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A Characterization of the Intuitionistic Propositional Calculus
A CHARACTERIZATION OF THE INTUlTIONISTlC PROPOSITIONAL CALCULUS*
D. H. J. DE JONGH We will present here a characterization of the intuitionistic piopositional calculus Pp from above, i.e. we will describe a property of Pp that no consistent propositional calculus stronger than Pp possesses. By a propositional calculus stronger than Pp (at least as strong as Pp) we understand one in which all formulas in Pp are provable and some others as well (and possibly some others as well), and which is closed under substitution and modus ponens. (Closure under substitution is, of course, guaranteed if no particular axioms are postulated, but only axiom schemata.) By a formula we understand a formula built up from PI, . . ., 8, (the atomic formulas) with the connectives &, v, 2 and l. We will follow the notation of Kleene [71. Lukasiewicz [ll] proposed the conjecture that Pp can be characterized from above by the property: for any formulas By23, if kpp BvB, then kpp B or kpp 23. This conjecture was disproved by Kreisel and Putnam [9], who showed that Pp+ the axiom schema (lB 3 23 v6) 2 (lB =I B) v (7 B 3 6) has the same property. In [8] Kleene proved a stronger property of Pp and he subsequently proposed to the author the conjecture that this property characterizes Pp from above. First, one defines a notion rlTBfor any sequence r of formulas, any formula B and any propositional calculus T, from the notion t - of ~ provability in T. Kleene states the definition in [8] in particular for the case that T is Pp (cf. [8] 0 4), and he proves (among other things) that, for each By%,(& if Blp, B and t-pp B 2 B v Q , then kppB 3 23 or 1-B 3 6. Kleene’s conjecture, which we will confirm in this presentation is: if T is a Part of the author’s typewritten doctoral dissertation. 211
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propositional calculus at least as strong as Pp, possessing the property if%lT%and k ~ 3%8 v 6 , then
!-T% 3
8 or I-T%
=I O;,
(*I
then T is Pp. Also we will give another characterization of P p from above by replacing (*) by if
%IT
% and I-T (%
3
8)& (83 a), then BIT8.
(**I
For this purpose we discuss some connections between pseudo-Boolean algebras and I-valuations (see for the definition of Z-valuations Beth [l], Kripke [lo], de Jongh [4]). If %(9,, . . ., P,,)is a formula we will write %*(al, . ., a,,) for the pseudo-Boolean algebraic term formed from ul,. . .,a,, with n, v , => and - in the same way as 8 from PI,. . ., P,, with &, v , 3 and 7 .A formula % is said to be valid in a pseudo-Boolean algebra A, iff %*(al, . . .,a,,) = 1 for all a l , . ., a,, E A .
.
.
THEOREM 1 (McKinsey and Tarski [12]). (a). kpp % i f % is valid in every pseudo-Boolean algebra. (b). kpp % if % is valid in every finite pseudo-Boolean algebra. DEFINITION 1. For any propositional calculus T at least as strong as P p we say that a pseudo-Boolean algebra A is a T-pseudo-Boolean algebra iff for each formula % such that t-T %, 8 is valid in A. THEOREM 2. For every propositional calculus T at least as strong as Pp, !-T % i f % is valid in every T-pseudo-Boolean algebra. PROOF.Immediate from results of Birkhoff [3] on equationally defined classes of algebras and the fact that T-pseudo-Boolean algebras can be defined by a system of equations, since pseudo-Boolean algebras can (cf. e.g. [14]). Another special case of a theorem of Birkhoff [3] is THEOREM 3. The class of all T-pseudo-Boolean algebras is closed under the formation of sub-algebras, homomorphisms and direct products. The following definitions are from [6] (mostly originally from [13]). DEFINITION 2. If a partially ordered set (V, d ) is a complete lattice, then a E V is called join-irreducible iff a > (8:p < a}. The set of all joinirreducible elements of V will be denoted by Vo.
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DEFINITION 3. A lattice V is called join-representable iff V is complete and completely distributive, and every a E V can be written as ct = {b: fi < a and p E Y o } . DEFINITION 4. A subset F of (V, < ) is called M-closed iff for all p , q E V, p E F a n d q < pimply q E F. The set of all M-closed subsets of a partially ordered set V will be denoted by V. Vis then complete and completely distributive.
