A system of relational syllogistic incorporating full Boolean reasoning

A system of relational sy llogist ic incorp orating full Bo olean reasoning ∗ Nik ola y Iv ano v naivan ov@gm ail.com Dimiter V ak arelo v dvak@f mi.un i-sof ia.bg F aculty of Mathematics and Informatics, Sofia University 5 James Bourchier Blvd., 116 4 Sofia, Bulga ria Abstract W e present a s ystem of relational syllogistic, based on classical propo si- tional logic, having primitives o f the following form: Some a are R -r elated to s ome b ; Some a are R -related to all b ; All a are R -r elated to s ome b ; All a a re R -r elated to all b . Such pr imitives formalize s ent ences fro m natura l langua ge like ‘ All students read some textb o oks’. Here a, b denote arbitra ry sets (of ob jects), and R denotes an ar bitrary binary r elation b etw een o b jects. The la nguage of the logic contains only v ariables denoting sets, determining the class of set terms, and v a riables denoting binary relatio ns betw een o b jects, determining the cla ss of relational terms. Both classes of ter ms are closed under the sta ndard Bo olean op erations. The set of r elational ter ms is also c losed under taking the conv erse of a relation. The results of the pap er are the c ompleteness theorem with resp ect to the intended semantics and the computational complexity of the satisfiability proble m. 1 In tro duc tion It is a w ell-kno wn f act th at the syllogistic w as the first formal theory of logic in tro du ced in An tiquity by Aristotle. It was presente d by Luk asiewicz in [ 13 ] as a quantifier-free extension of prop ositional logic, ha ving as atoms the expr essions A ( a, b ) (All a are b ) and I ( a, b ) (Some a are b ) and their negations E ( a, b ) def ⇔ ¬ I ( a, b ) and O ( a, b ) def ⇔ ¬ A ( a, b ), where a, b are set (class) v ariables interpreted in the natural l anguage b y noun phrases lik e ‘men’, ‘Greeks’, ‘mortal’. An example of an Aristotelian syllogism tak en fr om [ 13 ] is: “If all men are m ortal and all Greeks are men, then all Greeks are mortal”. The sp ecific axioms for A and I from [ 13 ] are (in a differen t logical notation) the follo wing: L1. A ( a, a ), L2. I ( a, a ), L3. A ( b, c ) ∧ A ( a, b ) → A ( a, c ), L4. A ( b, c ) ∧ I ( b, a ) → I ( a, c ). Th e only r ules ∗ A previous version of th is pap er has b een pub lished as [ 10 ]. 1 are Mo d us Po nens and substitution of a set v ariable with another set v aria ble. The standard seman tics of this language consists of in terpreting set v aria bles by arbitrary non-empty sets, A ( a, b ) as set-inclusion a ⊆ b , and I ( a, b ) as the o v erlap relation b etw een sets: a ∩ b 6 = ∅ . W edb erg introdu ced in [ 34 ] v ariations of the Aristotelian syllogistic with th e op eration of complemen tation a ′ on set v ariables interpreted as the Bo olean com- plemen t of the v ariable in a giv en universe. W edb erg’s system w ith unr estricted in terpretation on s et v ariables is based on the follo wing axioms (con taining only A and complementati on b ecause I ( a, b ) can b e d efined by ¬ A ( a, b ′ )): W1. A ( a, a ′′ ), W2. A ( a ′′ , a ), W3. A ( a, b ) ∧ A ( b, c ) → A ( a, c ), W4. A ( a, b ) → A ( b ′ , a ′ ). W5. A ( a, a ′ ) → A ( a, b ). Simple Henkin-style completeness and decidabilit y pro ofs for Luk asiewicz’s, W edb erg’s and some other classical syllogistic systems were giv en by Shepherds on in [ 32 ]. S hepherds on’s completeness pro ofs are b ased on the notion of p artially ordered set S with an op eration of complemen tation ‘ ′ ’ s atisfying the follo w ing axioms for all a, b ∈ S : a ′′ = a , a ≤ b → b ′ ≤ a ′ , and a ≤ a ′ → a ≤ b . Similar structures are n o w kno wn as orthop osets (see [ 20 ]). S hepherdson also men tioned in [ 32 ] systems conta inin g not only complement ation on set terms, b ut also Bo olean in tersection. W e call the v ariations of Ar istotelia n sy llogistic , mentio ned ab o v e, classical syllogistics . All suc h logics are based on pr op ositional logic, but w eak er systems, whic h do not contai n the prop ositional connectiv es or contai n only negation, h a v e also b een considered in the literature. F or instance, Moss in [ 19 , 20 ], motiv ated mainly with applicatio ns of syllogistics to natural languages, considers v arious syl- logistics of classical t yp e, based on languages conta ining primitiv es like A, I , E , O with or without complemen tation on set v ariables. Th e corresp onding axiomatic systems are b ased on a num b er of inference rules with finite sets of atomic premises. F or a long time classical syllogistic has b een considered only in introdu ctory courses on elemen tary lo gic. No wada ys, h o w ev er, syllogistic t heories, extended and mo dified in v arious wa ys, fi nd applications in d ifferen t areas, mainly in natural language theory [ 15 , 19 , 20 , 21 , 22 , 26 , 27 , 28 , 29 , 30 , 31 , 33 ], compu ter science and artificia l int elligence [ 3 , 1 7 , 11 , 23 ], ge neralized quantifiers [ 35 ], argumen tation theory [ 24 ], cognitiv e ps yc hology [ 12 , 25 ] and others (the list of references is fairly incomplete). Most of th e extend ed syllogistics generalize the standard syllogistic relations A ( a, b ), I ( a, b ), E ( a, b ) and O ( a, b ) using in their defin itions v arious n on- standard quant ifiers arising from natural language. Examples: ‘A t least 5 a are b ’, ‘Exactly 5 a are b ’, ‘Most a are b ’, ‘All except 2 a are not b ’, ‘Many a are not b ’, ‘Only a few a are n ot b ’, ‘Usually some a are not b ’, etc. Some of th e relations b et ween s ets a and b are determined b y certain relations b et wee n their m em b ers , expressible b y some ve rb s or v erb phrases in the natur al language. Examples: ‘All stud en ts read s ome textb o oks’, ‘Some p eople don’t lik e an y cat’, ‘Some v egetarians e a t some fish ’, ‘All veg etarians don’t lik e any meat’, ‘A t lea st 5 students read all textb o oks’, etc. Sy llogistic s studying suc h expressions are called by Moss an d Pratt-Hartmann [ 31 , 28 ] relational syllogistics . Aristotelian syllogistic and most of its extensions can b e considered as logics whic h fit the structure of natural language. Th eir p rimitiv es lik e Al l A ar e B , 2 Some A ar e B , Most A ar e B etc, can b e considered as relations b etw een classes (sets of ob j ects), and in th is sense syllogistic theories can b e treated as certain sp ecial theories of cla sses. On the other han d suc h primitiv es express kinds of quan tification stu died in th e theory of generalized quantifiers [ 4 , 35 ]. C om bining some f eatures from generalized quan tifier theory an d syllogistic reasoning, a new trend in logic has b een dev elop ed in recen t ye ars, called natur al lo gic , or lo gic f or natur al language with the aim to study logical form alisms whic h fi t w ell with the structure of natural language (see, for in stance, [ 16 ] and [ 6 ] for other r eferences). In this p ap er we introdu ce a quite ric h s ystem of relational syllogistic combining some seman tical ideas fr om the aforementioned pap ers on relational syllogistics and s ome tec hnical ideas from [ 1 , 2 ]. The language of the logic is similar to the language of Dynamic Logic and con tains b oth set v ariables and r elational v ariables from whic h w e construct complex terms. Both classes of terms are c losed with resp ect to all Bo olean op erations wh ile on relational terms w e also hav e the op eration ‘ − 1 ’ of taking the con v erse. W e ha v e fiv e atomic predicates from whic h we construct the set of f orm ulas using the prop ositional connectiv es: a ≤ b , ∃∃ ( a, b )[ α ], ∀∃ ( a, b )[ α ], ∃∀ ( a, b )[ α ], ∀∀ ( a, b )[ α ]. Here a, b are set terms and α is a r elational term. Th e semant ical structur es are the same as in Dynamic logic ( W , R , v ), where R is a mapping f rom relational v ariables to the set of b inary relations on W and v is a mappin g which assigns to eac h s et v ariable a sub set of W . T he seman tics of ∀∃ ( a, b )[ α ] is the follo wing: ( W , R , v )  ∀∃ ( a, b )[ α ] iff  ∀ x ∈ v ( a )  ∃ y ∈ v ( b )  xR ( α ) y  . The seman tics of the r emaining atomic f orm ulas is analogous. Linguistically these form ulas co ve r the examp les like ‘All studen ts r ead s ome textb o oks’, taking all com binations of ‘some’ and ‘all’, consid ering sub ject wide scop e reading. Ha ving the op eration α − 1 , we ma y also express in our language the ob ject wide scop e reading (see [ 21 ] for more details). By means of the Bo olean op erations on rela- tional terms w e ma y express “comp ound v erbs” like ‘to read bu t not to write’. Also by ‘ − 1 ’ we ma y express the passive v oice of the v erbs lik e ‘is r ead’. Similarly b y means of Boolean op erators on set terms w e may express comp ound noun s. Let u s note that the signs ∃∀ in ∃∀ ( a, b )[ α ], and sim ilarly in the other pr imitiv es, are not quantifiers on set or relational v ariables, but part of th e notation of our primitiv e sen tences. W e choose this notation j ust b ecause it corresp