Axiomatization of Aqvist’s CS - Logics

by

CHARLES PARSONS

(Columbia University)

Lennart Aqvist has proposed a number of semantical systems for the logic of preference or betterness and va1ue.l In this paper we present axiomatizations for these, systems and prove their com- pleteness.

Aqvist works with a binary sentential operator ‘P’ and two unary operators ‘G’ and ‘B’. ‘pPq’ may be read ‘it is better that p than that q’, ‘Gp’ as ‘it is good’that p’, ‘Bp’ as ‘it is bad that p’ . Other readings are possible which, however, suggest taking ‘p’ and ’q’ not as dummy sentences but as dummy terms for ‘states of affairs’: ‘p is preferable to q’ or perhaps ‘p is preferred toq’ (by an indeterminate subject). The formal work is neutral be- tween these interpretations.

The formal notation is as follows: We have a list p , q, r, p’, q‘, r’, p”, q”, r”, . . . of atomic ‘terms’. If 8 is a term, then so is - 8. If 8, f are terms, then so is (8 V f) (and perhaps (8 A f), (82t), and (8 = f)”.

In the systems CS{ (i=o, a, b, c, ca; j = o , e, II, t ) considered by Aqvist, if 8, f are terms, 8Pf is a formula, and if 8 is a term, G8 and BB are formulae.

We consider also variants CS:’ and CS3,” in which only atomic formulae 8Pf are admitted. In CS:‘, there is an additional atomic constant term o (for the ‘indifferent state of affairs’).

1 ’Chisholm-Sosa Logics of Intrinsic Betterness and Value’, Notis, vol. 2 (1%8),

a Whether these are primitive or introduced by definition can make a differ- ence, except in the systems CS; and CS;. However, this does not affect our discussion.

253-270.

44 CHARLES PARSONS

In all systems, if ‘21 is a formula so is - a; if (u, CZ? are formulae so are ((u V %), ((21 A %), (21 29), and ((u 3).

We replace parentheses by dots according to the conventions of Quine’s Methods of Logic.3 We abbreviate ‘ - 6 ’ as ‘3’ and, where d is atomic or the negation of an atomic term, we similarly abbreviate $ itself.

In CS;’ we have the definitions

D1 GB for $Po D2 B$ for oPB.

CSj and CS:’ are thus theories of preference and value; CS!” are pure theories of preference.

Aqvist’s semantics involves, for each term 6, assigning t o 6 and $ integers (positive, negative, or zero) as their values, with definite constraints on their relation. This is expressed by the following definitions:

A valuation of a set 9X of terms is a function which assigns t o each term t in 9X an integer V(t) so that the following hold:

(i) If t, i ~ 3 3 2 , then Vfi) = V(f ) . (ii) If d, t, 6 v t E 9X, then

min (V($), V(t)) I V($ v t) 2 max (V($), V(t)). (iii) (for CS;’). If o €332, then V(o) = 0.

If 0 E 332, then V(G) = 0.

For i=o, u, b, c, cu, a CS:-valuation is a valuation which satisfies the corresponding one of the following conditions (as- suming also $, i €332):

CSE: CS:: CS;: CS:: cs:,: V(i) = - V(6).

If V($) > 0 (V(6) < 0), then V( i ) i 0 (V(i) 2 0). If V($)+O, V(i)#O, then V($)= - V(6). v($) = O or v(%) =o. If V($) > 0 (V($) <O), then V ( i ) < O (V($) > 0).

A CS:a-valuation is clearly a CSZ-valuation which is also a CS:-valuation.

Revised edition, New York, Holt, 1959.

AXIOMATIZATION OF AQVIST’S cs - LOGICS 45

A valuation induces an assignment of truth-values to formulae composed from terms in 337, as follows:

(iv) (v)

(vi) (vii)

V(dPf) = T if V(6) > V(t) V(Gt) = T if V(t) > 0 V(Bt) =T if V(t) <O V ( 4 ) =T if V(’z1) =I, etc. according to the usual truth-tables.

(Note that (iii), D1, and D2, and (iv) insure that in CS? the defined formulae Gf and Bt get the truth-values which (v) and (vi) would give them.)

A CS:-valuation is a CSP-valuation satisfying the condition: If the mutual entailment of 6 and t is valid in the sense of Anderson and B e l n a ~ , ~ then V(6) = V(t).

A CSI-valuation is a CSP-valuation such that if d ~ t is a tautology, then V(6) = V(t).

