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An Approach to Constructive Mathematical Logic
AN APPROACH TO CONSTRUCTIVE MATHEMATICAL LOGIC
A. A. MARKOV Academy of Science, Moscow, USSR
According to the fundamental thesis of constructive mathematics we consider in this science merely the results of our constructions (called constructive objects) and our abilities of realizing these constructions. We admit the abstraction of potential realizability, i.e. we abstract from practical limitations of our abilities in space, time and material and we argue as if such limitations were absent. In constructive mathematical logic our first aim is to explain logical connectives applied to propositions in terms of logical connectives applied to actions. This presents no difficulties for conjunction, disjunction and existential quantifiers, since the applications of these connectives to actions are almost immediately clear. But the situation is different for implication: it is not immediately clear what we must do when we receive the order: “if you make action A then make action B” or something like that. In fact there are several possibilities of defining the meaning of irnplication. Two of them will be considered here. 1. We can explain ‘‘implies 9 9” [3 379711 as admissibility of the rule of passing from F to 9 in a certain calculus2. 2. We can explain ‘‘lS9’’ as the deducibility of 9 from 37 by means of a certain system of inference rules. Both these ways correspond to the naive use of implication in mathematics. The rigorous realizations of both these ideas are connected with extensions of the language used to formulate S and 59: if even this language contains already the implication sign, this sign is now introduced in an essentially new sense. A stairwise construction of mathematical logic is thus expedient. 1 2
We use here the polish notation. This idea is due to LORENZEN [1955].
284
A. A. MARKOV
Difficulties of a different kind arise in connection with the generality quantifier. The natural explanation of this quantifier includes the requirement of some “unique general method” of proving every instance of the general proposition in question. This leads to the acceptation of some, possibly transfinite, hierarchy of Carnap’s inference rules. If at the same time implication is introduced as deducibility, then we shall possibly have a transfinite hierarchy of implications. In the sequel I describe an attempt to build some first few floors of a semantjcal system of constructive mathematical logic according to the brief outline above. As basis we take a formal language L, for the so called “pure semiotics” i.e. for talking about strings of letters, their equality and inequality, their juxtaposition, their beginnings and ends etc. In L1 we use conjunctions, disjunctions, the existential quantifier, the restricted existential quantifiers (there exists a beginning (end) of..., there exists a string (letter) occurring in.. .), the restricted generality quantifiers (for all beginnings (ends) of ..., for all strings (letters) occurring in ...). In the description of L, we use the following signs : the sign of equality by definition “e”, the sign of graphical equality of strings in the alphabet of L, “T”, the sign of graphical inequality of such strings The meaning of other metalinguistic signs will be explained below. L, uses the alphabet
“+”.
A,$~ubc( )=#&vVg(
)I.
Strings (a), (aa), (aaa), ...
are calledforrnal letters [FI] ; strings ( b ) , (bb), (bbb),...
are called lettervariables [Lv] ; strings (c), (CC),
(CCC),
.*.
are called wordvariables [IWv]. F1 and the letter F are called constants [Cn]. Lv and Wv are called variables [Vr]. Vr 6 and f2 are similar if both are Lv or both are Wv. Cn and Vr are called atoms [At]. Nonempty strings composed of atoms are called terms Tr. They can also
285
CONSTRUCTIVE MATHEMATICAL LOGIC
be inductively defined by means of the following two generating rules: Tr 1
At are T r .
Tr 2
T - Tr, A - At TA - Tr
The signs V and 3 are called quuntiJers [Qu]. The signs & and v are calledjunctors [Jn]. The signs ( and ) are called restricters [Rs]. The signs = and # are called elementary signs [Es]. Strings of the form (ToU), where T and U are Tr, CT is an Es, are called elementaryformulae [Ef]. Formulae [Fr] are now inductively defined by means of the following five generating rules : Fr 1
Ef are F r .
