Zeitachr. 1. malh. Logik und Drundlayen d. Math. Bd. 11, S. 45-55 (1965)

TWO NON-HENKINIAN FRAGMENTS OF THE 2-VALUED PROPOSITIONAL CALCULUS WITH VARIABLE FUNCTORS

by ALAN ROSE in Nottingham

There has recently been given1) a method of formalising any fragment of the 2-valued propositional calculus with variable functors such that implication is definable in terms of the primitives. The object of the present paper is to formalise two further fragments of the propositional calculus with variable functors. These are the propositional calculus with E as the only primitive and the propositional calculus with E and N as primitives. The functor C is not, of course, definable in terms of E and N .

Our primitive rules of procedure for both formalisations will be the usual rules of substitutiona) and modified modus ponens, which we shall denote by R1 and R 2 respectively, and the rule

R3 If EPQ then EdPBQ.

Our axioms are the formulae A 1 4 given below, A4 being omitted in the case where E is the only primitive functor.

A 1 EEGE&p&qdErrE&EdpdqEErr, A2 EEdpdyEdErqdEpr, A3 EdpdEdpdEqq, A4 EEpNEqqNp.

Before commencing the completeness proofs we shall establish a theorem which is an analogue, for the E - N propositional calculus with variable functors, of the ~EsNIEWsKI-MIHAILESClJ3) Theorem. This theorem enables us to see easily that A 1-3 are tautologies, and hence that our formalisations are consistent. Syntactical

1) A. ROSE, Fragments du calcul propositionnel bivalent foncteurs variables. Comptes rendus 258 (1964), 1363-1365.

2) See, for example, J. EUKASIEWICZ, On variable functors of propositional arguments. Proc. Royal Irish Acad. 54 Sect. A (1951), 25-35. We regard a formula as a tautology if the formulae obtained by assigning any of the four t,ruth-tables of one argument to the variable functors occurring in thc formula are tautologics. It is easy to see that if R1 is applied to a tautology then the resulting formula is a tautology.

3, E. C. MIHAILESCU, Recherches sur la negation e t l'equivalence dans le calcul des proposi- tions. Ann. Sci. de l'Univ. Jassy, premiere partie 23 (1937).

46 ALAN ROSE

variables A , A , , A , , . . . , A , , , A , , , . . . , A , , , A , , , . . . will, throughout the paper, represent variable functors. If the syntactical variable P represents a propositional variable, P occurs in a formula Q and A Q is a subformula of a formula R we say that the particular occurrence of P in R is within the scope of the particular occur- rence of A . Thus, for example, in the formula

EdEpNENq

the first symbol E (reading from left to right) occurs within the scope of 6 and the second symbol N occurs within the scope of the second variable functor E , but p does not occur within the scope of the second symbol E and 6 does not occur within the scope of 6 .

Theorem 1. If the propositional variables and variable functors occurring in the formula P are P I , . . ., P,&, A , , . . .,A,, the subsets of the set {A,, . . .,A,} are 6,, . . . , 6,,,,, P, occurs in P outside the scopes of all variable functors of 8, ai, times (i = 1 , . . ., n; j = 1 , . . ., 2m), d k occurs in P outside the scopes of a.&? variable functors of 8, b k j times (k = 1 , . . . , m ; j = 1 , . . . , 2,) and N occurs in P outside the scopes of all variable functors of 8, c, times ( j = 1 , . . . , 2,) then P is a tautology if and only if the integers a,,, . . . , a,,,,,, . . ., a,,, . . ., anem, b,,, . . ., b,,,,,, . . . , bml , . . ., bnrZm, c,, . . . , C p are all even (0 being regarded as an even number).

Let us suppose first that P is a tautology. Let us assign truth-tables to the vari- able functors A , , . . . , A , in such a way that the truth-value of AP is independent of the truth-value of P if and only if A E 6,. Let the variable functors of 8, be Asl, . . ., dl,,j and let the formula Q be obtained from P by replacing each sub- formula of the form A b V R (1 5 v 5 u,) which is not a subformula of another sub- formula of this form by the propositional variable Pn+v, the variables P, , . . . , Pn+", all being distinct. Since P is a tautology the formula Q is a tautology also. I n Q

(1) Pi occurs ai, times (i = 1 , , . ., n ; j = 1 , . . ., 2"'). ( 2 ) P,,+, occurs bljv, times (v = 1 , . . ., ul; j = 1 , . . ., 2,).