A CSY-valuation is a CS7-valuation such that if d = f is a tautology and d and t contain the same atomic elements, then V(6) = V(t).6

1 Axiomatization of universally valid formulae

We first consider the problem of axiomatizing the system of all universally valid formulae. But all the systems below share the same logical framework: all tautologies as logical axioms, modus ponens and substitution of terms for ‘p’, ‘q’, ‘r’, etc. as logical rules of inference.

The system U” has as formulae those of CS:” and the axioms

p1 - W P )

p4 - ( P P B P5 -@Pp)

PZ pPq A qPr. 3pPr P3 ppq V qpp V : pPr = qPr . A . rPp = rPq

P6 pPr A qPr. 3 ( p v q) Pr P7 pPq ApPr. 3 p P ( q V r ) .

Anderson, Alan Ross, and Nuel D. Belnap Jr., ‘Tautological Entailments’, Philosophical Studies 13 (1962), 9-24. 6 Note that in each case the condition insures also VG) = V($

46 CHARLES PARSONS

The system U' has as formulae those of CS;' and, in addition to P1-7, the axioms

01 -(oP6) 02 -(6Po).

The system U has as formulae those of CSg and, in addition to P1-7, the axioms

G1 G2 B p A - B q . 2 q P p G3 - Gp A p P q . I B q G4 G p X - B p

Gp A - Gq. I p P q

U, U', and U" are intended to axiomatize the universally valid formulae of CS:, and CSO,', and CSg" respectively. This will follow easily from

LEMMA I. Let % be a finite set of terms such that

(a) I f ?€%then te% (b) If t €3 and t is not a negation then ? €3. (c) If @ v t € % then @e% and t ~ % . (d) (for the CS;') o €92 and 0 E %

Let A be an assignment of truth-values to atomic formulae @Pt (for the CSg, also G@ and Bd) with 6, t €3. Suppose every formula whose terms are in % and which is a substitution- instance of P1-7 is true under A. Let % be the closure of % under negation. Then A can be extended in a unique way to cover formulae with terms in 3 and to continue to make in- stances of P1-7 true. Consider now the relation @- t of terms of '% which holds iff

@Pt and tP@ are false under A. Then

classes is finite.

equivalence classes.

(i) - is an equivalence relation and the set 3 of equivalence

(ii) 'tPb is true under A' induces a linear ordering i<i of the

AXIOMATIZATION OF AQVIST'S cs - LOGICS 47

(iii) Any isomorphism of this ordering into the integers in- duces a valuation of the set %. That is, if cp is the isomorphism, we set V(t) = cp(i).

(iv) For formulae of U', if A makes 01 and 02 true there are isomorphisms with cp(o) = 0 and any such induces a valuation.

(v) For formulae of U, if A makes every instance of G 1 - 4 with terms in 92 true, there are isomorphisms with cp(g) < 0 iff Bd is true and cp6) > 0 iff G(t) is true, and any such isomorphism induces a valuation.

PROOF. The unique extendibility of A claimed in the statement is obvious. Every term in 3 is either in 92 or is 7 for some f €3. We must assign 1 to fP? and TPf . Then in view of P3, we assign to a formula UP? the same value as to uPt, and to @u the same value as to tPu. In view of G1 and G2, G? and B? get the same values as Gf and Bt.

I t follows immediately that if 2l is a formula whose terms contain no connectives except single negations, and 2l' is a substitution-instance of it with terms in 3, then 2l' gets the same truth-value under A as some instance with terms in 92. Thus A makes every instance of P1-3 (and G 1 - 4 where applicable) with terms in 3 true. Obviously A does so for every instance of P4--5. As for P6-7, we note that if d v f €3, then d v t €92, and so 8, t €92.

We introduce the abbreviation

D3 dSt for -(dPt)A -(tP$).

We make two preliminary observations: (I) Let d and t be terms, and let 0 and 2l' be formulae of U

(U', U") such that 2l' is like 2l except for containing occurrences of t a t some places where 2l contains occurrences of 6, except that these occurrences may not be as subterms of other terms (i.e. they must be in contexts uPd, dPu, Gd, Bd, etc.). Then the formula

@Sf3 .2 l=2l ' is provable in U (U', U").

48 CHARLES PARSONS

We use ( I-s 21) for ‘21 is provable in S’, omitting S where it is clear from the context.

It suffices to consider the case where ‘L1 is prime, since the re- maining cases follow by propositional logic. It suffices to show

(1) t-utj dSf3. uPd=uPt (2) FlJ# BSf 3 . dPu E fPu

I-u 6 3 3 . BB = Bt (3) t-u dSt3. Gd= Gt

The first two are immediate by substitution in P3 and proposi- tional logic. The last two are similarly immediate from G1 and G2 respectively.