F - Fr,
Fr2
-
a 9 3
Fr, a - Jn
- Fr
9- Fr, a - Vr I Q 9- F T *
Fr3
F - Fr, T - Tr, X
Fr4
- Wv, K
- Qu, p
- Rs
KP-TYXF- Fr
F - Fr, T - Tr, 5 - Lv, K 1~17'15 s- Fr
Fr5
- Qu
Parameters [Pr] of Fr are inductively defined by means of the generating rules 9 - Ef, a - Vr, Q occurs in F Pr 1 SZ - Pr o f 9
9- Fr, 0 - Vr, 52 - Pr o f F , Q + 0 a - Pr of 3 e F
Pr3 Pr4
Pr5
-
S - Fr, X - Wv, 7'- Tr, K - Qu, p - Rs, ______52 - Pr of rcpTpXF
- Pr
ofF,Q
F - Fr, X - Wv, T - Tr, IC - Qu, p - Rs, Q occurs in T ~
52 - Pr of x p T p X 9
+X
286
Pr6
Pr7
A. A. MARKOV
F-Fr,t-Lv,
T-Tr,~-Qu,O-ProfF,f2+t 52 - Pro f K(TI&%=
F - Fr, 5 - Lv, T - Tr, K - Qu, 52 occurs in T __ 52 - Pr of K J T I(9
Fl and Lv are called Zetterterms [Lt]. Substituents [St] of Vr are defined as follows: Lt are St of Lv, Tr are St of wv. Let now 52 be a Vr, T a St of a. The (metalinguistic) operator of substitution of T for 52 is defined inductively by means of the following rules. A - At
Sbl
U - Tr, A - At
Sb2
F," .UA,
e F: U.
F," LA-J
-
~
'1cpF2LUAp X 9 , if 52 is not a Pr of S different from X F,".KPUPX~
--~
;*
K ~ F ?LU-IpXF," LFJ, if 52 is a Pr of 9different from X and X does not occur in T; KPF," LU, pYF," LFtL9A-J, if 52 is a Pr of different from X and X occurs in T.
CONSTRUCTIVE MATHEMATICAL LOGIC
287
IFF LUJ( (9, if D is not a Pr of 9different from 5 ; IF," LU,I 5F: L9J, if SZ is a Pr of 9different from 5 and t does not occur in T; F,RLKIUI t.FJ*< K IFF LUAI qF," LF: L9JJ, if Q is a Pr of 9different from and 5 occurs in T . 'K
K
288
A. A. MARKOV
the potential realizability of choosing a Fb [Fe] 9’of 3 LT_r,so that F i LSJ will be true. Sn7. A Cf 3 IT1 5 9 , where 5 is a Lv, T a Ct, F-[F, expresses the potential realizability of choosing a F1 2 occurring in j LT_r, so that F2 LF,is true. Sn8. A Cf V p T p X F , where X i s a Wv, p a Rs, T a Ct, F an XF, expresses that every Fr FZ LF_l, where B is a Fb [Fe] of 3 LT_I,is true. Sn9. A Cf VlTl 5 9 , where 5 is a Lv, T a Ct, F a 510, expresses that every Fr F$ L9_r, where 2 is a Fl occurring in j LT_r,is true. The applicability of a given normal algorithm to a given word can be expressed by a closed formula of L,, and therefore truth is undecidable for closed formulae of L,. At the same time a semantically complete calculus C, dealing with such formulae can be constructed. C, has the three axiomschemes:
A1
( T = U ) [T,U - Ct; jLTJ
A2
(TZU ) CT, u - Ct; jLTJ 3LUJ. v l ~tlF [t - L v ; ~ tF]. -
A3
7
+
3LUJ.
It uses the following 20 inference rules R1 R2 R3 R4 R5
F.2 L S J ~[Q - Vr;
352%
9 - Av of 52; 3Y
R7 R8
,
Fc” 2 1 [ X - WV;9 - X F ; p - Rs,9- FI] . 3p9pXS X
R6
- QF]
FSLTJ
LSJ[T - c t ]
3pTpXJ
.