( 3 ) dk occurs b,, times (Ak E { A l , . . ., An,} - t",, j = 1, . . . , 2In).

(4) N occurs c, times (j = 1 , . . . ,2m). Let us first assign to all the functors A , of (3) the truth-table such that

A k P =T P.

Hence, by the LESNIEWSKI-MIHAILESCU Theorem, the integers a{, (i = 1 , . . ., n; j = 1 , . . . ,2,) , bljvj ( v = 1 , . . .,u,; j = 1 , . . ., 2,), cl' (j = 1 , . . ., 2,) are all even. Let us then assign to d k , for one particular value of k such that

{A, , * * . , A m } -8,s the truth-table of the functor N , the remaining assignments being unchanged. It follows a t once that the integer bki + cj is even. Since c, is even it follows a t once that bki is even. Since this has been proved for an arbitrary value of k [subject

NON-HENKINTAN FRAGMENTS OF TEE %VALUED PROPOSITIONAL CALCULUS 47

to the conditions of (3)] and the integers bb0, are even it follows a t once that the integers bki ( k = 1 , . . ., m ; j = 1 , . . ., 2"') are even. Thus the conditions for P to be a tautology are necessary.

We shall now show that the conditions are sufficient. We shall assume that the conditions are satisfied and show that, under an arbitrary assignment of truth- tables to A , , . . . , A,, P takes the truth-value T whatever the truth-values of P, , . . . , Pn . We may assume, without loss of generality, that, for all formulae R ,

A k R =T R ( k = 1 , . . ., t ) , A , R = , N R ( k = t + l , . . . , ~ ) ,

A k R = , E R R ( k = + 1 , . . . , 5 ; ) ,

A , R . = , N E R R ( k = 5; + 1 , . . . , m ) .

Let j be the integer (1 5 j 2m) for which

Lgf = {'q+17 * * * ? d m ~ }

and let S be the formula obtained from P by replacing each subformula of the form A k Q , starting from the innermost, by Q (k = 1 , . . . , t ) , N Q ( k = 5 + 1 , . . . , q), is a propositional variable distinct from P I , . . ., P,. It will be sufficient to show that S is a tautology. In S

E P , + , P , + , ( k = V + 1,...,5)OrNEPn,lP,+l(k=5+1, . . . ,m),where P n + 1

Pi occurs ai, times (i = 1 , . . . , n ) ,

P, .+ , occurs 22?= rl + bk times,

N occurs zi!bki + cr=tc+l bki + Cj times,

A , does not occur ( k = 1, . . . , m ) .

Hence, by the LESNIEWSKI-MIHAILESCU Theorem, S is a tautology.

As an example of the use of Theorem 1 we shall show that A1 is a tautology. Let 8, be the empty set and let

&,={6} , 8 ,={&} , 8 4 = { 6 , & } .

Hence, if the syntactical variables P I , P , , Pa, A , , A , represent p , q , r , 6 , E respect- ively, then

a,, = 2 , a,, = 2, a,, = 4 , b, , = 4 , bzl = 4 , c1 = 0;

a12 = 0 , a,, = 0 , a3, = 2, b, , = 4 , b, , = 2 , c2 = 0 ;

a,, = 0 , a,, = 0 , a,, = 2 , b , , =.2, b,, = 4 , c, = 0 ;

a14 = 0 , a,, = 0 , a,, = 0 , 614 = 2 , b, , = 2 , c, = 0 .

Since a,, , . . . , a,,, b , , , . . . , b,, , c,, . . . , c., are all even, A 1 is a tautology.

