Logics for the Relational Syllogistic

Logics for the Relational Sy llogist ic Ian Pratt-Hartmann La wrence S. Moss Abstract The Aristotelian syllogis tic cannot account f or the v a lidit y of man y infer- ences inv olving re lational facts. In this paper, w e inv estigate the prospects for providing a relational sy llogistic. W e identif y severa l fragments b ased on (a) whether negation is p ermitted on all nouns, including those in the sub ject of a sentence; and (b) whether the sub ject noun phrase may con- tain a relativ e clause. The logics we presen t are extensions of the classical syllogistic, and w e pay special attentio n to the qu estion of whether r e- ductio ad absur dum is needed . Thus our main goal is to derive results on the existence (or non- existence) of syllogistic pro of systems for relational fragmen ts. W e also determine the computational complexity of all our fragmen ts. Con ten ts 1 In tro duction 2 2 Preliminaries 3 2.1 Some syllo gistic fr agments . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Syllogistic rules and r e ductio ad absurd um . . . . . . . . . . . . . 8 3 S and S † : direct systems 12 4 R : a refutation complete system 18 4.1 There are no sound and co mplete syllogistic sy stems for R . . . . 18 4.2 A refutation-co mplete system for R . . . . . . . . . . . . . . . . . 2 0 5 R ∗ : an indirect system 26 6 R † and R ∗† : no indirect systems 30 7 Relation to other w ork 39 8 Conclusions 41 1 1 In tro duction Augustus de Morgan famously obser ved that t he Aris totelian syllo gistic can- not account for the v alidit y of even the most elementary inferences inv olving relational facts, fo r example (de Mo rgan [3], p. 1 14): Every man i s an anima l He who kil ls a man k ills an animal . (1) De Morgan was certainly not the ﬁrst to notice the pro blem o f relational in- ference: for example, it is given pro minen t tr eatment in the Port Roy al Logic (Arnauld [1], Ch. I I I). But wher eas the Port-Roy alists to o k such inferences to demonstrate at most the o ccasio nal need for in genious reformu lation, de Morgan saw in them clear e vidence that binary relations must b e governed by logical principles lying o utside the scop e of the traditional s yllogistic, a nd made o ne of the ﬁr st serio us attempts to extend syllo gism-like pr inciples to r elational judg- men ts (see de Morgan [4 ]). Ultimately , of course, de Morgan’s attempt was to be without issue: the metho ds he employed w ere r endered o bsolete at the end of the nineteenth century by the rise of predicate lo gic, which provided an expressive y et easily- understo o d appara tus for formalizing relationa l facts. (F or a mo dern reco nstruction of de Mo rgan’s work, see Mer rill [11].) F rom a co mputational p o in t of view, how ev er, expr essive p ow er is a do uble- edged sword: roughly sp eaking, the more expressive a languag e is, the har der it is to compute with. In the last decade, this trade-o ﬀ ha s led to renewed int erest in inexpr essive logic s, in which the problem of determining e nt ailments is algor ithmically decida ble with (in idea l cases) low complexity . The logic al fragments sub jected to this sort o f complexity-theoretic analysis hav e naturally enough tended to b e those which ow e their sa lience to the syntax of ﬁrst-o rder logic, f or example: the tw o-v aria ble fragment, the guarde d fragment, and v arious quantiﬁer-preﬁx fragments. But of co urse it is equa lly re asonable to consider instead log ics deﬁned in terms o f the syntax o f natur al languages . Perhaps, therefore, it is time to r eturn to where de Morgan and his contempora ries le ft oﬀ, but with a computational a genda in mind. As we s hall see, this lea ds to a rather diﬀerent view o f the syllogis tic tha n tha t defended by cer tain latter -day advocates of term-lo gic. This pap er investigates the following six logics: (i) S , which corresp o nds to the traditiona l syllogistic; (ii) S † , which extends S with negated no uns such as non-man or non-animal ; (iii) R , which extends S with transitive verbs such a s kills ; (iv) R † , which extends S with b oth these c onstructions; (v) R ∗ , which extends R by allowing sub ject no un phrases to co ntain rela tive clauses as in the co nclusion o f Ar gument (1); and (vi) R ∗† , which ex tends R ∗ with negated nouns. W e derive results on the existence (or non-existence) o f syllogistic pro of- systems for these logic s, paying particular regar d to the need for the r ule of r e ductio ad absur dum . Sp eciﬁca lly , we present sound a nd complete syllogistic systems for b oth S and S † that do not employ r e ductio (we call such systems dir e ct s yl lo gistic systems ). W e then show tha t, b y con trast, there is no sound and complete direct syllo gistic system for the fragment R ; how ever, we do pr esent a 2 direct syllog istic sys tem for this fra gment that is sound and r efutation-c omplete (i.e. b ecomes co mplete if r e ductio ad absur dum is a llow ed as a single, ﬁnal step). W e then consider indir e ct syl lo gistic systems —i.e. those in which the rule of r e ductio ad absur dum may b e employ ed at any p oint during the der iv ation. W e show that—unless PTime = NPTime —there is no direct syllo gistic system that is s ound and refutation-co mplete for the fragment R ∗ ; ho wev er, we do provide an indirect syllogistic system that is sound and complete for this fragmen t. Fina lly , we show tha t neither R † nor R ∗† has even an indirect syllo gistic system that is sound and co mplete. W e also obtain results on the computational complexity of determining v alidity in the ab ov e fr agments, a nd show how these r esults are re lated to the ex istence o f s ound a nd complete pro of-s ystems (of v ario us kinds) for the fragments in question. Spec iﬁcally , we s how that the problem of determining the v a lidity of sequents in any of the fragments S , S † or R is NLogSp ace -complete; the problem of determining the v alidity of sequents in the fra gment R ∗ is co- NPTime -complete; and the problem of deter mining the v alidity o f seq uents in either of the fra gments R † or R ∗† is ExpTime -complete. Thu s, although we a re heartened to lear n of interest in r elational syllog isms going back to de Morgan, our co ntribution sho uld not b e rea d as a historical reconstructio n. It is a pro duct of our own time, diﬀeren t fro m de Mo rgan’s in bo th motiv ation a nd outcome. 2 Preliminaries 2.1 Some syllogistic fragmen ts The fragmen ts S and S † Fix a countably inﬁnite set P . W e may assume P to contain v arious English common count-nouns s uch as man , animal etc . A unary atom is a n element of P ; a unary liter al is an expre ssion of either of the forms p o r ¯ p , where p is a unary atom. A unary literal is called p ositive if it is a unary a tom; otherwise, ne gative . W e use the (p o ssibly dec orated) v ar iables o , p , q to range ov er una ry a toms, and l , m , n to r ange ov er unary liter als. With these conv en tions, a n S - formula is a n expressio n of any of the for ms ∃ ( p, l ) , ∃ ( l, p ) , ∀ ( p, l ) , ∀ ( l, ¯ p ) . (2) W e provide Englis h glosses for S -formulas as follows: ∀ ( p, q ) Every p is a q ∀ ( p, ¯ q ) No p is a q ∃ ( p, q ) Some p is a q ∃ ( p, ¯ q ) Some p is not a q . (3) Note that S co nt ains the formulas ∃ ( ¯ p, q ) and ∀ ( ¯ p, ¯ q ), whic h are not glossed in (3); how ever, the semantics given b elow ensure that these formulas are lo g- ically eq uiv alent to ∃ ( q , ¯ p ) and ∀ ( q , p ), resp ectively . W e may re gard S a s the language of the tra ditional syllog istic. The syntax for S given a b ov e sug gests a natur al g eneralizatio n. An S † - formula is an ex pression of either o f the for ms ∃ ( l, m ) , ∀ ( l, m ) . (4) 3 W e pr ovide English glosse s for S † -formulas in the same wa y as for S , except that we sometimes r equire ne gate d sub jects: ∃ ( ¯ p, ¯ q ) Some non- p is not a q ∀ ( ¯ p, q ) Every non- p is a q . W e may rega rd S † the ex tension o f the traditiona l syllogistic with no un-level negation. The fragments R and R † Now ﬁx a c ountably inﬁnite set R , dis joint fr om P . W e may a ssume R to contain v arious Eng lish transitive verbs such as kil l , admire etc . A binary atom is an element o f R ; a binary liter al is an express ion of either o f the for ms r or ¯ r , where r is a binar y atom. A binary litera l is called p ositive if it is a bina ry atom; o therwise, ne gative . W e use the (p o ssibly decorated) v a riables r , s to ra nge over binary atoms, and t to r ange ov er binary literals. With these co nv en tions, a c-t erm is an ex pression of any of the forms l , ∃ ( p, t ) , ∀ ( p, t ) . (5) Thu s, a ll literals are , by deﬁnition, c-terms. W e use the v a riables c , d to ra nge ov er c-terms. With this conven tio n, an R - formula is an expre ssion of any of the forms ∃ ( p, c ) , ∃ ( c, p ) , ∀ ( p, c ) , ∀ ( c, ¯ p ); (6) Thu s, all S -formulas a re, by deﬁnition, R -formulas. W e gloss (no n-literal) c-terms using complex noun phra ses, as follows: ∃ ( q , r ) thing which r s some q ∀ ( q , r ) thing which r s every q (7) ∃ ( q , ¯ r ) thing whi ch do es n ot r every q ∀ ( q , ¯ r ) thing whi ch r s no q . (8) And we glos s R -for mu las inv olving such c-terms acc ordingly , thus: ∀ ( p, ∃ ( q , r )) Every p r s some q ∀ ( p, ∃ ( q , ¯ r )) No p r s every q ∃ ( p, ∃ ( q , r )) Some p r s some q ∃ ( p, ∃ ( q , ¯ r )) Some p do es n ot r every q ∀ ( p, ∀ ( q , r )) Every p r s every q ∀ ( p, ∀ ( q , ¯ r )) No p r s any q ∃ ( p, ∀ ( q , r )) Some p r s every q ∃ ( p, ∀ ( q , ¯ r )) Some p r s no q . In these glosses, q uantiﬁers in sub jects are assumed to hav e wide scop e; those in ob jects, na rrow s cop e. Also , the not in Some p do es n ot r e very q is ass umed to sco pe ov er the dir ect ob ject. Again, formulas of the forms ∃ ( c, p ) a nd ∀ ( c, ¯ p ), which are not glosse d here, will turn out to b e equiv alen t to formulas whic h ar e. W e may reg ard R a s the lang uage of the re lational syl lo gistic . Let us now extend R with noun-level negation just as we did with S . A n e-term is an expression of any o f the for ms l , ∃ ( l , t ) , ∀ ( l, t ) . (9) W e glo ss no n-literal e -terms along the lines of (7) and (8), but with (p os sibly) negated direct ob jects, for example: 4 ∀ ( ¯ q , r ) thing wh ich r s every no n- q ∀ ( ¯ q , ¯ r ) thing which r s no non- q . (Another wa y to glo ss the latter term w ould b e thing which only r s q s .) Thus, all c-terms ar e, by deﬁnition, e-terms. W e use the v ariables e , f to ra nge ov er e-terms. With this conv en tion, an R † - formula is an expression of any of the forms ∃ ( l, e ) , ∃ ( e, l ) , ∀ ( l , e ) , ∀ ( e, l ) , (10) with the obvious E nglish glosses. Note how ever that noun-level negation ma y be requir ed, b oth in sub jects and direct-o b jects: ∃ ( ¯ p, ∀ ( ¯ q , r )) Some non- p r s every non- q . W e may reg ard R † as the extension of the relationa l syllo gistic with no un-level negation. The fragment s R ∗ and R ∗† The rea der may hav e noticed that neither R nor R † can express the conclusion of Argument (1). W e now rectify this matter by introducing tw o additional frag ments which can. It will b e c onv enien t to e xtend the ‘bar’-nota tion (as in ¯ p and ¯ r ) to all e- terms. If l = ¯ p is a negative unary literal, then we take ¯ l to denote p ; and similarly for binary literals. If e is a n e -term o f the for m ∀ ( l , t ), w e deno te by ¯ e the co rresp onding e-ter m ∃ ( l , ¯ t ); if e is an e- term of the form ∃ ( l , t ), we denote by ¯ e the co rresp o nding e-ter m ∀ ( l, ¯ t ). T h us, for all e-terms e , we hav e ¯ ¯ e = e . Recalling tha t a c-term c is a n expr ession of any o f the forms l , ∃ ( p, t ) or ∀ ( p, t ), we call c p ositive if the liter al l o r t is p ositive; otherwise, negative. Evidently , if c is a pos itive c-term, then ¯ c is a nega tive c- term, and vice- versa. W e now deﬁne an R ∗ - formula to b e an expres sion of any of the forms ∃ ( c + , d ) , ∃ ( d, c + ) , ∀ ( c + , d ) , ∀ ( d, c + ) , (11) where c + ranges ov er p ositive c-ter ms a nd d over all c-terms (pos itive or neg- ative). These forms may then b e glossed by co mbin ing the glosses (3) and (7) in the o bvious wa y . Thus, fo r ex ample, the conclusion of Argument (1 ) may b e formalized in R ∗ as ∀ ( ∃ (man , kill) , ∃ (animal , kill)) Everything which k ills a man (is a thing which) kill s an animal . It is straightforward to check that, in pr oviding English glosses for R ∗ -formulas, noun-level negation (i.e. expressio ns such as non-man ) is not required. The study of R ∗ is motiv ated in par t by its close connection to the system describ ed in McAllester and Giv an [10], a nd in par t by the fact tha t, from b oth a pro of- theoretic and co mplexity-theoretic p oint o f view, it o ccupies an intermediate po sition, in a s ense that will b e made prec ise b elow. W e co me ﬁnally to the most ge neral of the f ragments studied here. Rec alling that the v a riables e and f range over e-ter ms, an R ∗† - formula is a n expres sion of either of the forms ∃ ( e, f ) , ∀ ( e, f ) . (12) These formulas receive the obvious glo sses, for example: 5 S R R ∗ S † R † R ∗† ⊆ ⊆ ⊆ ⊇ ⊇ ⊇ ⊇ S syllogistic S † syllogistic with noun-negation R relational syllogistic R † relational syllogistic with n oun-negation R ∗ relational syllogistic with complex sub jects R ∗† relational syllogistic with complex sub jects and nou n-negation Figure 1: The six s yllogistic frag ments: S , S † , R , R † , R ∗ and R ∗† . Expressio n V ar iables Synt ax unary atom o , p , q binary atom r , s unary literal l , m , n p | ¯ p binary literal t r | ¯ r po sitive c-term b + , c + p | ∃ ( p, r ) | ∀ ( p, r ) c-term c , d l | ∃ ( p, t ) | ∀ ( p, t ) e-term e , f l | ∃ ( l , t ) | ∀ ( l , t ) S -for mu la ∃ ( p, l ) | ∃ ( l , p ) | ∀ ( p, l ) | ∀ ( l, ¯ p ) S † -formula ∃ ( l, m ) | ∀ ( l , m ) R -formula ∃ ( p, c ) | ∃ ( c, p ) | ∀ ( p, c ) | ∀ ( c, ¯ p ) R † -formula ∃ ( l, e ) | ∃ ( e, l ) | ∀ ( l , e ) | ∀ ( e, l ) R ∗ -formula ∃ ( c + , d ) | ∃ ( d, c + ) | ∀ ( c + , d ) | ∀ ( d, c + ) R ∗† -formula ∃ ( e, f ) | ∀ ( e, f ) Figure 2: Syntax of syllogistic fra gments: quick reference g uide. ∀ ( ∃ ( animal , kill) , ∃ (man , kill)) Everything which k ills a non- animal (is a thing which) kills a non-man . W e may reg ard R ∗† as the re sult of extending R ∗ with no un-level neg ation. W e denote the set of S -formulas simply b y S , and similarly for S † , R , R † , R ∗ and R ∗† . Many of the results es tablished b elow apply to all o f these fra gments. Accordingly , we sa y that a syl lo gistic fr agment is any of the fragments S , S † , R , R † , R ∗ or R ∗† , and we use the v ariable F to range o v er the syllog istic fragments. Fig. 1 giv es an o verview of the syllogistic fragments, and F ig. 2 a quick reference guide to their s yntax. It will be conv enient to extend the bar- notation to formulas. If ϕ is an R ∗† - formula of the for m ∀ ( e, f ), we denote by ¯ ϕ the R ∗† -formula ∃ ( e, ¯ f ); if ϕ is a n R ∗† -formula of the form ∃ ( e, f ), we denote by ¯ ϕ the R ∗† -formula ∀ ( e, ¯ f ). It is easy to verify that, ¯ ¯ ϕ = ϕ , and that, fo r any syllo gistic fragment F , ϕ ∈ F implies ¯ ϕ ∈ F . 