onds d irectly to the seman tics of these p rimitiv es and in this wa y helps the reader to catc h m ore easily their meanin g. W e present a Hilb ert-st yle axiomatic system for the logic based on the axioms of p rop ositional logi c, Mo d us P onens and several ad ditional finitary inference rules satisfying some syn tactic restrictions. The list of axioms conta ins the finite list of axiom sc hemes for Bo olean algebra p lus a fin ite list of axiom sc hemes for the basic predicates. In this sen se our logic is a quan tifier-free fi rst-order system, based on prop ositional logic. W e will not treat in this pap er our pr imitiv e relations as generalized quantifiers. Logics with simila r rules, whic h in a sense imitate quan tification, and c anonical constructions for corresp ond ing completeness proofs are studied in [ 1 , 2 ]. W e adopt and modify these canonical tec h niques. There a re, h o w eve r, n ew difficulties, whic h 3 ha v e no analogs in [ 1 , 2 ]. That is why w e need to com bin e canonical constructions from [ 1 , 2 ] with a mo d ification of a copying construction from [ 5 , 7 , 8 ]. T he form ulas of our logic ha ve a translation int o Bo olean Mo dal Logic (BML) [ 5 , 7 ] extended with con verse on relational terms. W e obtain that the complexit y of the satisfiabilit y problem for the logic is the s ame as the complexit y of BML [ 14 ], i.e. NExpTime if the language contai ns an infin ite num b er of relational v ariables, and ExpTim e if only a finite num b er of r elational v ariables is a v ailable. The present pap er is an extended v ersion of the firs t auth or’s master’s thesis [ 9 ] and w as inspired by [ 31 ], esp ecially b y the pr esen tation of [ 31 ] b y Moss as an in vited lecture at the C onference “Adv ances in Mod al Logic 2008” [ 18 ]. The pap er is organized as follo ws. In section 2 , we in tro d uce the language an d semantics of our logic. In section 3 , w e list the axioms and in ference rules of our logical system. W e use the axioms for the cont act relation from [ 1 ] and some add itional axioms and inference rules whic h essent ially im itate quan tifiers in our quan tifier-fr ee language. In section 4 , we pro ve the completeness of our axiomatic system. The pro of uses s ome id eas fr om the completeness p ro ofs for mo dal logics of the con tact relation [ 1 ] and BML [ 5 , 7 ]. In section 5 , w e discuss th e complexit y of the s atisfiabilit y problem for the logic und er consideration and some of its fragmen ts. 2 Syn tax and seman tics 2.1 Language The language consists of the follo wing sets of s ym b ols: (1) an infi nite set V S of set v ariables; (2) the set constan ts 0 and 1; (3) a n on-empt y set V R of relational v ariables such that V R ∩ V S = ∅ ; (4) relational constants 0 R and 1 R ; (5) fun ctional sym b ols ∩ , ∪ and − for the o p erations meet, jo in and complemen t; (6) fun ctional sym b ol − 1 ; (7) relational symb ols ≤ , ∃ ∃ , ∀ ∃ , ∀∀ , ∃∀ ; (8) prop ositional connectiv es ∧ , ∨ , ¬ , → , ↔ ; (9) prop ositional constan ts ⊥ and ⊤ ; (10) the symb ols ‘(’, ‘)’, ‘[’, ‘]’, ‘,’. As the language is u niquely determined b y the pair (V S , V R ), we will also call (V S , V R ) a language. In the first tw o sections we w ill keep the language fixed. Set terms are built from the set constan ts and set v ariables b y means of the Bo olean connectiv es ∩ , ∪ and − . If V ⊆ V S , we will denote by T Set ( V ) the set of all set terms w ith v ariables f rom V . W e define the s et of relational terms T Rel ( X ) with v ariables in X ⊆ V R to b e the smallest set suc h that: (1) X ∪ { 0 R , 1 R } ⊆ T Rel ( X ); (2) If α ∈ T Rel ( X ) then  − α, α − 1  ⊆ T Rel ( X ); 4 (3) If { α, β } ⊆ T Rel ( X ) then { α ∩ β , α ∪ β } ⊆ T Rel ( X ). A tomic formulas ha ve one of th e forms a ≤ b ∃∃ ( a, b )[ α ] ∀∃ ( a, b )[ α ] ∀∀ ( a, b )[ α ] ∃∀ ( a, b )[ α ] , where a and b are set terms and α is a relational term. F orm ulas are b uilt from atomic formulas by means of the prop ositional connectiv es. W e will abbreviate ( a ≤ b ) ∧ ( b ≤ a ) as a = b and its negation as a 6 = b . If V ⊆ V S and R ⊆ V R , we will den ote by F orm( V , R ) the set of all form ulas with set v ariables from the set V and relational v ariables from R . 2.2 Seman tics Let W b e a set a nd let R : V R → P ( W 2 ) and v : V S → P ( W ) b e t wo functions 1 . R is a v aluation of the r elational v ariables, whic h maps ev ery relational v ariable to a relation on W . The v aluation v of th e set v ariables map s set v ariables to subsets of W . W e will call the p air ( W , R ) a f rame and the triple ( W, R , v ) a mo del. The set W is called th e domain of that frame or mo d el. W e extend the function R to the set of all relational terms b y defining R (0 R ) = ∅ and R (1 R ) = W 2 and interpreting the sym b ols ∩ , ∪ , − and − 1 b y inte rsection, union, complement in W 2 and taking the con verse of the relations on W . W e extend the fu nction v to the set of all set terms analogously . If M is a mo del and ϕ is a formula, w e will d enote the statemen t that ϕ is tr ue in M by M  ϕ . W e d efine the tru th and falsit y of atomic form ulas in a mo del ( W , R , v ) by the follo win g equiv alences: ( W , R , v )  a ≤ b ⇔ v ( a ) ⊆ v ( b ) ( W , R , v )  ∃∃ ( a, b )[ α ] ⇔  ∃ x ∈ v ( a )  ∃ y ∈ v ( b )  ( x, y ) ∈ R ( α )  ( W , R , v )  ∀∃ ( a, b )[ α ] ⇔  ∀ x ∈ v ( a )  ∃ y ∈ v ( b )  ( x, y ) ∈ R ( α )  ( W , R , v )  ∀∀ ( a, b )[ α ] ⇔  ∀ x ∈ v ( a )  ∀ y ∈ v ( b )  ( x, y ) ∈ R ( α )  ( W , R , v )  ∃∀ ( a, b )[ α ] ⇔  ∃ x ∈ v ( a )  ∀ y ∈ v ( b )  ( x, y ) ∈ R ( α )  . The definition is extend ed to the set of all formulas according to the standard meaning of the prop ositional connectiv es. 2.3 Relations with natural language seman tics Linguistically the relational v ariables are interpreted as transitiv e verbs, and the set v ariables – as coun t-nouns. The formulas a ≤ b and a ∩ b 6 = ∅ mean ‘Every a is a b ’ and ‘Some a is a b ’ r esp ectiv ely . T o illustrate the m eaning of the symb ols Q 1 Q 2 , let us interpret a as ‘man’, b as ‘animal’, and α as th e verb ‘to lik e’. W e denote the sub ject wid e scop e r eading and the ob ject wide scop e r eading of a sen tence . . . by (. . . ) sw s and (. . . ) ow s resp ectiv ely . 2 Then we ha ve the follo wing meanings: 1 W e denote by P ( X ) the p ow er set of the set X . 2 If the tw o readings are equiva lent, we omit the annotation. 5 ∃∃ ( a, b )[ α ] me ans S ome man like s some animal ∀∀ ( a, b )[ α ] me ans Every man lik es ev ery animal ∀∃ ( a, b )[ α ] me ans (Every man lik es some animal) sw s ∃∀ ( a, b )[ α ] me ans (Some man lik es ev ery animal) sw s . T o express the ob ject w ide scop e reading, we n eed the symbol − 1 whic h con- v erts a v erb into p assiv e vo ice. In our example α − 1 means ‘to b e liked’: ∀∃ ( b, a )[ α − 1 ] me ans (Some man lik es ev ery anim al) ow s ∃∀ ( b, a )[ α − 1 ] me ans (Ev ery m an lik es some animal) ow s . Bo olean connectiv es in set terms form alize negated nou ns and the connectiv es ‘and’ and ‘or’ b et w een nouns . The presence of Bo olean op erators in relational terms allo w s us to formalize natural language sentence s, whic h cont ain negated v erbs, as w ell as comp oun d predicates, such as ‘sees and h ears’ ( see ∩ hear ) and ‘sees, but is n ot seen’ ( see ∩ ( − see − 1 )). 3 Axioms and inference rules W e will use the f ollo win g notation: I f A is a formula or a term, then V Set ( A ) denotes the set of set v ariables w hic h occur in A . Also, V Set ( A 1 , . . . , A n ) = S n i =1 V Set ( A i ). The idea b ehind the list of axioms is the follo wing. S ince ∃∃ is the cont act relation from the mo dal logics of region-based theories of space [ 1 ], we use the same set of axioms for it. Th e truth of eac h of the other three relations Q 1 Q 2 is link ed to the truth of ∃ ∃ by the follo wing equiv alences: ( W , R , v )  ∀∃ ( a, b )[ α ] ⇔  ∀ p ⊆ W   v ( a ) ∩ p = ∅ ∨  ∃ x ∈ p  ∃ y ∈ v ( b )  ( x, y ) ∈ R ( α )   ( W , R , v )  ∀∀ ( a, b )[ α ] ⇔  ∀ p ⊆ W   v ( b ) ∩ p = ∅ ∨  ∀ x ∈ v ( a )  ∃ y ∈ p  ( x, y ) ∈ R ( α )   ( W , R , v )  ¬∃∀ ( a, b )[ α ] ⇔  ∀ p ⊆ W   v ( a ) ∩ p = ∅ ∨ ¬  ∀ x ∈ p  ∀ y ∈ v ( b )  ( x, y ) ∈ R ( α )   These equiv alences express the follo wing simple statemen t. If ϕ ( x ) is a prop- ert y of elemen ts x in some set W and A ⊆ W , then ( ∀ x ∈ A ) ϕ ( x ) is equiv alen t to ( ∀ X ⊆ W )  X ∩ A 6 = ∅ ⇒ ( ∃ x ∈ X ) ϕ ( x )  . Th us , w e exp ressed the un iv ersally quant ified prop erty ( ∀ x ∈ A ) ϕ ( x ) by the existen tially quan tified p rop erty ( ∃ x ∈ X ) ϕ ( x ) and a quan tification ov er sets. Substituting the appr opriate formulas in th e place of ϕ ( x ), we get the ab ov e equiv alences. The left-to-righ t direction of eac h of th ese equiv alences is a universal form ula. W e add it to the set of axioms. Th ese are the axioms ( AL 1 ), ( AL 2 ), ( AL 3 ) in the list b elo w. W e call th em linkin g axioms, b ecause th ey link r elation sym b ols Q 1 Q 2 and Q ′ 1 Q ′ 2 , which differ in the fir st or second quant ifier. The