Understanding Definability in First-Order Logic

CEU eTD Collection uevsr rfso li´ an eirRsac elwa Alfréd Rényi at Fellow Research Senior Ildikó Sain, Professor Supervisor: nprilfllmn fterqieet o h ereo atrof Master of degree the for requirements the of fulfillment partial In nesadn enblt in Definability Understanding eateto ahmtc n Applications and Mathematics of Department is-re Logic First-Order Joël Ravelomanantsoa-Ratsimihah eta uoenUniversity European Central nttt fMathematics of Institute uaet Hungary Budapest, umte to Submitted Science 2010 by CEU eTD Collection ilorpy45 36 24 16 2 Bibliography Logic 5 Hybrid Quantified in property Beth 5 Logic Modal Quantified in property Beth 4 Logic Order First in property Beth 3 terminology and notation General 2 Introduction 1 Contents of Table . icsin...... 43 ...... 36 ...... 24 . 39 ...... QHL 16 Discussion in . . theorems . . definability 5.3 Beth’s . . . and . . interoplation . Craig’s . . 8 . . . . 5.2 . . Logic . Hybrid . Quantified ...... 5.1 ...... 5 . . . quantified . . for . property . . . Beth . . of . . Failure ...... 4.2 . logic . . . modal . . Quantified ...... 4.1 ...... theorem . definability . . Beth . . . . 3.1 ...... logic . First-order . 2.2 functions and relations Sets, 2.1 1 S5B 28 ...... CEU eTD Collection e srcl rmaytxbo ntesbet(f ..[2,[0,[1,[] httealgebraic the that [2]) [11], [10], [12], e.g. (cf. subject version the on textbook any from of recall stone us corner let a is definability that explain Pigozzi Don [13]. in than implicitly more but 1905, in work, in (already Einstein’s factor [13] in basic book already a his is in definability Reichenbach, that Hans scientific explains that his 1920), remarkable in from is coming it motivations line have this well In might experience. he that indicates science of ology formulated theory. he definability a [15] about paper bringing the of In project the 1930’s. started the and in already subjects favourite Tarski’s Alfred in e.g. research, of fields theory Definability Introduction 1 Chapter nte oreo oiaincnb h ineigppr[] hr ilmBo and Blok Willem where [6], paper pioneering the be can motivation of source Another method- the in and sciences in research did Tarski logic, exploring before that, fact The of one was definability [14]), and [1] (e.g. works various in out pointed been has it As Cs ( M famodel a of ) sabac fmdlter hc a aiu plctosi several in applications various has which theory model of branch a is hoeia physics theoretical M ffis re oi sa ler h nvreo hc sthe is which of universe the algebra an is logic order first of , hoeia optrscience computer theoretical 2 eaiiytheory relativity leri logic algebraic hsie appears idea This . oilsrt this, illustrate To . , leri logic algebraic . CEU eTD Collection that property some whether determine of meaning the is all of collection r formula 1.1. Definition definitions. following the symbol predicate new a adding where by expressed often also is This of symbols all two interpreting are there that case the 1.2. Definition ( x 1 h trigpito enblt hoyi h olwn.Gvnater,hwto how theory, a Given following. the is theory definability of point starting The x , . . . , L enberelations definable T safis-re agaeand language first-order a is 0 ϕ seatylike exactly is ( n x r aifideatyb h same the by exactly satisfied are ) 1 x , . . . , esyta h theory the that say We esyta h theory the that say We ϕ n ,T T, in of ) T M | = 0 T ftemdl where model; the of L | = . ϕ ∀ xetta n curneof occurrence any that except uhta neeymdlof model every in that such M x ∀ L 1 L x := . . . 