6 Seman tics W e now pr ovide semantics for the fragmen t R ∗† (and hence f or any syllogistic fr agment). A st ructur e A is a triple h A, { p A } p ∈ P , { r A } r ∈ R i , whe re A is a non-empty set, p A ⊆ A , for every p ∈ P , a nd r A ⊆ A 2 , fo r every r ∈ R . The set A is called the domain o f A . Given a structure A , we extend the maps p 7→ p A and r 7→ r A to all e- terms by setting ¯ p A = A \ p A ¯ r A = A 2 \ r A ∃ ( l, t ) A = { a ∈ A | h a, b i ∈ t A for some b ∈ l A } ∀ ( l, t ) A = { a ∈ A | h a, b i ∈ t A for all b ∈ l A } . W e deﬁne the truth-relatio n | = b etw een struc tures a nd R ∗† -formulas by declar- ing A | = ∀ ( e, f ) if and only if e A ⊆ f A , and A | = ∃ ( e, f ) if and only if e A ∩ f A 6 = ∅ . If Θ is a set of formulas, we write A | = Θ if, fo r all θ ∈ Θ , A | = θ . Of course , this deﬁnes the tr uth-relation for fo rmulas of any syllogistic fragment F . A formula θ is satisﬁable if there exists A such that A | = θ ; a set of formulas Θ is satisﬁable if there exists A suc h that A | = Θ. If, for all structur es A , A | = Θ implies A | = θ , we write Θ | = θ . W e take it as uncontrov ersial that Θ | = θ constitutes a ra tional r econstruction of the pre-theore tic judgment that a co nclusion θ may b e v alidly inferred from pr emises Θ. F o r example, the v alid argument Some artist is a b eekeeper Every artist is a carpenter No b eekeeper is a dentist Some carpenter is not a dentist corres po nds to the v a lid sequent of S -formulas: {∃ (artist , be ekeeper) , ∀ (artist , carp enter) , ∀ (beekeep er , dent ist) } | = ∃ (carp enter , den tist) . (13) Likewise, the v a lid ar gument Some artist hates so me artist No b eekeeper hates any b eekeep er Some artist is not a b eekeep er corres po nds to the v a lid sequent of R -for mu las: {∃ (artist , ∃ (artist , hate)) , ∀ (beekeep er , ∀ (b eekeeper , hate)) } | = ∃ (artist , b eekeeper) . (1 4) W e follow mo dern pr actice in taking universal quantiﬁcation not to car ry exis- ten tial commitment: thus { ∀ ( p, q ) } 6| = ∃ ( p, q ). 7 Absurdit y , negation and iden tiﬁcations A simple chec k shows that, for any structure A and any e-ter m e , ¯ e A = A \ e A ; hence, A 6| = ∃ ( e, ¯ e ). W e refer to a formula of this fo rm as an absur dity . E ven the smallest syllo gistic fragment, S , contains absurdities, namely , those of the form ∃ ( p, ¯ p ). Where the frag ment F is clea r from context, we write ⊥ indiﬀerently to denote any a bsurdity in F . F o r any structure A and any R ∗† -formula ϕ , A | = ϕ if a nd only if A 6| = ¯ ϕ . Also, ϕ a nd ¯ ϕ b elong to the same syllog istic fragments. Thus, if F is a syllo gistic fragment and ϕ a n F -fo rmula, we may regar d ¯ ϕ a s the nega tion of ϕ . F o r any structure A a nd any e-terms e and f , A | = ∃ ( e, f ) if a nd only if A | = ∃ ( f , e ). Also, these formulas b elo ng to the same syllog istic fragments. In the seq uel, therefore, we iden tify such pair s of formulas, silently transforming one to the other as necessa ry . Similarly , for the pa ir of formulas ∀ ( e, f ) and ∀ ( ¯ f , ¯ e ). These identiﬁcations mak e no essential diﬀerence to the results derived b elow on syllo gistic pr o of-systems; how ev er, they gr eatly simplify their presentation and analysis. 2.2 Syllogistic rules and r e ductio ad absur dum Let F b e a syllo gistic fra gment. A derivatio n r elation | ∼ in F is a subset of P ( F ) × F , where P ( F ) is the p ow er set of F . F or rea dability , we write Θ | ∼ θ instead of h Θ , θ i ∈ | ∼ . W e s ay that | ∼ is sound if Θ | ∼ θ implies Θ | = θ , and c omplete ( for F ) if Θ | = θ implies Θ | ∼ θ . A set Θ o f F -formulas is inc onsistent ( with r esp e ct to | ∼ ) if Θ | ∼ ⊥ for some absurdit y ⊥ ∈ F ; other wise, c onsistent . A weak ening of c ompleteness called r efutation-c ompleteness will prove imp ortant in t he sequel: | ∼ is r efutation-c omple te if all unsatisﬁable sets Θ are inconsisten t. Completeness trivially implies refutation- completeness, but no t conv ersely . W e are primarily interested in deriv ation relations induced by tw o diﬀerent so rts of deductive sys tem: direct s yllogistic systems and indirect syllog istic systems . These we now pro ceed to deﬁne. Direct syl logisti c systems Let F b e a syllogis tic fr agment. W e employ the following terminology . A syl lo gistic ru le (sometimes, simply: rule ) in F is a pa ir Θ /θ , where Θ is a ﬁnite set (possibly empt y) of F -formulas, and θ an F - formula. W e call Θ the ante c e dents of the rule, a nd θ its c onse quent . The rule Θ /θ is sound if Θ | = θ . W e g enerally display rules in ‘natural- deduction’ st yle. F o r example, ∀ ( q , o ) ∃ ( p, q ) ∃ ( p, o ) ∀ ( q , ¯ o ) ∃ ( p, q ) ∃ ( p, ¯ o ) (15) where p , q a nd o ar e unary atoms, are syllogistic rules in S , co rresp o nding to the traditional syllog isms Darii a nd F erio , resp ectively . A substitu tion is a function g = g 1 ∪ g 2 , where g 1 : P → P a nd g 2 : R → R . If θ is a n F -formula, deno te by g ( θ ) the F -formula which re sults by replacing any ato m (unar y or binary) in θ by its imag e under g , and similar ly for s ets of form ulas. An inst anc e of a syllogistic rule Θ / θ is the s yllogistic rule g (Θ) /g ( θ ), whe re g is a substitution. 8 Syllogistic rules which diﬀer o nly with resp ect to re-naming of unar y or binary ato ms will b e infor mally reg arded as identical, beca use they hav e the same instances. Th us, the letters p , q and o in (15) function, in eﬀect, as variables ranging ov er una ry atoms. It is often co nv enien t to display s yllogistic rules using v a riables ranging o v er other t ypes of expressio ns, understanding that these are just more compact wa ys of writing ﬁnite collections of syllo gistic r ules in the oﬃcial se nse. F or exa mple, the tw o rules (15) may be more compactly written ∀ ( q , l ) ∃ ( p, q ) ∃ ( p, l ) (D1) where p and q rang e ov er unary ato ms, but l rang es over una ry liter als . Fix a s yllogistic fra gment F , and let X be a set of syllogistic r ules in F . Deﬁne ⊢ X to b e the s mallest deriv ation r elation in F satisfying: 1. if θ ∈ Θ, then Θ ⊢ X θ ; 2. if { θ 1 , . . . , θ n } /θ is a r ule in X , g a substitution, Θ = Θ 1 ∪ · · · ∪ Θ n , a nd Θ i ⊢ X g ( θ i ) for all i (1 6 i 6 n ), then Θ ⊢ X g ( θ ). It is s imple to show that the deriv ation r elation ⊢ X is sound if and only if each rule in X is sound. Informally , we imag ine chaining together instances of the rules in X to con- struct de rivations , in the o bvious way; and w e refer to the resulting pro of system as the dir e ct syl lo gistic system deﬁne d by X . W e g enerally display deriv ations in natural-deduction style. Thus, for example, if X is any rule set containing (D1), the deriv ation ∀ (beekeep er , dent ist) ∀ (artist , carp enter) ∃ ( artist , b eekeeper) ∃ (carp enter , beekeeper) (D1) ∃ (carp enter , den tist) (D1) establishes that {∃ (artist , be ekeeper) , ∀ (artist , carp enter) , ∀ (beekeep er , dent ist) } ⊢ X ∃ (carp enter , den tist) , which, g iven the soundness of (D1), en tails the v a lidit y (13) . Notice, inciden- tally , that the ﬁrst application of (D1) in the ab ove der iv atio n dep ends o n the silent identiﬁcation of ∃ ( p, q ) and ∃ ( q , p ). In the sequel, we r eason freely ab out deriv a tions in order to establish prop er ties of deriv ation r elations. Deriv atio n relatio ns deﬁned b y dir ect pro o f-systems are easily seen to have po lynomial-time complex it y . Lemma 2.1. L et F b e a syl lo gistic fr agment, and X a ﬁnite set of s yl lo gistic rules in F . The pr oblem of determining whether Θ ⊢ X θ , for a given set of F -formulas Θ and F - formula θ , is in P Time . 9 Pr o of. Let Σ b e the set of all atoms (unar y or binar y) oc curring in Θ ∪ { θ } , together with o ne additional binary atom r . W e ﬁrst obser ve that, if there is a deriv a tion of θ from Θ using the rules X , then there is suc h a deriv ation in volving only the ato ms o ccurring in Σ. F or, given an y deriv atio n of θ from Θ, uniformly replace a n y una ry atom that do e s not o ccur in Θ ∪ { θ } with one that do es. Similarly , uniformly replace any binary atom which do e s not occur in Θ ∪ { θ } with o ne which does (or with r in cas e Θ ∪ { θ } co nt ains no bina ry atoms). This pro cess obviously leaves us with a der iv atio n of θ fr om Θ, using the r ules X . T o prov e the lemma, let the the total num be r of symbols o cc urring in Θ ∪ { θ } b e n . Certa inly , | Σ | 6 n . Let X comprise k 1 pro of-rules , each of which contains a t most k 2 atoms (unary or binary ). The n umber o f rule insta nces inv o lving only atoms in Σ is bounded by p ( n ) = k 1 n k 2 . Hence, we need never consider deriv atio ns with ‘depth’ grea ter than p ( n ). Let Θ i be the set of formulas inv o lving only the atoms in Σ, and deriv able from Θ using a deriv ation of depth i o r les s (0 6 i 6 p ( n )). Evide n tly , | Θ i | 6 | Θ | + p ( n ). It is then straig ht forward to compute the success ive Θ i in tota l time b ounded by a p olyno mial function of n . Indirect s yllogis tic s ystems In addition to sy llogistic rules, we consider the traditional r ule of r e duct io ad absur dum (RAA), which a llows us to derive a prop osition ϕ (in so me syllogistic fra gment) by ﬁrst a ssuming ¯ ϕ and der iving an absurdity . Again, we display this rule, in na tural-deduction-s t yle, a s · · · [ ¯ ϕ ] i · · · [ ¯ ϕ ] i · · · . . . . ⊥ ϕ (RAA) i . The interpretation is as follows: if an absurdity has b een derived from the set of premises Φ together with the pre mise ¯ ϕ , then ϕ may b e derived from the premises Φ alo ne. The premise ¯ ϕ may b e us ed several times (including zero ) in the deriv a tion of the absurdit y; moreov er, Φ is allo wed to contain ¯ ϕ . W e say that the (zero or mo re) brack eted instances of the premise ¯ ϕ hav e b een dischar ge d by applica tion of (RAA). The num erical sup er script i simply allows us to keep track of which application of (RAA) was r esp onsible for the discharge of which (instances of a ) premise. Fix a s yllogistic fra gment F , and let X be a set of syllogistic r ules in F . Deﬁne  X to b e the s mallest deriv ation r elation in F satisfying : 1. if θ ∈ Θ, then Θ  X θ ; 2. if { θ 1 , . . . , θ n } /θ is a r ule in X , g a substitution, Θ = Θ 1 ∪ · · · ∪ Θ n , a nd Θ i  X g ( θ i ) for all i (1 6 i 6 n ), then Θ  X g ( θ ); 3. if Θ ∪ { ¯ θ }  X ⊥ , then Θ  X θ . It is ag ain simple to show that  X is sound if and o nly if each rule in X is s ound. 10 Informally , we imagine chaining together instances of the rules in X and the rule (RAA) to form indir e ct derivations in the o bvious wa y; and we refer to the resulting pro of system as the indir e ct syl lo gistic syst em deﬁne d by X . W e generally displa y indirect deriv a tions in natural-deduction st yle, as the following example illustrates. First, consider the (evidently sound) sy llogistic rules: ∀ ( q , ¯ c ) ∃ ( p, c ) ∃ ( p, ¯ q ) (D3) ∀ ( p, ∀ ( o, t )) ∃ ( q , o ) ∀ ( p, ∃ ( q , t )) ( ∀∀ ) . Here, as us ual, o , p and q range over unary ato ms, t over binar y literals and c ov er c-terms. In a ddition, w e gener alize the rule (D1) given ab ov e s o that the v ar iable l (ranging ov er literals) may be replaced by the v a riable c (ranging ov er c-terms). Then we hav e the following indirec t deriv ation. ∀ (bkpr , ∃ (bkpr , hate)) ∀ (art , ∀ (art , hate)) [ ∃ (art , bkpr)] 1 ∀ (art , ∃ (bkpr , hate)) ( ∀∀ ) [ ∃ (art , bkp r)] 1 ∃ (bkpr , ∃ (bk pr , hate)) (D1) ∃ (bkpr , b kpr) (D3) ∀ (art , bk pr) (RAA) 1 . Here, tw o insta nces of the premise ∃ (ar tist , b eekeeper) have been discharged by the ﬁna l a pplication o f (RAA). This der iv ation establishes that, if X is any set of rules containing (D3), ( ∀∀ ) a nd (D1) (genera lized as indicated), then {∀ (artist , ∀ (artist , hate)) , ∀ (beekeep er , ∃ (beekeep er , hate)) }  X ∀ (artist , b eekeep er) . Bearing in mind that ∃ (beekeep er , hate) = ∀ (b eekeeper , hate), and g iven the soundness of the s yllogistic rules employ ed, this entails the v a lidity (14). It is imp ortant to realize that (RAA) is not itself a syllog istic rule. In particular, a n application of (RAA) in general de cr e ases the set of pre mises of the deriv ations in which it fea tures. As we shall see b elow, the sp ecial sta tus of (RAA) is essential: indirect syllo gistic systems are in general mo re p owerful than direct syllo gistic systems. If ⊢ X is refutation-complete, then  X is complete. F or supp ose Θ | = θ . Then Θ ∪ { ¯ θ } is unsatisﬁable; hence, by the re futation-completeness of ⊢ X , Θ ∪ { ¯ θ } ⊢ X ⊥ ; hence, using a single, ﬁna l application of (RAA), Θ  X θ . W e stress, ho wev er, tha t (RAA) is not in g eneral restr icted to the ﬁnal step in a n  X -deriv ation; rather, it may be e mploy ed at a ny po int , and any num b er of times. In particular , the r easoning o f L emma 2.1 fails: in Section 5 we pr esent a set of sy llogistic rules R ∗ such that  R ∗ is co- NP Time -hard. W e conclude this dis cussion by showing that indirect pro o f-systems yield a version of Lindenbaum’s Lemma. Let F be any syllog istic fragment. W e say that a set of F -formulas Θ is F - c omplete if, for every F -formula θ , e ither θ ∈ Θ or ¯ θ ∈ Θ. If the fr agment F is clear from context, we sa y c omplete instead o f F - complete; notice howev er that F -completeness of a set of for mulas has nothing to do with the completeness o f a deriv a tion relation. 