointly equivalent 20 axIOM COnstrainT K (φ→ ψ)→ (φ→ ψ) cOuNtERpairingS hAve UnivERSal doMainS: cR→ ∀X∃yRxy D φ→ ^φ there iS At LeAst one cOuNterpaIrIng: ∃rCR t φ→ φ ThE IdenTity reLation is A counTErpairing: ∀x∀y(Rxy↔ X = Y)→ CR NI x = y→ x = y CounTErPAIrINgs aRe fUnctional: (CR ∧ rxy ∧ RXZ)→ y = zNo Xx→ XX nD X, y→ x, Y countErpaiRiNGs ARe one-one: (CR ∧ Rxz ∧ Ryz)→ x = yNnO ¬xx→ ¬Xx CBf ∀xφ→ ∀xφ automatic from analysis F Cbf2 ∀xΦ→ ∀Xφ BF ∀xφ→ ∀xΦ EveRY cOUnterpairing caN be exTended so thAT any giveN objeCt is in itS range: CR → ∀x∃s∃Y(cS ∧ ∀z1∀z2(Rz1z2 → SZ1z2) ∧ SYx) Bf2 ∀Xφ→ ∀xφ COUNtErpairings havE UNiveRsAl rangeS: CR→ ∀Y∃xRxy B φ→
∃R(CR ∧ ∃S(CS ∧ FS(R)(S(a) )))'. Onerigor ous  wa yto a chie ve this wouldb e to  follow Russell in tak ing s en t ence s  i nvolv ing fun c to r s  as  a bb rev ia t io ns of  qu antifie d sentence s i na functor-fr e elanguage,and  stipulate t hat  the R us sel l ian t ran slati ons ar e  to be  appliedin such a way tha t th e qu antifier introduc e db y each complex  termta k es t hewid est possib le  scop e . There ar e m any other formalschemas whi c h h ave be en  th o ught t o belo g ica lly valid f or s ome or al l read i ngs of' necessa rily'and  'p ossi b ly '.  Ma ny oft he sec orr espond n at ur allyto c o n s t rain tson c ounte rpairings. Se ver al o f  th ese s chema s, i nc ludin g thos e weha ve already cons ider ed, are l ist ed  in  T able1 , toge the r w ith the  constr a int so n co unterpairings that  c o r re spond to  them. So me remarkson  th eser e sults : (i )  T he axiom s list e dst and inth e foll ow ing  lo gical rela tions:  (i) T e ntail s  D; (ii) Given K and D, 5 an d  T a re join tly  equivalent  20A xiom  Con s tr ain t  K ( φ→ ψ) →( φ →  ψ) Counterpairing shave u niver sal domains:CR→ ∀x∃yRx y D  φ→ ^φTher e  i s  at least onecount erpairing: ∃RCR T  φ→ φThe iden tity rela t i on is acou nte rpa iri n g :∀x∀y(Rxy↔ x = y )→ C RNI x =y→x = yCou nte rpa iri ng s are fun ctional:  ( CR  ∧  R xy∧ Rxz ) → y = zN OXx→   XxND x, y→ x,  y Co unte rp ai r ing s are o n e- o n e: ( CR  ∧  Rxz  ∧Ry z)→ x  = y N NO¬Xx→ ¬ Xx CBF ∀ xφ→ ∀xφ  A ut omaticfrom Analysis  F  CBF2 ∀Xφ →∀X φ BF ∀ x  φ→ ∀xφEvery counterpairing ca n  be ext end ed so  tha t any giv enobject  is in its  range : CR→∀x∃ S ∃ y(CS∧ ∀z 1∀z 2( Rz1z2 → Sz 1 z 2)∧ Syx )BF2∀Xφ→  ∀Xφ Counterpairing s  ha ve universalran ges: C R→  ∀y ∃ xR x y B  φ → 
∃R(CR ∧_∃S(CS ∧_FS(R)(S(a))))'. One rigorous way_to achieve_this_would be_to_follow Russell in_taking sentences involving_functors as abbreviations of_quantified sentences in_a_functor-free language, and stipulate that the Russellian translations are to be applied in such_a_way that_the_quantifier_introduced by each complex term_takes the widest possible scope._There are_many other formal schemas which have been thought_to_be logically valid_for some or all readings of 'necessarily' and 'possibly'._Many of these correspond naturally to_constraints on counterpairings._Several_of_these schemas, including those_we have already considered, are listed_in Table 1, together with the_constraints on counterpairings that correspond to them._Some remarks on these results: (i)_The axioms listed stand in_the following_logical relations: (i) T entails_D; (ii) Given_K and_D, 5 and_T are jointly equivalent 20 Axiom_Constraint K (φ→_ψ)→ (φ→ ψ) Counterpairings have universal_domains:_CR→ ∀x∃yRxy D_φ→_^φ_There is_at least one_counterpairing:_∃RCR T_φ→_φ The identity relation is a_counterpairing:_∀x∀y(Rxy↔ x = y)→ CR NI x_= y→ x =_y_Counterpairings are functional: (CR_∧ Rxy ∧ Rxz)→ y_= zNO Xx→ Xx ND x,_y→ x,_y Counterpairings_are one-one: (CR ∧ Rxz ∧ Ryz)→ x = yNNO ¬Xx→_¬Xx CBF ∀xφ→ ∀xφ Automatic from_Analysis F CBF2 ∀Xφ→_∀Xφ BF_∀xφ→_∀xφ Every counterpairing_can_be extended_so that any given object is in_its range:_CR → ∀x∃S∃y(CS ∧ ∀z1∀z2(Rz1z2 →_Sz1z2) ∧ Syx) BF2_∀Xφ→_∀Xφ Counterpairings have universal ranges: CR→_∀y∃xRxy B φ→