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THEOREM 4. ([13], [6]).Every join-representable lattice V is isomorphic to p. THEOREM 5. (e.g. [2]). A complete and completely distributive lattice is a pseudo-Boolean algebra i f we define a * p = {y: a n y < S } .
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Every finite distributive lattice is complete, completely distributive, and join-representable (e.g. [2]). So theorem 5 implies that every finite distributive lattice is a pseudo-Boolean algebra V for some partially ordered set V. Since for every partially ordered set V, Vis a distributive lattice, there is a 1-1 correspondence between finite pseudo-Boolean algebras and finite partially ordered sets.
DEFINITION 5. If V is a partially ordered set, then V is T-admissible iff 7 is a T-pseudo-Boolean algebra. DEFINITION 6. A P.0.G.-set is a partially ordered set with a greatest element. THEOREM 6. (essentially in [6]). I f V is a P.0.G.-set, then there is the following correspondence between any I-valuation ( V , w ) and the pseudoBoolean algebra i? for all formulas ?I, '23, i f F = { p E V: w(p, 8 ) = l } and G = { p :w(p, '23) = l}, then (i). F n G = { p : w(p, 91 & '23) = l}, (ii). F u G = { p : w ( p , % v @ )= 11, (iii). F G = { p : w(p, 2l 3 23) = l}, 2l) = l}. (iv). - F = { p : w ( p , THEOREM 7. I f V i s a P.0.G.-set with greatest element m , then V is T-admissible ifffor all I-valuations ( V , w ) and all formulas 2l such that IT 8,w(m, 8)= 1. We will need a short resume of some results of [S].
DEFINITION 7.(a). An I-function is a function with domain a finite P.0.G.-set Vand range the set (0,l } with the property that, if q < p i n Vandf(p) = 1, thenf(q) = 1.
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(b). An I"-function f is a function with domain a P.0.G.-set V and range the set (0,l}" such that for all m (1 < m < n) the function f" defined on V byf"(p) = (f(p))(m) is an Z-function. We write F" for the set of all I"-functions and I; for the set of all Zfunctions, m, for the greatest element of D,, the domain o f f , and < for the partial ordering of the domain off. DEFINITION 8.(a). f, g E F" are congruent by cp iff cp is an isomorphism from D, onto DBsuch that f(p) = g(cp(p)) for all p E D,. (b). f is congruent to g (in symbols f = g) iff f is congruent to g by cp for some cp. DEFINITION 9. An n-ary Z-operator u is a function from F" into F with the properties: (i). D,,,, = D, for all f E F". (ii). iff = g by cp, then a(f) = a(g) by cp. If V is a partially ordered set and p E V , then we write V(p) for the set {p' E V:p' < p}. Iff E F" and p E D,, we write f, for the restriction off to D,(P), 9 < f iff (3P)b = f). DEFINITION 10. A function cp fiom the partially ordered set (V, < ) onto the partially ordered set (W, < is strongly isotone iff (i). for all p ' , p E V , if p' < p, then cp(p') < cp(p) and (ii). for all p ' , p E V , if cp(p') < cp(p), then for some p" < p, cp(p") = = cp(P'). DEFINITION ll.(a). I f f , g E F", then g is a reduced form off iff there is a strongly isotone function cp from D, onto D, such that, for all p E D,, dcp(P)) = f(P). (b). f is irreducible iff all reduced forms off are congruent tof. (c). If g is a reduced form o f f and g is irreducible, then we call g a normal form of f.
,
In [5] we proved that the normal form of an I"-function is unique up to congruence (theorem 2.3). Also we proved that, i f f is irreducible and g < then g is irreducible (lemma 2.2 Cor.).
DEFINITION 12. A normal Z-operator a is an Z-operator such that (9. for all f E F" and P E D , , (a(f,))(p) = (U(f))(P) and (ii). iff, g E F" and g is a reduced form off, then (u(g))(m,)= (u(f))(m,). DEFINITION 13. The normalized characteristic set C,*of a normal n-ary
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I-operator a is the set of all irreducible I"-functions f such that ( a ( f > ) ( m f= ) 1. In [5] it was proved that a normal Z-operator is uniquely characterized by its normalized characteristic set. DEFINITION 14. A normal Z-operator isJinite iff its normalized characteristic set contains only a finite number of congruence classes. In [S]we defined then Z-operators corresponding to the usual connectives in a natural way. These Z-operators were proved to be normal (and in the cases of & and 1finite) and we proved that, if we introduce definability in a natural way, then all finite normal Z-operators are definable from the Z-operators corresponding to the usual connectives (theorem 3.7). More in particular: THEOREM 8. Let g E F", g irreducible, me has k immediate predecessors q l , . . .,qk with respect to < # , for each i (1 < i < k) q, has k, immediate predecessors with respect to < #, and for all i and j (1 < i < k, 1 < j < k,) a , and a,, are respectively the normal Z-operators with normalized characteristic sets C: = {fE F":f < g,,} and C:, = {fE F": f < s,,,}. Then, in the case that for all m ( 1 < m < n ) f " ( p ) = 0 for some p < m,, the normal Z-operator a with normalized characteristic set C,*= {f E F":f < g } can be expressed as follows:
a = (with
u k
i= 1
u ail standing for a , i f k , kr
j= 1
(J u,,)
j = 1
u a, k
ki
(Ui 2
2
i=1
= 0).
The main theorem is a little bit stronger than we need to establish the results predicted earlier. We have not checked the intuitionistic validity of it. Probably the double negation will hold intuitionistically. THEOREM 9. If T is a consistent propositional calculus stronger than Pp, then for each integer r 2 2 there is a formula !X 2 Bl v . . .vBs ( s 2 r ) such that %I=% and IT % 3 Bi V . . . v Bs,but not I-T % 3 Bi, V . . . vBir for any proper subsequence ( i l ,. . ., ik) (k 2 1 ) of (1, . . ., s). OUTLINE OF PROOF. First we construct a finite P.0.G.-set ( W, 6 o, qo) having q l , . . ., qk as the immediate predecessors of qo (k 2 l), such that W is not T-admissible, but, for all i (1 < i < k), W(q,) is T-admissible.
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Then from this P.0.G.-set W we construct a P.0.G.-set ( V, 6 ,p o ) with p , , . . .,p , (s 2 r ) as immediate predecessors of p o such that V is not T-admissible, but for all j (1 Q j 6 s), V ( p j ) is. Then we construct an irreducible I"-function g on V for some n such that, for all m (1 6 m 6 n), g"(pi) = 0 for some i (1 6 i 6 s). Now assume a, a, ,. ., a, are the definable I-operators with normalized characteristic sets C,* = {fE F":f 6 g } and C z = {f E F": f 6 gp,} (1 6 i 6 s), and assume that %, 23, , . . .,Bsare the formulas corresponding to these definable Z-operators. We will show (a). I - T % I B , v . . . v B , , (b). not I-T % I Bilv . . . vBikfor any proper subsequence (il,. . ., i,) of (1,. . ., s), (c). %IT 8. (a). The crucial point of the proof is that the class of T-pseudo-Boolean algebras does not contain a pseudo-Boolean algebra on which % 3 Bl v . . . v B, is not valid (a 'counter-example' to this formula). More precisely, for any pseudo-Boolean algebra A on which 91 I> Bl v . . v Bs is not valid, the pseudo-Boolean algebra V is isomorphic to a sub-algebra of a homomorphism of A, and so theorem 3 implies that, since V is not a T-pseudo-Boolean algebra, A cannot be one, and therefore I-T % I B1v . . vB,. In effect, if al, . ., a, E A are such that
.
.
.
.