11 Lemma 2.2. Le t F b e a syl lo gistic fr agment, Θ a set of F -formulas and X a set of syl lo gistic rules in F . If Θ is c onsistent with r esp e ct t o  X , t hen ther e exists a c omplete set of F -formulas ∆ ⊇ Θ c onsistent with r esp e ct t o  X . Pr o of. Let ( ϕ n ) n ∈ N enum erate the formulas of F . W e deﬁne a sequence o f consistent sets ∆ n as follows. Let ∆ 0 = Θ. F o r n > 0, let ∆ n +1 = ( ∆ n ∪ { ϕ n } if ∆ n ∪ { ϕ n } is co nsistent ∆ n ∪ { ¯ ϕ n } otherwise. and let ∆ = S n ∆ n . W e show by inductio n that each ∆ n is consistent. F o r suppo se ∆ n is consistent, but ∆ n +1 inconsistent. Then ∆ n ∪ { ϕ n }  X ⊥ , so that ∆ n  X ¯ ϕ n using (RAA). In addition, ∆ n ∪ { ¯ ϕ n }  X ⊥ ′ (where ⊥ ′ is some absurdity). W e ta ke a deriv a tion establishing the latter ass ertion and re place each (undischarged) premise ¯ ϕ with a deriv ation of ¯ ϕ fr om ∆ n . This shows ∆ n to b e inconsis ten t, a co nt radiction. Thus, ∆ n is consistent, for a ll n . Since deriv a tions are ﬁnite, ∆ is consis ten t. Ob viously , ∆ is F -complete. 3 S and S † : direct systems In the pr evious section, we deﬁned o ur g eneral notions, including the fr agments S and S † . These are the s implest in our pap er, coming without verbs. This section provides sound a nd complete syllogistic log ics for them. Let S b e the following set of r ules, where p and q rang e ov er unary atoms, l ov er unary lit erals, and and ϕ , ψ ov er S -formulas. ∀ ( q , l ) ∃ ( p, q ) ∃ ( p, l ) (D1) ∀ ( p, q ) ∀ ( q , l ) ∀ ( p, l ) (B) ∀ ( p, ¯ p ) ∀ ( p, l ) (A) ∃ ( p, l ) ∀ ( p, q ) ∃ ( q , l ) (D2) ψ ¯ ψ ϕ (X) ∀ ( q , ¯ l ) ∃ ( p, l ) ∃ ( p, ¯ q ) (D3) ∀ ( p, p ) (T) ∃ ( p, l ) ∃ ( p, p ) (I) Rules (D1) and (D3) we have met a lready (the latter in a r ather more g eneral form); Rules (D2) and (B) a re new, but evidently versions of classica l syllo- gisms. Rule (X) is the classic al r ule of ex falso qu o d lib et : fro m a contradiction, the reasoner may hav e anything he pleases [quo dlib et]. This r ule is not to be confused with (RAA): (X) is a syllo gistic rule, in the technical sense of this pa - per ; (RAA) is not. Rules (A), (T) and (I) hav e no classica l counterparts. Rule (A) stems fro m the fact that if all p are non- p , then there are no p whats o ever; v acuo usly , then, all p ar e l . T o see tha t (T) is needed, note that without it there would b e no wa y to derive ∀ ( p, p ) from the empty set of premises. R ule (I) is se lf-explanatory . W e rema rk tha t Rule (D1) is actua lly redundant in this context. F or consider a ny instance {∀ ( q ′ , l ′ ) , ∃ ( p ′ , q ′ ) } / ∃ ( p ′ , l ′ ) o f (D1). If l ′ = o 12 is a pos itive litera l, then, using the identiﬁcation ∃ ( e, f ) = ∃ ( f , e ), we may re- write this instance as {∀ ( q ′ , o ) , ∃ ( q ′ , p ′ ) } / ∃ ( o, p ′ ), which is a n instance of (D2). On the other hand, if l ′ = ¯ o , then, using the identiﬁcation ∀ ( e, f ) = ∀ ( ¯ f , ¯ e ), we may re-wr ite the instance a s { ∀ ( o, ¯ q ′ ) , ∃ ( p ′ , q ′ ) } / ∃ ( p ′ , ¯ o ), which is an instance o f (D3). W e retain Rule (D1) for ease of use, and to make the relationship to the system R introduced in Section 4 .2 more tra nsparent. T ur ning now to the fr agment S † , let S † comprise the following syllo gistic rules, where l , m and n r ange ov er unar y litera ls, and ϕ , ψ ov er S † -formulas. ∃ ( l, n ) ∀ ( l, m ) ∃ ( m, n ) (D) ∀ ( l, m ) ∀ ( m, n ) ∀ ( l, n ) (B) ∀ ( l, ¯ l ) ∀ ( l, m ) (A) ∀ ( l, l ) (T) ∃ ( l, m ) ∃ ( l, l ) (I) ψ ¯ ψ ϕ (X) ∀ ( ¯ l , l ) ∃ ( l, l ) (N) The rules in S † are the natural g eneraliza tions of those in S to reﬂect the more liber al s yntax of S † , but with tw o small changes: ﬁrst, rules (D1)–(D3) have merged into a single r ule (D); sec ond, an extr a rule (N) has b een added. The soundness of (N) is due to the requirement that our struc tures hav e non- empt y domains. That S † really is an extension o f S is g iven by the following lemma. Lemma 3.1. Any instanc e of a rule in S † which involves only formulas in the fr agment S is an inst anc e of a rule of S , and c onversely. Pr o of. W e prove the ﬁrst statement b y considering the rules of S † one at a time. W e illustrate her e with Rule (D). Supp ose that all formulas in so me instance {∃ ( l , n ) , ∀ ( l , m ) } / ∃ ( m, n ) of this rule a re in S . If m is p ositive, then so is l (since ∀ ( l , m ) ∈ S ). In this case, our insta nce matches Rule (D2) in the system S . On the other ha nd, if m is neg ative, then n is p o sitive (since ∃ ( m, n ) ∈ S ). W e then use the identiﬁcations ∀ ( l , m ) = ∀ ( ¯ m, ¯ l ), ∃ ( l , n ) = ∃ ( n, l ), and ∃ ( m, n ) = ∃ ( n, m ), and re-write the rule-instance as { ∃ ( n, l ) , ∀ ( ¯ m, ¯ l ) } / ∃ ( n, m ), which (b earing in mind that n and ¯ m are unar y atoms) ma tches Rule (D3) in the sy stem S . The o ther rule s ar e dealt with similarly . No te in particular that all instances of Rule (N) inv olve a for mula lying outside the fra gment S , so that the sta tement ho lds trivia lly for this rule. The seco nd statement of the lemma is completely r outine. In view of Lemma 3.1, we hav e tak en the lib erty of using the same names—(B), (A), (T), (I) and (X)—for cor resp onding rules in S and S † . It is obvious that ⊢ S and ⊢ S † are sound. W e now pr ov e their completeness (for the fra gments for S and S † , resp ectively). It is conv enien t to prove the completeness of ⊢ S † , and then to derive the result for ⊢ S as a sp ecial ca se. Starting on the pro of In the rema inder of this section, then, a formula is an S † -formula unless o therwise stated. A universal formula is one of the form ∀ ( l , m ); a n existential formula is one of the form ∃ ( l , m ). If A a nd B ar e 13 structures with disjoint domains A and B , re sp ectively , denote by A ∪ B the structure with do main A ∪ B a nd interpretations p A ∪ B = p A ∪ p B for any unary atom p . (Note that S † features only unar y atoms.) Lemma 3.2. Supp ose Φ ∪ Ψ | = θ , wher e Φ is a set of universal formulas, Ψ a set of existent ial formulas, and θ a formula. 1. If Φ ∪ Ψ is satisﬁable and θ is universal, then Φ | = θ . 2. If Ψ 6 = ∅ and θ is existential, then ther e exists ψ ∈ Ψ such that Φ ∪ { ψ } | = θ . 3. If Ψ = ∅ and θ = ∃ ( l , m ) is existential, then Φ | = ∀ ( ¯ l , l ) and Φ | = ∀ ( ¯ m, m ) . Pr o of. In each ca se, assume the contrary . 1. There e xist structure s A | = Φ ∪ Ψ and B | = Φ ∪ ¯ θ . W e may assume that A ∩ B = ∅ . But then A ∪ B | = Φ ∪ Ψ ∪ { ¯ θ } , a contradiction. 2. F or every ψ ∈ Ψ, ther e exists a structure A ψ such tha t A ψ | = Φ ∪ { ψ , ¯ θ } . Again, we assume that the domains a re pa irwise disjoint. But then S ψ ∈ Ψ A ψ | = Φ ∪ Ψ ∪ { ¯ θ } , a contradiction. 3. If Φ 6| = ∀ ( ¯ l , l ), ther e exists a structur e A suc h that A | = Φ ∪ {∃ ( ¯ l, ¯ l ) } . Cho ose a ∈ ¯ l A , a nd let B be the s tructure obtained b y r estricting A to the singleton do main { a } . Then B | = Φ ∪ { ¯ θ } , a contradiction. A similar argument applies if Φ 6| = ∀ ( ¯ m, m ). W e wr ite Φ ⊢ BT A ϕ if there is a deriv ation in ⊢ S † of ϕ from Φ emplo ying only the r ules (B), (T) and (A). W e ca ll a set V o f liter als c onsistent if l ∈ V implies ¯ l 6∈ V (otherwise inc onsistent ); we c all V c omplete if l 6∈ V implies ¯ l ∈ V . Let Φ b e a se t o f universal formulas and V a set of liter als. Deﬁne S Φ to b e the set of literals: { m : Φ ⊢ BT A ∀ ( l, m ) for so me l ∈ V } . W e say that V is Φ- close d if V Φ ⊆ V . Lemma 3. 3 . L et Φ b e a set of universal formulas and V a set of liter als. Then V Φ is Φ -close d, and V ⊆ V Φ . Pr o of. Almost immediate, by r ules (B) and (T). Evidently , the union of any colle ction o f Φ-clos ed s ets o f literals is Φ-closed. Lemma 3.4. L et Φ b e a set of un iversal formulas and V a set of liter als. If V Φ is inc onsistent, then ther e exist liter als l , l ′ ∈ V such that Φ ⊢ BT A ∀ ( l, ¯ l ′ ) . 14 Pr o of. If m, ¯ m ∈ V Φ , pick l , l ′ ∈ V with Φ ⊢ BT A ∀ ( l, m ) a nd Φ ⊢ BT A ∀ ( l ′ , ¯ m ). Re-writing ∀ ( l ′ , ¯ m ) as ∀ ( m, ¯ l ′ ), we have a deriv a tion from Φ . . . . ∀ ( l, m ) . . . . ∀ ( m, ¯ l ′ ) ∀ ( l, ¯ l ′ ) (B) , as claimed. Lemma 3.5. L et Φ b e a set of universal formulas. A ny non-empty, Φ - close d, c onsistent set of liter als has a Φ -close d, c onsist en t, c omplete ext ension. Pr o of. Let V 0 be a non-empty , Φ-closed, co nsistent set of literals. Enumerate the set of a ll literals as l 1 , l 2 , . . . . F o r all i > 0, deﬁne V i +1 = ( V i if l i +1 ∈ V i  V i ∪ { ¯ l i +1 }  Φ otherwise. and deﬁne V = S i V i . Th us, V is Φ-closed, complete a nd includes V 0 . W e remark that, sinc e V 0 ⊆ V i , V i is non-empty for all i > 0. T o show that V is consistent, suppo se o therwise. Let i b e the least natura l num ber such that V i +1 is inconsistent. Hence, V i +1 =  V i ∪ { ¯ l i +1 }  Φ , and so by Lemma 3.4 there exist literals l , l ′ ∈ V i ∪ { ¯ l i +1 } such that Φ ⊢ BT A ∀ ( l, ¯ l ′ ). Since V i is Φ-closed and consistent, l and l ′ cannot bo th be in V i , and so we may as sume without loss of generality that l ′ = ¯ l i +1 , whence Φ ⊢ BT A ∀ ( l, l i +1 ). Now, either l ∈ V i or l = ¯ l i +1 . In the former ca se, w e hav e l i +1 ∈ V i , bec ause V i is Φ-closed; in the latter, let l ′′ be a ny literal in V i (whic h w e know to be non-empty). Then the inference ∀ ( ¯ l i +1 , l i +1 ) ∀ ( l ′′ , l i +1 ) (A) guarantees that, ag ain, l i +1 ∈ V i . Either wa y , V i +1 = V i , a contradiction. Theorem 3.6. The derivation r elation ⊢ S † is soun d and c omplete for S † . Pr o of. Soundness is routine. F or the converse, let Θ be a set of S † -formulas and θ a S † -formula, and suppo se Θ | = θ . By the co mpactness theorem for ﬁrst-or der logic, we may s afely a ssume that Θ is ﬁnite. Suppo se for the moment tha t Θ is satisﬁable, and write Θ = Φ ∪ Ψ, where Φ is a set of universal formulas and Ψ a set of existential formulas. W e co nsider thr ee case s: (1) θ is universal; (2) θ is existential and Ψ is no n-empty; and (3) θ is existential and Ψ is empty . In the r emainder of this pr o of, we simplify o ur notation to write ⊢ for ⊢ S † . Case (1): W rite θ = ∀ ( l 0 , m 0 ). B y Lemma 3 .2, Part 1, Φ | = θ . Let V 0 = { l 0 , ¯ m 0 } Φ . (16) Then V 0 is non-empty and Φ- closed, by Lemma 3.3. W e claim that V 0 is incon- sistent. F or s uppo se other wise. By Lemma 3.5, let V be a consistent complete 15 extension of V 0 , and deﬁne A to b e the structure w ith singleton domain { a } given by p A = ( { a } if p ∈ V ∅ otherwise, for e very atom p . It is easily seen that A | = Φ ∪ ¯ θ , a co nt radiction. So b y Lemma 3.4, there exis t literals l , l ′ ∈ { l 0 , ¯ m 0 } such that Φ ⊢ BT A ∀ ( l, ¯ l ′ ) . (17) By exchanging l and l ′ if necessar y , we hav e tw o sub-ca ses: (i) l = l 0 and l ′ = ¯ m 0 ; (ii) l = l ′ ∈ { l 0 , ¯ m 0 } . In sub-cas e (i), (17) s imply a sserts that Φ ⊢ θ . In sub-cas e (ii), we hav e o ne of the der iv atio ns fro m Φ . . . . ∀ ( l 0 , ¯ l 0 ) ∀ ( ¯ m 0 , ¯ l 0 ) (A) . . . . ∀ ( ¯ m 0 , m 0 ) ∀ ( l 0 , m 0 ) (A) , and so Φ ⊢ θ . Case ( 2): W rite θ = ∃ ( l , m ). By Lemma 3.2, Part 2 , there exists ψ = ∃ ( l 0 , m 0 ) ∈ Ψ such that Φ ∪ { ψ } | = θ . Set V 0 = { l 0 , m 0 , ¯ l } Φ . The s et V 0 m ust b e inconsistent. F or otherwise, we can easily construct, using a par allel ar gument to that emplo yed in Case (1 ), a structure A suc h that A | = Φ ∪ { ψ , ¯ θ } , contradicting the fact that Φ ∪ { ψ } | = θ . H ence, there exis t literals l 1 , l 2 ∈ { l 0 , m 0 , ¯ l } such that Φ ⊢ BT A ∀ ( l 1 , ¯ l 2 ). If l 1 and l 2 are both in { l 0 , m 0 } , then Θ is unsatisﬁable, contrary to h yp othesis. So assume, witho ut loss of g enerality , that l 2 = ¯ l . Th us, Φ ⊢ BT A ∀ ( l 1 , l ), and we have the following po ssibilities: (i) l 1 = l 0 ; (ii) l 1 = m 0 ; (iii) l 1 = ¯ l . Possibilit y (i) yields Φ ⊢ ∀ ( l 0 , l ). Possibilit y (ii) yields Φ ⊢ ∀ ( m 0 , l ). Possibilit y (iii) a lso yields Φ ⊢ ∀ ( l 0 , l ), via the deriv ation . . . . ∀ ( ¯ l , l ) ∀ ( l 0 , l ) (A) . In other words, we have pr ov ed: either Φ ⊢ ∀ ( l 0 , l ) or Φ ⊢ ∀ ( m 0 , l ) . (18) Replacing l by m in the a b ov e arg umen t yields, in ex actly the sa me wa y: either Φ ⊢ ∀ ( l 0 , m ) or Φ ⊢ ∀ ( m 0 , m ) . (19) Considering (18), we may a ssume, by transp osing l 0 and m 0 if neces sary , that Φ ⊢ ∀ ( l 0 , l ). This leav es us with the tw o p o ssibilities in (19). If Φ ⊢ ∀ ( m 0 , m ), 16 we hav e ∃ ( l 0 , m 0 ) . . . . ∀ ( l 0 , l ) ∃ ( l, m 0 ) (D) . . . . ∀ ( m 0 , m ) ∃ ( l, m ) (D); if, on the o ther hand, Φ ⊢ ∀ ( l 0 , m ), we hav e ∃ ( l 0 , m 0 ) ∃ ( l 0 , l 0 ) (I) . . . . ∀ ( l 0 , l ) ∃ ( l, l 0 ) (D) . . . . ∀ ( l 0 , m ) ∃ ( l, m ) (D) . (20) Either wa y , Θ ⊢ θ , as re quired. Case (3): W rite θ = ∃ ( l , m ). Since Θ = Φ | = θ , b y Lemma 3.2, Part 3, Φ | = ∀ ( ¯ l, l ) and Φ | = ∀ ( ¯ m, m ). By Cas e (1), Φ ⊢ ∀ ( ¯ l , l ) and Φ ⊢ ∀ ( ¯ m, m ). Therefore, we hav e the deriv ation from Φ . . . . ∀ ( ¯ l , l ) ∃ ( l, l ) (N) . . . . ∀ ( ¯ m, m ) ∀ ( l, m ) (A) ∃ ( l, m ) (D) , and Θ ⊢ θ . W e hav e now shown that, for Θ satisﬁable, Θ | = θ implies Θ ⊢ θ . It rema ins only to consider the case wher e Θ is unsatisﬁable. If s o, let Θ ′ ∪ { θ ′ } b e a minimal uns atisﬁable subset of Θ . (Remem b er, we a re a llow ed to assume that Θ is ﬁnite.) Hence, Θ ′ is satisﬁable, with Θ ′ | = ¯ θ ′ . By the previous ar gument, Θ ′ ⊢ ¯ θ ′ . Thu s, we have the deriv ation fro m Θ ′ ∪ { θ ′ } . . . . ¯ θ ′ θ ′ θ (X) , and Θ ⊢ θ . Theorem 3.7. The derivation r elation ⊢ S is sound and c omplete for S . Pr o of. Soundness is ob vious. Supp os e that Θ is a satisﬁable set of S -formulas and θ an S -formula such that Θ | = θ . By Theor em 3 .6, Θ ⊢ S † θ . W e must show that Θ ⊢ S θ . Since deriv ations (in ⊢ S † ) are ﬁnite, we may a ssume without los s of generality that Θ is ﬁnite. F or the mo men t, let us further assume that Θ is satisﬁable. Let p , q be any unary a toms. W e cla im tha t Θ 6| = ∀ ( ¯ p, q ). F or otherwise, by Lemma 3.2 Part 1 , there exis ts a subset Φ ⊆ Θ of u niversal formulas such that 17 Φ | = ∀ ( ¯ p, q ). Now let A b e any structure s uch that o A = ∅ for every unary atom o . S ince Φ ⊆ S , A | = Φ; but A 6| = ∀ ( ¯ p, q ), a contradiction. F urthermore, since ⊢ S † is sound, it follows of co urse that Θ 6⊢ S † ∀ ( ¯ p, q ). Consider any der iv atio n in ⊢ S † with premises Θ. W e cla im that, if this deriv a tion co n tains any formula of the form ∃ ( ¯ p, ¯ q ), then the ﬁnal conc lusion o f this der iv atio n is also of that form. F or, since Θ is, by assumption, s atisﬁable, (X) cannot b e used in the der iv ation; and the only other rules in S † with any premise of the for m ∃ ( ¯ p, ¯ q ) a re (I) and (D), thus: ∃ ( ¯ p, ¯ q ) ∃ ( ¯ p, ¯ p ) (I) ∃ ( ¯ p, ¯ q ) ∀ ( ¯ p, l ) ∃ ( ¯ q , l ) (D) . By the observ a tion of the previo us parag raph, the literal l in this instance o f (D) m ust b e nega tive. In either case , the consequent of the rule is o f the form ∃ ( ¯ p, ¯ q ), as cla imed. Now take any der iv ation of θ fro m Θ in ⊢ S † . (W e know that one exists.) Since θ ∈ S is deﬁnitely no t of the form ∃ ( ¯ p, ¯ q ), it follows from the previous t wo pa ragr aphs that this deriv ation cannot inv olve any fo rmula o f e ither of the forms ∀ ( ¯ p, q ) o r ∃ ( ¯ p , ¯ q ). That is, all the formulas are in S . By Lemma 3.1, Θ ⊢ S θ . This prov es the theorem in the case wher e the (ﬁnite) set of S -formulas Θ is satisﬁable. If Θ is unsatisﬁable, let Θ ′ ∪ { θ ′ } b e a minimal unsa tisﬁable subset of Θ. Hence, Θ ′ is sa tisﬁable, with Θ ′ | = ¯ θ ′ . By the r esult just esta blished, Θ ′ ⊢ S ¯ θ ′ . By a sing le application of (X), Θ ⊢ S θ . 4 R : a refutation complete system W e turn from S to R . W e exhibit a set R o f sy llogistic rules in R , and pr ov e that ⊢ R is a sound and r efutation -complete deriv ation re lation for R ; we also prov e that there is no ﬁnite set of rules X in R such that X is sound and c omplete for R . This highlights the impo rtance of (RAA) in obtaining complete logics. Our refutation-c ompleteness pro o f also implies that the problem of deter mining whether a given R -s equent is v alid is in NLogSp ace , a fact which is other wise not obvious. 4.1 There are no sound and complete syllogistic systems for R Theorem 4.1. Ther e exists no ﬁnite set X of syl lo gistic ru les in R such that ⊢ X is b oth sound and c omplete. Pr o of. Let X b e any ﬁ nite set of syllogistic rules for R , and supp ose ⊢ X is sound. W e show that it is no t c omplete. Since X is ﬁnite, ﬁx n ∈ N greater tha n the nu mber of a nteceden ts in any r ule in X . 18 Let p 1 , . . . , p n be dis tinct unary a toms and r a binar y atom. Let Γ be the following set of R -for mu las: ∀ ( p i , ∃ ( p i +1 , r )) (1 6 i < n ) (21) ∀ ( p 1 , ∀ ( p n , r )) (22) ∀ ( p, p ) ( p ∈ P ) (23) ∀ ( p i , ¯ p j ) (1 6 i < j 6 n ) (24) and let γ be the R -for mula ∀ ( p 1 , ∃ ( p n , r )). Observe that Γ | = γ . T o see this, let A | = Γ. If p A 1 = ∅ , then trivially A | = γ ; on the other hand, if p A 1 6 = ∅ , a simple induction using formulas (21) shows that p A i 6 = ∅ for all i (1 6 i 6 n ), whence A | = γ by (22) . F o r 1 6 i < n , let ∆ i = Γ \ {∀ ( p i , ∃ ( p i +1 , r )) } . Claim 4.2. If ϕ ∈ R and ∆ i | = ϕ , t hen ϕ ∈ Γ . It follows fr om this c laim that Γ 6⊢ X γ . F or , since no rule of X has more than n − 1 anteceden ts, any instance of those anteceden ts contained in Γ must be contained in ∆ i for some i . Le t δ b e the co rresp o nding instance o f the consequent of that r ule. Since ⊢ X is sound, ∆ i | = δ . By Claim 4.2, δ ∈ Γ. By induction on the num ber of steps in deriv ations, we se e that no deriv ation fr om Γ leads to a formula not in Γ. But γ 6∈ Γ. Pr o of of Claim. Certainly , ∆ i has a mo del, for instance the mo del A i given by: p 1 GF ED   / / p 2 / / · · · / / p i p i +1 / / · · · / / p n (25) Here, A = { p 1 , . . . , p n } , p A i j = { p j } for all j (1 6 j 6 n ), and r A i is indicated by the a rrows. All other atoms (unar y or binar y) a re a ssumed to have empty extensions. Note that there is no a rrow fr om p i to p i +1 . W e consider the v arious p ossibilities for ϕ in turn and chec k that either ϕ ∈ Γ o r there is a mo del of ∆ i in which ϕ is false. (i) ϕ is of the form ∀ ( p, p ). Then ϕ ∈ Γ by (23). (ii) ϕ is not of the form ∀ ( p, p ), a nd inv olves a t least o ne unary or binary atom other than p 1 , . . . , p n , r . In this ca se, it is straig htf orward to mo dify A i so as to obtain a mo del A ′ i of ∆ i such that A ′ i 6| = ϕ . Hencefo rth, then, we may assume that ϕ inv olves no atoms other than p 1 , . . . , p n , r . (iii) ϕ is of the form ∀ ( p j , p k ). If j = k , then ϕ ∈ Γ, b y (23). If j 6 = k , then A i 6| = ϕ by insp ection. (iv) ϕ is of the form ∀ ( p j , ¯ p k ). If j = k , then A i 6| = ϕ , since p A i j 6 = ∅ . If j 6 = k , then ϕ ∈ Γ, b y (24) and the identiﬁcation ∀ ( p j , ¯ p k ) = ∀ ( p k , ¯ p j ). (v) ϕ is of the form ∀ ( p j , ∀ ( p k , r )). If j = 1 and k = n , then ϕ ∈ Γ, by (22). So we may ass ume that either j > 1 or k < n , in which case, k 6 = j + 1 implies 19 A i 6| = ϕ , by insp ectio n. Hence, we may assume that ϕ = ∀ ( p j , ∀ ( p j +1 , r )), with j < n . Let B i,j be the structure o btained fr om A i by adding a second p oint b to the interpretation of p j +1 , and to which p j is not rela ted by r . In picture s: p 1 GF ED   / / p 2 / / · · · p j / / p j +1 / / p j +2 / / · · · / / p i p i +1 / / · · · p n b : : u u u u u u (This pictur e shows j + 2 < i . Similar pictures are p ossible in all other case s.) By insp ection, B i,j | = ∆ i , but B i,j 6| = ϕ . (vi) ϕ is o f the form ∀ ( p j , ∃ ( p k , r )). If k = j + 1, then ϕ ∈ Γ, by (21). Moreov er, if k 6 = j + 1, then, unless j = 1 and k = n , A i 6| = ϕ , b y inspection. Hence we may assume ϕ = ∀ ( p 1 , ∃ ( p n , r )). Let C i be the str ucture: p 1 / / p 2 / / · · · / / p i , with p C i j = ∅ for all j ( i < j 6 n ). Then C i | = ∆ i , but C i 6| = ϕ . (vii) ϕ is of either of the forms ∀ ( p j , ∀ ( p k , ¯ r )), ∀ ( p j , ∃ ( p k , ¯ r )). Deﬁne A ′′ i to be like A i except that r A ′′ i additionally contains the pair of p oints h p j , p k i . By insp ection, A ′′ i | = ∆ i , but A ′′ i 6| = ϕ . (viii) ϕ is o f the form ∃ ( p, c ). Let A 0 be a structure ov er any do main in which every atom has empty extension. Then A 0 | = ∆ i , but A 0 6| = ϕ . This also completes the pr o of of Theorem 4 .1. 4.2 A refutation-complete system for R Theorem 4.1 notwithstanding, we exhibit below a ﬁnite set R of rules in R , such that ⊢ R is sound and r efutation -complete. W e remind the reader that p and q range ov er unar y atoms, c over c- terms, and t over binary litera ls. ∃ ( p, q ) ∀ ( q , c ) ∃ ( p, c ) (D1) ∀ ( p, q ) ∀ ( q , c ) ∀ ( p, c ) (B) ∀ ( p, q ) ∃ ( p, c ) ∃ ( q , c ) (D2) ∀ ( p, p ) (T) ∃ ( p, c ) ∃ ( p, p ) (I) ∀ ( q , ¯ c ) ∃ ( p, c ) ∃ ( p, ¯ q ) (D3) ∀ ( p, ¯ p ) ∀ ( p, c ) (A) ∃ ( p, ∃ ( q , t )) ∃ ( q , q ) (I I) ∀ ( p, ∀ ( q ′ , t )) ∃ ( q, q ′ ) ∀ ( p, ∃ ( q , t )) ( ∀∀ ) ∃ ( p, ∃ ( q , t )) ∀ ( q , q ′ ) ∃ ( p, ∃ ( q ′ , t )) ( ∃∃ ) ∀ ( p, ∃ ( q , t )) ∀ ( q , q ′ ) ∀ ( p, ∃ ( q ′ , t )) ( ∀∃ ) 20 Rules (D1), (D2), (D3), (B), (A), (T) and (I) a re natural g eneraliza tions of their namesakes in S . In contrast, ( ∀∀ ), ( ∃∃ ), ( ∀∃ ) and (I I) express genuinely relational lo gical principles. In so me settings, these la st rules a re ca lled mono- tonicity principles . Because we seek only refutation-co mpleteness for ⊢ R , we do not need a version of the r ule (X). T o illustrate thes e rules, let n b e a n y integer grea ter tha n 1, let Γ ∗ = {∀ ( p i , ∃ ( p i +1 , r )) | 1 6 i < n } ∪ {∀ ( p 1 , ∀ ( p n , r )) } and let γ = ∀ ( p 1 , ∃ ( p n , r )). Noting that ¯ γ = ∃ ( p 1 , ∀ ( p n , ¯ r )), we hav e the deriv a- tion (shown her e for n > 3 ) ∀ ( p 1 , ∀ ( p n , r )) ∃ ( p 1 , ∀ ( p n , ¯ r )) ∃ ( p 1 , p 1 ) (I) ∀ ( p 1 , ∃ ( p 2 , r )) ∃ ( p 1 , ∃ ( p 2 , r )) (D1) ∃ ( p 2 , p 2 ) (I I) ∀ ( p 2 , ∃ ( p 3 , r )) ∃ ( p 2 , ∃ ( p 3 , r )) (D1) ∃ ( p 3 , p 3 ) (I I) . . . ∃ ( p n , p n ) ∀ ( p 1 , ∃ ( p n , r )) ( ∀∀ ) ∃ ( p 1 , ∀ ( p n , ¯ r )) ∃ ( p 1 , ¯ p 1 ) (D3), showing that Γ ∗ ∪ { ¯ γ } ⊢ R ⊥ . B y contrast, since Γ ∗ ⊆ Γ, where Γ is the set of formulas used in the pro o f o f Theo rem 4 .1, we know that, for any ﬁnite set X of syllogistic rules, n can b e ma de suﬃciently lar ge that Γ ∗ 6⊢ X γ . Starting on the pro of F or the remainder of this section, ﬁx a ﬁnite, non- empt y set Γ o f R -formulas. As usual, we tak e the (p ossibly decorated) v a riables p , q to ra nge over unary atoms , r ov er binary atoms, t over binary liter als, a nd c , d over c-terms. W e write c ⇒ d if c = d o r there exists a sequence of unary atoms p 0 , . . . , p k such that c = p 0 , ∀ ( p k , d ) ∈ Γ, and ∀ ( p i , p i +1 ) ∈ Γ for a ll i (0 6 i < k ). If V is a set o f c-terms, wr ite V ⇒ d if c ⇒ d for so me c ∈ V . Lemma 4.3. L et V b e a set of c-terms. 1. If V ⇒ c , t hen either c ∈ V or ther e exists p ∈ V su ch that Γ ⊢ R ∀ ( p, c ) ; 2. if V ⇒ p , then ther e ex ists p 0 ∈ V such that Γ ⊢ R ∀ ( p 0 , p ) ; 3. if p ⇒ c , t hen Γ ⊢ R ∀ ( p, c ) . Pr o of. Almost immediate, noting tha t R contains the rules (B) and (T). In the e nsuing lemmas , we show that, if Γ is co nsistent (wit h resp ect to ⊢ R ), then Γ is satisﬁable. As a ﬁrst step, we cr eate plent y of ob jects fro m which to 21 construct a potential mo de l of Γ. If 0 6 i 6 2 and V is a set o f c-terms with 1 6 | V | 6 2, let b V ,i denote some o b ject or other, and a ssume that the v ar ious b V ,i are pairw ise distinct. Now set B 0 = { b { p,c } , 0 | ∃ ( p, c ) ∈ Γ } . Lemma 4.4. L et b V , 0 ∈ B 0 , and let p, c ∈ V . Then Γ ⊢ R ∃ ( p, c ) . Pr o of. If p 6 = c , then V = { p, c } and ∃ ( p, c ) ∈ Γ by construction. If p = c , then V = { p, d } fo r some d , a nd we have the de riv a tion ∃ ( p, d ) ∃ ( p, p ) (I) . W e now deﬁne sets B 1 , B 2 , . . . inductively as follows. Supp ose B k has b een deﬁned. Let B k +1 = B k ∪{ b { p } ,i | 1 6 i 6 2 and, for some b V ,j ∈ B k and some t , V ⇒ ∃ ( p, t ) } . Let B = S 0 6 k B k . Evide nt ly , B is ﬁnite. (Indeed, | B | is bo unded by a linear function of | Γ | .) It is immediate from the co nstruction of B that, if b V ,i ∈ B 0 , then 1 6 | V | 6 2, V con tains at least one unar y atom p , a nd i = 0 . On the other ha nd, if b V ,i ∈ B k for k > 0, then V = { p } for some unar y a tom p , and i is either 1 o r 2 . The intuition her e is that the ele ment s of B 0 are witnesses for the exis tent ial for mulas of Γ, while the elements of B k +1 are the witnesses for existential c-terms s atisﬁed by elements of B k . Lemma 4.5. If b V ,i ∈ B , V ⇒ p and V ⇒ c , then Γ ⊢ R ∃ ( p, c ) . Pr o of. Let k b e the smallest num ber suc h that b S,i ∈ B k . W e pro ceed by induction on k . F o r the case k = 0 , we have b V ,i = b V , 0 ∈ B 0 . By L emma 4.3 Part 2, there exists q 1 ∈ S such that Γ ⊢ R ∀ ( q 1 , p ). By Lemma 4.3 Part 1, either c ∈ V o r there exists q 2 ∈ V such that Γ ⊢ R ∀ ( q 2 , c ). In the former cas e, Lemma 4.4 yields Γ ⊢ R ∃ ( q 1 , c ), so that we hav e the der iv atio n . . . . ∃ ( q 1 , c ) . . . . ∀ ( q 1 , p ) ∃ ( p, c ) (D2) . In the latter case, Lemma 4 .4 yields Γ ⊢ R ∃ ( q 1 , q 2 ), so that we hav e the de riv a tion . . . . ∃ ( q 1 , q 2 ) . . . . ∀ ( q 2 , c ) ∃ ( q 1 , c ) (D1) . . . . ∀ ( q 1 , p ) ∃ ( p, c ) (D2) . 