1 ntesm a u nepeigtesymbol the interpreting but way same the in 0 mdl nwhich in -models . . . r ups lothat also Suppose . { ∀ r k x ∀ n sdfial ntrso eti te oin?Suppose notions? other certain of terms in definable is x ∈ ϕ ( L n r M ω ( 0 ( r M x stefis-re agaeta egtfrom get we that language first-order the is : ( 1 x ϕ x , . . . , 1 uhthat such T T x , . . . , formula a 3 defines defines n tpe ( -tuples n T ) n ) ↔ od,hvn h aeeeet and elements same the having holds, ↔ M T r

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( V ( seuvln to equivalent is ) ⊆ t 1 = x 13 snecstu in true -sentences N esythat say We and ) 1 Fml ) = ) t M k , . . . , n , . . . , t ( ⊆ frua and -formula, M t U ( t M t ), V 0 ∀ ( eavcblr n let and vocabulary a be M x etovcblre n let and vocabularies two be ( ,w suethat assume we ), M . ϕ x sa is 1 n o vr symbol every for and ) n . . . ∈ ]or )] sa is M M t Fml ∀ mdl entc that notice we -model, x t and reduct hc sdntdby denoted is which , n M M ϕ t M t ϕ ( ter respectively. -theory ( V , N x n etne formula, sentence, Any . ϕ ,and ), 1 x , . . . , ϕ ϕ of are sa is ( ( k k N ( ( lmnayequiva- elementary x x T t otevocabulary the to x 1 1 snec.I the In -sentence. n ) ) 1 sa is ) k , . . . , k , . . . , x , . . . , . M s t -theory. ∈ and ( ( n Dom( x M x N n n r all are )for )) ) If )). N ea be t t be . ), CEU eTD Collection h olwn odtosaesatisfied: are conditions following the 2.13 Definition tion 2.11 Definition em 2.12. 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Lemma i)if (ii) (ii) (i) (i) if T T T ω < n a N T 1 a oe fadol feeyfiiesbe of subset finite every if only and if model a has a oe fadol if only and if model a has a , . . . , ϕ ∃ o vr formula every for , ϕ fadol if only and if yϕ M then n Trk-agtcriterion) (Tarski-Vaught If ( cmltns theorem) (completeness cmatestheorem) (compactness 4 a U ∈ ddcintheorem) (deduction eeetr submodel) (elementary 1 M a , . . . , N U N ( if , 4 M N ϕ ehave we ) ϕ M n T T ( o some for y , then a ⊆ { ∪ 1 ` ) a , . . . , N hnteeis there then ϕ σ M ϕ } n o every for and ( x T ≡ n 1 U x , . . . , ϕ fadol if only and if ) scnitn,and consistent, is N . nt ustof subset finite fadol if only and if . . . Let . . Let e eater n let and theory a be T Let esythat say We Let n b 15 y , T U ∈ T ) M ω < n eatheory, a be eater n let and theory a be n o all for and ( M ⊆ M ) T N o vr formula every for , T uhthat such M Then . . 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Let ϕ and ϕ θ uhthat such ea be ψ . t 2 T ter,rsetvl.W say We respectively. -theory, t and 1 snec and -sentence ϕ U are θ and inseparable θ t 0 ϕ ψ := ψ ea be . t ψ 1 fno if ∩ and t t 2 2 - . CEU eTD Collection ∀ Since θ and Proof. 3.2.1. Claim ntelnug of language the in t o all For theories of sequences increasing two construct will We respectively. ϕ of interpolant no is there that 1 x a h form the has ¬ ∧ ∪ 1 .()if (a) 3. if (a) 2. 1. 