M. Fattorosi-barnaba Graded Modalities. Iii (the Completeness and Compactness of $4 ~

We go on along the trend of [2] and [1], giving an axiomatization of $4 ~ and proving its completeness and compactness with respect to the usual reflexive and transitive Kripke models. To reach this results, we use techniques from [1], with suitable adaptations to our specific case. 1. The axioms of $4 ~ and their correctness We assume reader's familiarity with I-2] and 1-1], so that we shall use definitions, results and abbreviat ions from those two works, avoiding tedious repetitions whenever possible. The starting point is the language 5(' of [2] and [-1], and we axiomatize $4 ~ as a N L G M (see [1], Def. 2) by the usual inference rules and taking as axiom schemata Ax. 1-6 from [1] (that we repeat here for sake of readability), Ax.1. classical proposi t ional tautologies, Ax,2. M . + I ~ M , ~ , (neN) AX.3. L o (~ --+ fl) ~ (M, a ~ M , fl), (n E N) Ax.4. M!o(a/x f l )~( (M! , a/x M!,2fl)--.M!,l+,2(c~ v fl)), (nl, n2eN ) Ax.5. M 0 M . ~ ~ M , ~, (n E N) Ax.6. L o e ~ e, plus two more specific axiom schemata of a purely combinator ia l flavour (recall: if m, n E N then m l n means "m divides n" and m,~n means "m does not divide n"): Ax.7. Ax.8. 7 (M!. o~ A Lo(o~ ~ M!m~)) Mpm (fl A M! m fl /x M. -1 (o~ /x M. -1 fl)) Mpn o~ (m, nEN; m ~ 0; m,gn) (m, n, p e N ; m, n ~ 0). We may observe at once that m ~ 0 in Ax.7 is not an essential restriction because, owing to reflexivity (i.e. Ax.6), we derive Ax.7 with m = 0 as a theorem (all the unspecified notions, like ' theorem', 'model ' and so on, are to be referred to the system $4 ~ and to its models, that are the usual Kripke models with a reflexive and transitive accessibility relation) 9 In fact, taking T as a tautology, _1_ as its negation, and using P C ( = proposi t ional calculus) as a quick justification * The present work was carried out while the author had a grant of the Foundat ion "U. Bordoni". 100 M. Fattorosi-Barnaba, C. Cerrato of some obvious propositional steps, we have the following formal deduction 1. T Ax.1 2. L o T (N) 3. M o A_ ~ A_ PC 4. e ~ M o0~ Ax.6 5. (aA ~ M o a ) ~ • 4, PC 6. M o(aA 7 M oa)~-~l 3, 5, Eq 7. ( a ~ M ! o a ) ~ ~ ( ~ a A 7 Moa) Def. M!o, PC 8. L o ( ~ M ! o O O A M ! , a ~ L o ( ~ e A 7 M o e ) AMoO~ 7,(N),Def. M!, (with n > 0 so that 04Vn) Theor. 4-a) of [1], PC 9. L o ( ~ A 7 M oa) A M o ~ M o(0~^ 7 M o~) Ax.3, PC 10. L o ( a ~ M ! o ~ ) A M ! . ~ l 8, 9, 6, PC 11. 7 (M!,~ A L o ( ~ M ! o a ) ) 10, PC. The first job we have to do is to show the above axiomatization is correct, i.e. every theorem is valid in reflexive and transitive Kripke models: owing to 1-2] and [1], this amounts to show Ax.7 and Ax.8 are valid in such models. LEMMA 1. Ax.7 is valid in S4~ PROOF. To assume the contrary implies there exists a reflexive and transitive model (W, R, V) such that in a world we W one has (i) V (w, M !,, a) = 1 (ii) V (w, L o (ct ~ M!m ~)) = 1 where m, n e N and m•n. (i) yields ! (o) [Tw(~)l-n; (ii) assures that (oo) IT w, (a)[ = m, for every w' e T w (a) and transitivity gives (ooo) T w, (~)_ Tw(a ), for every w, w'e W such that wRw'. Furthermore, the accessibility relation R, restricted to T w (~), is an equivalence relation; in fact, one needs only to show R is symmetric: let w', w"e Tw (a) and w'Rw"; then T~,,(e) ~ Tw,(a ), by (ooo), and IZw,(~) l = IZw,,(~) l = m , by (oo), so that Tw,(e ) = T~,,(e); but w'e Tw,(e) (by reflexivity), so also w'e T~,,(e), i.e. w" Rw'. Now Tw(a) has n objects (by (o)), every R-class of its R-partition is of the form T~, (a) (w'e T~(~)) and every such class has m objects (see (oo)), so that m ln: a contradiction. [] LEMMA 2. Ax. 8 is valid in S4~ Graded modalities. I I I 101 PROOF. Assume the contrary: then there exists a S4~ (W, R, V) and a world we W such that (i) V[w, Mpm(fl A M!mfl A M , I (~ ^ Mm-~ fl))] = 1 (ii) V(w, Mp. ~) = 0 where n, m, p e N and n, m # O. Let us call 6 = fl ^ M! m fl ^ M,_ 1 (~t ^ Mm1 fi): so, by (i), ITw(6)l >pm>~O. Take w'eTw(6): for such a world w' one has (o) = 1 (oo) ITs, (~ A M,,-1 fl)[ >I n (ooo) I Tw, (fl)l = m. By hypothesis n > 0, so by (oo) we can choose a w"e T~, (~ ^ M m_ 1 fl) and this means that w'Rw". We want to show that (.) w"Rw'; by the choice of w", we have V(w", M,,-1 fl) = 1, so that IT~,,(/~)I >i m; on the other hand, by the transitivity of R, Tw,, (fl) --Tw, (fl) and, by (ooo), we have T~,,(fl) = Tw,(fl); taking into account (0) and the reflexivity of R, we conclude w'e Tw, (fl) = Tw,, (fl) so that w" Rw'. Let = be the following relation on Aw(= {w'eW: wRw'}; cfr. [2]): w l = w 2 iff wlRw 2 and w2Rw 1. Since R is reflexive and transitive = is an equivalence on Aw and we want to show that strictly more than p of its classes intersect Tw(6): let w'e T~(6) and [w'] its =-class; then Tw (6) c~ [w'] _ T w (fl) c~ [w'] ___ T~, (fl) so that, by (000), (**) T w (6) n [w'] ~< m. Since I Tw(6)l > pm (see above), one has necessarily that the number of the classes [w'] (w' e Tw (6)) is > p. Furthermore, by (.), Tw, (~ A Mr,_ 1 fl) [w'] SO that, taking into account (oo) and defining T ( e ) = {we W: V(w, ~ ) = 1}, one has I[w'] n T(~)[ I> [[w'] n T(~ A M,,-lfl)l >>[T~,(~ A M,,-lfl)l >>n. Since the number of the classes [w'] (w'e Tw(6) ) is > p, one has (***) ITw(e)l >~ IU {[w'] n T(e): w'e Tw(6)}[ > pn which contradicts (ii). [] 2 S tud ia Logica 2/88 102 M. Fattorosi-Barnaba, C. Cerrato At this point we may note that Ax.7 and Ax.8 can be substituted by the single axiom ax.9. Mpm(fl A M[mf l A M , I ( ~ ^ Mm-l f i ) ) -> M(p+l)n-1 ~ (m, n, p ~ N ; m, n ~ 0). In fact we shall show below one can deduce Ax.7 and Ax.8 from Ax.1, . . . , Ax.6, Ax.9 and that Ax.9 is valid in S4~ since Ax.1 . . . . . Ax.8 are complete w.r.t S4~ (w 2), one can conclude Ax. 1 . . . . , Ax.6, Ax.9 is an alternative (i.e. equivalent) axiomatization of $4 ~ LEMMA 3. {Ax.1 . . . . . Ax.6, Ax.9} ~Ax.7, Ax.8. PROOF. We begin with Ax.8: n r 0 implies (p + 1) n 1 ~> pn, so that from Ax.9, by Theor. 4-a) of [2] (and tautologies), one deduces Ax.8. As for Ax.7, firstly we show that (o) ~x4o Lo (a ~ M [ m a ) ~ L o [~ ~ a ^ M!, . a ^ M , . _ i (a A M m 1 C0] ; in fact what follows is (a synthesis of) a K4~ (we write r for L o (a-o M ! m 0 0 and Z for a ^ Mm-1 a): 1. , Lo Z) 2. ~ ~ L o (a --> M m_ 1 ~) 3. ~ ~ L o ( a ~O~ /x M!mO:) 4. ~ ~ L o L o (c~ ~ Z) 5. r ~ L o [ L o (a ~ Z) ^ (a ~ M m 1 ~)3 61 ~ ~ L o [ct ~ L o (ct ~ Z) ^ M m 1 0~] 7 . r ~ L o (a ~ M m 1 Z) 8. r ^ M!m~/x Mm-lX) Def. M!,,, PC, DR3(n = 0) Def. M!m, PC, DR3(n = 0) PC, DR3 (n = 0) 1, Ax.5, PC 2, 4, PC, Theor. 5-a) of [2] 5, PC, DR3 (n = 0) 6, Ax.3 (n = m-1 ) , PC, DR3 (n = 0) 3, 7, PC, Theor. 5-a) of [2], DR3 (n = 0) and (o) is proved. Now we can show the claim. Let 6 = L o ( ~ M I m C t ) A M ! , a where pm < n < (p+ 1)m (m, p e N ; m ~ 0), i.e. TAx.7 with the specified n; let y =ct /~ M!,,,a ^ M m l ( ~ ^ Mo~) and r /= Mpm 7 ~ Mtp+l)m-1 ~, i.e. Ax.9 with fl = ~ and m = n: we shall prove that I--K4 o 6 ~ 7 q. In fact what follows is (a synthesis of) a K4~