22 F o r the case k > 0, b V ,i ∈ B k implies V = { p k } for some p k , and 1 6 i 6 2. By cons truction of B k , there exist b W ,j ∈ B k − 1 , p k − 1 ∈ W and binar y ato m r , such that W ⇒ ∃ ( p k , r ). By inductive hypo thesis, Γ ⊢ R ∃ ( p k − 1 , ∃ ( p k , r )), and by Lemma 4.3 Part 3, Γ ⊢ R ∀ ( p k , p ), a nd Γ ⊢ R ∀ ( p k , c ). Ther efore, we have the deriv a tion . . . . ∃ ( p k − 1 , ∃ ( p k , r )) ∃ ( p k , p k ) (I I) . . . . ∀ ( p k , p ) ∃ ( p k , p ) (D1) . . . . ∀ ( p k , c ) ∃ ( p, c ) (D1) . Lemma 4.6. If b V ,i ∈ B , V ⇒ c , V ⇒ d , and c 6 = d , t hen ther e exists a unary atom p s u ch that either: (i) Γ ⊢ R ∃ ( p, c ) and Γ ⊢ R ∀ ( p, d ) ; or (ii) Γ ⊢ R ∃ ( p, d ) and Γ ⊢ R ∀ ( p, c ) . Pr o of. Suppose ﬁr st that c = q for some q . By Lemma 4.5, Γ ⊢ R ∃ ( q , d ), and by rule (T), Γ ⊢ R ∀ ( q , q ). Putting p = q then satisﬁes Condition (ii). On the other hand, if d = q for some q , then Co ndition (i) is sa tisﬁed, by a similar argument. Hence we may as sume that neither c nor d is a unary a tom. Since c 6 = d , we hav e either c 6∈ V or d 6∈ V , by construction of B . If the latter, then, by Lemma 4.3 Part 1, there exists p such that Γ ⊢ R ∀ ( p, d ). But now we hav e V ⇒ p a nd V ⇒ c , so that, by Lemma 4.5, Γ ⊢ R ∃ ( p, c ), and Condition (i) is satisﬁed. If, on the other hand, c 6∈ V , Condition (ii) is satisﬁed, by a similar argument. The set B will form the domain o f a structure B , deﬁned as follows. If p is a unary ato m, set p B = { b V ,i ∈ B | V ⇒ p } ; and if r is a binary atom, se t r B = {h b V ,i , b { p } , 1 i ∈ B 2 | V ⇒ ∃ ( p, r ) } ∪ {h b V ,i , b W ,j i ∈ B 2 | for some q , V ⇒ ∀ ( q , r ) a nd W ⇒ q } . The intuition is that the elements b { p } , 1 are w itnesses for the existential qua n- tiﬁers in c-terms of the form ∃ ( p, r ), while the elements b { p } , 2 are witnesses for the existential qua ntiﬁ ers in c-ter ms of the fo rm ∃ ( p, ¯ r ). Lemma 4 .7. If Γ is unsatisﬁable, then t her e exist an element b V ,i of B , a unary atom p and a c-term c , such that V ⇒ p , V ⇒ c , b V ,i ∈ p B and b V ,i 6∈ c B . Pr o of. Since Γ is unsatisﬁable, let ϕ ∈ Γ b e such that B 6| = Γ. If ϕ = ∃ ( p, c ), let V = { p, c } . T rivially , V ⇒ p and V ⇒ c . By construction o f B , b V , 0 ∈ B , and by constr uction of B , b V , 0 ∈ p B , whe nce (since B 6| = ϕ ) b V , 0 6∈ c B . If, on the other hand, ϕ = ∀ ( p, c ), there exists b V ,i ∈ B such that b V ,i ∈ p B and 23 b V ,i 6∈ c B . By cons truction of B , V ⇒ p ; and since ∀ ( p, c ) ∈ Γ, we hav e V ⇒ c , as require d. W e now prove the main Lemma, from which b oth the complexity and the refutation-completeness res ults follow. Lemma 4.8. If Γ is unsatisﬁable, then the fol lowing c ondition holds. ( C ) Ther e exist elements b V ,i , b W ,j of B , un ary atoms q , o , and binary atom r , such that one of the fol lowing is tru e: 1. V ⇒ q and V ⇒ ¯ q ; 2. V ⇒ ∃ ( q , ¯ r ) , V ⇒ ∀ ( o, r ) , and q ⇒ o ; 3. V ⇒ ∀ ( q , ¯ r ) , V ⇒ ∃ ( o, r ) , and o ⇒ q ; 4. V ⇒ ∀ ( q , ¯ r ) , V ⇒ ∀ ( o, r ) , W ⇒ q , and W ⇒ o . Pr o of. Let V , i , p and c b e a s in Lemma 4 .7. W e claim ﬁr st that c cannot be of the form q , ∃ ( q , r ) or ∀ ( q , r ). F or consider ea ch p ossibility in turn. If c = q , then, by the construction of B , V ⇒ c implies b V ,i ∈ q B , con tradicting b V ,i 6∈ c B . If c = ∃ ( q , r ), then, by the c onstruction of B , V ⇒ c implies b { q } , 1 ∈ B , b { q } , 1 ∈ q B , a nd h b V ,i , b { q } , 1 i ∈ r B , whence b V ,i ∈ ∃ ( q , r ) B , contradicting b 6∈ c B . If c = ∀ ( q , r ), then, since V ⇒ c , the cons truction of B ensures that, for any b W ,j ∈ q B , we ha ve W ⇒ q and hence h b V ,i , b W ,j i ∈ r B . That is, b V ,i ∈ ∀ ( q , r ) B , contradicting b 6∈ c B . Therefore, c is of o ne of the for ms ¯ q , ∃ ( q , ¯ r ), or ∀ ( q , ¯ r ). W e consider each po ssibility in turn, and show that one of the fo ur cases o f Condition ( C ) holds. If c = ¯ q , then b V ,i 6∈ c B means that b V ,i ∈ q B , s o that, b y construc tion o f B , V ⇒ q . B ut by assumption, V ⇒ c , a nd we ha v e Case 1 of Condition ( C ). If c = ∃ ( q , ¯ r ), then, since V ⇒ c , we ha v e, by the construction of B , b { q } , 2 ∈ B , and in fact b { q } , 2 ∈ q B . Since b V ,i 6∈ c B , we hav e h b V ,i , b { q } , 2 i ∈ r B . The construction of B then g uarantees that for some unary atom o , V ⇒ ∀ ( o, r ) and q ⇒ o ; thus we hav e Case 2 of Conditio n ( C ). If c = ∀ ( q , ¯ r ), then b V ,i 6∈ c B implies that there ex ists b W ,j ∈ B such that b W ,j ∈ q B and h b V ,i , b W ,j i ∈ r B . By construction of B , W ⇒ q , a nd, for some unary a tom o , either (a) V ⇒ ∃ ( o, r ), W = { o } a nd j = 1, or (b) V ⇒ ∀ ( o, r ) and W ⇒ o . 24 But these y ield Cases 3 a nd 4 of Condition ( C ), res p ectively . Lemma 4.9. The fol lowing ar e e quivalent: 1. Γ ⊢ R ⊥ ; 2. Γ is unsatisﬁable; 3. Condition ( C ) of L emma 4.8 holds. Pr o of. F or the implicatio n 1 ⇒ 2, we o bserve that ⊢ R is obviously sound. The implication 2 ⇒ 3 is Lemma 4.8. F or the implica tion 3 ⇒ 1, supp ose Condition ( C ) of L emma 4.8 holds. This condition has four cases: we consider each in turn, showing that Γ ⊢ R ∃ ( p, ¯ p ) for so me unary atom p . 1. V ⇒ q a nd V ⇒ ¯ q . By Lemma 4.5 we immediately have Γ ⊢ R ∃ ( q , ¯ q ). 2. V ⇒ ∃ ( q , ¯ r ), V ⇒ ∀ ( o, r ), q ⇒ o . By Lemma 4.3 Part 2 (or Part 3), Γ ⊢ R ∀ ( q , o ), and by Lemma 4.6, there exists p such that either: (i) Γ ⊢ R ∃ ( p, ∃ ( q , ¯ r )) and Γ ⊢ R ∀ ( p, ∀ ( o, r )); or (ii) Γ ⊢ R ∃ ( p, ∀ ( o, r )) and Γ ⊢ R ∀ ( p, ∃ ( q , ¯ r )). In Case (i), we then hav e . . . . ∃ ( p, ∃ ( q , ¯ r )) . . . . ∀ ( q , o ) ∃ ( p, ∃ ( o, ¯ r )) ( ∃∃ ) . . . . ∀ ( p, ∀ ( o, r )) ∃ ( p, ¯ p ) (D3) , while in Ca se (ii), we have . . . . ∃ ( p, ∀ ( o, r )) . . . . ∀ ( p, ∃ ( q , ¯ r )) . . . . ∀ ( q , o ) ∀ ( p, ∃ ( o, ¯ r )) ( ∀∃ ) ∃ ( p, ¯ p ) (D3). 3. V ⇒ ∀ ( q , ¯ r ), V ⇒ ∃ ( o, r ), o ⇒ q . By Lemma 4.3 Part 2 (or Part 3), Γ ⊢ R ∀ ( o, q ), and by Lemma 4.6, there exists p such that either: (i) Γ ⊢ R ∃ ( p, ∃ ( o, r )) and Γ ⊢ R ∀ ( p, ∀ ( q , ¯ r )); or (ii) Γ ⊢ R ∃ ( p, ∀ ( q , ¯ r )) and Γ ⊢ R ∀ ( p, ∃ ( o, r )). But then we ca n em ploy ex actly the same deriv ation patterns as for Cas es 2(i) and 2(ii), resp ectively . 4. V ⇒ ∀ ( q , ¯ r ), V ⇒ ∀ ( o, r ), W ⇒ q , W ⇒ o . By Lemma 4.6, there exists p such that Γ ⊢ R ∃ ( p, ∀ ( p 1 , u )) and Γ ⊢ R ∀ ( p, ∀ ( p 2 , ¯ u )), where u is either r or 25 ¯ r , a nd p 1 and p 2 are q and o in some or der. By Lemma 4.5, Γ ⊢ R ∃ ( o, q ), i.e. Γ ⊢ R ∃ ( p 1 , p 2 ). Thus we have the deriv ation . . . . ∀ ( p, ∀ ( p 2 , ¯ u )) . . . . ∃ ( p 1 , p 2 ) ∀ ( p, ∃ ( p 1 , ¯ u )) ( ∀∀ ) . . . . ∃ ( p, ∀ ( p 1 , u )) ∃ ( p, ¯ p ) (D3) . (26) Theorem 4. 10. The derivation r elation ⊢ R is sound and r efut ation-c omplete for R . Pr o of. Soundness is obvious. Refutatio n-completeness is the implication from 2 to 1 in Le mma 4.9. Theorem 4.11. The pr oblem of determining the validity of a se quent in any of the fr agments S , S † and R is NLogSp ace -c omplete. Pr o of. The lower bo und is obtained b y reduction of the reachabilit y pro blem for directed gr aphs to the v alidit y problem for S . If G = ( V , E ) is a directed graph, tak e the no des V to be unary atoms. Let Θ G = {∀ ( u , v ) | ( u, v ) ∈ E } . It is easy to chec k that Θ G | = ∀ ( u , v ) if a nd only if v r eachable from u . Recall that, by the Immerman-Szelep cs´ enyi The orem, NLogS p a ce is closed under complementation. The upp er b ound for S † is then immediate, since the problem of determining the satisﬁability of a given set of S † -formulas is (almost tr ivially) reducible to the problem 2SA T, which is well k nown to b e NLogSp ace -complete. (See, for example, Papadimitriou [16], pp. 185 and 398.) It remains only to establish the upp er b ound for R . F or this, it suﬃces to show that the pr oblem of determining the un satisﬁability of a given set Γ o f R -formulas is in NLogSp ace . Let Γ be a set of R -formulas. Le t B and B b e as deﬁned for Lemmas 4.4 – 4.9. Lemma 4.9 guarantees that Γ is unsatisﬁable if and only if there ex ist b S,i , b T ,j ∈ B , unary atoms q , o , and a binary atom r satisfying one of t he four cases in Condition ( C ) of Lemma 4.8. Nondeter ministically guess these V , W , i , j , q , o a nd r . This requires only lo garithmic spa ce, b ecause only the indices of the relev an t atoms need to b e enco ded, and the size o f B is linear in the num ber of fo rmulas in Γ. T o check that b V ,i , b W ,j ∈ B is ess ent ially a gra ph- reachabilit y pro blem, as are all the r equirements in the four cases o f Condition ( C ). Since g raph r eachabilit y is in NL ogSp ace , this proves the theorem. 5 R ∗ : an indirect system Next, we consider the fragment R ∗ . W e refer the reade r to Figure 2 for a review of the syntax of this fragment. The co mplexity-theoretic analys is of R ∗ has b een done for us. It is contained in the mo re expres sive logic (a lso inspired by natural la nguage s yntax) in vestigated in McAllester a nd Giv an [10], 26 whose s atisﬁability problem is shown to be NPTime -complete. Insp ection of McAllester a nd Giv a n’s NPTime -hardnes s pr o of ( op. cit. pp. 12– 14) reveals that it employs o nly formulas in o ur fra gment R ∗ ; hence that the lower b ound applies to R ∗ to o. In other words: Prop ositi on 5. 1 (McAllester and Giv an) . The pr oblem of determining the validity of a se quent in the fr agment R ∗ is c o- NPTime -c omplete. Prop os ition 5.1 is signiﬁca nt not least b eca use of what w e know ab out the complexity of sear ching for deriv ations in direct syllo gistic systems. Corollary 5.2. If PTime 6 = NP Time , then ther e exists n o ﬁnite set X of s yl lo- gistic rules in R ∗ such that ⊢ X is b oth sound and re futation-c omplete. Pr o of. Prop osition 5.1 and Lemma 2.1. Let R ∗ be the following s et of syllo gistic r ules in R ∗ . Our result is that the indir e ct deriv ation rela tion  R ∗ is complete. In the following, the v a riables b + , c + range ov er p o sitive c-terms, and d ov er c- terms. (As usua l, p, q range ov er unary atoms a nd r over bina ry atoms.) ∀ ( c + , c + ) (T) ∃ ( c + , d ) ∃ ( c + , c + ) (I) ∀ ( b + , c + ) ∀ ( c + , d ) ∀ ( b + , d ) (B) ∃ ( b + , c + ) ∀ ( c + , d ) ∃ ( b + , d ) (D1) ∀ ( b + , c + ) ∃ ( b + , d ) ∃ ( c + , d ) (D2) ∀ ( p, q ) ∀ ( ∀ ( q , r ) , ∀ ( p, r )) (J) ∀ ( p, q ) ∀ ( ∃ ( p, r ) , ∃ ( q , r )) (K) ∃ ( p, q ) ∀ ( ∀ ( p, r ) , ∃ ( q , r )) (L) ∃ ( q , ∃ ( p, r )) ∃ ( p, p ) (I I) ∀ ( p, ¯ p ) ∀ ( c + , ∀ ( p, r )) (Z) ∀ ( p, ¯ p ) ∃ (( ∀ ( p, r ) , ∀ ( p, r )) (W) Rules (D 1), (D2), (B), (T) and (I) are na tural generalizations of their namesak es in R . Rules (J), (K), and (L) embo dy logica l principles that a re intuitiv ely clear, yet not familiar when taken as sing le steps. If all p orcupines are bro wn animals , then everything which attack s all bro wn ani mals a ttacks all p orcupines (J), and everything which p hotographs some p orcupine photographs some bro wn animal (K ). And if some p orcupines are bro wn animal s , then everything which caresses al l p orcupines c ar esses so me bro wn animals (L). W e hav e already seen (II). Rule (Z) tells that if there a re no p o rcupines (say), then all far mers love all p or cupines. Rule (W) tells us tha t under this sa me as sumption, there is something which lov es all p orcupines (simply b ecause we assume the universe is non-empty , and everything in it v ac uously loves all p orc upines). If we did not assume tha t the universe o f a mo del is non-empty , then we would drop (W), and the completeness o f the r esulting system would b e prov ed the same wa y . 