48 ALAN ROSE

We note next that if the truth-values T, F are re-named 0, 1 respectively and the operations of addition and multiplication refer to the field J , , then, if P , Q , NP, EPQ take the truth-values x, y , n (x) , e (5, y) respectively,

n(x)=x+ 1, e ( x , y ) = z + y .

If, further, the formulae EANEPPAEPP, AEPP, AP take the truth-values a,P, y (a, B , x) respectively1) then

y(a,B,x) =ax + p . This may be established as follows.

the truth-value y (a , B , O ) . Hence Since EPP always takes the truth-value 0 it follows a t once that AEPP takes

y (a , p , 0) = p = a . 0 + B . , Since NEPP always takes the truth-value 1 it follows a t once that ANEPP

takes the truth-value y ( a , B , l ) . Hence the formula EANEPPAEPP takes the truth-value y (a , B , 1) + y (a , B , 0) and

y @ , B , 1) + y ( a , B , 0) = a .

y ( a , B , 1) = a + y ( a , p , 0) = 0L - 1 + B . Hence

We note that the formula, EAPAEQQ

takes the truth-value (a.+B) + p = a x .

We shall therefore make the definition

MPQA =df EAPAEQQ.

We shall now derive certain formulae in the formalisation of the system with E as the only primitive functor. Applying R 1 to A2 we obtain a t once

It follows a t once from a result of EUKASIEWICZ~) that every tautology which contains no variable functors is provable in the system. I n particular we can derive the formulae

F 1 EEpqEErqEpr .

F1 EPP, F 2 EEpqEEqrEpr , 1) Clearly, the truth-value of AP is determined uniquely by the truth-values of P , AEPP and

ANEPP. Since the truth-value of ANEPP is determined uniquely by the truth-values of AEPP, EANEPPAEPP it follows at once that the truth-value of AP is determined uniquely by the truth-values of P , AEPP and EANEPPAEPP.

a) See, for example, B. SOBOCII~KI, On the single axioms of protothetic. I, Notre Dame J. Formal Log. 1 (1960), 62-73. The reference to EUKASIEWICZ’S axiom is on p. 65.

NON-HENKlNIAN FRAQMENTS OB THE %VALUED PROPOSITIONAL CALCULUS 49

F 3 EEpqEErsEEprEqs,

F 4 EEPqEqp, F5 EEpEqrEEpqr , F6 EEEpqrEpEqr , F7 EEEpqqp, F 8 EEpqEEprEqr , F 9 EEppErr,

F 10 EEEpqErsEErsEEpEqs,

F11 EpEEqqp.

From F2, R 1 and R2 we deduce

R 4 If EPQ and EQR then E P R .

We shall now derive

R 5 If EPQ then EAPAQ.

If 6 does not occur in EPQ we may apply R 3 t o this formula and infer the formula E6P6Q. Applying R1 we then deduce the formula EAP4Q.

If 6 occurs in EPQ let A , be a variable functor, distinct from A , which does not occur in EPQ and let us apply R1 to this formula, substituting A , for 6. Let us denote tho formula thus derived by E R S . Applying R 3 we deduce the formula EGRSAY. Since 6 does not occur in ERS we may, using R1, derive the formula EdRAS. Applying R 1 again we may, since 4 and A , are distinct variable functors, deduce the formula E3PAQ.

From R5, R1 and R2 we deduce a t once the rule of substitutivity of equivalence, which we shall denote by R6.

From A 1 and R 1 we deduce the formula EE6 EEpsEqqdEqqEsE6pdEqqsEqq

which may be rewritten

From F8, A3, R 1 and R 2 we deduce the formula

F 12 EMMpq-zqdMMpqdps.

EEGp6EqqEGEGpdEqqdEqq which may be rewritten

F 13 EMpqdMMpqsqd. From A2 and R 1 we deduce

F 14 EEGpSqEGEpqSEpp.

F 15 EEdpSqEGEpqSErr . From F9, F14 and R6 we obtain

4 Ztschr. f . math. Logik

50 ALAN ROSE

From F15, F10, R1 and R 2 we then deduce the formula

EEdE~dErrEEdpdErrEdqdErr which may be rewritten

F 16 EMEpqrdEMprdMqrd.