(11) If 21 is provable in U (U‘, U”) by a proof in which, when the substitutions are pushed back into the axioms, only terms of 3 occur, then CZL is true under A.

For the result of pushing back the substitutions is a deduction of 21 by modus ponens from tautologies and substitution-in- stances of axioms whose terms are in 9 and which must therefore be true under A.

(i) Clearly BPd is false under A; hence - is reflexive. That it is symmetric is obvious. Evidently s - t iff dSf is true under A. By (I) and (11)

dSf 2 . BSU = fSu

is true under A. It is then immediate that if 3 - t and f -u then d N U . This shows that - is an equivalence relation. That 3 is finite is obvious, since if f E%, f - d for some d €3.

(ii) Since (1) and (2) above are true under A, ;<? is independent of the representatives d and f of the equivalence classes. Hence if i<? and i<i, then tPd and uPt are true under A. By P2 so is

(4) uPf A tPB . 3 up6

and therefore so is UP$. Hence ;<;.

then not i<i. Hence < is a partial ordering.

i<i nor i<$, d - f and hence $ =?.

Taking u = 6 in (4), since 6PB is false under A, we have if ;<;

That < is a linear ordering is now obvious, since if neither

AXIOMATIZATION OF AQVIST'S cs - LOGICS 49

(iii) Since 3 is finite and linearly ordered by <, there are infinitely many isomorphisms of it into the integers. If 'p is such an isomorphism, we can set V,(t) = cp(i).

For any t, clearly tP? and TPt are false under A; hence t -7; hence 'pfi) = 'pc); i.e. V,(t) = V$).

If d v t E%, then the formulae

dp(d V t) A tP(d V t) .I (d V t)P(d V t) (6 V t)Pd A (d V t)Pt .2 (d V t)P(d V t)

are true under A. Since (d V t)P(d V t) is false, it follows that we cannot have either

dvf<? and dvt<i or s<dvt and i<dvt.

Since < is a linear ordering, it follows that

whence Min (i, &dtcf<Max ($, i) where Min and Max are in the sense of <. This clearly implies min (V,(d), V,(t)) I VJd V t) 5 max (V,(d), V,(t)). This com- pletes the proof that V is a valuation of formulae of U" and thus proves (iii).

(iv) Since 01 and 02 are true under A, clearly o - 0. Hence for any isomorphism 'p, V,(o) = V,(O). But an isomorphism can be chosen with cp(i5) = 0.

(v) It is clear that any isomorphism satisfying the stated condi- tion induces a valuation; thus it suffices to show that such iso- morphisms exist.

If Gt is true for all t, clearly any isomorphism with 'p(?) > 0 for all d will do. Otherwise note that if Gt is false, then Gd is false for all d such that is?. For by G3 and G4 the formulae

(5) -GtAtPd . 2 B d (6) G d I - B d

are true under A; hence if ?<i, GI is false. But by (3) Gd is false ifd-t. 4-Theoria. 1:1970

50 CHARLES PARSONS

Let f be the greatest class in <-order containing a f with Gf false.

If Bf is true, it suffices to take 'p with cp(i) <O if isf and 'p(i)>O if f<6. Clearly (p(g)>O iff Gb is true. If 'p(i) <O then $<i; by (5) B i is true if i<t. If BB is true, then by (6) G8 is false an2 hence 'p($) I 0; by construction 'p($) < 0.

If Bf is false, it suffices to take 'p with 'p(E) = 0. If 'p(i) < 0 then i<i = t and by (5) BB is true. If BB is true then since by G2

Bd A -Bt .=)fPd

is true, $<i and q(i)<O. If 'p(8)>0 then clearly G8 is true; if Gd is true, we cannot have $<i by the above; hence t<i and 'p(i) > 0.

This proves (v).

THEOREM I . If 2l is a formula of U, it is universally valid if and only if it is provable in U. Similarly for U' and U".

PROOF. It is easy to see that the theorems of U, U', and U" are all universally valid, since the axioms can be verified by inspection to be such.

Conversely, suppose 2l. is not provable. Let 3 be the set of terms occurring in '21, augmented so that it satisfies the condi- tions (a)-(d) of lemma 1. Let G be the (finite) set of instances of axioms of the relevant system which have terms in 3. 6 does not truth-functionally imply 2l; hence there is an assignment A of truth-values to the formulae BPf (GB, Bd) with B, f €3, which makes 2l false and every formula of G true. Then lemma 1 yields a valuation which makes 0 false.

For let 'p be the isomorphism yielded by lemma 1, and let BPf be an atomic subformula of A.