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Claim ω ω {¬ ∪ { ∪ • • • Let (3), For Given T satisfied. n 3)(ep 2)ad(b)aestse,and T satisfied, are (3b)) and (2b) (resp. (3a) and T T ω T ϕ i i i Let Let T i +1 +1 +1 { ∪ ϕ i } ω i } T = = = = ϕ i σ ¬∀ i c i 6 and T T S θ } ea arbitrary an be T θ and x i i i h theories The h theories The i<ω 0 ω and and { ∪ ( (resp. { ∪ and x eabtaybtfie.Tetheories The fixed. but arbitrary be U T ϕ d = i ϕ , U i U i r hsnsc htte i o cu in occur not did they that such chosen are } i U T and ω x ω σ , U i ω (resp. and ) +1 r o neaal.Hne hr xssa exists there Hence, inseparable. not are ( i +1 c U ) ¬ and ¬ } ω = θ θ T T yasmlragmn,teei a is there argument, similar a By . (resp. = 0 U t U i ω yteddcinterm ehave we theorem, deduction the By . U 1 0 U i i and snec.W att hwta either that show to want We -sentence. i S +1 and T i ftecniini 2)(ep 2) sntsatisfied, not is (2b)) (resp. (2a) in condition the if ) T +1 i i<ω ω = σ and ∀ U r osrce nteovoswy ete aethe have then We way. obvious the in constructed are U U x U U 6∈ i ( i ω ϕ +1 x i i r ossetfrevery for consistent are i U ic every Since . { ∪ T r maximal. are → = i ω = . ψ and x θ U 18 i ,as ), } i and T fol h odto n(a rs.(b)is (2b)) (resp. (2a) in condition the only if ) { ∪ ω ¬ and σ U ψ ∀ x ω i 6∈ σ , ( T U x T i T ( i ω ¬ d and ω = and ) hnfrsome for Then . θ r inseparable. are } x fbt h odtosi (2a) in conditions the both if ) satuooy u hnwe then But tautology. a is ) U U i i r nt hoisfor theories finite are i r neaal.Therefore, inseparable. are 6 T t ω i t 0 0 , 0 0 -sentence . -sentence U i T σ , i ϕ ∈ sntconsistent. not is i ω < i T or ω θ or θ ψ 0 i uhthat such uhthat such nthat In . , ¬ σ σ ω < i ∈ = T ϕ ω i , . CEU eTD Collection or and ¬ ∃ symbol constant either Proof. 3.2.4. Claim U that follows It and x that li 3.2.5. 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x h 1 ( ,S ,n E, S, V, a x , n )) 2 ( = ) . ϕ ( σ x 1 x : x , . . . , i 1 S etwo be = . ∼ = x 2 T n ), ) , CEU eTD Collection that w 4.3. Lemma ∈ • • • W ρ ϕ Therefore ϕ exists ♦ that such ϕ 0 ( ( ( ψ ⊆ x x x h and ( S 1 1 1 σ x , . . . , x , . . . , x , . . . , h ρ w , ( S hr exists there , w a v 1 w , 0 i ) ∈ ∈ Let σ , . . . , i h σ n n n V W T r = ) = ) = ) ( v , : ups that Suppose . S a ∃ uhthat such S 1 i xψ a , . . . , ( r ∃ = ♦ w a ♦ ( xψ n 0 ψ x ( )implies )) ,x x, h 1 ( ∼ = σ ,R ,m D, R, W, ( x , . . . , x ψ ,x x, 1 n T containing 1 x , . . . , ( ) x , . . . , σ v h 1 0 ( S ⇒h ⇐⇒ ⇒h ⇐⇒ ⇒h ⇐⇒ ⇒h ⇐⇒ h ⇐⇒ yidcinhypothesis, induction By . x , . . . , a n w , 1 for ) ) ρ σ , . . . , n h 0 n .Asm that Assume ). S i : i ) ,a w, a σ T σ v, S n w , n r 1 and ( .W have We ). ⇒h ⇐⇒ ⇒h ⇐⇒ h ⇐⇒ v , w ρ ψ a , . . . , a i 0 1 ( ( ( i 1 ∼ = ayrlto ybl ehave We symbol. relation -ary ) a a x ♦ b nuto hypothesis] induction [by smnisof [semantics a , . . . , uhthat such σ , . . . , n 1 1 T 29 ) T ) ovrey since Conversely, )). x , . . . , r σ , . . . , T T S v ψ n ( = ups lota o vr finite every for that also Suppose . σ v , v , ∈ i ( w , a ( n h a i i ( 1 ∈ i ,S ,n E, S, V, i a a , . . . , 1 n r ) n ( .Since ). S σ , . . . , h a ) σ ψ ∃ S w ∈ i r n ψ xψ ( S ) : w , ( σ ∈ i ∃ ,a a, n ( S ( r x a ). i ,σ x, T ] ( ) i v r a 1 ♦ σ , ∼ = h σ T a , . . . , n T etwo be ( )) ( : a v , T a ψ S . 1 hn o n formula any for Then, . 1 ) 0 ( ) i σ , . . . , x σ , . . . , ∼ = σ b ento of definition [by n 1 b ento of definition [by − o some for ) x , . . . , T S5B 1 ψ hr exists there , : ( ( σ ( T a b definition] [by b definition] [by a srcue.Let -structures. ( [ n σ a n n ∼ = )) .Te there Then ). )) 1 : ) . σ , . . . , S S a w , ∈ ∼ = h T ρ D v S ¯ 0 ( T T v , 0 a w such ∈ v v n i ] ] ] )). V CEU eTD Collection Proof. 