From F 7 and R 1 we deduce the formula

EEEdpdEqqdEqqdp which may be rewritten

F 17 EEMpqddEqqdp.

We shall now give definitions of weak and strong no rm1 form. We first define the functors M , , M , , . . .; F, , F 2 , . . . by

MOP&= df P 3

M,+ iPQA1- . . A , + 1 =df MM,P&A, . . . A,,QAy + 1 (Y 1 0 , 1 , . . .), (Y = 1 , 2 , . . .). F$’QA,. . . A,, =di M,- ,A1EPP&A2. . . A ,

I. If P is a propositional variable then M T Q A , . . . A , is an A-formula (Y = 0 ,

11. The formula F,,PQA,. . . A , is an A-formula (Y = 1 , 2 , . . .). 111. A formula cannot be an A-formula except in virtue of I or of 11. IV. If P,, . . ., P,, are A-formulae then the formula’) En-’P1 . . . P, is in weak

V. A formula cannot be in weak normal form except in virtue of IV.

Let us now consider an arbitrary particular method of enumerating the variable functors of our system. We shall refer to this as the standard ordering and, with respect to this ordering, we shall define strong normal form.

1 , . . .).

normal form (n = 1 , 2 , . . .) .

I. If P is a propositional variable and A f precedes Ai + in the standard ordering (i = 1, . . ., Y - 1) then M,PA, . . . A , is a B-formula (Y = 0, 1, . . .). [Since Ai precedes di + (i = 1 , . . . , Y - 1) i t follows a t once that the variable functors A , , . . . , A , are all di~t~inct.1

F a d , . . . A , , is a B-formula (Y = 1, 2 , . . .). 11. If di precedes Ai + 1 in the standard ordering (i = 2 , , . ., Y - 1) then

111. A formula cannot be a B-formula except in virtue of I or of 11. IV. If P,, . . . , P,, are B-formulae then E n - PI . . . P,, is in strong normal form. V. A formula cannot be in strong normal form except in virtue of IV.

l ) En-’ denotes n - 1 consecutive symbols E ( ‘ 1 ~ = 1 ,2 , . . .). Similar notations are used later in the paper.

NON-EENKINIAN FRAGMENTS OF TEE 2-VALUED PROPOSITIONAL CALCULUS 51

In order to establish the completeness of our formalisation we shall prove three lemmas.

Lemma 1. To each formula P there corresponds a formula Q in weak normal form such that EQP i s provable.

We shall establish the lemma by strong induction on the') length, I , of P . If 1 = 1 then P is a propositional variable. Thus Q is P, the formula EPP being obtained from F1 and R1.

We now assume the lemma for all positive integers less than 1 and deduce it for 1. We shall divide the proof of the induction step into two cases, tho case where the principal connective of P is E and the case where it is a variable functor.

If P is of the form ERS then

l (R) 5 J ( S ) < Z(P) and i t follows a t once from our induction hypothesis that there exist formulae T , U in weak normal form such that the formulae

ETR, EUS

E m - I T 1 . . . T,, E n - ' U , . . . U , respectively, where T , , . . . , T,, U , , . . ., U,, are A-formulae. From F3 and R1 we obtain the formula

EETREE USEETUP.

are provable. Since T , U are in weak normal form they are of the forms

Thus, using R2, we deduce the formula

i.e. EETUP,

E7j1 .+ IT, . . . T,En - U , . . . U,P.

E m + " T 1 . . . T , U 1 . . . U,P. Using F5 and R 6 we then deduce

Thus the required formula Q is

E m - k n - l T I . . . T,U, . . . U, . If P is of the form AR then

1 (R) < 1 (PI and, by our induction hypothesis, there exists a formula S in weak normal form such that the formula

is provable. ESR

1) We define the length of P to be the number of symbols occurring in it, repeated symbols being reckoned according to their multiplicities. Thus, for example, the formula EpESqddq is of length 8.

