xy)'. Like Analysis E, Analysis F needs to be supplemented with further axioms if we want to recover a reasonable modal logic, validating inferences like S1 and S2. In both cases, we can recover as much of orthodox modal logic as we want by imposing such axioms (see section 9). From my point of view, Analysis F has one big advantage over Analysis E, namely the fact that it can very easily be extended to cover sentences in which plural terms and free variables occur in the scope of modal operators. All we need to do is to define 'RXY' to abbreviate the following conjunction: (i) for any x that is one of X, there is some y that is one of Y such that Rxy and (ii) for any y that is one of Y, there is some x that is one of X such that Rxy. We can then use Analysis F as it stands to analyse modal sentences involving plural terms: we simply allow the terms t1,..., tn to be plural as well as singular, understanding that the variables vi on the right-hand side should be made plural or singular according as the original terms ti were. 9 Constraints on counterpairings Since modal operators are context-dependent, the answer to the question what it is for some ordered pairs to be a counterpairing must vary from one context to another. But it is plausible that certain schematic principles involving modal operators are valid across all contexts; 17 these will correspond to constraints which all legitimate senses of 'counterpairing' must meet. The most central such principle is K: (K) (φ→ ψ)→ (φ→ ψ). This does not follow from Analysis F by itself. For example, '((Fa∧b = b) → Fa) → ((Fa ∧ b = b) → Fa)' is false if some counterpairings map a to objects that are not F, but none of these counterpairings have b in their domain. Given Analysis F, the most straightforward way to guarantee the truth of every instance of (K) is to lay it down as an axiom, true across all contexts, that counterpairings have universal domains: whenever R is a counterpairing, ∀x∃yRxy. It is easy to see that this validates (K). We don't, strictly speaking, need something quite this strong to validate (K). It would be enough,
xy)'. Like Analysis E, Analysis F needs to be supplemented with further axioms if we want to recuperate a fair modal logic, validating inferences like S1 and S2. In both lawsuit, we can reclaim as much of orthodox modal logic as we want by enforce such axioms (see part 9). From my point of view, Analysis F suffer one big advantage over Analysis E, namely the fact that it can very well be extended to cover sentences in which plural terms and free variables occur in the setting of modal operators. All we need to do is to specify' RXY' to abbreviate the following conjunction: (i) for any x that is one of X, there be some y that is one of Y such that Rxy and (ii) for any y that is one of Y, there is some x that is one of X such that Rxy. We can then use Analysis F as it stands to analyse modal prison term involving plural terms: we simply allow the terms t1, ..., tn to be plural as well as singular, understanding that the variables vi on the right - hand side should be made plural or singular according as the original terms ti were. 9 Constraints on counterpairings Since modal operator are context - dependent, the answer to the motion what it is for some order pairs to be a counterpairing must vary from one context to another. But it is plausible that certain conventional principles involve modal operators are valid across all contexts; 17 these will represent to constraints which all legitimate senses of' counterpairing' must meet. The most central such rationale is K: (K) (φ→ ψ)→ (φ→ ψ). This does not follow from Analysis F by itself. For example,' ((Fa∧b = b) → Fa) → ((Fa ∧ b = b) → Fa)' is false if some counterpairings map a to objects that are not F, but none of these counterpairings have b in their domain. establish Analysis F, the most straightforward way to guarantee the truth of every case of (K) is to lay it down as an axiom, true across all context, that counterpairings have universal domains: whenever R is a counterpairing, ∀x∃yRxy. It is easy to experience that this validates (K). We don't, rigorously speaking, need something quite this strong to validate (K). It would be enough,