PI* * B: u . . . u @:(a,,.. ., a,) z
1,
then Vcan be proved to be isomorphic to the sub-algebra of the relativization of A with respect to %*(a,, . .,a,) generated by the images of u l , . . ., CI, under the natural homomorphism on that relativization. (b). If t # ( i l , ..., i,) ( O Q t G s ) , then wecanprove that % I > B ~... , vv B i k __ is not valid on V ( p , ) ; then, since V ( p J is a T-pseudo-Boolean algebra, not I - T % n B i 1 v . . . v B i k . (c). This is easy to prove with the help of theorem 9 and the definition of IT.
.
For our second characterization we give first an equivalent expression for 8.
%IT
DEFINITION 15. a is a connectedz-operator iff a is normal and for allf, g E C,* there exists an h such that f 6 h and g Q h. Of the next theorem we again did not check the intuitionistic validity.
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THEOREM 10. For any formula 8 is connected.
a, 8Ipp 8 ITthe I-operator corresponding
to
From this theorem and the proof of theorem 9 theorem 1 1 follows almost immediately.
THEOREM 11. If T is a consistent propositional calculus at least as strong as Pp, and for each 8,'Bi f 9 l I ~8 and t-T (83 '23) & (23
3
a), then BIT8,
(**)
then all theorems of T are provable in Pp. REFERENCES
[l]E. W. BETH,Semantische Begriindung der derivativen Implikationslogik, Archio. J Math. Logik u. Grundl. Forsch. 7 no. 1, 2 (1961) 23-28. [2]G. BIRKHOFF, Lattice theory, New York (1948). [3]G.BIRKHOFF, On the structure of abstract algebras, Proc. Cambridge Phil. SOC.29 (1935) 433-454.
[4]D. H.J.
DE
JONGH,Recherches sur les I-valuations, rapp. no 17 Contr. Euratom
010-60-12, CETN (1962).
[5] D. H. J. DE JONGH,Investigations on the intuitionistic propositional calculus, dissertation (typewritten), University of Wisconsin (1968). [6] D. H. J. DE JONGHand A. S. TROELSTRA, On the connection of partially ordered sets with some pseudo-Boolean algebras, Indag. Math. 28 no. 3 (1966). [7] S. C. KLEENE,Introduction to metamathematics, Amsterdam (1952). [8] S. C. KLEENE,Disjunction and existence under implication in elementary intuitionistic formalisms, J . Symb. Logic. 27 (1962) 11-18. [9] G. KREISELand H. PUTNAM, Eine Unableitbarkeitbeweismethode fur den intuitionistischen Aussagenkalkiil, Archio J Math. Logik u. Grundl. Forsch. 3 no. 3, 4 (1957) 74-78.
[lo] S. A. KRIPKE,Semantical analysis of intuitionistic logic I, in Formal systems and recursive functions, eds. J. N. Crossley and M.A.E. Dummett, Amsterdam (1965) 92-130.
[ll]J. LUKASIEWICZ, On the intuitionistic theory of deduction, Indag. math. 14 (1952) 202-2 12.
[12]J. C. MCKINSEYand A. TARSKI.Some theorems about the sentential calculi of Lewis and Heyting, J . Symb. Logic 13 (1948) 1-5. [13]G. N. RANEY,Completely distributive complete lattices, Proc. Am. Math. SOC.3 (1952) 667-680.
[I41 H. RASIOWAand R. SIKORSKI,The mathernarics of metamathematics, Warszawa (1963).