righ t-to-left directions of the equiv alences are not univ ersal form ulas. Since w e do not ha v e quan tifiers in our language, we cannot wr ite th ese conditions as 6 axioms. Instead, we imitate th em by inference r ules with a sp ecial v ariable, corre- sp ond ing to the quanti fied v ariable p in the ab o ve equ iv alences, using a tec hn ique from [ 1 ]. These are the r ules ( R 1 ), ( R 2 ), ( R 3 ) from the list b elo w. W e call them linking rules. W e w ill also use a rule wh ose only p urp ose is to deriv e all formulas of the form a 6 = 0 → ∃∃ ( a, a )  ( α − 1 1 ∪ − α 1 ) ∩ ( α − 1 2 ∪ − α 2 ) ∩ · · · ∩ ( α − 1 k ∪ − α k )  . These formulas state that the v aluation of an y relational term of the form α − 1 ∪ ( − α ) m ust b e reflexiv e. The fact that they are theorems is prov ed in Lemma 4.10 and is used in Prop osition 4.15 . The set of axioms consists of th e f ollo win g group s of formulas: (1) A s ound and complete set of axiom schemes for prop ositional calculus; (2) A s et of axioms for Bo olean algebra in terms of the relation ≤ ; (3) Axioms f or equalit y: Q 1 Q 2 ( a, b )[ α ] ∧ a = c → Q 1 Q 2 ( c, b )[ α ] ( A = 1 ) Q 1 Q 2 ( a, b )[ α ] ∧ b = c → Q 1 Q 2 ( a, c )[ α ] ( A = 2 ) (4) Axioms f or ∃∃ : a = 0 ∨ b = 0 → ¬∃ ∃ ( a, b )[ α ] ( A 0) ∃∃ ( a ∪ b, c )[ α ] ↔ ∃∃ ( a, c )[ α ] ∨ ∃∃ ( b, c )[ α ] ( A ∪ 1 ) ∃∃ ( a, b ∪ c )[ α ] ↔ ∃∃ ( a, b )[ α ] ∨ ∃ ∃ ( a, c )[ α ] ( A ∪ 2 ) (5) Linking axioms: ∀∃ ( a, b )[ α ] → a ∩ c = 0 ∨ ∃∃ ( c, b )[ α ] ( AL 1 ) ∀∀ ( a, b )[ α ] → b ∩ c = 0 ∨ ∀∃ ( a, c )[ α ] ( AL 2 ) ¬∃∀ ( a, b )[ α ] → a ∩ c = 0 ∨ ¬ ∀∀ ( c, b )[ α ] ( AL 3 ) (6) Axioms f or 0 R and 1 R : ¬∃∃ ( a, b )[0 R ] ( A 0 R ) ∀∀ ( a, b )[1 R ] ( A 1 R ) (7) Axioms f or ∩ , ∪ , − and − 1 in relational terms: ∀∀ ( a, b )[ α ∩ β ] ↔ ∀∀ ( a, b )[ α ] ∧ ∀∀ ( a, b )[ β ] ( A ∩ ) ∃∃ ( a, b )[ α ∪ β ] ↔ ∃∃ ( a, b )[ α ] ∨ ∃∃ ( a, b )[ β ] ( A ∪ ) ∀∀ ( a, b )[ − α ] ↔ ¬ ∃∃ ( a, b )[ α ] ( A − ) ∃∃ ( a, b )[ α − 1 ] ↔ ∃∃ ( b, a )[ α ] ( A − 1 ) Inference rules: 7 (1) ϕ, ϕ → ψ ⊢ ψ ( M P ) (2) Sp ecial rules imitating quan tifiers: If p ∈ V S \ V Set ( ϕ, a, b ) then ϕ → a ∩ p = 0 ∨ ∃∃ ( p, b )[ α ] ⊢ ϕ → ∀∃ ( a, b )[ α ] ( R 1) ϕ → b ∩ p = 0 ∨ ∀∃ ( a, p )[ α ] ⊢ ϕ → ∀∀ ( a, b )[ α ] ( R 2) ϕ → a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ α ] ⊢ ϕ → ¬ ∃∀ ( a, b )[ α ] ( R 3) a ∩ p = 0 ∨ ∃∃ ( p, p )[ α ] ⊢ a = 0 ∨ ∃∃ ( a, a )  α ∩ ( β − 1 ∪ − β )  ( RS ) The v ariable p is called the sp ecial v ariable of the r ule. The n otions of pro of and th eorem are d efined in the stand ard wa y . W e will denote by Thm(V S , V R ) the set of all theorems in the language (V S , V R ). Prop osition 3.1. Al l the or ems ar e true in al l mo dels. Pr o of. All axioms are true in all m o dels a nd the rule o f MP preserv es truth in ea ch mo del. Eac h of the sp ecial ru les preserve s v alidit y in eac h frame, that is: if the premise is true in all v aluations on a giv en frame, then so is the conclusion. T o illustrate the pro of sys tem, we will show a p ro of of the form ula ∃∀ ( a, b )[ α ] → ∀∃ ( b, a )[ α − 1 ] . Let p, q ∈ V S , p 6 = q and { p, q } ∩ V Set ( a, b ) = ∅ . ⊢¬∀∀ ( p, b )[ α ] ∨ b ∩ q = 0 ∨ ∀∃ ( p, q )[ α ] b y ( AL 2 ) ⊢¬∀∀ ( p, b )[ α ] ∨ b ∩ q = 0 ∨ p ∩ a = 0 ∨ ∃∃ ( a, q )[ α ] b y ( AL 1 ) ⊢ a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ α ] ∨ b ∩ q = 0 ∨ ∃∃ ( q , a )[ α − 1 ] b y ( A − 1 ) ⊢¬∃∀ ( a, b )[ α ] ∨ ∀ ∃ ( b, a )[ α − 1 ] b y ( R 3 ) and ( R 1 ) 4 Completeness 4.1 Plan of t he c ompleteness pro of First w e review the definition of theories and the construction of maximal theo- ries fr om consisten t sets of formulas in the presence of sp ecial rules of inference, whic h imitate quanti fiers (for d etails, see [ 1 ]). W e do not ha v e b ound v ariables in form ulas, b ut w e will thin k of some of the v ariables as b eing b ound by u niv ersal quan tifiers. Th at is why w e define a theory as a s et of formulas together with a set of u n b ound v ariables. The set of f orm ulas will not b e closed under arb itrary applications of the sp ecial r ules, but only und er applications of instances of these rules, in wh ic h the s p ecial v ariable is among the u niv ersally b ound v ariables. T o build a mo del of a consisten t set of form ulas, we first need to extend it in to a maximal thery . W e require that suc h theories conta in f or eac h formula exactly one of th e f orm ula itself or its n egation, b ut w e also require an analog of Henkin’s condition – if the th eory con tains the negation of the conclusion of s ome 8 instance of a s p ecial ru le (wh ic h is existen tial), it s hould also cont ain a negation of the premise of that rule (for some sp ecial v ariable, w hic h ma y b e thought of as a witness for th at existen tial form ula). Our construction of the canonical mo del is b ased on the Stone repr esen tation theorem for Bo olean algebras. It b uilds the p oints in the mo d el as ultrafilters in the Bo olean algebra of set terms. This giv es us the correct interpretatio n of the Bo olean op er ators on set terms without fur ther effort. The pr oblem is that we do not obtain automatically the in tended int erpr etation of the Bo olean op erators on relational terms. W e explain ho w we deal with this prob lem in su bsection 4.4 , after we introdu ce th e n ecessary notation. 4.2 Theories Definition 1. Let Γ ⊆ F orm(V S , V R ) and ϕ ∈ F orm(V S , V R ). W e will write Γ ⊢ 0 ϕ w hen there is a pro of of ϕ f rom Γ, w hic h do es n ot use the sp ecial r ules (that is, a proof using only ( M P )). Γ is called consisten t if Γ ∪ Thm (V S , V R ) 0 0 ⊥ . Definition 2 (Th eory) . Let Γ ⊆ F orm(V S , V R ) and let V ⊆ V S . W e sa y that the pair ( V , Γ) is a theory in the language (V S , V R ) when the f ollo win g conditions hold: (1) Thm(V S , V R ) ⊆ Γ ; (2) If ϕ, ϕ → ψ ∈ Γ then ψ ∈ Γ; (3) Let P ( q ) b e a pr emise of a linking ru le, where q ∈ V S is the sp ecial v ariable of the rule. Let C b e the conclusion of th at rule, q ∈ V S \  V ∪ V Set ( C )  and P ( q ) ∈ Γ. Then C ∈ Γ. W e sa y that the theory ( V , Γ) is consisten t if ⊥ / ∈ Γ. W e sa y that the theory ( V , Γ) in the language (V S , V R ) is a go o d theory if | V | < | V S | . The theory ( V , Γ) is called complete if it is consisten t and f or eac h f orm ula ϕ in its language we h a v e either ϕ ∈ Γ or ¬ ϕ ∈ Γ. The theory ( V , Γ) in th e language (V S , V R ) is called ric h if f or eac h linking ru le with premise P ( q ) ∈ F orm(V S , V R ) and conclusion C the follo win g implication holds: C / ∈ Γ ⇒ ( ∃ q ∈ V S )  P ( q ) / ∈ Γ  . (The conclusion C un iquely determines P ( q ) up to a substitution of q with another set v ariable.) Lemma 4.1. F or every c onsistent set of formulas Γ 0 ther e exists a c onsistent the ory T = ( V , Γ) with Γ ⊇ Γ 0 . Pr o of. Let T =  V S ,  ϕ ∈ F orm(V S , V R )   Γ 0 ∪ T hm(V S , V R ) ⊢ 0 ϕ  . Notation. W e define a relation ⊆ b etw een theories in the same language: ( V 1 , Γ 1 ) ⊆ ( V 2 , Γ 2 ) def ⇔ V 1 ⊆ V 2 ∧ Γ 1 ⊆ Γ 2 . W e will wr ite ϕ ∈ ( V , Γ) if ϕ ∈ Γ. W e fix a language (V S , V R ) and introd uce the follo wing notation: 9 Notation. If Γ is a set of form ulas and ϕ is a formula, Γ + ϕ def =  ψ ∈ F orm(V S , V R )   ϕ → ψ ∈ Γ  . If T = ( V , Γ ) is a theory and ϕ is a form ula, T ⊕ ϕ def =  V ∪ V Set ( ϕ ) , Γ + ϕ  . Lemma 4.2. If T = ( V , Γ) is a go o d the ory and ϕ is a formula then: (1) T ⊕ ϕ is a go o d the ory, T ⊆ T ⊕ ϕ and ϕ ∈ T ⊕ ϕ ; (2) T ⊕ ϕ is inc onsistent ⇔ ¬ ϕ ∈ Γ ; (3) If P ( q ) and C ar e the pr emise and c onclusion of a linking rule and the the ory T ⊕ ¬ C is c onsistent, then ther e is a set variable q ∈ V S \  V ∪ V Set ( C )  , such that T ⊕ ¬ C ⊕ ¬ P ( q )  is a go o d c onsistent the ory. Pr o of. Straigh tforw ard verificatio n. Lemma 4.3 (Lind en baum) . Every go o d c onsistent the ory T 0 = ( V 0 , Γ 0 ) i n a lan- guage (V S , V R ) with | V R | ≤ | V S | is c ontaine d in a c omplete rich the ory T = ( V , Γ) . Pr o of. Let T 0 = ( V 0 , Γ 0 ) b e a goo d consisten t theory . Let κ = | V S | and let F orm(V S , V R ) = { ϕ α | α < κ } . W e will b uild a sequen ce of theories { T α } α<κ with the follo w ing prop erties: (1) T α is a go o d consistent theory; (2) ¬ ϕ α ∈ T α or ϕ α ∈ T α +1 ; (3) If ϕ α ∈ T α +1 , ϕ α = ¬ C and C is the conclusion of a lin king rule, th en there is a s et v ariable q , suc h that th e negated pr emise of the rule ¬ P ( q ) b elongs to T α +1 . Supp ose that T β ha v e b een defined for β < α . W e will define T α . W e consider the follo w ing cases: (1) α = β + 1 for some β and T β = ( V β , Γ β ) has already b een d efined. W e need to consider t wo p ossib ilities for the theory T β ⊕ ϕ β : (a) T β ⊕ ϕ β is consistent . W e hav e t wo cases dep end ing on ϕ β : i. ϕ β do es not ha ve the form of a negat ed conclusion of a linkin g rule. In this case we define T α = T β ⊕ ϕ β . ii. ϕ β = ¬ C and C is a conclusion of a linking rule. L et P ( q ) b e the premise of that rule. Acc ordin g to Lemma 4.2 there is a s et v ariable q ∈ V S \  V β ∪ V Set ( ϕ β )  suc h that T β ⊕ ϕ β ⊕ ¬ P ( q ) is a go o d consistent theory . W e c ho ose such a v ariable q and define T α = T β ⊕ ϕ β ⊕ ¬ P ( q ). (b) T β ⊕ ϕ β is inconsisten t. Then Lemma 4.2 tells us that ¬ ϕ β ∈ Γ β . W e define T α = T β . 