xy)'. Pike Analysis E, Analysis F needs to be supplemxnted wjth furtfer axioms if we want to reclvwr a eeasonable modal logic, validatijg inferwncew like S1 anv S2. In both caacs, we ran recover as kuch of ordhodox modal lmgkc as we want by imposing such axioms (see sevtlon 9). From my pjint jf vjvw, Analysis F has one big advanfage ovtr Analysis E, nameky the fact that it can vegy ewsily be extended ho cover sebtensws in which olural terms and free bariables occur in the scope of modak operatorw. Qll fe need to vo is no define 'RXY' to abbraviate yhe following gonjuictiin: (i) for any x that iv one of X, there if some y dhct is one of Y such tyar Rxy and (ii) wir xny y tgat is onx of Y, thers is some x that is one of X sicr that Rxy. We dan thqn use Analysis F as it stands to analyse mosal sentences involving plural terms: we simppy allow ehe terms t1,..., tn to be plural as well as singular, utdersvavdiun gyah the variables vi on the right-hand side shoujs ne made plural ov singular accordimg ax the original terms ti were. 9 Constraints on cougterpqirings Synce modal operators are contexr-dependent, tke qnswer to the quescion what it is gor spme ordered pairs to be a ckunterpairijg must vzfy from one contdxt tm another. But it is plausifle that rertann schemxtic princyples invopving modal operators arf valnd acsoss all clntexts; 17 these will correspond vp constraintx fhibh all leyitimabe senses of 'cotnterpairing' most meet. Che mort central such pcinciple is H: (K) (φ→ ψ)→ (φ→ ψ). This gles not follmw from Wnaltsis F by igrelf. For exampke, '((Fa∧b = b) → Fa) → ((Fa ∧ v = b) → Fa)' is false lf soos counterpairinyr map a to objecys ghae wrx not X, but none ox thdse vountdrpairings mavd b on their domain. Givet Anzlysis F, the most xtvaightforqard way to guarantee the truth of everj insvance uf (K) os jo lay it down as an axiom, true across apl gontexts, that coukterkairings hare universal domains: whenever R is a couiterpairing, ∀x∃yRxy. It is easy to see that tkix validates (K). We don't, strhctly speaking, need womething quite tmis strong to validate (K). It fould be enough,
xy)'. Like Analysis E, Analysis F needs supplemented further axioms we want to validating like S1 and In both cases, can recover as much of orthodox logic as we want by imposing such axioms (see section 9). From my of view, Analysis F has one big advantage over Analysis E, namely the that can easily extended to cover sentences in which plural terms and free variables occur in the scope of operators. All we need to do is to 'RXY' to abbreviate the conjunction: (i) for any x is of X, is y is one of such that Rxy and (ii) for any y that is one of Y, there is some x is one such that We then Analysis F as to analyse modal sentences involving plural allow the terms t1,..., tn to be plural well as understanding that the variables vi on right-hand side should be made plural or singular as the original terms ti were. 9 Constraints on counterpairings Since modal operators are context-dependent, to the question what is for some pairs be counterpairing vary from context to another. But it is plausible that certain schematic principles modal operators are valid across all contexts; 17 these will constraints all legitimate senses 'counterpairing' must meet. The central principle is K: (K) (φ→ This from F itself. For example, '((Fa∧b b) → Fa) → ((Fa b = b) → counterpairings map a to objects that are not but none of these counterpairings have b their domain. Given Analysis F, the most straightforward way to guarantee the of every (K) is to lay it down as an true across all contexts, counterpairings have universal domains: whenever R is a counterpairing, It easy to that this validates We don't, strictly need something quite to validate It be
xy)'. Like Analysis E, Analysis F nEeds to be suPplemEntEd wItH furTher Axioms if we want TO recOver a reasonable modal loGic, vaLiDAtinG InFerenCes like s1 AnD s2. in bOtH cAseS, wE CaN recoVer As much oF orthodox mOdaL lOgic as we want BY iMposing sucH axIoms (see sectiOn 9). FRom my pOiNt oF View, ANalYsis F Has one BIg advaNtage over anALysis E, NAmely thE FAcT thaT it can very easily bE ExTEnded to cover seNtenceS iN WhICH plUraL terms and fReE variABles occUR iN THE scOPe of modal operAtors. All we nEEd tO do is tO dEfiNE 'RXY' to AbbreViATe tHe following ConjUnction: (i) fOr any x THat is onE Of X, therE is somE y tHat Is onE Of y sUch ThAT RxY AnD (ii) FOr aNy y that iS oNe Of Y, thEre iS SOME x thAt iS one Of X suCh that Rxy. We caN thEn usE anaLysis f as it StanDs To anaLyse moDal seNtEnces involving pLuraL terms: we sImpLy AllOw The teRMs t1,..., tn tO be PluRal as weLl as sinGUlaR, uNDERsTanding that the variAbLES vI on the riGht-hanD SiDe SHould be mAdE plUral OR SinguLar aCCoRding as tHe origINaL tErms ti wErE. 9 ConstRaIntS on CountERpaiRings SInce modaL operATors are context-DEpendent, the anSWeR TO tHE queStiOn what it is fOr soME ordEred PAiRs tO Be a coUnterPaIRiNG must vary from one conTeXt to anOther. but it is plausiBle that cerTAIN schematIc prINcIPles involving mOdal oPerators arE Valid acrOss alL contextS; 17 these wilL COrresponD to ConStrAinTS WhIch all legitimATE senSeS of 'counTerPairing' MusT meEt. THe mOsT central sUch princIpLe Is k: (K) (Φ→ ψ)→ (φ→ Ψ). This DOes not foLlOw fRoM AnAlysiS f by itsElf. FoR exaMpLe, '((fA∧b = b) → fa) → ((Fa ∧ b = b) → FA)' Is FALse iF sOmE couNteRpAirinGs maP A to Objects That are noT F, bUT nonE oF tHese couNterpairings hAvE b in their dOmAin. given ANALysis F, thE most straightforward way TO guaranTee The trUth oF every insTanCe of (K) iS to LAy it doWn as an Axiom, TrUe aCROss alL COnTexTs, That counteRPAirIngs hAvE uniVersal dOmains: whenever R is a COunTerpairing, ∀x∃yRXy. IT is eASY tO seE ThAT thIs VAliDATes (K). We don't, stricTly speakinG, nEEd Something qUIte ThIs stronG to valiDate (K). iT would bE enough,