V,(BPf) = T iff V,(8) > V,(f) iff i<i iff 8Pt is true under A.

and in the case of U, if G8 is an atomic subformula of %,

V,(G8) = T iff V,($) > 0 iff q(4) > 0 iff G8 is true under A

AXIOMATIZATION OF AQVIST'S cs - LOGICS

atomic subformula of 2l.

51

V,(Bd) = T iff B1 is true under A similarly, if B$ is an

I t follows by the truth-tables that V,(2l)=T iff 2l is true under A. Since 9 is false under A, V,(2l) =I, q.e.d.

COROLLARY. U and U' are inessential extensions of U".

2 Axiomatization of the CSp The systems CSO,' and CSE" are obtained from U' and U" respec- tively by adding the axiom

P8 p P q 2 . p P p V qPq.

CS8 is obtained from U by dropping G4 and adding

G5 GpI-GP G6 B p I - B j

PROOF. (7) is obvious by P4-5 and (I).

PPP A ppp .I p p p By P2

and (8) follows by P1. By P8

qPp 1. qpq v PPp

In the case of all such claims, the proofs will be such that (11) is applicable, even where additional axioms are involved. This will be taken for granted, and its verification in each case is left to the reader. The essential point is that no proofs from here on depend on P6 or P7. Application of (11) will be tacit.

52 CHARLES PARSONS

and by D3

(since its expansion is a tautology). Hence (9) follows by (8). qsii = - 14pa

By P8

and by (8)

and hence (10) follows by D3.

PPq 3. PPp v qpq

pPq 3. p p p v qpq

(gal PpP= - C P m

By (7) and P8

and by (10)

and then (11) follows by D3. PPq A qsq .I - @Pp)

(12) is proved similarly, using (9).

LEMMA 3. The following are provable in CSg’.

(15) PPP 3 - COPP) (16) jd’p2 -@PO) (17) PPO ZI - @PO) (18) OPPI -COP#)

PROOF. By 01 and 02, oS0. Then substitute ‘0’ for ‘q’ in (9), (lo), (11), (12).

LEMMA 4. G4 is provable in CS;.

PROOF. By G1 and G2 (19) (20)

I- Gp A - G3. ~ P P P t Bp A - BP . I P P p

and hence G4 follows by (8), G5, and G6. (The proof of (8) did not require P8.)

LEMMA 5. Let 52 and A be as in lemma 1. Suppose that in addition A makes true every instance with terms in 3 of P8 (or, for CSE, G5, G6). Then the valuations of lemma 1 can be so constructed as to be CSg-valuations.

AXIOMATIZATION OF AQVIST'S cs - LOGICS 53

PROOF. The cases of CSO, and CSO,' are easy; any V, constructed by the procedure of lemma 1 is a CSg-valuation.

CSO,: If V,(f)>O then Gf is true under A, and hence by G5, G i is false under A; hence V,@) 50. Similarly if V,(t) <O, Bt is true, B j is false, and therefore V,(?)20.7

CSO,': V,(o) = V&) = 0; hence if V,(t) > 0, fPo is true under A; hence by (17) PO is false; hence not O<?; hence i<O; therefore V,@) 1 0. Similarly, by (18), if V,(t) < 0 then V,@) 23.

For CSO,", we consider cases.

CASE I . There is a term uo in 3 such that u0 -Go . We can choose 'p so that cp(uo)=O, and by (11) and (12) the case is just like that of CSO,'.

CASE 2. Otherwise. We note that by P8 and (8) if i<i then 8Pg and ?Pf cannot both be true. Hence if 8P3 and iPt are true, then t 1 8 .

Let d1, . . ., 8, be all the equivalence classes in <-order. Since in this case for each f either tPi or ?Pt holds, and if fPi is true so is tPt, there are classes & and 8, such that B i is the last containing a t with 7Pf true and is the first containing a f with fP? true. By the above i 5 j , but if i < rn < j for some rn we have case 1. Hence i = j or i + l = j .

Clearly there are isomorphisms 'p with cp(i,) 1 0 if k i and cp(ik) 2 0 if k2 j. If 'p is such, then V, is a CSE-valuation.

For suppose V,(t)>O. Then ii<i and hence i€'t is false and tPi true. Then TP? is true and i5ki. Hence V,e) 50.

If V,(f) < 0 then i<i,, tPi is false and ?Pt true. Then jP? is true and &<f. Hence V,@)20.

- -

= -

This shows P8 provable in CSE. To see it directly note that by G5, G6, (191, and (20)

F GP3PPF t- BqxGPq

and by G3, t- pPq 3. Gp v Bq. In CS;', P8 could be replaced by (17).