4.4. Theorem ϕ and Here ( Proof. x ( 1 x • • • 1 = . x , . . . , p ϕ and that such uhthat such exists ϕ ♦ implies ϕ x egv h oneeapei 9.Let [9]. in counterexample the give We ewl rv yidcino h opeiyof complexity the on induction by prove will We ( ( ( ψ sapooiin(ulr eainsmo)and symbol) relation (nullary proposition a is 2 x x x ), ( 1 1 1 ρ x , . . . , x , . . . , x , . . . , σ ϕ ( n a ) w h ge on agree ( = 1 h S ihfe variables free with ) 0 S ρ , . . . , w , ∈ ψ h quantified The w , σ σ n n n = ) = ) = ) i ∧ W : : i χ S S ♦ ( ,and ), r uhthat such ∃ ϕ a ♦ a w ( xψ n ( ∼ = 1 0 ψ x a ψ ) ovrey since Conversely, )). a , . . . , 1 ( 1 ∼ = ( ( x , . . . , x a , . . . , T a ϕ ,x x, ϕ 1 1 2 T x , . . . , ϕ a , . . . , 1 and def = v 1 def = = 0 n S5B x , . . . , yLma4.2, Lemma By . ehave we , ¬ n h x n ¬ p S .Sm ro si em 4.2. Lemma in as proof Same ). ) σ 1 p ψ n x , . . . , → n w , fadol if only and if → .Asm that Assume ). ). osnthv h ehproperty. Beth the have not does r rva.Frteohrcss ehave we cases, other the For trivial. are n ♦ 0 { i .Sm ro si em 4.2. Lemma in as proof Same ). ∀ a ∃ 1 x n a , . . . , x ( h rx ψ ( n o n tuple any for and 30 T rx ( v , a → ρ 1 T ∧ − 0 a , . . . , n i 1 } = h T : ( h ( = { p T ¬ v , T h ψ ϕ v , p S ¬ → r i n ( ρ ∼ = 1 ρ → .B h supin hr is there assumption, the By ). ϕ , w , 0 0 sa nr eainsymbol. relation unary an is hrfr,teeexists there Therefore, . i ( S a ϕ ϕ 2 rx i rx 1 } gi,tecases the Again, . , ( ) ♦ ρ ρ , . . . , )) )) h ψ eteter uhthat such theory the be ( h T a ( a . v , σ 1 1 ψ ) a , . . . , ( ρ , . . . , ( i a a ( 1 ♦ 1 a ) a , . . . , σ , . . . , n ) Hence, )). n ψ ( ∈ i a ( ρ n ( n )) ( a D .Te there Then ). a ¯ . 1 n n ) ) Since )). ϕ ρ , . . . , , ( x h 1 v T x , 0 v , ( ∈ 2 a = ) i n V )) σ ρ CEU eTD Collection etesto aua numbers natural of set the be Proof. 4.4.2. Claim have we h have We is that Proof. 4.4.1. Claim all that such w r n h mlctdfiaiiyof definability implicit the and ¯ S w a 0 ∈ w , nfc,w have we fact, In . w ∈ Since Since Hence, W D 0 ∈ ¯ i ecntutan construct We Let h W : S uhta o all for that such h a π h S h p w , , a S S ( h ∈ S w , a a S ¬ → h ∈ w , w , 0 ) ,π i, r ¯ ∈ i w , = 0 w 6= r ¯ h proposition The proposition The 0 i 0 w r ¯ i i i 0 ra h w a fadol if only and if i ϕ ,R ,m D, R, W, and :

∩ hs o all for Thus, . i i ϕ sfiie Let finite. is for ∃ r 0 = ϕ ¯ p 2 w x 1 , h h 0 , ( S hr exists there S fadol if only and if h i . rx , S h 1 = w , 1 w , a S5B S , w , ∧ and 2 ∈ w , i 0 i , i 0 r ¯ .Gvnayworld any Given 2. D 6 i ea be -structure 0 ¯ w ♦ ( h i p p p ¬ S if , N def = p W sntepiil enbein definable explicitly not is in definable implicitly is follows. a w , ( π permutation A . p ¬ ∀ → ∈ S5B w h p x sa setal nt emtto on permutation finite essentially an is mle htfrno for that implies p i S a eteset the be ( D ∈ rx mle htfrsome for that implies ¯ → rx ∈ w , if , ra srcueadlet and -structure W ) htmasfrall for means That )). D ¯ i ra S → . a : uhta ¯ that such ,i particular in ), 31 h = ∈ ra ,a w, ( r ¯ p h w then ,R ,m D, R, W, ∈ i ¬ → then , w of π m r h rx S w W on ( w r a ∩ w , )wihmasta hr exists there that means which )) ) e ¯ let , 6∈

T s¯ is D r ¯ h ¯ i h w S i . . r ¯ S r 0 w w scalled is w , T w fvocabulary of r w , = 0 w w hti ¯ is that , . ijitfo ¯ from disjoint ∈ 0 ∅ 0 ( i be ∈ i , p W W ¬ → ¯ , T ¬ hr exists there , r setal ﬁnite essentially p o some for w r ra w → sdson from disjoint is D ∩ ,i particular in ), ¯ ra