D. H. J. DE JONGH We will present here a characterization of the intuitionistic piopositional calculus Pp from above, i.e. we will describe a property of Pp that no consistent propositional calculus stronger than Pp possesses. By a propositional calculus stronger than Pp (at least as strong as Pp) we understand one in which all formulas in Pp are provable and some others as well (and possibly some others as well), and which is closed under substitution and modus ponens. (Closure under substitution is, of course, guaranteed if no particular axioms are postulated, but only axiom schemata.) By a formula we understand a formula built up from PI, . . ., 8, (the atomic formulas) with the connectives &, v, 2 and l. We will follow the notation of Kleene [71. Lukasiewicz [ll] proposed the conjecture that Pp can be characterized from above by the property: for any formulas By23, if kpp BvB, then kpp B or kpp 23. This conjecture was disproved by Kreisel and Putnam [9], who showed that Pp+ the axiom schema (lB 3 23 v6) 2 (lB =I B) v (7 B 3 6) has the same property. In [8] Kleene proved a stronger property of Pp and he subsequently proposed to the author the conjecture that this property characterizes Pp from above. First, one defines a notion rlTBfor any sequence r of formulas, any formula B and any propositional calculus T, from the notion t - of ~ provability in T. Kleene states the definition in [8] in particular for the case that T is Pp (cf. [8] 0 4), and he proves (among other things) that, for each By%,(& if Blp, B and t-pp B 2 B v Q , then kppB 3 23 or 1-B 3 6. Kleene’s conjecture, which we will confirm in this presentation is: if T is a Part of the author’s typewritten doctoral dissertation. 211
212
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propositional calculus at least as strong as Pp, possessing the property if%lT%and k ~ 3%8 v 6 , then
!-T% 3
8 or I-T%
=I O;,
(*I
then T is Pp. Also we will give another characterization of P p from above by replacing (*) by if
%IT
% and I-T (%
3
8)& (83 a), then BIT8.
(**I
For this purpose we discuss some connections between pseudo-Boolean algebras and I-valuations (see for the definition of Z-valuations Beth [l], Kripke [lo], de Jongh [4]). If %(9,, . . ., P,,)is a formula we will write %*(al, . ., a,,) for the pseudo-Boolean algebraic term formed from ul,. . .,a,, with n, v , => and - in the same way as 8 from PI,. . ., P,, with &, v , 3 and 7 .A formula % is said to be valid in a pseudo-Boolean algebra A, iff %*(al, . . .,a,,) = 1 for all a l , . ., a,, E A .
.
.
THEOREM 1 (McKinsey and Tarski [12]). (a). kpp % i f % is valid in every pseudo-Boolean algebra. (b). kpp % if % is valid in every finite pseudo-Boolean algebra. DEFINITION 1. For any propositional calculus T at least as strong as P p we say that a pseudo-Boolean algebra A is a T-pseudo-Boolean algebra iff for each formula % such that t-T %, 8 is valid in A. THEOREM 2. For every propositional calculus T at least as strong as Pp, !-T % i f % is valid in every T-pseudo-Boolean algebra. PROOF.Immediate from results of Birkhoff [3] on equationally defined classes of algebras and the fact that T-pseudo-Boolean algebras can be defined by a system of equations, since pseudo-Boolean algebras can (cf. e.g. [14]). Another special case of a theorem of Birkhoff [3] is THEOREM 3. The class of all T-pseudo-Boolean algebras is closed under the formation of sub-algebras, homomorphisms and direct products. The following definitions are from [6] (mostly originally from [13]). DEFINITION 2. If a partially ordered set (V, d ) is a complete lattice, then a E V is called join-irreducible iff a > (8:p < a}. The set of all joinirreducible elements of V will be denoted by Vo.
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DEFINITION 3. A lattice V is called join-representable iff V is complete and completely distributive, and every a E V can be written as ct = {b: fi < a and p E Y o } . DEFINITION 4. A subset F of (V, < ) is called M-closed iff for all p , q E V, p E F a n d q < pimply q E F. The set of all M-closed subsets of a partially ordered set V will be denoted by V. Vis then complete and completely distributive.
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THEOREM 4. ([13], [6]).Every join-representable lattice V is isomorphic to p. THEOREM 5. (e.g. [2]). A complete and completely distributive lattice is a pseudo-Boolean algebra i f we define a * p = {y: a n y < S } .
u
Every finite distributive lattice is complete, completely distributive, and join-representable (e.g. [2]). So theorem 5 implies that every finite distributive lattice is a pseudo-Boolean algebra V for some partially ordered set V. Since for every partially ordered set V, Vis a distributive lattice, there is a 1-1 correspondence between finite pseudo-Boolean algebras and finite partially ordered sets.