3
CONSTRUCTIVE MATHEMATlCAL LOGlC
289
R9 R10 R11 R12 R13
R15 R16
R17 Rl8 R19 R20 Implication applied to closed formulae of L, is then introduced as admissibility of a rule in C , . This implication is formalized in a language L,. which permits to build implications applied to formulae of L, and to form repeatedly conjunctions of formulae already obtained. The alphabet A2 of L2 is obtained from A, by adjoining the implication sign =I : A 2 e A , 2 . F1, Lv, Wv, Cn, Vr, At, Tr, Es, Qu, Ef, Lt, St, Ct, Fw, Fb, Fe, are defined as in L,. Fr of L, are now called formulae of first degree [FI d]. where F and 9 are Fld, are called implications. Strings of the form =I S9?,
290
A. A. MARKOV
Formulae [Fr] are now inductively defined by the three generating rules: Frl Fld are F r . Fr2
Im are F r .
Fr3
F - Fr, 9 - Fr &.%9 - Fr ~~~
Parameters [Pr] of Fr are inductively defined by means of the three rules:
9- F l d ; SZ - Pr of F in L, SZ-Prof@
Pr 1
Pr 3
@,F? - Fr; SZ - Pr o f 9
~-
~~~~
SZ - Prof &93,52 - Pr o f & 9 F
Let 52 be a Vr, T a St of a. The operator of substitution of T for Q[FF] is defined by means of nine rules Sbl-Sb9, where: Sbl-Sb7 are as in L1 with F l d instead of Fr; Sb8 is as Sb4 in L, with F l d instead of Fr and 2 in the role of a; Sb9 is as Sb4 in L, with & in the role of a. Cf and SZF are defined as in L, . V1 of Ct and Av of Vr are defined as in L,. Let 9 and 9 be Cf of L,. Let C' be the calculus, obtained from C, by adjoining to the list of inference rules of C, the rule F/9 of passing from F to 9'.We say that this rule is admissible in C, if there is an algorithm transforming every proof in C', into a proof of the same formula in C1. We define now the meaning of Cf in L, as follows. For Cf which are F 1d : as in L, . An Im 199[F,9-Cf of L,] expresses the admissibility of the rule F/9'in C,. A Cf &FF? [@, F? - Cf] expresses that both these Cf are true. No semantically complete calculus is possible for L,. A system of inference rules R'l-R'9 valid in L, can nevertheless be proposed. R'1 R'2
R'3
99' [F, 3 - Cf of L,] 9
@,3 ~~
3
~
.
F9,3 9 2 [ 2- Cf of L,] . 3 92
9
CONSTRUCTIVE MATHEMATICAL LOGIC 3
R’4
29 1
P 9 , 3 9%
R‘5
R’6
x39
&X9
[Z,9 - Cf of L,].
R’7 R’8
R‘9
3
F g LYJ9 for every AvP of Q ____ [a - Vr, Y - QFof L,]. 3 3QY9
-__
In this system the rule R 9 is of an essentially different nature in comparison with the other rules R‘l-R’8. Like Carnap’s rule for the generality quantifier this rule permits to pass to the Cf below the dash as soon as we have a unique general method of establishing every instance of the Cf above the dash. We intend now to introduce implication “+” applied to closed formulae f and X of L, as deducibility of .X from $ by means of the rules R’I-R‘9 with permission to use also arbitrary true closed formulae of L,. In the exact definition it will be appropriate to separate the applications of R’9. The definition will be inductive. % f X will mean that X is deducible from f and some true formula of L2 by means of the rules R l - R 8 . %$X will mean that there are 9, 9 and Q, satisfying the following conditions : 1. Y i s Q F o f L , ; 2. 3 is Cf of L,; 3. we possess a general method of establishing S f 3 F ; L9,9? for every Av B of 52;
4. %&$33QY9X. Here i is an arbitrary natural number. One can suppose that transfinite induction can also be necessary. Happily the facts are much more simple. Even the implication ‘‘A” gives nothing new in comparison with “%”. For an exact formulation of the corresponding result we need an algorithm 9, transforming every string of the form i $ X [$,Z - Cf of L,] into a closed formula of L,. This algorithm