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m 34 ∗ -model. . ∧ ( c ϕ ), m ϕ ∗ S ) ∗ ∗ [ ( w r := ∗ ∗ := ) ← h ,R ,m D, R, W, M w m 0 ∗ ∗ ] ( , := r ), h W i ∪ sa is ,m D, ¯ w S5B ∗ with ∗ i srcueof -structure fsimilarity of w 0 ∗ ”. CEU eTD Collection oml oepes(..) ntenx eto,w ilcnie netnino QML, of extension an consider will holds. we property section, Beth that next such the enough expressive In (4.2.1). express to formula p fadol f(..)i aifid u since But satisfied. is (4.2.1) if only and if ntecutrxml ie ntepofo hoe .,framodel a for 4.4, Theorem of proof the in given counterexample the In 35 p sntepiil enbe hr sn modal no is there definable, explicitly not is h S w , 0 i , h S w , 0 i CEU eTD Collection hs e ybl r called vocabulary. are the in symbols already new symbols These propositional any from distinct symbols propositional operators. satisfaction and Nominals Logic Hybrid Quantified 5.1 Hybrid [3]. on structures. based is modal section of logic This expansions hybrid tools. are quantified four structures of these with language QML the of Let expansion the worlds. be over nominals, (QHL) variables and tools: world, four named uses a at logic hybrid QML, over the of quantify operators, nor vocabularies satisfaction what worlds the name about with cannot reason Starting one to logic, possible modal them. is In it which worlds. particular in at logics happens modal of extension are logics Hybrid Logic Hybrid Quantified in property Beth 5 Chapter ↓ bne onm olsadt setta oml strue is formula a that assert to and worlds name to -binder nominals hycnb oprdwt osat.While constants. with compared be can They . Let 36 NOM easto ulr eainsmosor symbols relation nullary of set a be CEU eTD Collection olwn e ye fformulas: of types variables, new existing following The already formulas. the are unlike variables Again, new worlds. those over range will variables new Those The ¯ where clause: following the state unique unique a exists there nominal a for impose We named world @ formula The satisfaction new the @ introduce also operators We formulas. are we However, nominals constants, worlds. unlike for that name notice are nominals universes, in individuals for name are constants • • • h • • • o aifcini oesfrfruaivligtestsato prtr @ operators satisfaction the involving formula for models in satisfaction For Let ↓ if if every if for -binder. n S S ϕ ϕ ϕ w , stedntto of denotation the is n safruaand formula a is and formula a is and formula a is := ∈ i α n w NOM n nee ynmnl.W hnhv w e ye fformulas: of types new two have then We nominals. by indexed h ∈ ,R ,m D, R, W, @ . n scle h eoainof denotation the called is WVAR ϕ Let n ϕ sra “at read is , [ k n WVAR ] w safruaand formula a is ⇒h ⇐⇒ saformula, a is ∈ def i W n α α n easto aibe ijitfo h aibe earayhave. already we variables the from disjoint variables of set a be eaqatfidmdlsrcue For structure. modal quantified a be ob nepee sasnltn hti o every for is that singleton, a as interpreted be to n ∈ ∈ ∈ uhthat such S n in NOM WVAR WVAR , , n ¯ ϕ S i n nutvl tmasta formula that means it intuitively and ” , hn@ then , w ϕ then , @ then , [ m ∈ ↓ k bne steaaoosof analogous the is -binder ], ( W n n = ) 37 in ↓ and n ϕ α .ϕ α. S ϕ { sas formula. a also is w . k safrua and formula, a is } saformula. a is valuation. a olwn h emnlg f[] the [3], of terminology the Following . n ∈ ∃ ete aethe have then We . NOM ϕ , od tthe at holds m n ( n n eadd we , ∈ ) NOM ⊆ W . , CEU eTD Collection auto.W aetefloigclauses: following the have We valuation. then of elements to valuation extend we for vocabulary the n expand to need first we that For where where namely, ˜ of Interpretation d osat˜ constant a add h • h • h • • • • • nodrt en rt namdlfrfrua novn h el nrdcdsymbols, introduced newly the involving formulas for model a in truth define to order In egv h diinlcassnee o tnadtasaino n oml nQHL. in formula any of translation standard a for needed clauses additional the give We α k ∈ ( (@ α n ( ↓ k M S S S ∗ ∗ α w WVAR α n .