4*

52 ALAN ROSE

Using R6 we obtain the formula

i. e.

and from F 17 and R 1 we deduce

EASAR,

EASP

EEMSTAAETTAS, where T is a propositional variable not occurring in 8. Using R 4 we then obtain

Since S is in weak normal form it is of the form En-%', . . . S,,

EME"-'S, . . . S,TAEn -'MS,TA . . . MS,,T4,

EEMSTAAETTP.

where S, , . . . , X, are A-formulae. Using F 16, R 1 and R 6 we deduce

i.e.

Using R6 again we deduce

E M S T A E ~ ~ - ' M S ~ T A . . . MS,TA.

E"+'MS,TA . . . MS,TAAETTP. If Si is of the form

we may then use FB, R1 and R 6 to deduce MUiViAi (i = 1 , . . ., n)

E 1 ' + ' M S I V I A . . . MSnV,',,A4ETTP. Since MSIV,A , . . . , MS,V%A, AETT are A-formulae the required formula Q is

EnMSIVIA . . . MSnV,4AETT. Lemma 2. To each A-formula P there corresponds a B-formula Q such that EQP

Since P is an A-formula it is of the form

is provable.

M,RSA, . . . A,,, where R is a propositional variable or of the form AETT. Let i, , . . . , i, be a permuta- tion of the iiitegers 1 , , . . , Y such that Aij does not precede Ail in the standard ordering ( j = 1 , . . ., v - 1) and let jl (= l ) , . . ., jk be the integers such that

Ail,+ 1 > * * . , A i J m + l - '

denote the same variable functor but A; , A i

4 0 4o+l

denote different variable ,functors (w = 1 , . . ., k - 1) and Ai , . . ., Aiy denote the same variable functor. The formula Jk

MkRSAi,l . . . A6 5k

NON-HENKINIAN FRAGMENTS OF THE %-VALUED PROPOSITIONAL CALCULUS 53

is thus a B-formula. We shall denote this formula by Q and ,show that EQP is prov- able.

Using F1, F12, R1 and R6 we deduce the formula

E M , R S A i l . . . divM,,RSA,. . . A,,,

i.e. EM$S5.il . . . Ai,P.

Using F13, F4, R1 and R6 we then deduce

EMkRS5, . . . Ai P, J1 f k

i. e. EQP.

Lemma 3. To each formula P there correspmds a formula Q i n strong normal form such that EQP is provable.

By Lemma 1 there exists a formula R in weak normal form such that E R P is provable. Thus, since we have derived R4, i t will be sufficient to show that to each formula R in weak normal form there corresponds a formula Q in strong normal form such that EQR i s provable.

Since R is in weak normal form i t is of the form

En- 'R , . . . R,,

where R , , . . ., R, are A-formulae. By Lemma 2 there exist B-formulae Q1, . . ., Q, such that the formulae EQ,Ri ( i = 1 , . . ., n ) are provable. Using F 1 and R6 we obtain the formula

En&, , . . Q, ,E ' l - lR1 . . . R,,

E7", . . . Q,,R. i.e.

Since Q1, . . . , Q,& are B-formulae the required formula Q is

En-'&, . . .Q,,.

Theorem 2. The formalisation of the E-propositional calculus with variable functors is complete.

Let P be a tautology of the system. By Lemma 3 there exists a formula Q in strong normal form such that EQP is provable. Since the formalisation is consistent EQP is a tautology and i t follows a t once that Q is a tautology in strong normal form.

Since F4, F5, F6 and R6 have been derived in the formalisation we may assume, without loss of generality, that Q is of the form

Q 1 * . . Qni+n, Emi-n-1

where Q1, . . . , Q, are B-formulae by I and Qrn+', . . ., Q, are B-formulae by 11.

54 ALAN ROSE

Thus (i) Qi is of the form

M,,RiSiAi, . . . Ai E , 9

where Ri is a propositional variable and A i , precedes Ai, + , in the standard order- ing (j = 1 , . . ., ki - 1) ( i = 1 , . . ., m ) .