292
A. A.MARKQV
can be inductively defined by the conditions :
L%LLF9,+3F9, &F329p3
&F29,
.%LA3 9F9J+& V $9 3 9-9, 9%
2
29,e & 2 .# v f 9
9LA 9& X S Je &B+
3
&F#9,
&X, gL>&LfJ,
*
2 LL&X5?AA 9Lf 2-9 LA
Here 9, 9, S, 3 can be arbitrary Cf of L, ; # can either be an arbitrary 9 - C f of L,]; X , 9, A can Cf of L, or a Fr of L, of the form 399[2F, be arbitrary Cf of L2 such that & X 9 is not a formula of L,. We see easily that the result of application of W to a string of the form k 4 ? 9 [ c . X , 8 - C f of L,] is everytimes a CF of L, independent of i. Now we can state the THEOREM 1. Whatever be the closedf o r m u l a e X a n d 9 of L, and the natural number i, we have L X 9 if and only i f W ‘ , s % 9 J holds. “The deducibility of 9from 3? of rank i” is thus expressible by a formula of L, independent of i. In the sequel we omit the rank index i over the arrow. It is natural to define the deductional negation of a closed formula X of L, by means of 7XS-r .X(F # 5). It follows from Theorem 1 that we have 7 -19-if and only if F is true. Here F is an arbitrary Cf of L, and the meaning of the ordinary negation sign i is defined in terms of implication 3 : 19-=+39.7 # 7). Since the applicability of a normal algorithm to a word can be expressed by a closed formula of L,, this may be considered as the proof of a form of the principle of constructive selection: if the hypothesis of the nonapplicability of a normal algorithm to a word is reduced ad absurdum by means of certain inference rules then the algorithm does apply to this word. Another consequence of Theorem 1 is the deduction theorem which states that we have +99,if and only if zS9, where S and 9 are Cf of L,.
CONSTRUCTIVE MATHEMATICAL LOGIC
293
The deductive implication can now be formalized in a language L, permitting to form implications, applied to formulae of L,, and conjunctions of formulae already obtained. The construction of consecutive languages L, can be so continued that implication in every language L,(n 2 3) will have a deductive meaning. The sequence of languages L, so obtained can be incorporated into one language L,, in which implications and conjunctions can be formed without restrictions. The operator W can be extended to L, by means of the stipulations B L+%LYj $9 L+ W L X j W L L Y A j ,
W L&X-Yj
&9LX-JW L 2 - J .
Now this operator converts every closed formula of L, into a closed formula of L,. The language L, is thus reduced to L,. Let us now introduce the generality quantifier V. In language formulae are built from formulae of L, by successive left adjoinings of strings of the form VQ [Q-variable]. V Q X where sf? is an Q-formula means that we possess a method for proving every formula F: LX,,where 9is an arbitrary admissible value of Q. A natural system of inference rules R"I-R"I3 can be proposed for L,+, : R"l-R"5 identical with R'l-R'5. R"6-R"8 as Rr6-R'8 with the difference that X and 9 are now Cf of L,.
R"9
VO
96 [Q - V r , Y - Q F o f L , , 9 - Cf of L,]. 3 3896 3
R"10 R"11 R"12
R"13
Fg LJVJ for every Av P of 8 __ __ . ~ _ _ _
VQJV
Here R"13 is Carnap's rule, corresponding to the semantics of the generality quantifier. In virtue of R"13 we could replace R'9 by the simpler rule R"9. Now we can introduce a hierarchy of deductive implications for Cf of
294
A. A. MARKOV
7 f Y will mean that .X is deducible from 2 and a list of true formulae of L,+
by means of R”l-R”12.