54 CHARLES PARSONS

THEOREM 2. Let 21 be a formula of CSX. Then ‘z1 is provable in CSX iff 21 is CSX-valid. Similarly for CSX’ and CSX”.

PROOF. Again it is easy to see that the axioms are CSX-valid. Completeness follows from lemma 5, as in the proof of theorem 1.

COROLLARY. CSE and CSX’ are inessential extensions of CSX”.

The approach to axiomatizing the remaining systems is the same as that to CSX: to find axioms such that, in a situation like that of lemma 5, the requirement that the assignment A should make all instances of them true constrains the relation < so that an isomorphism cp will exist so that that V is a valuation of the re- quired sort. The axioms are as follows (in addition to those of U, U’, or U”):

CS:, CS“,: G5, G6 (G4 therefore redundant) G7 p P q A Gq A B p . I q P p G8 p P q A B p A Gq . 3 q P p

CS:”: P8 and P9: p P q A q P r A r‘Pp A - (Tpr) A - (?-’Pi‘) .3 qpp P10: p P q A r P p A qPr’ A - (rPi) A - (T’Pr’) . ~ q P p P11: P P q A @‘q A - (q@) Ap‘Pq‘ A q’pq’ A -($Pis’).

P12: P P q A p P p A - (q@) Ap’Pq’ A\’@’ A - (i’pp’)

P13: p p q A p P p A -(qpp) Ap’Pq’ A p ’ P p ‘ A -(q‘Pp’).

CS: - CS;’: G9 Gp V B p .2 - ( G P V B p ) (G4 is now redundant.) CS;”: P148 psq A q S r . V . p S q A q S i . V . p S q A qSr

3. qSq’ v qsp’ v p s q ’ v psp‘

3 . pSp’ v p s q ’ v qsp’ v qsq‘

3. qSp’ v qsq’ v p s p ’ v psq’

. V . psi A q s i . V . psq Aqsr . V . p s q A q s r

. V . pSq A q S r . V . p S q A q S r

8 P14 could not be replaced by

P14’ psq v psq v psq v pq. For we can give a CSX-valuation in which every instance of P14’ is true but P14 is false. Let VCp) = V(q) = 1, V@) = V(f) = ->, V(4) = V(r) =0, and let V(@ v t) = V(8). Hence P14 is not provable in CSX with the additional axiom P14’, and hence that system is not complete for CS:-valuations.

AXIOMATIZATION OF AQVIST'S cs - LOGICS 55

CS;, CS:,: G10 G p 3 B p G11 B p 3 G p

CS:": P8 P15 PPq A - (jf'q) . 3 qf') P16 PPq A - ( p P p ) . 2 q P p CSL: G10, G11 P17 p P q 3 q P j CS::: P17, 03 p P o 2 o P p (02 is now redundant.) CS;;: P8, P17

Then we have

THEOREM 3. For i=a , b, c, ca, the systems CS:, CSY', and CSP" are sound and complete for CS:-valuations. CS: and CSP' are inessential extensions of CSP".

PROOF. Soundness is easy to verify in each case. In each case, the proof of completeness proceeds by a lemma corresponding to lemma 5. The case i = a is the most complex and we leave it for the moment.

i = b. Obviously G9 implies G5 and G6 (and therefore P8), and in the situation of lemma 5 if A makes every instance of G9 true, 'p will be such that for each f, 'p(i)=O or cpfi)=O, whence V , is a CS:-valuation.

In the case of CS:", we show by induction on n that if every instance of P14 holds under A, then if B,, . . ., 8 n are any terms in%, there - is an equivalence class 9X such that for each i, I i ~ 9 X or B i €357. If 6,, . . ., 8, represent all the equivalence classes of I .., f A A - i, then if (p(9X) = 0, V, is a CSfvaluation.

The construction clearly yields a CS:-valuation if all instances of G10 and G11 are true under A. By lemma 6 below, CS: is a conservative extension of CS;"; hence CS;" is also complete.

The completeness of CSZa, CS;: and CS:a" can be seen by a much simplified version of the construction sketched below for cso,.

56 CHARLES PARSONS

Consider now the case i = a . Let 9? and A be as in lemma 1, and suppose every instance of an axiom of CSO,, CSO,’, and CS:” with terms in 9? is true under A.

We call an equivalence class 5% of - zero if (for CSE) it contains a f such that Gf and Bt are false, or (for CSO,’) it contains o,

or (for CSO,”) it contains a t with f -7, or 8, f with 8 - t, $<:, f<i. (The last case holds iff Z28t is true, where Z,8t abbreviates 6st A iP8 A tpi)

From the proof of lemma 5 it is clear that - has a t most one zero class. We shall show

(*) There is a single equivalence class %)I such that whenever i<i and 2%; then either 8 €%)I, f ~ % , ? €%)I, or ~ E ~ I . % ) I is the zero class if there is one.