ϕ α. w , w , w , def ( = def ( = ) ϕ samdlmdl ysml nuto eaanhv o any for have again we induction simple By model. modal a is steasgmn hc iesfrom differs which assignment the is ∈ ) i i i ∗ w w ) W ∗ def ( = ∗ ∗ ↓ def @ α = and o,let Now, . ˜ = ˜ = .ϕ α. [ α ϕ k ϕ n ϕ n α n ] ∗ ∗ ,and ), [ [ ). n in [˜ k w [ ⇐⇒ ntecrepnigfis-re oe sdn nteovosway, obvious the in done is model first-order corresponding the in α k ] def ∈ 0 ∗ ] t ← ∗ ⇒h ⇐⇒ ← NOM ⇒h ⇐⇒ o aibe,if variables, For . def def w k S n ˜ ( ∗ ])[ a ], M := , = ) S w S ∗ k , h w , ,R ,m D, R, W, ← w ϕ ( α , i WVAR n ˜ ) ⇐⇒ i ], ϕ n ˜ [ M ϕ k hrfr,if Therefore, . α M w α [ ∗ k ] 38 i , = ∈ ,and ], ∗ eastructure, a be AR V W n k M ϕ nyi that in only t ∗ , ∗ [ w ihnwcntns o ahnominal each for constants: new with ∗ hnad˜ add then , ← α w ∈ 0 ∗ ] . SVAR w k w α ∈ ( α α W = ) n if and , svral in variable as , α w ∈ . t -sentence WVAR k savaluation a is and , t ∗ ϕ Now, . k a CEU eTD Collection em 5.1. Lemma following the need will we theorem interpolation fact. Craig’s prove interpo- To Craig’s making holds. by theorem property Beth’s lation for failure the repairs Logic Hybrid Quantified the- definability Beth’s and interoplation Craig’s 5.2 compactness completeness, the use theorems. can deduction we and interpolants), (like involved are formulas new omlssc oeo the of none such formulas n oso that show to Proof. α of vocabulary the in o oti the contain not structure a h h S ,R ,m D, R, W, i)if (ii) i 1 i if (i) . . . srpae by replaced is w , ycnieigtetasaino yrdfrua nofis-re ns sln sno as long as ones, first-order into formulas hybrid of translation the considering By o i) if (ii), For i ↓ o i,let (i), For α rm nQHL in orems ϕ ϕ θ l θ . ( →↓ n i → S ( a , 1 α 0 n , . . . , 1 θ Let ihnominals with 6 α α , . . . , ϕ ( w n ↓ n α 1 i →↓ . . . 1 i , ∈ Then, . n , . . . , n α ϕ h l 1 1 W S ) n , . . . , ↓ → . . . { l α → 0 .Asm that Assume ). α w , ϕ, 1 n valuation a and l θ . . . θ . l ϕ ↓ ( i ↓ ) n n then , then , ( α i l 1 α ↓ socrin occur ’s α 1 n , . . . , enmnl.Let nominals. be 1 l ϕ α . . . n θ . α , . . . , Therefore . 1 l θ . n , . . . , ( α ↓ ( ↓ 1 α α l ϕ α , . . . , n the and ) α 1 l l θ . →↓ α , . . . , ). 1 l h . . . ϕ ( S k α uhthat such Let . α 1 w , uhthat such l ↓ α , . . . , ) 1 39 h l . . . α i S ,ta sfraystructure any for is that ), → n l θ . θ 0 i w , ( ↓ uhta hyd o cu in occur not do they that such ϕ α ϕ ( ϕ l α α i ) 1 ecnepn h structure the expand can We . hnteeeit structure a exists there then , } n α , . . . , and 1 l θ . α , . . . , i S h n any and S 0 ( θ α ( := w , θ n 1 ( α , . . . , 1 l n i ) n , . . . , w l ) 1 be n , . . . , → for ↓ w θ α l ϕ ( ) 1 n l i and , . hc seuvln to equivalent is which ) ∈ . . . 1 1 = l n , . . . , ) W ↓ eqatfidhybrid quantified be l , . . . , , α S l h θ . l S ) : ( w , nwiheach which in Since . α h ,R ,m D, R, W, ϕ 1 i α , . . . , ewant We . 