27 All the rules of R a re deriv able in  R ∗ . F o r example, here is a pro of o f (A): [ ∃ ( p, ¯ d )] 1 ∃ ( p, p ) ( I ) ∀ ( p, ¯ p ) ∃ ( p, ¯ p ) ( D1 ) ∀ ( p, d ) ( RAA ) 1 The rest of this section is devoted to a pro of that  R ∗ is sound and complete for R ∗ . Starting the pro of T o prove completeness, we need only show that every consistent se t Γ in the fra gment R ∗ is satisﬁable. Also, b y Lemma 2.2, w e may assume that Γ is R ∗ -complete. F or the remainder of this sectio n, ﬁx some R ∗ - complete set o f formulas Γ which is co nsistent with res pec t to  R ∗ . W e s implify our notation to wr ite ⊢ for  R ∗ . W e sha ll construct a str ucture A a nd prov e that it satisﬁes Γ. First, let C + be the se t o f p ositive c- terms. The n we deﬁne A by: A = {h c 1 , c 2 , Q i ∈ C + × C + × {∀ , ∃} : Γ ⊢ ∃ ( c 1 , c 2 ) } p A = {h c 1 , c 2 , Q i ∈ A : Γ ⊢ ∀ ( c 1 , p ) or Γ ⊢ ∀ ( c 2 , p ) } h c 1 , c 2 , Q 1 i r A ( d 1 , d 2 , Q 2 ) iﬀ e ither (a) for s ome i , j , and q ∈ P , Γ ⊢ ∀ ( c i , ∀ ( q , r )) and Γ ⊢ ∀ ( d j , q ); or else (b) Q 2 = ∃ , and for some i and q ∈ P , d 1 = d 2 = q , and Γ ⊢ ∀ ( c i , ∃ ( q , r )) . Note that the set A is non-empty . F or let p ∈ P . If Γ ⊢ ∃ ( p, p ), then h p, p, ∀i ∈ A . Otherwise, Γ ⊢ ∀ ( p, ¯ p ), a nd so for a ll binary atoms r , Γ ⊢ ∃ ( ∀ ( p, r ) , ∀ ( p, r )) by (W). Thus h c, c, ∀i ∈ A , where c is ∀ ( p, r ). Lemma 5.3. F or al l c ∈ C + , c A = {h d 1 , d 2 , Q i ∈ A : either Γ ⊢ ∀ ( d 1 , c ) , or Γ ⊢ ∀ ( d 2 , c ) } . Pr o of. The re sult for c a una ry atom is immediate. W e often s hall use the resulting fact that if Γ ⊢ ∃ ( p, p ), then p A 6 = ∅ ; it co ntains b oth h p, p, ∀i and h p, p, ∃i . The main w ork concerns c-terms of the form ∀ ( p, r ) a nd ∃ ( p, r ). W e remark that a ll c-terms referr ed to in this pro of ar e p ositive . W e b egin with c = ∀ ( p, r ). Let h d 1 , d 2 , Q i ∈ ∀ ( p, r ) A . Now, either Γ ⊢ ∃ ( p, p ) or Γ 6⊢ ∃ ( p, p ). If the former, then h p, p, ∀i ∈ p A . B y the s emantics o f our fragment, h d 1 , d 2 , Q i r A h p, p, ∀i . By the structure of A , ther e are i a nd q giving the deriv ation fr om Γ as in the tree on the left b elow: . . . . ∀ ( d i , ∀ ( q , r )) . . . . ∀ ( p, q ) ∀ ( ∀ ( q , r ) , ∀ ( p, r )) ( J ) ∀ ( d i , ∀ ( p, r )) ( B ) . . . . ∀ ( p, ¯ p ) ∀ ( d j , ∀ ( p, r )) (Z) . 28 This shows that Γ ⊢ ∀ ( d i , ∀ ( p, r )). On the other hand, if Γ 6⊢ ∃ ( p, p ), we use the assumption that Γ is complete to asser t that Γ ⊢ ∀ ( p, ¯ p ). And then we have the deriv a tion from Γ o n the right ab ove, for b oth j . Conv ersely , ﬁx i and supp o se that Γ ⊢ ∀ ( d i , ∀ ( p, r )). W e cla im that h d 1 , d 2 , Q i belo ngs to ∀ ( p, r ) A . F or this, take any h b 1 , b 2 , Q ′ i ∈ p A so that Γ ⊢ ∀ ( b j , p ) for some j . (W e are thus using b 1 and b 2 to ra nge over positive c-ter ms, just as the c ’s and d ’s do.) Then p , i and j show tha t h d 1 , d 2 , Q i r A h b 1 , b 2 , Q ′ i . T his for a ll elements of p A shows tha t h d 1 , d 2 , Q i ∈ ∀ ( p, r ) A . W e next prove the statement o f o ur lemma for c = ∃ ( p, r ). Let h d 1 , d 2 , Q i ∈ ∃ ( p, r ) A . Thus we hav e h d 1 , d 2 , Q i r A h b 1 , b 2 , Q ′ i for s ome h b 1 , b 2 , Q ′ i ∈ p A . W e ﬁr st consider ca se (a) in the deﬁnition o f o ur structure A : there ar e i , j , and q so that Γ ⊢ ∀ ( d i , ∀ ( q , r )) and Γ ⊢ ∀ ( b j , q ). W e hav e Γ ⊢ ∃ ( b 1 , b 2 ), since h b 1 , b 2 , Q ′ i ∈ A . F ur ther, let k b e such that Γ ⊢ ∀ ( b k , p ). W e show the desired conclus ion using a deriv a tion from Γ: . . . . ∀ ( d i , ∀ ( q , r )) . . . . ∃ ( b 1 , b 2 ) ∃ ( b k , b j ) . . . . ∀ ( b j , q ) ∃ ( b k , q ) ( D1 ) . . . . ∀ ( b k , p ) ∃ ( q , p ) ( D1 ) ∀ ( ∀ ( q , r ) , ∃ ( p, r )) ( L ) ∀ ( d i , ∃ ( p, r )) ( B ) This concludes the work in cas e (a). In ca se (b), Q ′ = ∃ , there is some q ∈ P such that b 1 = b 2 = q , and for so me i , Γ ⊢ ∀ ( d i , ∃ ( q , r )). Again we hav e Γ ⊢ ∀ ( q , p ). So we hav e a der iv a tion from Γ as follows: . . . . ∀ ( d i , ∃ ( q , r )) . . . . ∀ ( q , p ) ∀ ( ∃ ( q , r ) , ∃ ( p, r )) ( K ) ∀ ( d i , ∃ ( p, r )) ( B ) A t this po int , w e know that if h d 1 , d 2 , Q i ∈ ∃ ( p, r ) A , then Γ ⊢ ∀ ( d i , ∃ ( p, r )) for s ome i . W e now verify the conv erse. Let h d 1 , d 2 , Q i ∈ A , and ﬁx i such that Γ ⊢ ∀ ( d i , ∃ ( p, r )). Then Γ ⊢ ∃ ( d 1 , d 2 ). W e thus hav e a der iv ation fro m Γ: . . . . ∃ ( d 1 , d 2 ) ∃ ( d i , d i ) ( I ) . . . . ∀ ( d i , ∃ ( p, r )) ∃ ( d i , ∃ ( p, r )) ( D1 ) ∃ ( p, p ) ( I I ) This go es to show that h p, p, ∃i ∈ A . By the co nstruction of A , h d 1 , d 2 , Q i r A h p, p, ∃i , and h p, p, ∃i ∈ p A . So h d 1 , d 2 , Q i ∈ ∃ ( p, r ) A . This completes the pr o of. 29 Lemma 5.4. A | = Γ . Pr o of. The pro of is b y c ases on the v arious for mula types in R ∗ . Using the fact that formulas ∃ ( e, f ) and ∃ ( f , e ) are identiﬁed, and s imilarly for ∀ ( ¯ e, ¯ f ) and ∀ ( f , e ), we may take all R ∗ -formulas to hav e one of the for ms: ∀ ( c + , d + ) , ∀ ( c + , d + ) , ∃ ( c + , d + ) , ∃ ( c + , d + ) , where c + and d + range ov er p ositive c-terms. In the rema inder of the pro of, we omit the + -sup erscripts for cla rity: i.e. c and d rang e over p o sitive c-terms. Let ϕ ∈ Γ be ∀ ( c, d ). Using (B) and Lemma 5.3, we see that c A ⊆ d A . Let ϕ ∈ Γ b e ∀ ( c, ¯ d ). Suppo se tow ards a contradiction that A 6| = ϕ . Let h b 1 , b 2 , Q i ∈ c A ∩ d A . Let i and j b e s uch that Γ ⊢ ∀ ( b i , c ) a nd Γ ⊢ ∀ ( b j , d ). Then us ing (B), Γ ⊢ ∀ ( b i , ¯ d ). And since Γ ⊢ ∃ ( b i , b j ), w e use (D1) to see that Γ ⊢ ∃ ( d, ¯ d ). So Γ is inconsis ten t, a contradiction. If ϕ ∈ Γ is ∃ ( c, d ), then ( c, d, ∃ ) ∈ A . Indeed, ( c, d, ∃ ) ∈ c A ∩ d A , by Rule (T) and Lemma 5 .3. Finally , consider the cas e when ϕ ∈ Γ is of the for m ∃ ( c, ¯ d ). Then, using (I), Γ ⊢ ∃ ( c, c ), so h c, c, ∀i ∈ A . Supp os e tow ards a cont radiction that A | = ∀ ( c, d ). Then h c, c, ∀i ∈ d A . But then we hav e Γ ⊢ ∀ ( c, d ), by Lemma 5.3 again. One application of (D2) now shows tha t Γ ⊢ ∃ ( d, ¯ d ). Thu s we have a contradiction to the consis tency of Γ. Hence, we hav e sho wn that any consistent set Γ of R ∗ -formulas has a mo del. Therefore: Theorem 5.5. The derivation r elation  R ∗ is soun d and c omplete for R ∗ . 6 R † and R ∗† : no indirect systems Our last fra gments are R † and R ∗† . W e o p en with some s imple complexity results, showing that these frag men ts hav e no direct sy llogistic pro of-system which is b oth so und a nd refutation- complete. After this, we strengthen the result to show that ther e are no in direct syllog istic pro of-sy stems, either. The pro of of this negative result is similar to what we saw earlier in Theo rem 4.1 for R , but the a rgument here is mo re intricate. Lemma 6.1. The pr oblem of determining the validity of a se qu ent in R † is ExpTime -har d. Pr o of. The logic K U is the basic mo dal logic K together with an additional mo dality U (for “universal”), whose semantics are g iven by the standard re la- tional (Kripke) se mantics, plus | = w U ϕ if and only if | = w ′ ϕ for all worlds w ′ . The satisﬁability problem for K U is ExpTime -hard. (The proo f is an easy adap- tation of the cor resp onding result for propositiona l dynamic logic; see, e.g. Harel 30 et al. [7]: 216 ﬀ.) It suﬃces, ther efore, to reduce this pr oblem to satisﬁa bility in R † . Let ϕ b e a formula of K U . W e ﬁrst tr ansform ϕ into an equisa tisﬁable set of fo rmulas T ϕ ∪ S ϕ of ﬁrst- order logic; then we translate the formulas of T ϕ ∪ S ϕ int o an equisatisﬁa ble set of R † -formulas. T o simplify the notation, we shall take unary a toms (in R † ) to b e unary predicates (in ﬁrs t-order logic); similarly , we tak e binary a toms to do double duty as binary pr edicates. L et r and e b e binary atoms. F o r a n y K U -formula ψ , let p ψ be a unar y atom, and deﬁne the set of ﬁrst-order form ulas T ψ inductively as follows: T p = ∅ (wher e p is a pr op osition letter) T ψ ∧ π = T ψ ∪ T π ∪ {∀ x ( p ψ ( x ) ∧ p π ( x ) → p ψ ∧ π ( x )) , ∀ x ( p ψ ∧ π ( x ) → p ψ ( x )) , ∀ x ( p ψ ∧ π ( x ) → p π ( x )) } T ¬ ψ = T ψ ∪ {∀ x ( p ¬ ψ ( x ) → ¬ p ψ ( x )) , ∀ x ( ¬ p ¬ ψ ( x ) → p ψ ( x )) } T  ψ = T ψ ∪ {∀ x ( p  ψ ( x ) → ∀ y ( ¬ p ψ ( y ) → ¬ r ( x, y ))) , ∀ x ( ¬ p  ψ ( x ) → ∃ y ( ¬ p ψ ( y ) ∧ r ( x, y ))) } T U ψ = T ψ ∪ {∀ x ( p U ψ ( x ) → ∀ y ( ¬ p ψ ( y ) → ¬ e ( x, y ))) , ∀ x ( ¬ p U ψ ( x ) → ∃ y ( ¬ p ψ ( y ) ∧ e ( x, y ))) } . Now let S ϕ be the co llection of ﬁve ﬁrst-order formulas ∃ x ( p ϕ ( x ) ∧ p ϕ ( x )) , ∀ x ( ± p ϕ ( x ) → ∀ y ( ± p ϕ ( y ) → e ( x, y ))) . (Although the ﬁrst for m ula lo oks like it has a r edundant conjunct, we state it in this wa y only to make our work b elow a little easier .) W e claim tha t the mo dal formula ϕ is satisﬁable if and only if the set o f ﬁr st-order fo rmulas T ϕ ∪ S ϕ is satisﬁable. F or let M be any (K ripke) mo de l o f ϕ over a frame ( W, R ). Deﬁne the ﬁrst-order structure A with domain W , by setting r A = R , e A = A 2 , and p A ψ = { w | M | = w ψ } , for a ny subformula ψ of ϕ . It is then easy to chec k that A | = T ϕ ∪ S ϕ . Conversely , suppose A | = T ϕ ∪ S ϕ . W e build a Kripke structure M ov er the fra me ( A, r A ) by setting, for any pro po sition letter o mentioned in ϕ , M | = a o if and only if a ∈ p A o . A straightforw ard structur al induction establishes that for any subformu la ψ of ϕ , M | = a ψ if a nd only if a ∈ p A ψ . The formula ∃ x ( p ϕ ( x ) ∧ p ϕ ( x )) ∈ S ϕ then ensures that ϕ is sa tisﬁed in M . Now, all o f the formulas in T ϕ ∪ S ϕ are of one o f the forms ∀ x ( ± p ( x ) → ± q ( x )) ∀ x ( ± p ( x ) → ∀ y ( ± q ( y ) → ± r ( x, y ))) (27) ∃ x ( p ( x ) ∧ p ( x )) ∀ x ( ± p ( x ) → ∃ y ( ± q ( y ) ∧ r ( x, y ))) (28) ∀ x ( p ( x ) ∧ q ( x ) → o ( x )) . (29) Notice that for mulas of the forms (27) and (28) tra nslate (in the o bvious sense) directly into the fragment R † ; those o f form (29 ), by contrast, do not. The next step is to eliminate formulas o f this last type. Let o ∗ be a new unary r elation symbol. F or θ ∈ T ϕ ∪ S ϕ of the form (29), 31 let r θ be a new binary atom, a nd deﬁne R θ to b e the s et of formulas ∀ x ( ¬ o ( x ) → ∃ z ( o ∗ ( z ) ∧ r θ ( x, z ))) (30) ∀ x ( p ( x ) → ∀ z ( ¬ p ( z ) → ¬ r θ ( x, z ))) (31) ∀ x ( q ( x ) → ∀ z ( p ( z ) → ¬ r θ ( x, z ))) , (32) which ar e all of the forms in (27) or (28). It is easy to chec k that R θ | = θ . F or suppo se (for cont radiction) that A | = R θ and a satisﬁes p and q but not o in A . By (30), there exists b such that A | = r θ [ a, b ]. If A 6| = p [ b ], then (31) is false in A ; on the other hand, if A | = p [ b ], then (32) is false in A . Thus, R θ | = θ a s claimed. Co nv ersely , if A | = θ , expand A to a structure A ′ by interpreting o ∗ and r θ as follows: ( o ∗ ) A = A r A θ = {h a, a i | A 6| = o [ a ] } . W e chec k that A ′ | = R θ . F ormula (30) is true, b ecause A ′ 6| = o [ a ] implies A ′ | = r θ [ a, a ]. F ormula (31) is true, b eca use A ′ | = r θ [ a, b ] implies a = b . T o see that F ormula (32) is true, supp ose A ′ | = q [ a ] and A ′ | = p [ b ]. I f a = b , then A | = o [ a ] (since A ′ | = θ ); that is , either a 6 = b o r A | = o [ a ]. By co nstruction, then, A ′ 6| = r θ [ a, b ]. Now let T ∗ ϕ be the result o f replacing all formulas θ in T ϕ of form (29) with the co rresp o nding trio R θ . (The binary atoms r θ for the v arious θ ar e assumed to b e distinct; how ever, the same unary atom o ∗ can b e used for all θ .) By the previo us parag raph, T ∗ ϕ ∪ S ϕ is sa tisﬁable if a nd only if T ϕ ∪ S ϕ is satisﬁable, and hence if a nd o nly if ϕ is satisﬁable. But T ∗ ϕ ∪ S ϕ is a set of formulas of the forms (27) and (28), and can evidently b e transla ted into a set of R † -formulas satisﬁed in exactly the sa me s tructures. Moreov er, this set can be computed in time b ounded by a p olynomial function o f k ϕ k . This c ompletes the reduction. W e note the following fa ct. (W e omit a detailed pro of, since subseq uent developmen ts do no t hinge on this result.) Lemma 6.2. The pr oblem of determining the validity of a se quent in R ∗† is in ExpTime . Pr o of. T rivia l ada ptation of Pr att-Hartmann [17], Theorem 3, which consider s a fragment obta ined b y adding rela tive clauses to the relatio nal sy llogistic. Theorem 6.3. The pr oblem of determining t he validity of a se quent in either of the fr agments R † and R ∗† is E xpTime -c omplete. Pr o of. Lemmas 6.1 and 6.2. Corollary 6. 4 . Ther e exists no ﬁn it e set X of syl lo gistic rules in either R † or R ∗† such that ⊢ X is b oth sound and re futation-c omplete. 