For CS: and CSO,’, (*) is easy to verify. We can derive

which insures that the zero class will satisfy (*) and will exist if there is any d, t with f P 8 true, SP? false

Similarly, using P9 and P10, we can derive in CS:”

pPq A - (qPp) A . r s f V Z p ‘ : 2 . p s r V pSr V qSr V qSr

which insures that if there is a zero class it will satisfy I*). How- ever, - - it is possible to have no zero class and yet have 8, t with i<t and j<t, for example if % contains only p , q, p , q, pPq, pPp, qPq are true, and p P q and qPp are false.

Suppose now there is no zero class. Let GI, . . ., @.,be the equiv- alence classes in <-order; let i and j be as in the proof of lemma 5, case 2. This is the case where j = i + 1. We shall show that either d i or Bi+, is the required %)I.

Suppose that i<t and -. We must have either tP? or ZP8 true by P8. If ;<:, one of tPf and 8 P j must hold. Hence if tPj and 8P8 are both true, $ -? and is the zero class. Hence we can assume either fP? or i P 8 is false, and since there is no zero class, that one of 8 P j or ?Pf is true.

AXIOMATIZATION OF AQVIST’S cs - LOGICS 57

hold. (Call this a type I case.) Then either 8 - a i + , or t - G i . Otherwise ii+,<i and t<ii, and the formula

(a) Suppose i<t, i5:, and

_ _ tP8 A 8 ~ 8 , + , A B~ P F A - ( i i + l P@i+J A - (8i ~ i i ) . 28Pt

would be false, but it is an instance of P9. (b) Similarly, by P10, if $<f, $<;, and i<? hold (call this a

(c) In fact either 8 - G i + , in every type I case, or t - G i in every such case. For suppose 8 -di+,, t ?j 8 , but 8’ and t’ are a type I case with V - 8i and 8’ + bi+,.

type I1 case), then either i - 8 i + l or t-&. -

-

By P11 the formula

tP8 A $ P i A - (Bpi) A t’P8‘ A $ ’ P i r A - (3Pf’) . 3 . BSB’ v 8St. v ts8f v tst;

is true, so its consequent is true. But 8S8’ and %’ contradict the hypothesis, and $S? and ?S$’ imply Z,$? and Z&‘? respectively.

(d) Similarly, by P12, one of 8-&+, or t -8 i holds in every type I1 case.

(e) if the first alternative holds in both (c) and (d), 8i+1 is the required %; if the second holds in both, then d i is the required %. In all other cases an instance of P13 can be seen to be con- tradict e d .

Thus (it) is proved. Now we construct ‘p so that V, is a CSZ- valuation. Let zl, . . ., in be the equivalence classes in order, and let i, be %. We set ‘p(&) = 0 and proceed by induction. (If there is no %, let dj be as above, and let ‘p(ii)= - 1 and v(ii+l)= + 1.)

For i#j, we can suppose &+$junless $ - & for every $-Pi. In the latter case we call ij blank.

Let k , = I, = j (k , = i and I, = i + 1 if % does not exist). Suppose now that k,, I, have been defined, 1 I k , I Im I n ; V ( i i ) has now been defined for k, I i I 1, such that

(i) - m I cp(ii) I m (ii) ‘p is an isomorphism of ikm, . . ., &, into

[-m, ml.

58 CHARLES PARSONS

(iii) If c p ( i t ) is defined, T(Zi)= -cp($i) or v($i)=O or

(iv) cp(irm)= -m or cp(i,,)= +m. q(ii) =o.

CASE I . ik,-,, both not blank. We set cp(@Km-l) = - m- 1;

(~(i I m + J = + m + 1, k m + l = k m - 1, I,+, = l m + 1.

(i), (ii), and (iv) are obvious. (iii) carries over if k m I i I Im. Suppose f - @ l m + l . Then by (*) either 5-6) or f -@I ,+~ . By (iii) (p(%,+,) was undefined a t m; hence %m+l<i~,,,. (!I,+, - $1 is excluded.) If dim +,xi km-l then @ k m - l < i l n , +l; but ~m-l<%,,+l is impossible, and ik,-l<iIm implies by (iii) ikm-l - - L%, which - contradicts the non-blankness of i k m - l . Hence @lm+l $km-l, t - @hm-l . 1.e. if y(i) = + m + 1, then cp(i) = - m- 1, or cpe) = 0. Similarly if 'p(t) =

-m- 1, then cpci>=O or cp@)= -m-1.