DEFINITION 5. If V is a partially ordered set, then V is T-admissible iff 7 is a T-pseudo-Boolean algebra. DEFINITION 6. A P.0.G.-set is a partially ordered set with a greatest element. THEOREM 6. (essentially in [6]). I f V is a P.0.G.-set, then there is the following correspondence between any I-valuation ( V , w ) and the pseudoBoolean algebra i? for all formulas ?I, '23, i f F = { p E V: w(p, 8 ) = l } and G = { p :w(p, '23) = l}, then (i). F n G = { p : w(p, 91 & '23) = l}, (ii). F u G = { p : w ( p , % v @ )= 11, (iii). F G = { p : w(p, 2l 3 23) = l}, 2l) = l}. (iv). - F = { p : w ( p , THEOREM 7. I f V i s a P.0.G.-set with greatest element m , then V is T-admissible ifffor all I-valuations ( V , w ) and all formulas 2l such that IT 8,w(m, 8)= 1. We will need a short resume of some results of [S].
DEFINITION 7.(a). An I-function is a function with domain a finite P.0.G.-set Vand range the set (0,l } with the property that, if q < p i n Vandf(p) = 1, thenf(q) = 1.
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(b). An I"-function f is a function with domain a P.0.G.-set V and range the set (0,l}" such that for all m (1 < m < n) the function f" defined on V byf"(p) = (f(p))(m) is an Z-function. We write F" for the set of all I"-functions and I; for the set of all Zfunctions, m, for the greatest element of D,, the domain o f f , and < for the partial ordering of the domain off. DEFINITION 8.(a). f, g E F" are congruent by cp iff cp is an isomorphism from D, onto DBsuch that f(p) = g(cp(p)) for all p E D,. (b). f is congruent to g (in symbols f = g) iff f is congruent to g by cp for some cp. DEFINITION 9. An n-ary Z-operator u is a function from F" into F with the properties: (i). D,,,, = D, for all f E F". (ii). iff = g by cp, then a(f) = a(g) by cp. If V is a partially ordered set and p E V , then we write V(p) for the set {p' E V:p' < p}. Iff E F" and p E D,, we write f, for the restriction off to D,(P), 9 < f iff (3P)b = f). DEFINITION 10. A function cp fiom the partially ordered set (V, < ) onto the partially ordered set (W, < is strongly isotone iff (i). for all p ' , p E V , if p' < p, then cp(p') < cp(p) and (ii). for all p ' , p E V , if cp(p') < cp(p), then for some p" < p, cp(p") = = cp(P'). DEFINITION ll.(a). I f f , g E F", then g is a reduced form off iff there is a strongly isotone function cp from D, onto D, such that, for all p E D,, dcp(P)) = f(P). (b). f is irreducible iff all reduced forms off are congruent tof. (c). If g is a reduced form o f f and g is irreducible, then we call g a normal form of f.
,
In [5] we proved that the normal form of an I"-function is unique up to congruence (theorem 2.3). Also we proved that, i f f is irreducible and g < then g is irreducible (lemma 2.2 Cor.).
DEFINITION 12. A normal Z-operator a is an Z-operator such that (9. for all f E F" and P E D , , (a(f,))(p) = (U(f))(P) and (ii). iff, g E F" and g is a reduced form off, then (u(g))(m,)= (u(f))(m,). DEFINITION 13. The normalized characteristic set C,*of a normal n-ary
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I-operator a is the set of all irreducible I"-functions f such that ( a ( f > ) ( m f= ) 1. In [5] it was proved that a normal Z-operator is uniquely characterized by its normalized characteristic set. DEFINITION 14. A normal Z-operator isJinite iff its normalized characteristic set contains only a finite number of congruence classes. In [S]we defined then Z-operators corresponding to the usual connectives in a natural way. These Z-operators were proved to be normal (and in the cases of & and 1finite) and we proved that, if we introduce definability in a natural way, then all finite normal Z-operators are definable from the Z-operators corresponding to the usual connectives (theorem 3.7). More in particular: THEOREM 8. Let g E F", g irreducible, me has k immediate predecessors q l , . . .,qk with respect to < # , for each i (1 < i < k) q, has k, immediate predecessors with respect to < #, and for all i and j (1 < i < k, 1 < j < k,) a , and a,, are respectively the normal Z-operators with normalized characteristic sets C: = {fE F":f < g,,} and C:, = {fE F": f < s,,,}. Then, in the case that for all m ( 1 < m < n ) f " ( p ) = 0 for some p < m,, the normal Z-operator a with normalized characteristic set C,*= {f E F":f < g } can be expressed as follows:
a = (with
u k
i= 1
u ail standing for a , i f k , kr
j= 1
(J u,,)
j = 1
u a, k
ki
(Ui 2
2
i=1
= 0).