32 q 1 p 2 p 1 q 1 q 1 p n q 1 p i +1 o 2 o 1 p 1 p 2 q 2 p i q 2 p i +1 q 2 p n q 2 o 2 o 3 q 1 p i p 1 q 2 q 2 q 1 ... ... ... ... r r s s r r r a i a i +1 a n r r a 2 r a 1 r r r r r r r r r r u 0 u 1 v 1 v 2 a ′ 1 a ′ 2 a ′ i a ′ i +1 a ′ n u 3 u 4 r r u 2 Figure 3 : The str ucture A ( n ) . Every element inside the dotted b ox is re lated by r to u 2 . Pr o of. It is a s tandard result that PTime 6 = ExpTime . The result is then im- mediate by Lemmas 2.1 and 6.1. Of course, Cor ollary 6.4 le av es o p en the p ossibility that there exist indir e ct syllogistic systems that ar e sound and co mplete for R † and R ∗† . T o show tha t there do not, strong er methods are req uired. That is the task of the rema inder of this section. Starting on the pro of Let n > 2; let o 1 , o 2 , o 3 , q 1 , q 2 , p 1 , . . . , p n be unary atoms, and r , s bina ry atoms; and let A ( n ) be the set { a 1 , . . . , a n , a ′ 1 , . . . , a ′ n , u 0 , u 1 , u 2 , u 3 , u 4 , v 1 , v 2 } . W e take the s tructure A ( n ) , with domain A ( n ) , to b e as depicted in Fig. 3. Here, member ship of a n element in p A ( n ) (where p is any unary atom) is indicated by writing p in a b ox next to that element. The extensions of the binary atoms r and s a re depicted similarly : we indica te that a pair o f elements is in r A ( n ) by writing r ne xt to an arrow b etw een those elements (and likewise for s ). The r -lab elled arr ow fro m the dotted b ox to the element u 2 is to be in terpreted a s follows: for any element a inside the do tted box, h a, u 2 i ∈ r A ( n ) . Thus, A ( n ) contains tw o ‘ r -chains’ of elements satisfying, successively , p 1 , . . . , p n : the elements o f the ﬁrst chain additionally satisfy q 1 but not q 2 ; those of the second chain additiona lly sa tisfy q 2 but not q 1 . The terminal elements of the tw o r -chains are related, by s , to the elements v 1 and v 2 . There are also some additional ‘out-lier’ elements, which ensure the tr uth of v arious existential for mulas in A : for ex ample, u 1 ensures (together with u 0 ) the truth of ∃ ( o 1 , ∃ ( p 1 , ¯ r )); u 2 ensures the truth of ∃ ( q 1 , q 2 ), a nd so on. In the sequel, we deﬁne s tructures containing additiona l elements, which ma ke almost the sa me R ∗† -formulas true as A ( n ) . These la tter structur es will then b e used to show that there can b e no sound and complete indirect syllo gistic s ystem for 33 R ∗ or R ∗† . T o av oid notational clutter, we drop the sup er scripts in A ( n ) (and in rela ted constructions) where the v a lue of n do es not matter. By ins pe ction, we have A | = ∀ ( o 1 , ∃ ( q 1 , r )) (33) A | = ∀ ( o 1 , ∀ ( ¯ p 1 , ¯ r )) (34) A | = ∀ ( q 1 , ∀ ( ¯ q 1 , ¯ r )) (35) A | = ∀ ( q 2 , ∀ ( ¯ q 2 , ¯ r )) (36) A | = ∀ ( p i , ∃ ( p i +1 , r )) (1 6 i < n ) (37) A | = ∀ ( q 1 , ∀ ( o 3 , ¯ s )) ( 38) A | = ∀ ( q 2 , ∀ ( ¯ o 3 , ¯ s )) ( 39) A | = ∀ ( p n , ∃ ( o 2 , s )) . (40) F o r any structur e C , denote by Th( C ) the set of R † -formulas true in C , and denote b y Th ∗ ( C ) the set of R ∗† -formulas true in C . Recall that a set of form ulas Φ is R † -complete (o r R ∗† -complete) if, for every R † -formula (resp ectively , R ∗† - formula) ϕ , either ϕ ∈ Φ or ¯ ϕ ∈ Φ. T rivially , for any C , Th( C ) is R † -complete, and Th ∗ ( C ) is R ∗† -complete. Let γ b e the R † -formula given by γ = ∀ ( o 1 , ∃ ( ¯ q 2 , r )) . Noting that o A 1 = { u 0 } , h u 0 , a 1 i ∈ r A , and a 1 6∈ q A 2 , we hav e A | = γ (i.e. γ ∈ Th( A )). Let Γ ( n ) be obtained from T h( A ( n ) ) by reversing the truth-v alue of γ , and simila rly for Γ ∗ ( n ) . That is: Γ ( n ) =  Th  A ( n )  \ { γ }  ∪ { ¯ γ } Γ ∗ ( n ) =  Th ∗  A ( n )  \ { γ }  ∪ { ¯ γ } . Again, we dr op the ( n )-s upe rscript when the v alue o f n do es not matter. Thu s, Γ is R † -complete and Γ ∗ is R ∗† -complete. Lemma 6.5. Γ ( and t her efor e Γ ∗ ) is uns atisﬁ able. Pr o of. Noting that ¯ γ is the formula ∃ ( o 1 , ∀ ( ¯ q 2 , ¯ r )), from (33) a nd (34), w e see that, in a ny mo del B | = Γ, there exists b ∈ p B 1 ∩ q B 1 ∩ q B 2 . F rom (3 5)–(37), it then follows that, in a ny mo del B | = Γ, there exists b ∈ p B n ∩ q B 1 ∩ q B 2 . But, from (38) –(40), no mo del B | = Γ ca n hav e a ny element b ∈ p B n ∩ q B 1 ∩ q B 2 . This prov es the lemma. Fixing n > 2 , for any i (1 6 i < n ), let B ( n ) i = A ( n ) ∪ { b 1 , . . . , b i , u 5 } , and consider the s tructure B ( n ) i , with domain B ( n ) i , depicted in Fig. 4. W e employ the same notationa l conv en tions as in Fig. 3 . In particular, the r -lab elled arrows from the do tted b oxes are to b e interpreted a s follows: for any element a inside either of the dotted b oxes, h a, u 2 i ∈ r B ( n ) i . Again, we drop s uper scripts where 34 q 1 p 2 p 1 q 1 q 1 p n q 1 p i +1 o 2 o 1 p 1 o 1 p 1 q 2 q 1 q 2 q 1 p 2 q 2 q 1 p i p 2 q 2 p i q 2 p i +1 q 2 p n q 2 o 2 o 3 q 1 p i p 1 q 2 q 2 q 1 ... ... ... ... ... r r s s r r r a i a i +1 a n r r a 2 r a 1 r r r r r r r r r r r r r r r u 0 u 1 v 1 v 2 a ′ 1 a ′ 2 a ′ i a ′ i +1 a ′ n b 1 b 2 b i r u 5 u 3 u 4 r r u 2 Figure 4: The structure B i . Every element inside either of the dotted b oxes is related by r to u 2 . the v alue of n do es no t matter. The s tructure B i contains a co py of A , but has an additional r -chain whose elements satisfy b oth q 1 and q 2 ; notice, how ev er, that this a dditional r -chain stops a t the i th element. W e employ the following terminology . An existential formula is one of the form ∃ ( e, f ). If ϕ = ∃ ( e, f ) and C a structure, then a witness for C | = ϕ is any a ∈ C such that a ∈ e C and a ∈ f C . Now, w e saw ab ove that A | = γ ; by contrast B i | = ¯ γ ( ¯ γ is a n e xistential formula, and u 5 is a witnes s). Let δ i be the R † -formula given by δ i = ∀ ( p i , ∃ ( p i +1 , r )) . W e o bserved in (37) that A | = δ i . By contrast, B i | = ¯ δ i ( ¯ δ i is an existential formula, and b i is a witness). Ho wev er, it turns o ut that γ and δ i are the only diﬀerences b etw een A and B i as far as R ∗† is concerned: Lemma 6.6. F or al l i (1 6 i < n ) : Th  B ( n ) i  =  Th  A ( n )  \ { γ , δ i }  ∪ { ¯ γ , ¯ δ i } ; similarly, Th ∗  B ( n ) i  =  Th ∗  A ( n )  \ { γ , δ i }  ∪ { ¯ γ , ¯ δ i } . 35 Pr o of. Since γ , δ i , ¯ γ , ¯ δ i are all in R † , the ﬁr st of these statements follows in- stantly fro m the second. W e pro ceed, therefore, to establish the second state- men t. W e ﬁrs t pr ov e the statement for the case i = 1. Notice that here, the third r -‘chain’ in B ( n ) 1 contains just one element, b 1 . Our initial goal, then, is to show that Th ∗ ( B ( n ) 1 ) = (Th ∗ ( A ( n ) ) \ { γ , δ 1 } ) ∪ { ¯ γ , ¯ δ 1 } . The ba sic intuit ion is s imple: on the one ha nd, we may chec k the ca ses n = 2 a nd n = 3 by brute force; on the other, w e see from Fig . 4 that, once the ends of the ﬁrst t wo r -chains are suﬃciently distant from the elements u 5 and b 1 , extending thos e r -chains further will have no eﬀect on the diﬀerences b etw een the fo rmulas made true in A ( n ) and B ( n ) 1 . T o make this idea rigorous, w e ﬁrst establish three simple claims. W e employ the following terminolog y . An ex ist ential e-term is one o f the for m ∃ ( l , t ). (As usual, l rang es over unary literals, and t ov er binary literals.) If e = ∃ ( l, t ), C is a structure, and a ∈ C , then a witness for a ∈ e C is any b ∈ C s uch tha t b ∈ l C and h a, b i ∈ t C . Claim 6.7. L et a b e any element of A ( n ) ( n > 2) , and e any e-t erm. Then a ∈ e B ( n ) 1 if and only if a ∈ e A ( n ) . Pr o of. W e pr ov e the only- if dir ection by induction on n . The if-direction then follows by consider ing the e-term ¯ e . The case n = 2 is c heck ed by brute force. (W e used a computer.) Assume the claim is true for n = m > 2; we show that it is true for n = m + 1. Let a ∈ A ( m +1) . F o r contradiction, supp o se a ∈ e B ( m +1) 1 , but a 6∈ e A ( m +1) . Then e is existential, and a ∈ e B ( m +1) 1 has a witness in B ( m +1) 1 \ A ( m +1) . W r iting e = ∃ ( l, t ), there exists b ∈ { u 5 , b 1 } such that b ∈ l B ( m +1) 1 and h a, b i ∈ t B ( m +1) 1 . In that case , if a ∈ A ( m ) , we o bviously hav e a ∈ e B ( m ) 1 , whence a ∈ e A ( m ) by inductive hypothes is, whence a ∈ e A ( m +1) , since A ( m ) is a sub-mo del o f A ( m +1) —a cont radiction. Thus, a is either a m +1 or a ′ m +1 . But, by insp ectio n of Fig. 4 (b ea ring in mind i = 1), it is eas y to see that, for a ∈ { a m +1 , a ′ m +1 } , if a ∈ e B ( m +1) 1 with a witnes s in { u 5 , b 1 } , then a ∈ e A ( m +1) with a witness in { u 0 , u 2 , a 1 , a ′ 1 } — again, a contradiction. Claim 6.8. L et ϕ b e any existent ial formula other than ¯ δ 1 , and let A 0 = { a 1 , a ′ 1 , u 2 } . If b 1 is a witness for B ( n ) 1 | = ϕ , t hen ther e exists a ∈ A 0 such that a is a witness for A ( n ) | = ϕ . Pr o of. By induction on n . The case s n = 2 and n = 3 are chec k ed by brute force. (W e used a co mputer.) No te in pas sing that the condition that ϕ is not ¯ δ 1 is required for these cases. Suppos e now the c laim holds for n = m > 3; we show tha t it holds for n = m + 1 . If e is any e-ter m, let ˆ e b e the r esult of replacing any o c currence of the unary ato m p m +1 by the unary ato m p m . W riting ϕ = ∃ ( e, f ), let ˆ ϕ = ∃ ( ˆ e, ˆ f ). Note tha t, sinc e m > 3 and ϕ 6 = ¯ δ 1 , we 36 hav e ˆ ϕ 6 = ¯ δ 1 . The following facts are o b vious (see Figs. 3 and 4): a ∈ e A ( m +1) iﬀ a ∈ ˆ e A ( m ) for all a ∈ A 0 b 1 ∈ e B ( m +1) 1 iﬀ b 1 ∈ ˆ e B ( m ) 1 . It follows that, if b 1 is a witness for B ( m +1) 1 | = ϕ , then b 1 is a witness for B ( m ) 1 | = ˆ ϕ , whence, by inductive hypothesis, there exists a ∈ A 0 such that a is a witness fo r A ( m ) | = ˆ ϕ , whence there exis ts a ∈ A 0 such that a is a witness for A ( m +1) | = ϕ . This completes the inductio n. Claim 6.9. L et ϕ b e any existen tial formula other than ¯ γ , and let A 0 = { u 0 , u 2 , u 3 } . If u 5 is a witness for B ( n ) 1 | = ϕ , then ther e exists a ∈ A 0 such that a is a witness for A ( n ) | = ϕ . Pr o of. Iden tical in structure to the pr o of of Claim 6.8. W e can now prov e Lemma 6.6 in the case i = 1. Claim 6.7 shows that an y existential for mula true in A ( n ) is true in B ( n ) 1 , a nd more ov er, that if ϕ is a n existential fo rmula tr ue in B ( n ) 1 but false in A ( n ) , then either either u 5 or b 1 is a witness. But these p os sibilities ar e ruled o ut—except in the ca ses ϕ = ¯ γ and ϕ = ¯ δ 1 —b y Cla ims 6.8 a nd 6.9, resp ectively . Finally , if ϕ is universal (and not equal to γ or δ 1 ), then ¯ ϕ is exis tent ial (a nd no t equal to ¯ γ o r ¯ δ 1 ). The fore going analysis shows that A ( n ) and B ( n ) 1 agree o n the truth v alue o f ¯ ϕ , henc e o n the truth v a lue of ϕ . W e hav e thu s proved the lemma for i = 1 and a ll v alues of n . Fix n > 3 , and consider now the structure B ( n ) 2 , which diﬀer s fro m B ( n ) 1 only in that the third r -chain has b een extended fr om one to t wo elements. By insp ection of Fig. 4, the only eﬀect on the s et of sen tences ma de true is to restor e the truth of δ 1 and to falsify δ 2 . That is: Th ∗  B ( n ) 2  =  Th ∗  B ( n ) 1  \ { ¯ δ 1 , δ 2 }  ∪ { δ 1 , ¯ δ 2 } =  Th ∗  A ( n )  \ { γ , δ 2 }  ∪ { ¯ γ , ¯ δ 2 } . Pro ceeding in the same wa y , we hav e, for all i (1 6 i < n − 1), Th ∗  B ( n ) i +1  =  Th ∗  B ( n ) i  \ { ¯ δ i , δ i +1 }  ∪ { δ i , ¯ δ i +1 } =  Th ∗  A ( n )  \ { γ , δ i +1 }  ∪ { ¯ γ , ¯ δ i +1 } . This prov es the lemma. Fixing n > 3, for a ll i and j (1 < i < j < n ) deﬁne ∆ ( n ) i,j = Γ ( n ) \ { δ i , δ j } ∆ ∗ ( n ) i,j = Γ ∗ ( n ) \ { δ i , δ j } . Again ( n )-sup erscr ipts are omitted where p os sible for clarity . 37 Lemma 6.1 0. L et θ b e an R † -formula and θ ∗ an R ∗† -formula. F or al l i , j (1 < i < j < n ) , if ∆ i,j | = θ , t hen θ ∈ ∆ i,j . Likewise, if ∆ ∗ i,j | = θ ∗ , then θ ∗ ∈ ∆ ∗ i,j . Pr o of. W e prove the ﬁrst sta temen t only; the pr o of o f the s econd is similar. Given the equatio ns Γ ( n ) =  Th  A ( n )  \ { γ }  ∪ { ¯ γ } and ∆ ( n ) i,j = Γ ( n ) \ { δ i , δ j } , Lemma 6.6 yields B i | = ∆ i,j ∪ { ¯ δ i , δ j } (41) B j | = ∆ i,j ∪ { δ i , ¯ δ j } . (42) Certainly , then, ∆ i,j is satisﬁable; hence, the only θ we need cons ider a re those such that neither θ no r ¯ θ is in ∆ i,j . By the completeness of Γ ( n ) , this e n tails θ ∈ { δ i , ¯ δ j , ¯ δ i , δ j } . In the ﬁrst tw o cases, (41) s hows that ∆ i,j 6| = θ ; in the third and fourth ca ses, (4 2) shows the same. Lemma 6. 1 1. If X is a ﬁnite set of syl lo gistic rules in R † , t hen ther e ex ists n such t hat, for any absur dity ⊥ in R † , Γ ( n ) 6⊢ X ⊥ . If X is a ﬁn ite set of syl lo gistic rules in R ∗† , then t her e exists n such that, for any absur dity ⊥ in R ∗† , Γ ( n ) 6⊢ X ⊥ . Pr o of. W e prove the lemma for R † ; the pro of for R ∗† is similar . Since the nu mber of rules in X is ﬁnite, let k b e the la rgest num ber of antecedents in any of these rules, and ﬁx n = k + 2. Since no instance of a rule of X ha s more than n − 2 anteceden ts, any such s et of anteceden ts included in Γ ( n ) m ust also b e included in ∆ ( n ) i,j for some 1 < i < j < n . Let θ b e the conseq uent o f this rule-ins tance. Since ⊢ X is sound, ∆ ( n ) i,j | = θ , whence θ ∈ ∆ ( n ) i,j ⊆ Γ ( n ) , by Lemma 6.1 0. A simple induction o n the leng ths of pro ofs then shows that any formula der ived from Γ ( n ) is in Γ ( n ) . But Γ ( n ) contains no abs urdity . W e now hav e the promised str engthening of Coro llary 6.4. In some sens e, Lemma 6.11 has done all the work; f or (RA A) is, in the con text of a R † -complete (or R ∗† -complete) set o f pr emises, essentially redundant. Theorem 6.12. Ther e exists no ﬁn ite set X of syl lo gistic rules in either R † or R ∗† such that  X is b oth sound and c omplete. Pr o of. W e pr ov e the lemma for R † ; the pro of for R ∗† is almos t iden tical. Sup- po se X is a ﬁnite set of syllo gistic r ules in R † , with  X sound. Let n and Γ ( n ) be as in Lemma 6.11, a nd let ⊥ b e any absurdity in R † . Thus, by Lemma 6.5, Γ ( n ) | = ⊥ , but, by L emma 6.11, Γ ( n ) 6⊢ X ⊥ . It suﬃces to show that Γ ( n ) 6  X ⊥ . Suppo se then, for co nt radiction, tha t there is an indirect deriv atio n of an absurdity from Γ ( n ) , us ing the rules X . Consider such a der iv ation in which the num ber k of applica tions of (RAA) is minimal. Since Γ ( n ) 6⊢ X ⊥ , we know that k > 0. Consider the last applica tion of (RAA) in this der iv atio n, which derives an R † -formula, say , ¯ θ , discharging a premise θ . That is, there is a n (indirect) der iv a tion of some absurdity ⊥ ′ from Γ ( n ) ∪ { θ } , emplo ying fewer 38 than k applicatio ns of (RAA). By minimality of k , θ 6∈ Γ ( n ) , and so, by the R † - completeness of Γ ( n ) , ¯ θ ∈ Γ ( n ) . But then we can replace our o riginal deriv a tion of of ¯ θ with a tr ivial der iv atio n, so obtaining a deriv ation o f ⊥ fr om Γ ( n ) with few er than k applications of (RAA), a contradiction. The upshot: unlike in the case of R ∗ , the full p ow er of r e ductio doe s not help with R † and R ∗† . 