- -

- -

- CASE 2. ilm+l is blank, ;,,-, is not. As before, B l m s @ k m - l . If $lm+l - @km-l, then ilm+, is not blank, contrary to hypothesis. We set k m + 1 = k m , I m + 1 = I m + l and ~ ( ~ z , + J = + m + l - (i)-(iv) are clear.

-

CASE 3. @ k m - l is blank, and cp(irm-l) = - m- 1. (i)-(iv) follow as in case 2.

is not. We set I,n+l = I,, k,,,,, = k,- 1

CASE 4. i k m - l and Slm+l are both blank. We set k m + I = k m - l ,

I m + l = I m + l , cp(ix,-J=-m-l, ~ p ( i t , + J = + m + l . (i)-(iv) are ob- vious.

NOTE. It could be that k , = l while I m < n , or k,> 1 while Im=n. Then the remaining classes are all blank, and we follow the proce- dure of cases 2 and 3.

Clearly an m is eventually reached so that k,= 1 and I, =n. Then 'p is completely defined on 9, and (iii) implies that V, is a C Sz-valuation.

AXIOMATIZATION OF AQVIST’S cs - LOGICS 59

3 The systems CS:, CS;, CS:

These systems can be axiomatized by adding to the systems CSq simple axiom schemata CS:: If 8 and t entail each other in the sense of Anderson and

Belnap, then t-dSt. CSP: If $ = t is a tautology and 8 and t have the same atomic

constituents, then I- $St CS;: If 8 = t is a tautology, then t- $St.

Systems CS; and CS{” can be obtained by adding these schemata to CSq’ and CSP”. Perhaps in the case of CSY’, $ and t should be allowed to differ with respect to whether o occurs.

4 Definability

We now consider the question, discussed by Aqvist in !j 8 of his paper,‘ whether any of the operators, P, G, B is definable in terms of the others. This question is resolved by theorems4 and 5 below.

In CS: and stronger systems, pPp intuitively expresses Gp. For if VCp)>O then V@)<O and V(pPp)=T; if VCp)>V@), then V(p) > 0 and V@) < 0. Similarly, PPp expresses Bp. In fact we have

LEMMA 6. If in CS:“ we define

D5 Gt for t P i D6 Bt for iPt

then the axioms of CS: are provable.

PROOF. G1 is expanded as

(21) p p p A - (qpq). IpPq. By P8 and (8)

(22) I- qPp 2 . qpq v - (pPp)

(23) BY (1)

t- pPp A pSq .2 qPp

60 CHARLES PARSONS

By P16 and 17)

(24) I- qP@ A - (qpq) . ZpPq.

G2 is derived by a similar argument. G3 becomes

- (ppp) A ppq .2 $‘q

(21) follows truth-functionally from (22)-(24), by D3.

which is immediate from P8. G4 expands to (8). G10-11 become

pPp 3 p P p PPp 2 ply

which are immediate from (7).

LEMMA 7. Let CSP have as formulae those of CS; which lack B and as axioms G1, G5, P1-7 and the following:

G12 pPq 2 . Gp V qPq G13 - Gp ApPq .2 - Gij

Then with the definition

D7 Bf for tPf A - Gt the axioms of CS; are provable in CS,”G.O

axioms P1-7, G2, G6, and If C S t B has as formulae those of CS! which lack G and as

G14 pPq3. BqVpPp G15 -BqApPq . I - B p

then with the definition

D8 Gf for fPiA - BF the axioms of CS! are provable in CStB.

Note that G12 is CSE-valid and G13 is not. The result of dropping G13 from CSgG is an axiomatization of the CSE-valid formulae not containing B. G12 cannot be replaced for this purpose by P8. For G1 and G5 are valid if Gf is interpreted as true whenever V(f) > k for some fixed k 2 0. G12 and G14 do not together imply G3.

AXIOMATIZATION OF AQVIST’S cs - LOGICS 61

PROOF. In CS:G, it suffices to proveG2-3, G6andG9. G2expandsto

(25) PPPA -GpA - ( q P q A -Gq) . 3 q P p We note that G12 and G13 yield

- G p A p P q .I. qPqA -Gq which is G3.