10 (2) α = S α . W e define V α = S { V β | β < α } , Γ α = S { Γ β | β < α } and T α = ( V α , Γ α ). It is easy to verify the three prop erties of T α stated ab o v e b y induction on α . W e defin e V = S { V α | α < κ } , Γ = S { Γ α | α < κ } and T = ( V , Γ). By the prop erties of T α for α < κ it easily follo ws that T is a complete r ic h th eory . The Lind en baum lemma is only applicable to go o d theories. That is w h y we will also need th e follo wing lemma: Lemma 4.4. L et T 0 = ( V , Γ 0 ) b e a c onsistent the ory in a language (V S 0 , V R ) and let V S ⊇ V S 0 with | V S | > | V S 0 | b e an extension of V S 0 with a set V S \ V S 0 of new set variables. Then ther e is a go o d c onsistent the ory T = (V S 0 , Γ) in the language (V S , V R ) such that Γ 0 ⊆ Γ . Pr o of. Define Γ =  ϕ ∈ F orm (V S , V R )   ( ∃ ψ ∈ Γ 0 )  ψ → ϕ ∈ Thm(V S , V R )  . It is straightfo rward to chec k that T = (V S 0 , Γ) h as the desired prop erties. Corollary 4.5. (1) Every c onsistent set of f ormulas is c ontaine d in a go o d c on- sistent the ory i n an extension of the language with a set of new set variables. (2) Every c onsistent set of formulas is c ontaine d in a c omplete rich the ory in an extension of the language with a set of new set variables. A complete rich th eory is also called a maximal theory . 4.3 Bo olean algebras of classes of terms Let S b e a maximal theory . W e will asso ciate w ith S some equiv alence relations in T Set (V S ) and T Rel (V R ) and will sho w th at the equiv alence classes form Bo olean algebras with resp ect to some naturally defined op erations. 4.3.1 The Bo olean algebra of classes of set terms W e will asso ciate with S a Bo olean algebra of classes of set terms. W e define th e relations 4 and ≈ on T Set (V S ): a 4 b def ⇔ a ≤ b ∈ S a ≈ b def ⇔ ( a 4 b ∧ b 4 a ) . The r elation ≈ is an equiv alence relation. W e denote b y [ a ] the equiv alence class of a . W e d enote b y Cl S the set of all equiv alence classes. W e d efine a relation ≤ on Cl S : [ a ] ≤ [ b ] def ⇔ a 4 b . W e define the op erations ∩ , ∪ and − on Cl S : [ a ] ∩ [ b ] def = [ a ∩ b ] [ a ] ∪ [ b ] def = [ a ∪ b ] − [ a ] def = [ − a ] . The r elation ≤ and the op erations ∩ , ∪ and − are w ell-defined. The six-tuple  Cl S , ∩ , ∪ , − , [0] , [1]  is a Bo olean algebra. 11 4.3.2 The Bo olean algebra of classes of relational terms W e define the relations 4 and ≈ on the set of all relational terms: α 4 β def ⇔  ∀ a, b ∈ T Set (V S )  ∃∃ ( a, b )[ α ] → ∃∃ ( a, b )[ β ] ∈ S  α ≈ β def ⇔ ( α 4 β ∧ β 4 α ) . The intuitio n b ehind this definition is that in every mo d el ( W , R, v ) of S the follo w ing implication must hold for arbitrary relational terms α and β : α 4 β ⇒ R ( α ) ⊆ R ( β ). The relation ≈ is an equiv alence relation. W e d enote by [ α ] the equiv alence class o f α . W e den ote b y Cl R the set o f all equiv alence classes. W e define a relation ≤ on Cl R : [ α ] ≤ [ β ] def ⇔ α 4 β . W e defin e the op erations ∩ , ∪ , − and − 1 on Cl R : [ α ] ∩ [ β ] def = [ α ∩ β ] [ α ] ∪ [ β ] def = [ α ∪ β ] − [ α ] def = [ − α ] [ α ] − 1 def = [ α − 1 ] . Prop osition 4.6. The r elation ≤ and the op e r ations ∩ , ∪ , − and − 1 on Cl R ar e wel l-define d. The six-tuple (Cl R , ∩ , ∪ , − , [0 R ] , [1 R ]) is a Bo ole an algebr a and for arbitr ary r elational terms α and β we have the e quiv alenc e α 4 β ⇔ α ∪ β ≈ β . Pr o of. See App end ix A . Lemma 4.7. [1] ≤ [0] ⇔ [1 R ] ≤ [0 R ] . Pr o of. ( → ) Let 1 = 0 ∈ S and a, b ∈ T Set (V S ). Then a = 0 ∈ S and b = 0 ∈ S . By ( A 0 ), ¬∃ ∃ ( a, b )[1 R ] ∈ S , and hen ce ∃∃ ( a, b )[1 R ] → ∃∃ ( a, b )[0 R ] ∈ S . ( ← ) Assume that [1 R ] ≤ [0 R ]. Then ∃∃ (1 , 1)[1 R ] → ∃∃ (1 , 1)[0 R ] ∈ S . By ( A 0 R ), ¬∃∃ (1 , 1)[0 R ] ∈ S , and hence ¬∃∃ (1 , 1)[1 R ] ∈ S . By ( A 1 R ), ∀∀ (1 , 1)[1 R ] ∈ S , hence ∀ ∀ (1 , 1)[1 R ] ∧ ¬ ∃∃ (1 , 1)[1 R ] ∈ S . Using ( AL 1 ) and ( AL 2 ), we conclude that 1 = 0 ∈ S . 4.3.3 The Bo olean algebra of symmetric classes of relational terms W e define an op eration − 1 on P (Cl R ): F or eac h V ⊆ Cl R V − 1 def =  [ α ] − 1   [ α ] ∈ V  . Lemma 4.8. (1) If α ∈ T Rel (V R ) then  α − 1  − 1 ≈ α . (2) If x ∈ Cl R then  x − 1  − 1 = x . If V ⊆ Cl R then  V − 1  − 1 = V . (3) L et V ⊆ Cl R . If V is a filter, then so is V − 1 . If V is an ultr afilter, then so is V − 1 . 12 W e will call x ∈ Cl R symmetric if x = x − 1 . Similarly , we will call V ⊆ Cl R symmetric if V = V − 1 . Lemma 4.9. The set of symmetric classes of r elational terms is a Bo ole an sub- algebr a of (Cl R , ∩ , ∪ , − , [0 R ] , [1 R ]) . Lemma 4.10. If a is a set term and α 1 , α 2 , . . . , α k ar e r elational terms, then the formula a = 0 ∨ ∃ ∃ ( a, a )  ( α − 1 1 ∪ − α 1 ) ∩ ( α − 1 2 ∪ − α 2 ) ∩ · · · ∩ ( α − 1 k ∪ − α k )  is a the or em. Pr o of. Let p 1 , p 2 , . . . , p k b e differen t s et v ariables, whic h do not o ccur in a . By ( A 1 R ), ( AL 2 ) and ( AL 1 ), we ha ve ⊢ p 1 ∩ · · · ∩ p k ∩ a = 0 ∨ ∃∃ ( p 1 ∩ · · · ∩ p k ∩ a, p 1 ∩ · · · ∩ p k ∩ a )[1 R ] ⊢ p 1 ∩ · · · ∩ p k ∩ a = 0 ∨ ∃∃ ( p 1 , p 1 )[1 R ] b y item 1 in Lemma A.3 ⊢ p 2 ∩ · · · ∩ p k ∩ a = 0 ∨ ∃ ∃ ( p 2 ∩ · · · ∩ p k ∩ a, p 2 ∩ · · · ∩ p k ∩ a )  α − 1 1 ∪ − α 1  b y ( RS ) Similarly w e obtain ⊢ p 2 ∩ · · · ∩ p k ∩ a = 0 ∨ ∃∃ ( p 2 , p 2 )  α − 1 1 ∪ − α 1  ⊢ p 3 ∩ · · · ∩ p k ∩ a = 0 ∨ ∃ ∃ ( p 3 ∩ · · · ∩ p k ∩ a, p 3 ∩ · · · ∩ p k ∩ a )  ( α − 1 1 ∪ − α 1 ) ∩ ( α − 1 2 ∪ − α 2 )  Con tinuing in the same wa y , w e arriv e at ⊢ a = 0 ∨ ∃ ∃ ( a, a )  ( α − 1 1 ∪ − α 1 ) ∩ ( α − 1 2 ∪ − α 2 ) ∩ · · · ∩ ( α − 1 k ∪ − α k )  . 4.4 Canonical construction Let S b e a maximal theory . W e will p ro v e that S h as a mo d el. W e denote by M ∅ the mo del  ∅ , V R × {∅} , V S × {∅}  . Lemma 4.11. If S is a maximal the ory and 1 = 0 ∈ S , then M ∅  S . Pr o of. As S is a maximal theory , it suffices to prov e the equiv alence M ∅  ϕ ⇔ ϕ ∈ S for atomic formulas ϕ . (1) Clearly , all formulas in the form of a ≤ b b elong to S and are true in M ∅ . (2) ϕ is ∃ ∃ ( a, b )[ α ]. Then M ∅ 2 ϕ . By ( A 0 ) ϕ / ∈ S . (3) ϕ is ∀∃ ( a, b )[ α ]. Then M ∅  ϕ . F or the sak e of con tradiction sup p ose that ϕ / ∈ S . There exists a set v ariable p for wh ic h a ∩ p = 0 ∨ ∃ ∃ ( p, b )[ α ] / ∈ S . This is a con tradiction, since a ∩ p = 0 ∈ S . 13 (4) ϕ is ∀ ∀ ( a, b )[ α ] or ∃∀ ( a, b )[ α ]. F ollo ws from the ab o v e an d Lemma A.1 . W e will n o w consider the case when 1 = 0 / ∈ S . W e denote by Ult S the set of u ltrafilters of the Bo olean algebra  Cl S , ∩ , ∪ , − , [0] , [1]  . Similarly , we d enote by Ult R the set of u ltrafilters of the Bo olean algebra (Cl R , ∩ , ∪ , − , [0 R ] , [1 R ]) . Since [1]  [0], the set Ult S is non-empt y . By Lemma 4.7 , we h a v e also Ult R 6 = ∅ . If Q is a qu an tifier and F ( a ) is a statemen t ab out set terms, su c h that a ≈ b implies F ( a ) ⇔ F ( b ), w e will u se  Q [ a ] ∈ Cl S  F ( a ) as an abbreviation for ( Qx ∈ Cl S )( ∃ a ∈ x ) F ( a ). W e will also use a sim ilar notation f or statement s ab out relational terms. Notation. If a ∈ T Set (V S ), we den ote by [ a ) =  x ∈ Cl S   [ a ] ≤ x  the smallest filter con taining [ a ]. S imilarly , if α ∈ T Rel (V R ), we den ote b y [ α ) =  x ∈ Cl R   [ α ] ≤ x  the smallest fi lter con taining [ α ]. W e will explain the id eas whic h lead us to the defi nition of the canonical mo d el of S . W e ma y attempt to d efine the mo del as M 0 = ( W 0 , R 0 , v 0 ), where: (1) W 0 = Ult S ; (2) F or eac h relational term α let R 0 ( α ) = n ( U 1 , U 2 ) ∈ Ult S 2     ∀ [ a 1 ] ∈ U 1  ∀ [ a 2 ] ∈ U 2  ∃∃ ( a 1 , a 2 )[ α ] ∈ S  o ; (3) F or eac h set v ariable p let v 0 ( p ) = { x ∈ W 0 | [ p ] ∈ x } . Then, for arb itrary set terms a, b and an arbitrary relational term α we hav e: a ≤ b ∈ S ⇔ v 0 ( a ) ⊆ v 0 ( b ) Q 1 Q 2 ( a, b )[ α ] ∈ S ⇔  Q 1 x ∈ v ( a )  Q 2 y ∈ v ( b )  ( x, y ) ∈ R 0 ( α )  . There is, ho we ver, a problem with this mo del. The function R 0 ma y not follo w the correct seman tics of the Bo olean op erators – we d o not necessarily h a v e R 0 ( α ∩ β ) = R 0 ( α ) ∩ R 0 ( β ) and R 0 ( α ) ∩ R 0 ( − α ) = ∅ . T o bu ild the canonical mo del, we first define a rela tion R 0 V on W 0 for e ac h relational ultrafilter V , such that the v aluation of ea ch r elational term α in M 0 will b e a union of such r elations. F or eac h V ⊆ Cl R w e d efine a relation R 0 V ⊆ P (Cl S ) 2 : R 0 V = n ( U 1 , U 2 ) ∈ P (Cl S ) 2     ∀ [ α ] ∈ V  ∀ [ a 1 ] ∈ U 1  ∀ [ a 2 ] ∈ U 2  ∃∃ ( a 1 , a 2 )[ α ] ∈ S  o . 