The main theorem is a little bit stronger than we need to establish the results predicted earlier. We have not checked the intuitionistic validity of it. Probably the double negation will hold intuitionistically. THEOREM 9. If T is a consistent propositional calculus stronger than Pp, then for each integer r 2 2 there is a formula !X 2 Bl v . . .vBs ( s 2 r ) such that %I=% and IT % 3 Bi V . . . v Bs,but not I-T % 3 Bi, V . . . vBir for any proper subsequence ( i l ,. . ., ik) (k 2 1 ) of (1, . . ., s). OUTLINE OF PROOF. First we construct a finite P.0.G.-set ( W, 6 o, qo) having q l , . . ., qk as the immediate predecessors of qo (k 2 l), such that W is not T-admissible, but, for all i (1 < i < k), W(q,) is T-admissible.
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Then from this P.0.G.-set W we construct a P.0.G.-set ( V, 6 ,p o ) with p , , . . .,p , (s 2 r ) as immediate predecessors of p o such that V is not T-admissible, but for all j (1 Q j 6 s), V ( p j ) is. Then we construct an irreducible I"-function g on V for some n such that, for all m (1 6 m 6 n), g"(pi) = 0 for some i (1 6 i 6 s). Now assume a, a, ,. ., a, are the definable I-operators with normalized characteristic sets C,* = {fE F":f 6 g } and C z = {f E F": f 6 gp,} (1 6 i 6 s), and assume that %, 23, , . . .,Bsare the formulas corresponding to these definable Z-operators. We will show (a). I - T % I B , v . . . v B , , (b). not I-T % I Bilv . . . vBikfor any proper subsequence (il,. . ., i,) of (1,. . ., s), (c). %IT 8. (a). The crucial point of the proof is that the class of T-pseudo-Boolean algebras does not contain a pseudo-Boolean algebra on which % 3 Bl v . . . v B, is not valid (a 'counter-example' to this formula). More precisely, for any pseudo-Boolean algebra A on which 91 I> Bl v . . v Bs is not valid, the pseudo-Boolean algebra V is isomorphic to a sub-algebra of a homomorphism of A, and so theorem 3 implies that, since V is not a T-pseudo-Boolean algebra, A cannot be one, and therefore I-T % I B1v . . vB,. In effect, if al, . ., a, E A are such that
.
.
.
.
PI* * B: u . . . u @:(a,,.. ., a,) z
1,
then Vcan be proved to be isomorphic to the sub-algebra of the relativization of A with respect to %*(a,, . .,a,) generated by the images of u l , . . ., CI, under the natural homomorphism on that relativization. (b). If t # ( i l , ..., i,) ( O Q t G s ) , then wecanprove that % I > B ~... , vv B i k __ is not valid on V ( p , ) ; then, since V ( p J is a T-pseudo-Boolean algebra, not I - T % n B i 1 v . . . v B i k . (c). This is easy to prove with the help of theorem 9 and the definition of IT.
.
For our second characterization we give first an equivalent expression for 8.
%IT
DEFINITION 15. a is a connectedz-operator iff a is normal and for allf, g E C,* there exists an h such that f 6 h and g Q h. Of the next theorem we again did not check the intuitionistic validity.
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INTUITIONISTIC PROPOSITIONAL CALCULUS
THEOREM 10. For any formula 8 is connected.
a, 8Ipp 8 ITthe I-operator corresponding
to
From this theorem and the proof of theorem 9 theorem 1 1 follows almost immediately.
THEOREM 11. If T is a consistent propositional calculus at least as strong as Pp, and for each 8,'Bi f 9 l I ~8 and t-T (83 '23) & (23
3
a), then BIT8,
(**)
then all theorems of T are provable in Pp. REFERENCES
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