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Let ψ and ↓ l , . . . , θ T b if (b) b if (b) aen interpolant. no have α i o h aeo otaito,asm htteeeit a exists there that assume contradiction, of sake the For a h form the has ϕ 1 and i {¬ ω < i , . . . and n 0 hrfr,b em 5.1, lemma by Therefore, . ψ 0 ϕ T ψ T U ϕ ∈ ↓ } i i i i i hnw have we Then . i +1 { ∪ N α a h om@ form the has a h om@ form the has r inseparable. are a h om@ form the has and l 0 and , and . ϕ t ∀ 1 0 i x ψ } and θ 1 U i 0 and ω < i , ( . . . i c 1 { ∪ t c , . . . , ∀ 2 0 U epciey uhta o all for that such respectively, , x ψ i l θ i eeueain falclosed all of enumerations be r neaal then inseparable are } 0 ϕ n n ( l x ∃ ∃ n {¬ r neaal then inseparable are n , ♦ 1 xσ xσ { x , . . . , σ 1 ψ ϕ θ n , . . . , ( ( } } x x and and and ) and ) = = ϕ l U T α , ¬ ϕ 0 l 0 ↓ 0 ψ i ,where ), ⊆ 1 41 ⊆ +1 α ψ ϕ α , . . . , 1 i i T U +1 . . . ∈ ∈ ¬ 1 1 θ ⊆ ∈ U ⊆ T ↓ i requivalently or i ϕ l +1 T T 0 U α ) i hn@ then c i 2 ψ 2 l hn@ then i ∈ 0 . . . hn@ then . i . . . ∈ ∀ ∈ T ψ x i C +1 U 1 otaitn h atthat fact the contradicting , ω < i t 1 . . . i n and , snecsi n l closed all and in -sentences +1 n for ♦ n σ σ , n ∀ ( ( : x 0 c i c ) ∧ l ) t θ 1 = 0 0 θ ∈ 0 ∈ -sentence @ ( x U n T 1 n , . . . , 0 i ψ x , . . . , i σ +1 +1 emyassume may We . ∈ o some for o some for T , l i θ α , +1 n separating i 1 o some for α , . . . , ∈ c c N ∈ ∈ C for t C l 2 0 ϕ , ) - , CEU eTD Collection h expand can if one fact, In (4). with only worried in occur not x 5.3.4. Claim t 5.3.3. Claim that follows it theorem, Compactness the by li 5.3.2. Claim 1 S ) o n osatsymbol constant any For . ∈ 0 w , • U • Given o 3 n 4,teconstant the (4), and (3) For Since ehv h olwn liswoepof r xcl iei h rtodrcase. order first the in like exactly are proofs whose claims following the have We Let T e oa model translation modal standard a for adopted have we notation the use we 0 ω b if (b) M i ( T sn 3,w a hncntutafis-re submodel first-order a construct then can we (3), Using . 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Logic, Modal Quantified in expressed be cannot (4.2.1) condition The ee eseta aigtecretsaeo vlainusing evaluation of @ state with current the naming that see we Here, ∃ The language. evaluations. original semantic for relevant are of worlds reachable use introduce only can logic: One modal of required. locality is to quantifier order some in of of However, use use logic. the the hybrid language, in original nominals the of in use impossible. stay the is worlds by back naming achieved go logic, is can modal worlds we In Naming that quantifiers. and using objects by name language rely to original constants mainly the new we to introduce logic, can order we first that for fact theorem the interpolation on Craig’s of proof the In portant. Logic. agae uheitnilfruacnte ewtesdi h yrdlnug ythe by language hybrid the in witnessed @ be sentence then closed can formula existential Such language. constant new then one, x ∗ hrl,frtetheories the for Thirdly, that prove to Order order First in Secondly, of compactness and completeness the on property. heavily Beth relies achieve to proof needed the steps Firstly, the highlight to like would we Now uhthat such ↓ α bne ep hslclpoet fmdllgcadalw st obc othe to back go to us allows and logic modal of property local this keeps -binder nbe st xrs 421 ntelanguage. the in (4.2.1) express to us enables x ∀ ∗ oqatf vrwrd.Ufruaey h s uhqatfirwl oethe lose will quantifier such use the Unfortunately, worlds. over quantify to a ee oa niiulo ol.Tefis aei el yitouiga introducing by dealt is case first The world. a or individual an to refer can c ∃ h eodcs sdatb nrdcn e nominal new a introducing by dealt is case second The . x ∗ n σ ♦ ( x n ∗ 0 si h hoy