14 No w for eac h α ∈ T Rel (V R ) we hav e: R 0 ( α ) = W 2 0 ∩ [  R 0 V   V ∈ Ult R ∧ [ α ] ∈ V  . W e should rep lace R 0 V with another relation R V defined for eac h V ∈ Ult R , s uc h that R V ′ ∩ R V ′′ = ∅ for different V ′ , V ′′ ∈ Ult R . The u niv erse of our mo del will consist of a num b er of copies of W 0 . If ( x, y ) ∈ R 0 V ′ ∩ R 0 V ′′ for different V ′ , V ′′ ∈ Ult R , we will hav e some copies x ′ , x ′′ of x and some copies y ′ , y ′′ of y , suc h that ( x ′ , y ′ ) ∈ R V ′ and ( x ′′ , y ′′ ) ∈ R V ′′ . Notation. Let F 1 and F 2 b e filters in the Bo olean algebra of Cl S and let G b e a filter in the Boolean algebra of C l R . W e will use the follo wing notation: I F 1 ,F 2 = n [ α ] ∈ Cl R     ∃ [ a 1 ] ∈ F 1  ∃ [ a 2 ] ∈ F 2  ∃∃ ( a 1 , a 2 )[ α ] / ∈ S  o I G,F 2 = n [ a 1 ] ∈ Cl S     ∃ [ α ] ∈ G  ∃ [ a 2 ] ∈ F 2  ∃∃ ( a 1 , a 2 )[ α ] / ∈ S  o I F 1 ,G = n [ a 2 ] ∈ Cl S     ∃ [ a 1 ] ∈ F 1  ∃ [ α ] ∈ G  ∃∃ ( a 1 , a 2 )[ α ] / ∈ S  o It is easy to c hec k that the I ’s are ideals in the resp ectiv e Bo olean algebras. Lemma 4.12. L et F 1 and F 2 b e filters in the Bo ole an algebr a of Cl S and let G b e a filter in the Bo ole an algebr a of Cl R . If ( F 1 , F 2 ) ∈ R 0 G , then ther e ar e U 1 , U 2 ∈ Ult S and V ∈ Ult R such that F 1 ⊆ U 1 , F 2 ⊆ U 2 , G ⊆ V and ( U 1 , U 2 ) ∈ R 0 V . Pr o of. W e use the equ iv alences ( F 1 , F 2 ) ∈ R 0 G ⇔ F 1 ∩ I G,F 2 = ∅ ⇔ G ∩ I F 1 ,F 2 = ∅ ⇔ F 2 ∩ I F 1 ,G = ∅ and apply the separation theorem for filter-ideal pairs in Bo olean algebras three times. Let us first exclude the symbol − 1 from the language. T o constru ct the relations R V , w e n eed the follo wing lemma: Lemma 4.13. L et ( U 1 , U 2 ) ∈ Ult S 2 . Then: (1) ( U 1 , U 2 ) ∈ R 0 [1 R ) . (2) Ther e is a V ∈ Ult R such that ( U 1 , U 2 ) ∈ R 0 V . Pr o of. (1) Supp ose this is not tru e. Since S is a complete theory ,  ∃ [ a 1 ] ∈ U 1  ∃ [ a 2 ] ∈ U 2  ¬∃∃ ( a 1 , a 2 )[1 R ] ∈ S  . By the a xiom for 1 R , ∀∀ ( a 1 , a 2 )[1 R ] ∈ S . Using the linking axio ms, w e deriv e a 1 = 0 ∨ a 2 = 0 ∈ S . Since S is a complete theory , a 1 = 0 ∈ S or a 2 = 0 ∈ S , hence [ a 1 ] = [0] or [ a 2 ] = [0]. Th is is a contradict ion, as U 1 and U 2 are ultrafilters. Thus, ( U 1 , U 2 ) ∈ R 0 [1 R ) . (2) By the pr evious item, ( U 1 , U 2 ) ∈ R 0 [1 R ) . By Lemma 4.12 , there is a V ∈ Ult R suc h that ( U 1 , U 2 ) ∈ R 0 V . 15 F or eac h pair ( U 1 , U 2 ) ∈ Ult S 2 w e c ho ose one V ∈ Ult R , su c h that ( U 1 , U 2 ) ∈ R 0 V , and denote it b y V U 1 ,U 2 . The canonical mo del M = ( W , R , v ) corresp ond ing to S is defin ed as follo ws: The d omain is W = Ult S × Ult R . If x ∈ W , we d enote by x 1 and x 2 its fi rst and second comp onent resp ectiv ely . F or eac h p ∈ V S w e d efine v ( p ) =  x ∈ W   [ p ] ∈ x 1  . It is easy to c hec k that for all set terms a we hav e v ( a ) =  x ∈ W   [ a ] ∈ x 1  . F or eac h V ∈ Ult R w e d efine a relation R V ⊆ W 2 : R V = n ( x, y ) ∈ W 2     ( x 1 , y 1 ) ∈ R 0 y 2 ∧ V = y 2  ∨  ( x 1 , y 1 ) / ∈ R 0 y 2 ∧ V = V x 1 ,y 1  o . That is, if there should b e a p air  ( x 1 , ) , ( y 1 , )  3 in R V , we put all pairs  ( x 1 , ) , ( y 1 , V )  there; if R V should not conta in a p air  ( x 1 , ) , ( y 1 , )  , we put all pairs  ( x 1 , ) , ( y 1 , V )  in R V x 1 ,y 1 . This simple construction suffices to pro v e the completeness of the pro of system without ( A − 1 ) and ( RS ) for the language with out − 1 . When we include th e symb ol − 1 , ho we ver, we need something more sophisti- cated. T he pr oblem is that w e should ensu re that R V 1 ∩ R V 2 = ∅ for V 1 6 = V 2 while at the same time preserving the p rop erty stated in the f ollo win g lemma: Lemma 4.14. If V ⊆ Cl R , then R 0 V − 1 =  R 0 V  − 1 . The decision where to pu t p airs of p oints should not b e made indep endently for ( x, y ) and ( y , x ). W e should ha ve ( x, y ) ∈ R V ⇔ ( y , x ) ∈ ( R V ) − 1 . T o this end , w e will replace the first disjunct in the ab o v e defin ition of R V with a cond ition, in wh ic h V do es not dep end solely on y 2 , b ut on a symmetric function of x 2 and y 2 . T o define su c h a function, w e num b er the elements of Ult R with ord inals and define a symmetric b inary op eration ⊖ on them. This is Definition 3 b elo w. W e ha v e t w o different definitions of ⊖ – for fin ite | Ult R | and for infin ite | Ult R | . W e will denote b y V α the elemen t of Ult R n umb ered w ith α . W e will tak e the s econd comp onent of eac h p oin t of W to b e the n umb er (o rdin al) of a rela tional u ltrafilter rather than the ultrafilter itself. Let ( x, y ) ∈ W 2 and n = x 2 ⊖ y 2 . First we consider the case x 2 6 = y 2 . If ( x 1 , y 1 ) ∈ R 0 V n \ R 0 V − 1 n , w e p ut ( x, y ) in R V n and ( y , x ) in R V − 1 n . If ( x 1 , y 1 ) ∈ R 0 V − 1 n \ R 0 V n , w e pu t ( x, y ) in R V − 1 n and ( y , x ) in R V n . In the case when ( x 1 , y 1 ) ∈ R 0 V n ∩ R 0 V − 1 n , w e n eed to c ho ose one of ( x, y ) and ( y , x ) and th en pu t the c hosen pair in R V n , w hile the other one sh ould go to R V − 1 n . If x 2 = y 2 or ( x 1 , y 1 ) / ∈ R 0 V n ∪ R 0 V − 1 n , we p ut the pair ( x, y ) in the relation corresp ondin g to some relational ultrafilter V x 1 ,y 1 , wh ic h we choose among those V ∈ Ult R for whic h ( x 1 , y 1 ) ∈ R 0 V . The reason wh y w e treat the case x 2 = y 2 along with ( x 1 , y 1 ) / ∈ R 0 V n ∪ R 0 V − 1 n rather th an p utting ( x, y ) in R V n , is that we cannot guaran tee th at V x 2 ⊖ x 2 is symm etric. But w e can c ho ose V x 1 ,x 1 to b e symmetric according to the follo win g p rop osition: 3 The symbol ‘ ’ here denotes an arbitrary element of U lt R . 16 Prop osition 4.15. If U ∈ Ult S , then: (1) ( U, U ) ∈ R 0 V ⇔  ∀ [ α ] ∈ V  ∀ [ a ] ∈ U  ∃∃ ( a, a )[ α ] ∈ S  . (2) Ther e exists a symmetric V ∈ Ult R such that ( U, U ) ∈ R 0 V . Pr o of. See App end ix B . Definition 3. L et κ =   Ult R   . W e consider tw o cases for κ : (1) κ < ω . Let Ult R =  V i   1 ≤ i ≤ κ  with ( i 6 = j ⇒ V i 6 = V j ). W e d enote R w = { 0 , 1 , . . . , 2 κ } . F or m, n ∈ R w w e d efine m ⊕ n = ( m + n ) mo d (2 κ + 1) m ⊖ n = min  ( m − n ) mo d (2 κ + 1) , ( n − m ) mo d (2 κ + 1)  . W e ha ve 0 ≤ m ⊖ n = n ⊖ m ≤ κ for all m, n ∈ R w . Also, ( m ⊕ n ) ⊖ m = n for arbitrary m ∈ R w and 1 ≤ n ≤ κ . W e defin e an irrefl exiv e relation ⋖ on R w m ⋖ n ⇔ ( n − m ) mo d (2 κ + 1) < ( m − n ) mo d (2 κ + 1) , suc h that for all d ifferen t m, n ∈ R w either m ⋖ n , or n ⋖ m . W e ha ve m ⋖ m ⊕ n for arbitrary m ∈ R w and 1 ≤ n ≤ κ . (2) κ ≥ ω . Let Ult R =  V α   0 < α < κ  with ( α 6 = β ⇒ V α 6 = V β ). W e d enote R w = κ . F or α, β ∈ R w w e d efine α ⊕ β = α + β 4 and α ⊖ β = ( α − β if β < α, β − α otherwise . Again, w e ha ve µ ⊖ ν = ν ⊖ µ for all µ, ν ∈ R w , and ( µ ⊕ ν ) ⊖ µ = ν f or arbitrary µ ∈ R w and 0 < ν < κ . As in the pr evious case, w e defin e a relation ⋖ on R w , whic h in this case is just the usu al strict total order: µ ⋖ ν ⇔ µ < ν . W e h a v e µ ⋖ µ ⊕ ν for arb itrary µ ∈ R w and 0 < ν < κ . The domain of the canonical mo del is W = Ult S × R w . F or eac h p ∈ V S w e d efine v ( p ) =  x ∈ W   [ p ] ∈ x 1  . W e c h o ose a set  Ult 2 S  0 ⊆ Ult S 2 suc h that for ea ch ( U 1 , U 2 ) ∈ Ult S 2 it con tains exactly one x ∈  ( U 1 , U 2 ) , ( U 2 , U 1 )  . F or eac h pair ( U 1 , U 2 ) ∈ Ult S 2 w e choose one V ∈ Ult R , suc h that ( U 1 , U 2 ) ∈ R 0 V ∧ ( U 1 = U 2 ⇒ V = V − 1 ) , and denote it by V U 1 ,U 2 . 4 Here ‘+’ d enotes ordinal addition. If β < α , we den ote by α − β the un ique ordinal γ such that β + γ = α . 17 F or eac h V ∈ Ult R w e d efine a relation R V ⊆ W 2 : R V = ( ( x, y ) ∈ W 2       x 2 6 = y 2 ∧ ( x 1 , y 1 ) ∈ R 0 V x 2 ⊖ y 2 ∩ R 0 V − 1 x 2 ⊖ y 2 ∧   x 2 ⋖ y 2 ∧ V = V x 2 ⊖ y 2  ∨  y 2 ⋖ x 2 ∧ V = V − 1 x 2 ⊖ y 2    ∨  x 2 6 = y 2 ∧ ( x 1 , y 1 ) ∈ R 0 V x 2 ⊖ y 2 \ R 0 V − 1 x 2 ⊖ y 2 ∧ V = V x 2 ⊖ y 2  ∨  x 2 6 = y 2 ∧ ( x 1 , y 1 ) ∈ R 0 V − 1 x 2 ⊖ y 2 \ R 0 V x 2 ⊖ y 2 ∧ V = V − 1 x 2 ⊖ y 2  ∨  x 2 = y 2 ∨ ( x 1 , y 1 ) / ∈ R 0 V x 2 ⊖ y 2 ∪ R 0 V − 1 x 2 ⊖ y 2  ∧   ( x 1 , y 1 ) ∈  Ult 2 S  0 ∧ V = V x 1 ,y 1  ∨  ( y 1 , x 1 ) ∈  Ult 2 S  0 ∧ V = V − 1 y 1 ,x 1   ! ) . Lemma 4.16. (1) S  R V   V ∈ Ult R  = W 2 . (2) V ′ 6 = V ′′ implies R V ′ ∩ R V ′′ = ∅ . (3) R V − 1 =  R V  − 1 ; (4) ( x, y ) ∈ R V implies ( x 1 , y 1 ) ∈ R 0 V ; (5) If ( U 1 , U 2 ) ∈ R 0 V ν , then for e ach µ ∈ R w it holds that  ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  ∈ R V ν . Pr o of. The fi rst four items may b e easily verified by considerin g the four cases in the defin ition. W e p ro v e the last one. Let ( U 1 , U 2 ) ∈ R 0 V ν and µ ∈ R w . Note that µ ⋖ µ ⊕ ν and hence µ 6 = µ ⊕ ν . C onsider th e pair  ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  . W e hav e ( µ ⊕ ν ) ⊖ µ = µ ⊖ ( µ ⊕ ν ) = ν . Th ere are t wo p ossibilities: • ( U 1 , U 2 ) ∈ R 0 V ν ∩ R 0 V − 1 ν . As µ ⋖ µ ⊕ ν , w e hav e  ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  ∈ R V ν . • ( U 1 , U 2 ) ∈ R 0 V ν \ R 0 V − 1 ν . Then  ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  ∈ R V ν . F or eac h α ∈ V R w e d efine R ( α ) = S  R V   V ∈ Ult R ∧ [ α ] ∈ V  . Lemma 4.17. F or e ach term α ∈ T Rel (V R ) R ( α ) = [  R V   V ∈ Ult R ∧ [ α ] ∈ V  . Pr o of. F or eac h ( x, y ) ∈ W 2 w e denote by V ( x, y ) the un ique V ∈ Ult R suc h that ( x, y ) ∈ R V . W e need to pro ve that R ( α ) =  ( x, y ) ∈ W 2   [ α ] ∈ V ( x, y )  . This can b e pro ve d b y s tructural indu ction on α . 