7 Relation to other w ork Systems rel ated to S and S † Mo dern treatments of the the syllogistic can all b e tra ced back to Luk asie wicz [8], where a logic is presented in which formu- las of the forms (2) are treated as ato ms in a propo sitional calculus. Luk a siewicz provides a colle ction o f a xiom-schemata whic h, together with the usual axioms of prop ositiona l logic, yields a co mplete pr o of-system for the r esulting languag e (a strict sup erset of S ). The co mpleteness of this s ystem was sho wn independently by W esterst ˚ ahl [20]; note that it diﬀers from the syllogistic fra gments cons id- ered in this pap er, in that it is embedded within pro p ositional logic. Other commentators, for exa mple, Smiley [18], Cor coran [2] and Ma rtin [9], o b jecting to Luk a siewicz’ exegesis o f Ar istotle, provide pro of-sys tems in the form o f syllo- gistic r ules similar to those of S , ag ain pr oving completeness of their resp ective systems. These sy stems are not dir e ct sy llogistic sys tems, since they all employ r e ductio ad absur dum . Our emphasis on dir ect syllo gistic systems, and es pe- cially the formulation o f r efutation-c ompleteness is new, and motiv ated by the results we hav e obtained on relational extensions of the syllo gistic. In addition, the authors mentioned a b ov e do not tr eat the fragment S † . The completeness of S itself app ears as Theore m 6.2 in Mos s [14], where v ario us fragments of S are also axioma tized. The completeness of S † is proved in Moss [12], using a somewhat diﬀerent trea tmen t. Systems related to R The ﬁrst pr esentation of a complete pro of-system for a fra gment close to the r elational syllog istic seems to b e Nishihar a, Mor ita, and Iwata [15]. This log ic is in eﬀect a rela tional version o f L uk as iewicz’, in that formulas roughly similar to those of the forms (6) are trea ted as atoms of a prop ositiona l calculus. The author s provide axio m-schemata which, together with the usual axioms of prop ositiona l logic, y ield a complete pro of-s ystem for the langua ge in question. Actually , the prop o sitional atoms in this la nguage are allow ed to feature n -ar y predicates for all n > 1. How ever, the ra ther strange restrictions on quantiﬁer-scop e (existen tials m ust always o utscop e univ ersals ), mean that this language is primar ily o f interest for a toms featuring o nly unary and binary pr edicates; these atoms (and their nega tions) then essentially cor re- sp ond the formulas of our fragment R . W e men tion in pa ssing that Nishihara et al. ’s lang uage includes individual constants; but in practice, this lea ds to no useful increase in expres sive p ow er, and we ignore this fea ture. Be cause it inv o lves such an expr essive fragment, which includes full prop o sitional logic, 39 their pro of certainly yields no upp er co mplexity b ound compa rable to that of Theorem 4.11. A logic inspired b y the s ystem of Nishihara et al may b e found in Mo ss [13]. Roughly sp eaking , that logic is the negatio n-free fr agment o f R , cor resp onding to sentence-forms involving the words some and al l , but not no . How ev er, it extends R in that it allows bo th readings of sco p e-ambiguous sentences. F or example, a s entence like E very painter admi res some artist has both a sub ject wide scop e reading and a sub ject narr ow scop e re ading. The fact that the sub ject wide scop e reading o f lo gically implies the s ub ject narrow reading is then a r ule of inference in Moss’ system. No c ontradictions are p ossible in the sys tem. Moss [13] also analyses a syllogistic logic with negated nouns (not verbs) and only using Al l . So in our notation, its formulas ar e ∀ ( l , m ) and ∀ ( l , ∀ ( m, r )), with l and m unar y literals. In addition to rules we hav e seen, it uses the following additional rules which might be taken to b e fo rms of the law of the excluded middle: ∀ ( n, ∀ ( q , r )) ∀ ( ¯ n, ∀ ( q , r )) ∀ ( p, ∀ ( q , r )) ∀ ( p, ∀ ( n, r )) ∀ ( p, ∀ ( ¯ n, r )) ∀ ( p, ∀ ( q , r )) . The system is co mpleted by a r ule with thre e premises: ∀ ( p, ∀ ( n, r )) ∀ ( o, ∀ ( q , r )) ∀ ( ¯ o, ∀ ( ¯ n, r )) ∀ ( p, ∀ ( q , r )) . McAllester and Giv an’s fragmen t W e have already had o ccasio n to men- tion the results of McAllester and Giv an [10] in connection with our fra gment R ∗ . McAllester and Giv an present a “Montagovian syntax” for a ﬁrst-order lan- guage ov er a signa ture of unar y and bina ry predica tes and individual consta nt s, together with (what they ca ll) its “quantiﬁer-free” fragment. In fac t, this la tter fragment is lik e our fragment R ∗ , e xcept that its ‘cla ss-terms’ (the equiv a lent of our c-terms) ca n b e nested to arbitr ary depth. Thus, in McAllester and Giv a n’s language, the for mula ∀ ( ∃ ( ∃ ( man , kill ) , kill ) , ∃ ( ∃ ( animal , kill ) , kill )) expresses the prop osition that, as de Morga n migh t hav e put it, he who kills one who kills a man kills one who kills a n animal. How ever, the ability to embed c-terms to arbitrar y depth is ea sily seen not to co nfer any essential increas e in ex pressive p ow er o f sets of fo rmulas, since deeply nested class-terms can alwa ys b e ‘deﬁned out’ by intro ducing new unar y predicates. (This is r eﬂected in the fact that the sa tisﬁability pro blem for R ∗ is no diﬀerent from that of McAllester and Giv an’s fragment.) In terestingly , McAllester and Giv an show that the satisﬁability of a set Γ of R ∗ -formulas can b e determined in p oly nomial time if, for each class -term c o ccur ring in Γ, Γ con tains either the form ula ∃ ( c, c ) or the formula ∀ ( c, ¯ c ). (Such formula sets are said to determine existentials ). McAllester a nd Giv an provide a syllogistic- like system for this fragment, which is complete for sets of for mulas which determine existentials. 40 A sub-frag ment o f the McAllester and Giv a n fragment was consider ed in Moss [13]. In o ur ter ms, the formulas would b e o f the for ms ∀ ( c, d ) and ∃ ( c, d ), where c and d ar e class -terms o f the forms p (an atom), ∃ ( c, r ), or ∀ ( d, r ). Thu s the system lacks negation. The rules ar e versions of rules we hav e se en in Section 5: (T), (I), (B), (D), (J), (K), (L), and (I I); it also employs a r ule allowing for reasoning-by-cases, which is not a syllogistic rule in the sense of this pap er. An informal example sho ws tha t the system allo ws non-trivia l infer ences, and inferences with mo re than one verb. All p orcupines are mammals All who resp e ct all mamma ls resp ect all p orcupines All who dislike al l who resp ect all p orcupines dislike all wh o resp ect all mammals . Mo dern reviv als of term lo gic In re cent decades, v arious logicians hav e challenged the dominant paradig m of quantiﬁcation-theory by s eeking to reha- bilitate term lo gic —essen tially , the extension of the traditional pr esentation of the syllogistic to the ca se of p olyadic relations (se e, e.g. Sommers [19], Engle- bretsen [5]). Bro adly , the strategy ado pted by these term-lo gicians has be en to stress the expr essiveness of the new term-lo gical syntax, and its a bility to repre- sent all that is repr esented in qua nt iﬁcational logic. (F or a very clear acc ount, see, Michael Lo ckw o o d’s Appendix G to Sommers [19], pp. 426–4 56.) By adopt- ing the new, term-log ical framework—so g o es the arg ument—w e obtain a formal system with all the expres sive pow er of ﬁr st-order log ic, but with the a dded (al- leged) adv a ntage o f greater ﬁdelity to the structure of na tural languag e. The outlo ok of the present pap er is rather diﬀerent, ho wev er. T rue, the frag men ts considered her e could b e s imply and elegantly expressed using the syntax of term-logic. The resulting formulas w ould exhibit o nly cosmetic diﬀerence s to the s yntax in troduce d in Section 2; and, of co urse, the asso cia ted pro o f-systems would b e, mo dulo these cosmetic diﬀerences, unaﬀected. How ever, we have bee n a t pa ins to str ess the inexpr essiveness of the fragments studied a b ov e, bec ause with inexpr essiveness co mes low computational complexity . And from the p o int o f view of computational co mplexit y , the iss ue of the ruling syntactic r´ e gime—predicate lo gic or term lo gic—is immaterial. 8 Conclusions In this pap er, we hav e in v estigated the av ailability of syllogism-like pro of-sys tems for v a rious extensions of the tra ditional syllog istic, with sp ecia l emphasis on the need for the rule of r e ductio ad absur dum ; in addition, we have derived tigh t complexity b ounds for all the lo gics inv estigated. These logics are: (i) S , which co rresp o nds to the tr aditional syllogistic; (ii) S † , whic h extends S with negated nouns; (iii) R , whic h extends S w ith transitive verbs; (iv) R † , which ex tends S with bo th these co nstructions; (v) R ∗ , which extends R by allo wing sub ject noun phrases to contain relative clauses; and (vi) R ∗† , which extends R ∗ with neg ated nouns. The inclusion relations betw een 41 S R R ∗ S † R † R ∗† F O 2 ⊆ ⊆ ⊆ ⊇ ⊇ ⊇ ⊇ ⊆ S direct, complete NLogSp ace S † direct, complete NLogSp ace R direct, refutation complete NLogSp a ce R † not even ind irect ExpTime R ∗ indirect, complete Co- NPTime [10] R ∗† not even ind irect ExpTime FO 2 NExpTime [6] Figure 5: The six fragments studied in this paper together with the tw o-v aria ble fragment FO 2 of ﬁrst-o rder logic. The table shows st r ongest p ossible results o n the existence o f s yllogistic systems, together with tig ht complexity b ounds. See Section 8 for an explanation. these sys tems, toge ther with the familiar two-variable fr agment of ﬁr st-order logic, are shown in Figure 5. The a sso ciated table lis ts these frag ments together with the r esults we ob- tained on the ex istence of syllo gistic systems and the co mplexity of determining v alidity of sequents. Rega rding the existence of syllo gistic s ystems, we showed that: (i) S and S † bo th have sound and complete dir ect syllo gistic systems (i.e. s ystems containing no rule of r e du ct io ad absur dum ); (ii) R has a sound and r efutation-c omplete direct syllogis tic s ystem (i.e. one which b eco mes com- plete if r e ductio ad absur dum is allow ed as a single, ﬁnal step), but no sound and complete direc t syllogistic sys tem; (iii) R ∗ has a sound and complete indir e ct syllogistic system (i.e. one allowing unr estricted use of r e ductio ad absur dum ), but—unless PTime = NPTime —no sound and refutation-complete direct syllo- gistic system; (iv) neither R † nor R ∗† has even an indirect syllogistic system that is so und and co mplete. Regarding complexity , we showed that: (i) the problem of deter mining the v alidit y o f a sequent in any o f S , S † or R is NLogSp ace - complete; (ii) the problem of determining the v a lidit y of a sequent in R ∗ is co- NPTime -co mplete; (iii) the problem of determining the v alidity of a s equent in either of R † or R ∗† is ExpTime -complete. References [1] An toine Arnauld. L o gic, or, the Art of Thinking (“The Port-R oyal L o gic”) . Bobbs-Merr ill, 19 64. tr. J. Dick oﬀ a nd P . James (ﬁrst published 16 62). [2] John Corc oran. Completeness of an a ncient logic. Journal of Symb olic L o gic , 37 (4):696–7 02, 19 72. [3] A. de Mo rgan. F ormal L o gic: or, the c alculus of infer enc e, ne c essary and pr ob able . T aylor and W alton, London, 1 847. [4] Augustus de Mor gan. On the s yllogism, Part IV. T r ansactions of the Cambridge Philosophi c al So ciety , 1 0:331– 357, 186 0. [5] George Englebr etsen. Thr e e Lo gicia ns . V an Gorcum, Assen, 198 1. 42 [6] E. Gr¨ adel, P . Kolaitis, and M. V ardi. O n the decision pro blem for tw o- v ar iable ﬁrst-or der lo gic. Bul letin of S ymb olic L o gic , 3(1):53– 69, 19 97. [7] Da vid Harel, Dexter Ko zen, and Jerz y Tiuryn. D ynamic L o gic . MIT P ress, Cambridge, MA., 2000. [8] J. Luk asiewicz . Aristotle’s Syl lo gistic . Clarendon Press, Oxfor d, 2nd e di- tion, 19 57. [9] John N. Martin. Ar istotle’s na tural deduction r evisited. History and Phi- losophy of L o gic , 1 8(1):1–1 5, 199 7. [10] David A. McAllester and Rob ert Giv an. Natur al language synt ax and ﬁrst- order inference . Artiﬁcial Intel ligenc e , 56:1– 20, 19 92. [11] Daniel D. Mer rill. Augustus De Mor gan and t he L o gic of R elations . Kluw er Academic Publisher s, Dordrecht , 19 90. [12] Lawrence S. Mo ss. Syllogistic lo gic with complements. Ms. submitted for publication, F ebruary 200 7. [13] Lawrence S. Moss. Syllogistic lo gics with verbs. Journal of L o gic and Computation , to app ea r. [14] Lawrence S. Mos s. Completeness theo rems for s yllogistic fra gments. In F. Ha mm a nd S. Kepser, editor s, L o gics for Lingu ist ic St ructur es . Mo uton de Gruyter, to app ear , 20 08. [15] Nor itak a Nishihara, Kenichi Mor ita, and Shigeno ri Iwata. An extended syllogistic system with verbs and prop er nouns, a nd its completeness pro of. Systems and Computers in Jap an , 21(1):7 60–7 71, 199 0. [16] Chr istos H. Papadimitriou. C omputational Complexity . Addison-W es ley , Reading, MA, 1994 . [17] Ia n Pra tt-Hartmann. F ragments o f la nguage. Journ al of L o gic, L anguage and Information , 13 :207– 223, 200 4. [18] T.J . Smiley . What is a syllog ism? Journal of Philo sophic al L o gic , 2:135 – 154, 19 73. [19] F red Sommers. The L o gic of N atur al L anguage . Clarendon Pres s, Oxford, 1982. [20] Dag W esterst ˚ ahl. Aristotelian syllogisms and gener alized quantiﬁers. Studia L o gic a , XL VII I(4 ):577–5 85, 19 89. 43

Logics for the Relational Syllogistic

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