G6 is immediate from (8). By G1 and G5 we have

(26) GP’PPP = - CPPP) by (73

so that by G3

(27) I- PPP A - GP A - (qPq A - Gq) . 3 - (PPq)

by (1) (28) I- pPp A pSq . 3 p P q

By G13 t- - G p A j j P q . 3 -Gq

and hence by G5

(29) t- p P p A - G p A Gq . 3 - (pSq). By G12

pPq 3 . Gp V qPq

and hence by (28) ppp A - Gp A - (qpq) . 3 - (psq)

and (25) follows by (27), (29) and D3. G9 follows easily from G5, G6, (I), and D7. The proof of the result concerning CS:B is similar. These positive results are the best possible. B cannot be defined

in terms of P and G in CS:. Suppose there were a formula 2, not containing B, such that

B p = 2 is true in every CS:-valuation. We can suppose 2 contains no atomic elements but ‘p’, for our hypothesis implies that

B p = 2

62 CHARLES PARSONS

remains valid if ‘p’ is substituted for each other letter. We may also suppose that the only terms occurring in S are ‘p’, ‘j’, ‘p Vp’, and P A P , for any other would be tautologically equivalent to one of these and can be replaced by it.

Thus 3 is a truth-functional combination of formulae dPf and Gd, where each of 8, t is one of the above four terms.

The same is true of -3 A -Gp, which we call%.% is true in a CSf-valuation if and only if V(p)=O. We show that this is im- possible.

Suppose 3 to be in complete disjunctive normal form and consider the formula

(30) p S ( p vp) ~ p P p ~ p S ( p ~ p ) A Gp. Every disjunct of % is either CSi-equivalent to (30) or CSi-

incompatible with it. (30) CSb-implies G(p Vp), - Gp, - G(p A p ) and in view of the transitivity and irreflexibility of P and the substitutivity of S, either sPt or -(dPt) for each 8, tE{p,p, PVP, P A P > .

Hence either (a) 3 is CSk-incompatible with (30) (b) % is CSi-equivalent to the disjunction of (30) and a for-

mula %’ CSi-incompatible with (30). In case (a), set V@)=V(pVp)=l, V(p)=V(pAp)=O. V i s a

CSf-valuation with V(p)=O which makes (30) true a n d 3 false, which violates the condition on %.

In case (b), set V@) = VCp Vp) = 1, V(p) = VCp A p ) = - 1. Then V is a CSf-valuation which makes 3 true because it makes (30) true. Since V(p) # 0, the condition on % is again violated.

Similar reasoning shows that in CSf, G is not definable from B and P.

We can see in the same way that neither G nor B is definable in CS: from P alone. Suppose that for a formula 0 of CS:”

G p = O is CS:-valid. We can again suppose 2l to contain no atomic term ex- cept p and p, p, p v p, p A p as only terms. We consider the formula

(31) P~P”PSCpVP)APS(P~P).

AXIOMATIZATION OF AQVIST'S cs - LOGICS 63

If '21 is in complete disjunctive normal form then every alternand of 2l is either CSt-equivalent to (31) or CSt-incompatible with it. Hence either (a) '21 is CSi-incompatible with (31), or (b) '21 is CSi-equivalent to the alternation of (31) and a formula CSt- incompatible with (31).

In case (a) set Vlp) = Vlp Vp) = 1; V@) = Vlp A p ) = 0. Then V is a CS:-valuation which makes Gp true, (31) true, and hence '21 false.

In case (b) set Vlp) = Vlp Vp) = 0; V@) = Vlp A p ) = - 1. Then V is a CS:-valuation which makes (31), and hence '21, t k e , and which makes Gp false.

Summarizing, we have

THEOREM 4. In the systems CS; and CS!, j = o , e, v, t, G is not definable from B and P nor is B definable from G and P.

In the systems CS;, G is definable from B and P, and B from G and P, but neither G nor B is definable from P alone.

In the systems CS: and CS:,, G and B are definable from P. It is also easy to prove

THEOREM 5. In none of the CS: is P definable from G and B.

PROOF. It suffices to prove this for the cases CS:, and CS: since every other system is a subsystem of one of these.

Suppose that for some formula % not containing P

Let t,, . . ., fa be terms expressing those truth-functions of p and q which are true whenp and q are both true; letf,+ibe?i, i = 1 , . . ., 8. We can suppose that % is in complete disjunctive normal form and that its only terms are t,, .: ., f16.

Consider the formula

(33) G(tJ A . . . A G(tJ A B(tJ . . . A B(tlJ -B(tl) A . . . A -B(t8) A - G(tJ . . . A - G(tlB).

If (33) is a disjunct of 3 we can falsify (32) by setting V(ti) = 1 if i = l , ..., 8, V(ti)=-1 if i=9, ..., 16.

64 CHARLES PARSONS

If (33) is not a disjunct of S we falsify (32) by setting VCp)=2, V@) = - 2, V(ti) = 1 for i = 1, . . ., 8 and = - 1 for i = 9, . . ., 16, other- wise.

The proof for the case of CS: is similar.

Received on December 27, 1968.