18 Notation. F or a ∈ T Set (V S ) we denote h ( a ) =  U ∈ Ult S   [ a ] ∈ U  . F or α ∈ T Rel (V R ) w e d enote h ( α ) =  V ∈ Ult R   [ α ] ∈ V  . Lemma 4.18. If a, b ∈ T Set (V S ) and α ∈ T Rel (V R ) , then: ∀∃ ( a, b )[ α ] ∈ S ⇔  ∀ U ∈ h ( a )  ∀ [ c ] ∈ U  ∃∃ ( c, b )[ α ] ∈ S  . Pr o of. ( → ) Let ∀∃ ( a, b )[ α ] ∈ S , U ∈ h ( a ) and [ c ] ∈ U . Then a ∩ c 6 = 0 ∈ S . By ( AL 1 ), ∃∃ ( c, b )[ α ] ∈ S . ( ← ) Ass ume that ∀∃ ( a, b )[ α ] / ∈ S . Since S is a rich theory , there is a set v ariable p suc h that a ∩ p = 0 ∨ ∃∃ ( p, b )[ α ] / ∈ S . Hence a ∩ p = 0 / ∈ S and ∃∃ ( p, b )[ α ] / ∈ S . T hen [ a ] ∩ [ p ] 6 = [0] and there is an ultrafilter U ⊇  [ a ] , [ p ]  . Lemma 4.19. F or e ach f ormula ϕ ∈ F orm(V S , V R ) the fol lowing e quivalenc e holds: ϕ ∈ S ⇔ M  ϕ . Pr o of. The pro of is b y indu ction on the s tructure of ϕ . Since S is a maximal theory , we need to consider exp licitly only the cases w here ϕ is an atomic form ula. (1) ϕ is a 1 ≤ a 2 . By the S tone representat ion theorem for Bo olean algebras, a 1 ≤ a 2 ∈ S ⇔ h ( a 1 ) ⊆ h ( a 2 ) ⇔ v ( a 1 ) = h ( a 1 ) × R w ⊆ h ( a 2 ) × R w = v ( a 2 ) . (2) ϕ is ∃ ∃ ( a 1 , a 2 )[ α ]. ( → ) Let ∃∃ ( a 1 , a 2 )[ α ] ∈ S . Th en  [ a 1 ) , [ a 2 )  ∈ R 0 [ α ) . By Lemma 4.12  ∃ U 1 ∈ h ( a 1 )  ∃ U 2 ∈ h ( a 2 )  ∃ V ∈ h ( α )  ( U 1 , U 2 ) ∈ R 0 V  . Let V = V µ . Then ( U 1 , 0) ∈ v ( a 1 ), ( U 2 , µ ) ∈ v ( a 2 ) and R V ⊆ R ( α ). By item 5 in Lemma 4.16 ,  ( U 1 , 0) , ( U 2 , µ )  ∈ R V . Th is shows that M  ∃∃ ( a 1 , a 2 )[ α ]. ( ← ) Let M  ∃∃ ( a 1 , a 2 )[ α ]. Then  ∃ x ∈ v ( a 1 )  ∃ y ∈ v ( a 2 )  ∃ V ∈ h ( α )  ( x, y ) ∈ R V  . Th us , we ha ve [ a 1 ] ∈ x 1 , [ a 2 ] ∈ y 1 , [ α ] ∈ V , and item 4 in Lemma 4.16 giv es us ( x 1 , y 1 ) ∈ R 0 V . Hence ∃∃ ( a 1 , a 2 )[ α ] ∈ S . (3) ϕ is ∀ ∀ ( a 1 , a 2 )[ α ]. ∀∀ ( a 1 , a 2 )[ α ] ∈ S ⇔ ¬∃ ∃ ( a 1 , a 2 )[ − α ] ∈ S ⇔ ∃∃ ( a 1 , a 2 )[ − α ] / ∈ S ⇔ M 2 ∃∃ ( a 1 , a 2 )[ − α ] ⇔ M  ∀∀ ( a 1 , a 2 )[ α ] (4) ϕ is ∀ ∃ ( a 1 , a 2 )[ α ]. ( → ) Let ∀∃ ( a 1 , a 2 )[ α ] ∈ S . By Lemma 4.18  ∀ U 1 ∈ h ( a 1 )  ∀ [ c ] ∈ U 1  ∃∃ ( c, a 2 )[ α ] ∈ S  . 19 Hence  ∀ U 1 ∈ h ( a 1 )    U 1 , [ a 2 )  ∈ R 0 [ α )  . By Lemma 4.12  ∀ U 1 ∈ h ( a 1 )  ∃ U 2 ∈ h ( a 2 )  ∃ V ∈ h ( α )  ( U 1 , U 2 ) ∈ R 0 V  . By item 5 in Lemma 4.16 ,  ∀ U 1 ∈ h ( a 1 )  ∃ U 2 ∈ h ( a 2 )  ∃ ν ∈ R w  ∀ µ ∈ R w    ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  ∈ R V ν ∧ V ν ∈ h ( α )  and hence  ∀ U 1 ∈ h ( a 1 )  ∀ µ ∈ R w  ∃ U 2 ∈ h ( a 2 )  ∃ ν ∈ R w    ( U 1 , µ ) , ( U 2 , µ ⊕ ν )  ∈ R ( α )  Therefore  ∀ x ∈ v ( a 1 )  ∃ y ∈ v ( a 2 )  ( x, y ) ∈ R ( α )  . ( ← ) Assume that M  ∀∃ ( a 1 , a 2 )[ α ]. Then  ∀ x ∈ v ( a 1 )  ∃ y ∈ v ( a 2 )  ∃ V ∈ h ( α )  ( x, y ) ∈ R V  . By item 4 in Lemma 4.16 ,  ∀ x ∈ v ( a 1 )  ∃ y ∈ v ( a 2 )  ∃ V ∈ h ( α )   ( x 1 , y 1 ) ∈ R 0 V  . As R w 6 = ∅ ,  ∀ U 1 ∈ h ( a 1 )  ∃ U 2 ∈ h ( a 2 )  ∃ V ∈ h ( α )  ( U 1 , U 2 ) ∈ R 0 V  . Hence  ∀ U 1 ∈ h ( a 1 )  ∀ [ c ] ∈ U 1  ∃∃ ( c, a 2 )[ α ] ∈ S  . Then, by Lemma 4.18 we obtain ∀∃ ( a 1 , a 2 )[ α ] ∈ S . (5) ϕ is ∃ ∀ ( a 1 , a 2 )[ α ]. ∃∀ ( a 1 , a 2 )[ α ] ∈ S ⇔ ¬∀ ∃ ( a 1 , a 2 )[ − α ] ∈ S ⇔ ∀∃ ( a 1 , a 2 )[ − α ] / ∈ S ⇔ M 2 ∀∃ ( a 1 , a 2 )[ − α ] ⇔ M  ∃∀ ( a 1 , a 2 )[ α ] Theorem 4.20 (Completeness) . If Γ ⊆ F orm(V S , V R ) , then Γ is c onsistent ⇔ Γ has a mo del . Pr o of. ( → ) Let Γ b e a consistent set of formulas. By item 2 in Corollary 4.5 there is a maximal th eory S wh ic h conta ins Γ. S has a mo del whic h is also a m o del of Γ. ( ← ) Let M  Γ an d let ∆ =  ϕ ∈ F orm (V S , V R )   M  ϕ  . By Prop osition 3.1 w e h a v e Thm(V S , V R ) ⊆ ∆. As ( M P ) preserves th e truth in ev ery mo del, the set ∆ is closed un der ( M P ). Sin ce M 2 ⊥ , ⊥ / ∈ ∆. Hence Γ ⊆ ∆ is consisten t. 20 5 Complexit y Before w e consider the complexity of the satisfiability problem, w e fi rst note th at our logic is a fragment of Bo olean Mo dal Logic (BML)[ 5 , 7 ] extended with a sym b ol ‘ − 1 ’ for the con v erse of the accessibilit y relation. BML is a m u ltimo dal log ic, whose language contai ns t wo t yp es of v ariables – a set of relatio nal v ariables (ato mic mo dal parameters) and an infi nite set of prop ositional v ariables. The set of m o dal parameters consists of the set of re- lational v ariables, the relational constant 1 and all their Bo olean com b inations. The set of formulas is the smallest set whic h con tains the prop ositional v ariables and is closed u nder prefixing a formula b y a b o x or diamond mo dalit y as we ll as connecting formula s with the pr op ositional op erators. A mo del for BML is a triple M = ( W, R, v ), w here W 6 = ∅ is th e domain, R is a function, which assigns to eac h atomic mo dal parameter a relation on W , and v assigns to eac h pr op ositional v ariable a sub set of W . R is extended to all mo d al parameters according to the standard interpretatio n of the Boolean op erators as set int ersection, union and complement, interpreting 1 as th e universal r elation W 2 . W e hav e the standard m eaning of the mo d al op erators: ( M , x )  h α i ϕ ⇔ ( ∃ y ∈ W )  ( x, y ) ∈ R ( α ) ∧ ( M , y )  ϕ  ( M , x )  [ α ] ϕ ⇔ ( ∀ y ∈ W )  ( x, y ) ∈ R ( α ) ⇒ ( M , y )  ϕ  . W e consider the extension of the language of BML with a s ym b ol ‘ − 1 ’ in mod al parameters. W e in terpret it as taking the con ve rse of the relation: R ( α − 1 ) =  R ( α )  − 1 . The formula s of our language ha ve equiv alent s in th is extension of the language of BML: a ≤ b is equiv alent to [1]( a → b ) ∃∃ ( a, b )[ α ] is equiv alent to h 1 i  a ∧ h α i b  ∀∃ ( a, b )[ α ] is equiv alent to [1]  a → h α i b  ∀∀ ( a, b )[ α ] is equiv alent to [1]  a → [ − α ] ¬ b  ∃∀ ( a, b )[ α ] is equiv alent to h 1 i  a ∧ [ − α ] ¬ b  . The satisfiabilit y problem for our log ic is decidable in NEx pTime, since the form ulas are translatable (in p olynomial time) into the NExp Time-decidable t wo - v ariable fragmen t of first-order predicate logic. W e argue that the complexit y is the same as the complexit y of BML, whic h is pro ve d b y Lutz and S attler [ 14 ] to b e NExpTime if the language con tains an infinite n umb er of relational v ariables, and ExpT ime if only a fi nite n umb er of r elational v ariables is av ailable. Also, the complexit y d o es not dep end on whether we allo w − 1 in the language. In th e case of an infinite num b er of relational v ariables, the lo wer NExp Time b ound is p ro v ed in [ 14 ] by a reduction fr om an NExp Time-complete tiling problem. The BML f orm ula, used to enco de the tiling, is a conju nction of a formula, whic h describ es the initial condition for t he problem, and several conjun cts, whic h ensure that eve ry mo d el satisfying the formula is indeed a tiling. All conjun cts but the one for the initial condition can b e translated into our fragmen t. Th e formula for the initial condition can b e r eplaced by a formula from our fragmen t, such that the whole conjun ction is equisatisfiable with the original one. In the case o f a finite num b er of relat ional v ariables, the lo we r ExpTime b ound of BML follo ws from the Exp Time-completeness of K u (the basic mo d al logic 21 enric hed w ith the universal mo d alit y ). Ho wev er, the intersect ion of K u with our fragmen t is also ExpTime-hard , hen ce the ExpTime-hardn ess of our logic. The up p er ExpTime b ound f or BML is pro ved in [ 14 ] b y redu ction to the satisfiabilit y problem for the basic m ultimo d al logic enriched with the universal mo dalit y . Th e same reduction is applicable in th e presence of − 1 , and multimod al K u enric hed with − 1 is also Exp Time-complete. These high complexities are d ue to the presence of ∀ ∃ and ∃∀ in the language. If w e remo v e these s ym b ols from the language, the resulting logic has an NP- complete s atisfiabilit y problem, as it p ossesses th e p olysize m o del p rop erty . T his can b e prov ed by selection of p oin ts f rom a mo del in the wa y it is done in [ 2 ] for the dynamic logics of the region-based theory of discrete sp aces. 6 Concluding remarks The firs t completeness pro of for a non-classical relational syllogistic (i.e. on e that con tains relational te rms ) was giv en b y Nishih ara, Morita, an d Iwata in [ 22 ]. Their fragmen t conta ins v ariables for p rop er nouns and n -ary relational terms c losed only under complemen tation and do es n ot allo w Bo olean op erations on set terms. Later w orks on relational syllogistics, d ev oted mainly to the computational complexit y pr oblems, are McAllester and Giv an [ 15 ] and Pratt-Hartmann [ 26 , 27 , 28 , 30 ]. Th e pap er b y Moss and Pratt-Hartmann [ 31 ] is devot ed b oth to complete axiomatiza tions and some computational complexit y r esults. A su ccessor of [ 31 ] is Moss [ 21 ], dev oted to axiomatiza tions and completeness pro ofs for a num b er of relational syllogistics. Our logic differs in expressiv eness from all systems