xy)'. Like Analysis E, Ana lysis F ne eds t o b e s up plem ente d with further axio ms if we want to recov er are a sona b le  moda l logic ,  v a l ida ti ng  in fe r en ces l ike  S1 and  S2. In bo thca ses, we canr ec over as mu chof orthodoxmod al log ic  as we wa ntby im posing such a xioms (se es ection 9). Fro m my  poi nt of view, Analy s is F has one bigadvant ag e  o v e r A nal ysis E, na me ly th e  fact t h at i t  ca n  very easilybe extended tocoverse nte n ces in  whic hp lur al terms an d fr ee variab les oc c ur in t h e scope  of mo dal  op erat o rs .All  w e  ne e dtod o i s to def in e'RXY'  toa b b r evia tethefollo wing conjunct ion : (i )  fo r any  x th at i sone o f X, t hereis  some y that is  one  of Y suc h t ha t R xy  and( ii) fo r a nyy thatis oneo f Y ,t h e re  is some x that is  o n e  o f X such  thatR xy .W e can th en  us e An a l ysisF as it  standsto ana l ys emodal s en tences  i nvo lvi ng pl u ralterms:  we simp ly al l ow the terms t 1 ,..., tn to b e  p l u ra l  aswel l as singul ar,u nder stan d in g t h at th e var ia b le s  vi on the right-ha nd  sideshoul d be made plu ral or sin g u l ar accor ding as the original t ermsti were. 9 Constrai nts o n counte rpairings S ince mod alope rat ors a re  context-depe n d ent,  t he answ erto theque sti onwha tit is for  some or de re dpa irs  to b e  a count er pai ri ngmustv ary fr om on e co nt ex t  to  anothe r .B u t it  i splau sib le  that  cer t ain  schema tic princ ipl e s in vo lv ing mod al operatorsar e valid ac ro ssall co n t exts; 17  these will correspondt o const rai nts w hich  all legi tim ate se nse s  of 'c ounter pairi ng ' m u s t mee t .  T hemo st central s uch  prin ci pleis K: ( K) (φ→ ψ)→ (φ→  ψ ).This does not  fo llow f ro m A n al y sis  F byi t self. For examp le, '((Fa ∧b =b) → Fa) → (( Fa  ∧ b =b) → F a)' i s  falseif some c ounterpai ri ngsm a p a  to object s that a re not F, but n o ne  of t hes e coun te rpa iring s have b i n the ir dom ai n. Giv en An al ysis F,the most straightforwar d wayto gu ara ntee thetru t h o f every i nsta nce of (K)  is  to  layitd own a s an ax iom ,  true  acr o ss all co n te xts , th at counterp a i r ing s hav e u n iversa l do mains: whenever R is a counterpa irin g ,  ∀x ∃yR x y. I tis easy to see  th at t his vali da tes (K). We  don't,st r ictly  speak ing, n eed som e t hi n g quit e th isstrong to  va li d ate (K) .It wouldbe e no ugh,
xy)'. Like_Analysis E,_Analysis F needs to_be supplemented_with_further axioms_if_we want to_recover a reasonable_modal logic, validating inferences_like S1 and_S2._In both cases, we can recover as much of orthodox modal logic as we_want_by imposing_such_axioms_(see section 9). From my_point of view, Analysis F_has one_big advantage over Analysis E, namely the fact_that_it can very_easily be extended to cover sentences in which plural_terms and free variables occur in_the scope of_modal_operators._All we need to_do is to define 'RXY' to_abbreviate the following conjunction: (i) for_any x that is one of X,_there is some y that is_one of Y such that_Rxy and_(ii) for any y that_is one of_Y, there_is some x_that is one of X such_that Rxy. We_can then use Analysis F as_it_stands to analyse_modal_sentences_involving plural_terms: we simply_allow_the terms_t1,...,_tn to be plural as well_as_singular, understanding that the variables vi on_the right-hand side should_be_made plural or singular_according as the original terms_ti were. 9 Constraints on counterpairings_Since modal_operators are_context-dependent, the answer to the question what it is for some_ordered pairs to be a counterpairing_must vary from one_context to_another._But it is_plausible_that certain_schematic principles involving modal operators are valid_across all_contexts; 17 these will correspond to_constraints which all legitimate_senses_of 'counterpairing' must meet. The most_central such principle is K: (K)_(φ→ ψ)→ (φ→ ψ). This_does_not_follow from Analysis F by_itself. For example, '((Fa∧b = b)_→ Fa) →_((Fa ∧ b = b) → Fa)'_is_false if some counterpairings map a_to_objects that are not F, but_none_of_these counterpairings have b in_their domain. Given Analysis F, the_most straightforward way to guarantee the truth of every_instance of (K)_is to lay it down_as_an_axiom, true across all contexts, that counterpairings have universal domains:_whenever R_is a counterpairing,_∀x∃yRxy. It is easy to see that this validates (K)._We don't, strictly speaking, need something quite_this strong to validate (K). It would be enough,