fw rnlt oa oml noafirst-order a into formula modal a translate we If theory. the in is ) ∧ @ n 0 σ h T S . ω ∗ w , S and 0 i 421 fadol if only and if (4.2.1) { U ↓ ϕ ω ∗ } α. hr sawtesfrec xseta quantifier existential each for witness a is there , and ♦ ∀ 44 x {¬ ( rx ψ → } r neaal,the inseparable, are @ α ¬ rx ) . ↓ α n eern akt it to back referring and ↓ n bne a im- was -binder 0 ntehybrid the in CEU eTD Collection Bibliography 1 .Ad´k,J .Mdraz n .Németi, I. Madarász, and X. Andréka, J. H. [1] 4 .H ega,R .Mdu,adD .Pgzi(eds.), Pigozzi L. D. and Maddux, D. R. Bergman, H. C. [4] Marx, M. and Blackburn, P. Areces, C. [3] Sain, I. Németi, and Andréka, I. H. [2] 6 .J lkadD .Pigozzi, L. D. and Blok J. W. [6] Beth, E.W. [5] 7 ..CagadHJ Keisler, H.J. and Chang C.C. [7] Rényi Institute. Logic sorted rc e.A o.5,15,p.330–339. pp. 1953, 56, vol. A, Ser. Proc. science p. computer xi+292 1990, in Berlin, notes Verlag, lecture science. computer in algebra logic modal tified science) of 396 18) vi+78. (1989), tde nUieslLgc iküe BslBso–ttgr) 2008. Birkhäuser (Basel–Boston–Stuttgart), Logic, Universal in Studies , rpit(01,I ilso eaalbefo h oepg fthe of page home the from available be soon will It (2001), Preprint , nPdasmto nteter fdefinition of theory the in method Padoa’s On naso ueadApidLogic Applied and Pure of Annals , oe theory Model lerial logics Algebraizable nvra leri oi ddctdt h unity the to (dedicated logic algebraic Universal 45 earn h neplto hoe nquan- in theorem interpolation the Repairing ot oln,1990. Holland, North , enblt fNwUiessi Many- in Universes New of Definability eor mr ah Soc. Math. Amer. Memoirs , 124 leri oi n universal and logic Algebraic 20) o -,287–299. 1-3, no. (2003), eel kd Wetensch. Akad. Nederl. , o.45 Springer– 425, vol. , 77, CEU eTD Collection 1]A Tarski, A. [15] Sain, I. [14] Reichenbach, H. [13] Monk, D. J. [12] Németi, I. Andréka, and H. Tarski, A. Monk, D. J. Henkin, L. [11] Tarski, A. and Monk, D. J. Henkin, L. [10] [9] Fine, K. [8] an oedti:i eu hlspiu,Vl3,13,p.637–639. pp. 1934, Vol.37, Philosophique, Revue con- in version detail: more Polish tains original The 1986, Birkhäuser (Basel–Boston–Stuttgart), eds.), main) logic edition German original 1920. the in from Reichenbach published M. by Translated 1960, Press, ifornia http://www.jigpal.oupjournals.org. IGPL the of pp. vi+323 1981, Berlin, Springer-Verlag, 883, vol. Mathematics, in Notes Lecture 1985. and 1971 Amsterdam, Holland, Logic bolic logic philosophical n[](90,209–226. (1990), [4] In , olce aes oue – See .Gvn n ap .McKenzie, N. Ralph and Givant R. (Steven 1–4 Volumes Papers, Collected , , ehsadCagspoete i pmrhssadaagmto nalgebraic in amalgamation and epimorphisms via properties Craig’s and Beth’s alrso h neplto em nqatfidmdllogic modal quantified in lemma interpolation the of Failures oe hoyfrmdllgcpr TeD ed it distinction dicto re/de De IThe part logic modal for theory Model oemtoooia netgtoso h enblt fcnet i ger- (in concepts of definability the on investigations methodological Some 44 17) o ,201–206. 2, no. (1979), 8 h hoyo eaiiyadApir Knowledge priori A and Relativity of Theory The nItouto oClnrcStAlgebras Set Cylindric to Introduction An 7 Jl 00,n.4 4–9,Eetoial vial as: available Electronically 449–494, 4, no. 2000), (July 17) o ,125–156. 1, no. (1978), 46 yidi lersPrsIadII and I Parts Algebras Cylindric yidi e Algebras Set Cylindric nvriyo Cal- of University , oi Journal Logic , ora fSym- of Journal , ora of Journal , North– , ,