(ii) Qa is of the form F,,,UiViAi, . . . Aik, where A i , precedes Ai, + in the standard

Let Wi be obtained from Qi by replacing Si by W (i = 1, . . . , m) or by replac-

ordering ( j = 2 , . . ., ki - 1) (i = m + 1 , . . ., n).

ing Ui and Vi by W (i = m + 1 , , . . , n) . It follows at once that the formula

is a tautology.

We shall show next that if there are exactly Ni integers j such that W , is Wi (i = 1, . . . , m + n) then the integers N l , . . . , Nm + ,, are even. Let us suppose that the integers N , , . . . , N , +

Nil, - . ., N l y ,

En' + n - 1 w 1 * * - Wm+n

are not all even, the odd integers being

i j < i j+l ( j = 1 , . . ., M - 1). where

By the LESWIEWSKI-MIHAILESCU Theorem, if P , , . . . , Pm + denote propositional variables and P A , P, denote the same propositional variable if and only if W A is W,, the formula

Em + nPl . . . P , + ,,EM - 'Pil . . . Pix is a tautology. Hence the formula

E m + n W 1 . . . W m + , r E M - ' W i . . . Widl

is a tautology and i t follows a t once that the formula

E M - l W a1 - . * W$y a tautology. Let K be the greatest integer such that

i , 2 m. Thus Wij is

and W i is

M,, RijWAijl . . . Ail,.% ( j = 1 , . . . , K ) 1

1 F,, W W A g l . . . Aijki1 ( j = K + 1 , . . ., M ) .

0 Hence if Ri takes the truth-value xj ( j = 1 , . . . , M ) and AijmX takes the truth- value miw x + /3 ,,,, where x is the truth-value of X ( 0 = 1, . . . , k. * j = 1 , . . . , M ) the tautology

takes the truth-value

1

'1 '

w - l w i l . . * ITty

cjg_i u f i - m j E , , x f + ~ ~ B + 1 - * * aik,l&l.

NON-HENIUNIAN FFGAQMENTS OF THE 2-VALUED PROPOSITIONAL CALCULUS 55

Let U = min(ki, + 1 , . . ., ki, + 1, k i a + l , . . ., k , M )

and let us suppose thiit the vth of the above M summands contains exactly U factors. Let the summands which are identical with the vth be the v,th, . . ., v,th (v, = v) . Thus z is odd. Every summand, other than the v,th, . . . , v,th will contain at least one factor which does not occur in the vth summand. Thus the value of thr sum when the value 1 is assigned to these U variables and the value 0 is assigned to all other variables will be 1. Hence the formula E M U 1 W i , . . . W i , is not a taut- ology and we should have a contradiction. Thus if there are exactly Ni integers j such that W , is (i = 1 , . . . , m + n) then the integers N , , . . . , N?,, +,, are all even.

It follows a t once that, with the notation used above, the formula

P I - . P m + n E?I& 4- n - 1

is a tautology. Hence this formula is derivable by means of F1, R1 and R 2 and we may, by further use of R1, derive the formula

w,. . . W m + n . Em + n - 1

Using F9, R1 and R6 we deduce E n r + n - l Q 1 . . . Qm+n, i.e. Q . Since Q and EQP are provable we can deduce P by a single application of R2. Thus the theorem is proved.

Theorem 3 . The formalisation of the E - N-propodional calculus with variable ficnctors is complete.

Let P be a tautology of the system and let Q be the formula obtained by replac- ing each subformula of P of the form N R , starting from the innermost, by ENESSR, whew 8 is a propositional variable not occurring in P. Since

N R =T ENESSR it follows a t once that Q is a tautology. Let U be the formula obtained from Q by replacing each occurrence of the subformula NESS by S.

Since Q is a tautology it follows a t once from Theorem 1 that i f d is a subset of the set of variable functors occurring in Q then the number of occurrences of N not within the scopc of any variable functor of &' is even. Hence the number of occurrences of S in U not within the scope of any variable functor of d is even also. It then follows easily from Theorem 1 that U is a tautology. Since N does not occur in U i t follows at, once from Theorem 2 that U is provable. Using R 1 we deduce Q . Hence, using A4. R1 and RG we can deduce P. Thus the theorem is proved.

(Eingegangen am 26. Miirz 1964)