of relational syllogistic men tioned ab o ve . One of the reasons is that we ha v e quite ric h language based on b oth class terms and relational terms, while the other lo gics are b ased on languages that are w eak er than our sy stem, or in comparable with it, some of them d ealing only with atomic formulas. Su c h is, for instance, the system of McAllester and Giv an [ 15 ] and some systems stu died in Moss and Pr att-Hartmann [ 31 ] and Moss [ 21 ]. The fragment of our language, w hic h conta ins only t wo relational terms α and − α and all s et terms are v ariables or negated v ariables, coincides with the language of the system R † studied by Moss and Pratt-Hartmann in [ 31 ]. In the p resen t pap er w e ha ve p ro ve d the completeness of a syllog istic logic with a set of binary relations closed und er the Boolean op erations and under taking the con v erse. Th e completeness pro of can b e generalized to the case of n -ary r elations for arbitrary n , w hic h will co v er th e case of n -transitiv e verbs. W e also plan to study extensions of our logic with several kind s of n ominals making it p ossible to co ver sen tences from natural languag e lik e ‘So crates is a man’, ‘S o crates is mortal’. The construction of the canonical mo del in our logic is sim ilar to that for BML. I t is also p ossible to use the construction of the r elations R V from R 0 V to pro ve the completeness of BML extended with a symbol − 1 for the con v erse of the accessibilit y r elation. 22 Ac kno wledgemen ts The authors would lik e to thank Ian Pratt-Hartmann and the anonymous referees for v aluable comments on the p ap er. This w ork wa s su pp orted b y the Eu rop ean So cial F un d thr ough the Hu- man Resource Dev elopmen t Op erational Programme 2007–2013 under con tract BG051PO00 1-3.3.04/ 27/ 28.08. 2009, by the pro ject DID02/32 /2009 of Bulgarian Science F und and by Sofia Universit y under con tract 136/201 0. References [1] Balbiani, Ph., Tinchev, T. & V ak ar elov, D. (2007 ). Moda l Logics for region-bas ed theory o f space. F undamenta Informaticae, 81, 29–8 2 [2] Balbiani, Ph., Tinchev, T. & V a k are lov, D. (200 7). Dynamic lo gics o f the regio n- based theory of discrete spaces. Jour nal of Applied Non- Classical Logics, 17, 39– 61 [3] F er ro, A., O mo deo, E. G. & Schw a rtz, J . T. (2006 ). Decision pro ce dures for ele- men tar y sublanguag es of set theory . I. Multilevel syllo gistic and so me extensio ns. Communications on pure and Applied Mathematics , 33 , 5 99–6 08 [4] V an der Do es, J . & V a n Eijck, J. (1996 ). Ba sic q uantifier theory . (In: Quantifiers, Logic, a nd Languag e, 1–4 5, CSLI, Stanford.) [5] Gargov, G., Passy , S. & Tinchev, T. (1987). Mo dal En vironment for Boolea n sp ecula - tions. (In D. Skordev (Ed.), Ma thematical Lo gic a nd its Applications (pp. 25 3–26 3). New Y ork: Plenum Pr ess.) [6] V an Eijck, J . (2007). Natur al Lo gic for Natural Lang uage. (In: T en Cate, B. & Zeev at, H. (Eds.), Logic, Languag e, and Co mputation (pp. 21 6–230 ), Spr inger.) Presented at 6 -th International Tbilisi Sympo sium on Lo gic, La nguage, and Com- putation, B atumi, Georgia , September 12 –16, 20 05. [7] Gargov, G. & Passy , S. (1990). A note on Bo o lean Mo dal Logic. (In: P . Petk ov (Ed.), Mathematical Logic (pp. 2 99–30 9). New Y ork: Plenum Pres s.) [8] Goranko, V. (1990). Completeness a nd incompleteness in the bimo dal base L (R,-R). (In: P . Petk ov (Ed.), Mathema tical Logic (pp. 3 11–3 26). New Y ork: Ple n um P ress.) [9] Iv anov, N. (20 09). Rela tional Syllogistics . Master ’s thesis (in Bulg arian), So fia Univ er sity [10] Iv a nov, N. & V ak are lov, D. (2012 ). A system o f relationa l syllog istic incor po rating full Bo olean reasoning. Jo urnal of Logic, Language a nd Information, 21, 433-45 9. (Av ail- able at http:/ /link .spri nger.com/article/10.1007/s10849- 012- 9165- 1 ) [11] Khayata, M. Y., Pacholczyk, D. & Ga rcia, L. (2002). A Qualita tive Appro ach to Syllogistic Reasoning. Annals o f Mathematics and Artificia l Int ellige nce, 34 , 131-1 59 [12] Leevers, H. J. & Harris, P . L. (2000). Co un terfac tual Syllogistic Reaso ning in Normal 4-Y ea r-Olds, Children with Learning Disa bilities, and Children with Autism. Jour nal of E xp erimental Child Psychology , 76, 64–8 7 [13] Luk asiewicz, J. (1957). Aris totle’s Syllogistic fr om the standp oint of mo der n for mal logic, 2 nd ed. (Oxford: Clarendo n Press ) 23 [14] Lutz, C. & Sattler, U. (200 1). The complexity of r easoning with B o olean mo dal logic s. (In F. W olter, H. W a nsing, M. de Rijke, & M. Zakha ryasc hev (Eds.), Adv ances in Mo dal Logic volume 3 (pp. 329 –348 ). Stanfor d: CSLI P ublications.) [15] McAllester, D. A. & Giv an, R. (1992 ). Natural la nguage synt ax and first-o rder inference. Ar tificial Intelligence, 56, 1– 20 [16] Purdy , W. C. (1991). Studies on Natural La nguage. Notre Dame J ournal of F or mal Logic, 3 2, 409–4 25 [17] Rayside, D. & Kontogiannis, K. (20 01). On the Syllogistic Structure o f Ob ject- Oriented Pro gramming. (Presented at the 23rd International Conference on Softw are Engineering (ICSE’01)) [18] Moss, L. S. (2008, Septem be r). Relational syllogistic logics and o ther connections be- t ween mo dal logic and natural log ic. (Presented at A iML, Nancy; based on work with Ian P ratt-Hartma nn; av ailable at http:/ /aiml 08.loria.fr/talks/moss.pdf .) [19] Moss, L. S. (200 8). Completeness theorems for sy llogistic fragments. (In F. Hamm & S. Keps er (Eds.), Lo gics for Linguistic Structures (pp. 143– 174). B erlin, New Y ork: Mouton de Gruyter.) [20] Moss, L. S. (2007 ). Syllogis tic Logic with Co mplement s. Retrie ved from Indiana Univ er sity W eb site: http:/ /www. india na.edu/ ~ iulg/m oss/c omp2. pdf [21] Moss, L. S. (20 10). Syllogis tic lo gics with verbs. Jo urnal o f Lo gic and Computation, 20, 94 7–96 7 [22] Nishihara , N., Morita, K. & Iwata. S. (1990 ). An extended syllogis tic system with verbs and prop er nouns, a nd its completeness pro o f. Systems and Co mputers in Japan, 2 1, 760–7 71. [23] Orlowsk a , E . (199 8). St udying Incompleteness o f Information: A Clas s of Informatio n Logics. In K. K ijania-Pla cek & J. W ole ´ nski (Eds.), The Lvow-Warsaw Schol l and Contemp or ary Philosophy (pp. 28 3–300 ). Dordrech t: K lu wer. [24] Pfeifer, N. (200 6). Contempora ry syllogistics: Compa rative and quantit ative syllo- gisms. (In G. Krenzeba uer & G. J. W. Dor en (Eds.), Argumentation in Theorie und Praxis : Philosophie und Didaktik des Argumentierens (pp. 57–71 ). Wien: LIT.) [25] Politzer, G. (200 4). Some precurso rs o f c urrent theories of syllogistic reaso ning. (In K. Manktelow & M.-C. Chung (Eds.), Psy chology o f rea soning. The oretical a nd historical pe rsp ectives (pp. 214 –240). Hov e: Psychology Pr ess.) [26] Pra tt-Hartmann, I. (20 05). Complexity of the Two-V ariable F r agment with Coun ting Quantifiers. Journal of Logic, Lang uage and Informa tion, 1 4, 36 9–395 [27] Pra tt-Hartmann, I. (2004). F ragments of language . Jo urnal o f Log ic, Langua ge and Information, 13, 207–2 23 [28] Pra tt-Hartmann, I. (2008). On the computationa l complexity of the num eric ally definite syllogistic and rela ted log ics. Bulletin of Sy m b olic Log ic, 1 4, 1–28 [29] Pra tt-Hartmann, I. (200 9). No syllogis ms for the numerical syllo gistic. (In La n- guages: from F ormal to Natura l, vol. 5 533 of LNCS (pp. 192 –203). Springer.) [30] Pra tt-Hartmann, I. & Third, A. (2006). Mo re fragments of la nguage. Notre Dame Journal of F ormal Log ic, 4 7, 151– 177 24 [31] Pra tt-Hartmann, I. & Moss, L. S. (2009). Lo gics for the relational syllog istic. The Review o f Symbolic Logic, 2, 647 –683 [32] Shepherdso n, J. (1956). On the interpretation of Aristo telian syllog istic. Jo urnal of Symbolic Logic, 21, 137–1 47 [33] Thorne, C. & Ca lv anese, D. (200 9). The Data Complexity of the Syllogistic F rag - men ts of English. (In Pr o ceedings of the 17th Amsterdam co llo quium conferenc e o n Logic, la nguage and meaning 200 9 (pp. 11 4–123 ).) [34] W edb er g, A. (1948). The Aristotelian theory of c lasses. Ajatus, 1 5, 299– 314 [35] W es terst ˚ ahl, D. (1989 ). Aristotelian syllogis ms and g eneralized q uantifiers. Studia Logica, 48, 577– 585 A Pro of of Prop osition 4.6 In the pr o of of Prop osition 4.6 we will need th e f ollo win g lemmas. If Q is a quan tifier, we will denote by Q the dual quantifier. Lemma A.1. L et a and b b e set terms and let α b e a r elational term. Then, for arbitr ary qu antifiers Q 1 and Q 2 the formula Q 1 Q 2 ( a, b )[ − α ] ↔ ¬ Q 1 Q 2 ( a, b )[ α ] is a the or em. Pr o of. One of these fou r formulas is an axiom. It remains to pro ve 6 implications. In the f ollo win g pro ofs p and q are differen t set v ariables, whic h do not o ccur in the terms a and b . W e kn o w that su c h v ariables exist, since V S is infi nite. (1) ⊢∀∃ ( a, b )[ α ] → a ∩ p = 0 ∨ ∃∃ ( p, b )[ α ] b y ( AL 1 ) ⊢∀∃ ( a, b )[ α ] → a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ − α ] b y ( A − ) ⊢∀∃ ( a, b )[ α ] → ¬∃ ∀ ( a, b )[ − α ] b y ( R 3 ) (2) ⊢¬∃ ∀ ( a, b )[ − α ] → a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ − α ] by ( AL 3 ) ⊢¬∃∀ ( a, b )[ − α ] → a ∩ p = 0 ∨ ∃∃ ( p, b )[ α ] by ( A − ) ⊢¬∃∀ ( a, b )[ − α ] → ∀∃ ( a, b )[ α ] b y ( R 1 ) (3) ⊢¬∃ ∃ ( a, b )[ − α ] → a ∩ p = 0 ∨ ¬∀∃ ( p, b )[ − α ] b y ( AL 1 ) ⊢¬∃∃ ( a, b )[ − α ] → a ∩ p = 0 ∨ b ∩ q = 0 ∨ ¬∀ ∀ ( p, q )[ − α ] b y ( AL 2 ) ⊢¬∃∃ ( a, b )[ − α ] → a ∩ p = 0 ∨ b ∩ q = 0 ∨ ∃ ∃ ( p, q )[ α ] b y ( A − ) ⊢¬∃∃ ( a, b )[ − α ] → b ∩ q = 0 ∨ ∀∃ ( a, q )[ α ] b y ( R 1 ) ⊢¬∃∃ ( a, b )[ − α ] → ∀∀ ( a, b )[ α ] by ( R 2 ) (4) ⊢¬∃ ∀ ( a, b )[ α ] → a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ α ] by ( AL 3 ) ⊢¬∃∀ ( a, b )[ α ] → a ∩ p = 0 ∨ ∃∃ ( p, b )[ − α ] b y item 3 ⊢¬∃∀ ( a, b )[ α ] → ∀∃ ( a, b )[ − α ] b y ( R 1 ) 25 (5) ⊢∀∀ ( a, b )[ α ] → b ∩ p = 0 ∨ ∀∃ ( a, p )[ α ] b y ( AL 2 ) ⊢∀∀ ( a, b )[ α ] → b ∩ p = 0 ∨ ¬∃∀ ( a, p )[ − α ] b y item 1 ⊢∀∀ ( a, b )[ α ] → b ∩ p = 0 ∨ ∀∃ ( a, p )[ − − α ] by item 4 ⊢∀∀ ( a, b )[ α ] → ∀ ∀ ( a, b )[ − − α ] b y ( R 2 ) ⊢∀∀ ( a, b )[ α ] → ¬∃ ∃ ( a, b )[ − α ] b y ( A − ) (6) ⊢∀∃ ( a, b )[ − α ] → a ∩ p = 0 ∨ ∃ ∃ ( p, b )[ − α ] by ( AL 1 ) ⊢∀∃ ( a, b )[ − α ] → a ∩ p = 0 ∨ ¬∀∀ ( p, b )[ α ] b y item 5 ⊢∀∃ ( a, b )[ − α ] → ¬∃∀ ( a, b )[ α ] b y ( R 3 ) Lemma A.2. L et α, β ∈ T Rel (V R ) and let B = T Set (V S ) . Then the fol lowing c onditions ar e e qu i valent: (1) ( ∀ a, b ∈ B )  ∃∃ ( a, b )[ α ] → ∃∃ ( a, b )[ β ] ∈ S  (2) ( ∀ a, b ∈ B )  ∀∃ ( a, b )[ α ] → ∀∃ ( a, b )[ β ] ∈ S  (3) ( ∀ a, b ∈ B )  ∀∀ ( a, b )[ α ] → ∀∀ ( a, b )[ β ] ∈ S  (4) ( ∀ a, b ∈ B )  ∃∀ ( a, b )[ α ] → ∃∀ ( a, b )[ β ] ∈ S  Pr o of. W e w ill pr o v e ( 1 ) → ( 2 ). Assume that item 1 is true and su pp ose that there are set terms a and b su c h that ∀ ∃ ( a, b )[ α ] → ∀∃ ( a, b )[ β ] / ∈ S . Sin ce S is a ric h theory , there is a set v ariable p su c h that ∀∃ ( a, b )[ α ] → a ∩ p = 0 ∨ ∃ ∃ ( p, b )[ β ] / ∈ S . Hence ∀∃ ( a, b )[ α ] → a ∩ p = 0 ∨ ∃∃ ( p, b )[ α ] / ∈ S . This is a con tradiction, since the last formula is a th eorem. ( 2 ) → ( 3 ) The pr o of is analogous to the previous on e. ( 3 ) → ( 4 ) Assume that item 3 is true. By Lemma A.1 ( ∀ a, b ∈ B )  ¬∃∃ ( a, b )[ − α ] → ¬∃∃ ( a, b )[ − β ] ∈ S  . F or the sake of con tradiction supp ose that there are set terms a and b su c h that ∀∃ ( a, b )[ − β ] → ∀ ∃ ( a, b )[ − α ] / ∈ S . Since S is a ric h theory , there is a set v ariable p suc h th at ∀∃ ( a, b )[ − β ] → a ∩ p = 0 ∨ ∃ ∃ ( p, b )[ − α ] / ∈ S . Hence ∀∃ ( a, b )[ − β ] → a ∩ p = 0 ∨ ∃ ∃ ( p, b )[ − β ] / ∈ S . This is a con tradiction, since the last formula is a th eorem. Hence ( ∀ a, b ∈ B )  ∀∃ ( a, b )[ − β ] → ∀∃ ( a, b )[ − α ] ∈ S  . 26 Using Lemma A.1 again, w e conclud e that ( ∀ a, b ∈ B )  ¬∃∀ ( a, b )[ β ] → ¬∃∀ ( a, b )[ α ] ∈ S  , whic h imp lies item 4 . ( 4 ) → ( 1 ) The pr o of is analogous to the previous on e. Lemma A.3. If a , b , c , d ar e set terms and α , β ar e r elational terms, then the fol lowing formulas ar e the or ems: (1) ∃∃ ( a, b )[ α ] ∧ a ≤ c → ∃∃ ( c, b )[ α ] and ∃∃ ( a, b )[ α ] ∧ b ≤ c → ∃∃ ( a, c )[ α ] (2) ∀∀ ( a, b )[ α ] ∧ c ≤ a → ∀∀ ( c, b )[ α ] and ∀∀ ( a, b )[ α ] ∧ c ≤ b → ∀∀ ( a, c )[ α ] (3) ∀∀ ( a, b )[ α ] ∧ ∀∀ ( c, d )[ β ] → ∀∀ ( a ∩ c, b ∩ d )[ α ∩ β ] (4) ∀∀ ( a, b )[ α ] ∧ ¬∃∃ ( c, d )[ α ∩ β ] → ∀∀ ( a ∩ c, b ∩ d )[ − β ] Pr o of. (1) W e will pro ve the first one. ⊢∃∃ ( a, b )[ α ] ∧ a ≤ c →  ∃∃ ( a, b )[ α ] ∨ ∃∃ ( c, b )[ α ]  ⊢∃∃ ( a, b )[ α ] ∧ a ≤ c → ∃∃ ( a ∪ c, b )[ α ] ∧ a ∪ c = c by ( A ∪ 1 ) ⊢∃∃ ( a, b )[ α ] ∧ a ≤ c → ∃∃ ( c, b )[ α ] b y ( A = 1 ) (2) F ollo ws from the previous item and Lemma A.1 . (3) ⊢∀∀ ( a, b )[ α ] ∧ ∀∀ ( c, d )[ β ] → ∀∀ ( a ∩ c, b ∩ d )[ α ] ∧ ∀∀ ( a ∩ c, b ∩ d )[ β ] b y item 2 ⊢∀∀ ( a, b )[ α ] ∧ ∀∀ ( c, d )[ β ] → ∀∀ ( a ∩ c, b ∩ d )[ α ∩ β ] by ( A ∩ ) (4) W e will pro ve item 4 . Let p and q b e different set v ariables wh ic h d o not o ccur in a , b , c and d . ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] ∧ ∀ ∀ ( p, q )[ β ] → ∀∀ ( a ∩ p, b ∩ q )[ α ∩ β ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] b y item 3 ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] ∧ ∀ ∀ ( p, q )[ β ] → a ∩ p ∩ c = 0 ∨ b ∩ q ∩ d = 0 b y ( AL 1 ) and ( AL 2 ) ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] → a ∩ c ∩ p = 0 ∨ b ∩ d ∩ q = 0 ∨ ¬∀ ∀ ( p, q )[ β ] ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] → a ∩ c ∩ p = 0 ∨ b ∩ d ∩ q = 0 ∨ ∃∃ ( p, q )[ − β ] b y Lemma A.1 ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] → b ∩ d ∩ q = 0 ∨ ∀∃ ( a ∩ c, q )[ − β ] by ( R 1 ) ⊢∀∀ ( a, b )[ α ] ∧ ¬∃ ∃ ( c, d )[ α ∩ β ] → ∀∀ ( a ∩ c, b ∩ d )[ − β ] by ( R 2 ) 27 Pro of of Prop osition 4.6 . The w ell-definition of ≤ is ob vious. The w ell- definition of ∪ follo ws from ( A ∪ ). The w ell-defin ition of ∩ and − follo ws from ( A ∩ ), ( A − ), Lemma A.1 and Lemma A.2 . The w ell-definition of − 1 follo w s from ( A − 1 ). W e need to verify the follo wing prop erties for arb itrary relational terms α , β and γ : (1) α 4 α , ( α 4 β ∧ β 4 γ ) ⇒ α 4 γ , ( α 4 β ∧ β 4 α ) ⇒ α ≈ β (2) α ∩ β 4 α , α ∩ β 4 β , ( γ 4 α ∧ γ 4 β ) ⇒ γ 4 α ∩ β (3) α 4 α ∪ β , β 4 α ∪ β , ( α 4 γ ∧ β 4 γ ) ⇒ α ∪ β 4 γ (4) 0 R 4 α , α 4 1 R (5) α ∩ ( β ∪ γ ) 4 ( α ∩ β ) ∪ ( α ∩ γ ) (6) α ∩ − α 4 0 R (7) 1 R 4 α ∪ − α ( 1 ) f ollo ws directly from the definition of the r elation 4 . ( 2 ) follo ws from ( A ∩ ). ( 3 ) follo ws analogously from ( A ∪ ). ( 4 ) follo ws from ( A 0 R ) and ( A 1 R ). W e will pro ve the remaining three theorems. Let a, b ∈ T Set (V S ). ( 5 ) W e will mak e u se of item 4 in Lemma A.3 . Let p , q ∈ V S , p 6 = q and { p, q } ∩ V Set ( a, b ) = ∅ . ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  ∧ ¬ ∃∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  → ∀∀ ( a, b )[ α ] ∧ ∀ ∀ ( a, b )[ β ∪ γ ] ∧ ¬∃∃ ( p, q )[ α ∩ β ] ∧ ¬ ∃∃ ( p, q )[ α ∩ γ ] b y ( A ∩ ) and ( A ∪ ) ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  ∧ ¬ ∃∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  → ∀∀ ( a, b )[ β ∪ γ ] ∧ ∀∀ ( a ∩ p, b ∩ q )[ − β ] ∧ ∀ ∀ ( a ∩ p , b ∩ q )[ − γ ] b y Lemma A.3 ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  ∧ ¬ ∃∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  → ∀∀ ( a, b )[ β ∪ γ ] ∧ ¬∃ ∃ ( a ∩ p, b ∩ q )[ β ] ∧ ¬ ∃∃ ( a ∩ p, b ∩ q )[ γ ] b y ( A − ) ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  ∧ ¬ ∃∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  → ∀∀ ( a, b )[ β ∪ γ ] ∧ ¬∃ ∃ ( a ∩ p, b ∩ q )[ β ∪ γ ] b y ( A ∪ ) ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  ∧ ¬ ∃∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  → a ∩ p = 0 ∨ b ∩ q = 0 by ( AL 1 ) and ( AL 2 ) ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  → a ∩ p = 0 ∨ b ∩ q = 0 ∨ ∃ ∃ ( p, q )  ( α ∩ β ) ∪ ( α ∩ γ )  ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  → b ∩ q = 0 ∨ ∀ ∃ ( a, q )  ( α ∩ β ) ∪ ( α ∩ γ )  b y ( R 1 ) ⊢∀∀ ( a, b )  α ∩ ( β ∪ γ )  → ∀∀ ( a, b )  ( α ∩ β ) ∪ ( α ∩ γ )  b y ( R 2 ) 28 ( 6 ) ⊢∀∀ ( a, b )[ α ∩ − α ] → ∀∀ ( a, b )[ α ] ∧ ∀∀ ( a, b )[ − α ] b y ( A ∩ ) ⊢∀∀ ( a, b )[ α ∩ − α ] → ∀∀ ( a, b )[ α ] ∧ ¬∃∃ ( a, b )[ α ] b y ( A − ) ⊢∀∀ ( a, b )[ α ∩ − α ] → a = 0 ∨ b = 0 b y ( AL 1 ) and ( AL 2 ) ⊢∀∀ ( a, b )[ α ∩ − α ] → ¬ ∃∃ ( a, b )[ − 0 R ] b y ( A 0 ) ⊢∀∀ ( a, b )[ α ∩ − α ] → ∀ ∀ ( a, b )[0 R ] b y Lemma A.1 ( 7 ) ⊢¬∃ ∃ ( a, b )[ α ∪ − α ] → ¬∃∃ ( a, b )[ α ] ∧ ¬∃∃ ( a, b )[ − α ] by ( A ∪ ) ⊢¬∃∃ ( a, b )[ α ∪ − α ] → ¬∃∃ ( a, b )[ α ] ∧ ∀∀ ( a, b )[ α ] b y Lemma A.1 ⊢¬∃∃ ( a, b )[ α ∪ − α ] → a = 0 ∨ b = 0 by ( AL 1 ) and ( AL 2 ) ⊢¬∃∃ ( a, b )[ α ∪ − α ] → ¬∃ ∃ ( a, b )[1 R ] b y ( A 0 ) ⊢∃∃ ( a, b )[1 R ] → ∃∃ ( a, b )[ α ∪ − α ] The equiv alence α 4 β ⇔ α ∪ β ≈ β follo ws f rom ( A ∪ ). B Pro of of Prop osition 4.15 (1) This is obvious and is u sed only to shorten the notation. (2) Let I = n [ α ] ∈ Cl R    α ≈ α − 1 ∧  ∃ [ a ] ∈ U  ∃∃ ( a, a )[ α ] / ∈ S  o F = n [ α ] ∈ Cl R    α ≈ α − 1 and there exists a non-empt y finite set { α 1 , α 2 , . . . , α k } ⊆ T Rel , suc h that ( α − 1 1 ∪ − α 1 ) ∩ · · · ∩ ( α − 1 k ∪ − α k ) 4 α o The set I is an id eal in t he Bo olean alge br a of symmetric classes of relati onal terms. By Lemma 4.10 , F is a filter in that algebra and F ∩ I = ∅ . By the separation th eorem for fi lter-ideal pairs, there exists an ultrafilter F ′ ⊇ F in the Bo olean algebra of symmetric classes of r elational terms, such that F ′ ∩ I = ∅ . Let V =  x ∈ Cl R   x − 1 ∩ x ∈ F ′  . V has the follo w ing prop erties: • V ∈ Ult R . W e need to c hec k the follo wing: (a) [1 R ] ∈ V . (b) If x ∈ V , y ∈ Cl R , and x ≤ y , then y ∈ V . (c) If x, y ∈ V , then x ∩ y ∈ V . 29 (d) If x ∈ Cl R , then either x ∈ V , or − x ∈ V . Supp ose otherw ise. Then x − 1 ∩ x / ∈ F ′ and ( − x − 1 ) ∩ − x / ∈ F ′ . Hence  x − 1 ∩ x  ∪  ( − x − 1 ) ∩ ( − x )  =  x − 1 ∪ − x  ∩  ( − x − 1 ) ∪ x  / ∈ F ′ , whic h contradict s F ⊆ F ′ . (e) [0 R ] / ∈ V . • V = V − 1 . • ( U, U ) ∈ R 0 V . Since F ′ ∩ I = ∅ , w e h a v e  ∀ [ α ] ∈ V  ∀ [ a ] ∈ U  ∃∃ ( a, a )  α ∩ α − 1  ∈ S  and hence ( U, U ) ∈ R 0 V . 30