2 f . X will mean that we have an Q-formula Jlr of L,+l such that bfF$ is provable for every admissible value B of SZ, and that X is by means of deducible from f , V O N and a list of true formulae of L,,
R’ 1-R’ 12. 7 $ X , where p is a limit ordinal, will mean that we have 2 f . X for some ci
A. A. MARKOV Academy of Science, Moscow, USSR
According to the fundamental thesis of constructive mathematics we consider in this science merely the results of our constructions (called constructive objects) and our abilities of realizing these constructions. We admit the abstraction of potential realizability, i.e. we abstract from practical limitations of our abilities in space, time and material and we argue as if such limitations were absent. In constructive mathematical logic our first aim is to explain logical connectives applied to propositions in terms of logical connectives applied to actions. This presents no difficulties for conjunction, disjunction and existential quantifiers, since the applications of these connectives to actions are almost immediately clear. But the situation is different for implication: it is not immediately clear what we must do when we receive the order: “if you make action A then make action B” or something like that. In fact there are several possibilities of defining the meaning of irnplication. Two of them will be considered here. 1. We can explain ‘‘implies 9 9” [3 379711 as admissibility of the rule of passing from F to 9 in a certain calculus2. 2. We can explain ‘‘lS9’’ as the deducibility of 9 from 37 by means of a certain system of inference rules. Both these ways correspond to the naive use of implication in mathematics. The rigorous realizations of both these ideas are connected with extensions of the language used to formulate S and 59: if even this language contains already the implication sign, this sign is now introduced in an essentially new sense. A stairwise construction of mathematical logic is thus expedient. 1 2
We use here the polish notation. This idea is due to LORENZEN [1955].
284
A. A. MARKOV
Difficulties of a different kind arise in connection with the generality quantifier. The natural explanation of this quantifier includes the requirement of some “unique general method” of proving every instance of the general proposition in question. This leads to the acceptation of some, possibly transfinite, hierarchy of Carnap’s inference rules. If at the same time implication is introduced as deducibility, then we shall possibly have a transfinite hierarchy of implications. In the sequel I describe an attempt to build some first few floors of a semantjcal system of constructive mathematical logic according to the brief outline above. As basis we take a formal language L, for the so called “pure semiotics” i.e. for talking about strings of letters, their equality and inequality, their juxtaposition, their beginnings and ends etc. In L1 we use conjunctions, disjunctions, the existential quantifier, the restricted existential quantifiers (there exists a beginning (end) of..., there exists a string (letter) occurring in.. .), the restricted generality quantifiers (for all beginnings (ends) of ..., for all strings (letters) occurring in ...). In the description of L, we use the following signs : the sign of equality by definition “e”, the sign of graphical equality of strings in the alphabet of L, “T”, the sign of graphical inequality of such strings The meaning of other metalinguistic signs will be explained below. L, uses the alphabet
“+”.
A,$~ubc( )=#&vVg(
)I.
Strings (a), (aa), (aaa), ...
are calledforrnal letters [FI] ; strings ( b ) , (bb), (bbb),...
are called lettervariables [Lv] ; strings (c), (CC),
(CCC),
.*.
are called wordvariables [IWv]. F1 and the letter F are called constants [Cn]. Lv and Wv are called variables [Vr]. Vr 6 and f2 are similar if both are Lv or both are Wv. Cn and Vr are called atoms [At]. Nonempty strings composed of atoms are called terms Tr. They can also
285
CONSTRUCTIVE MATHEMATICAL LOGIC
be inductively defined by means of the following two generating rules: Tr 1
At are T r .
Tr 2
T - Tr, A - At TA - Tr
The signs V and 3 are called quuntiJers [Qu]. The signs & and v are calledjunctors [Jn]. The signs ( and ) are called restricters [Rs]. The signs = and # are called elementary signs [Es]. Strings of the form (ToU), where T and U are Tr, CT is an Es, are called elementaryformulae [Ef]. Formulae [Fr] are now inductively defined by means of the following five generating rules : Fr 1
Ef are F r .
F - Fr,
Fr2
-
a 9 3
Fr, a - Jn
- Fr
9- Fr, a - Vr I Q 9- F T *
Fr3
F - Fr, T - Tr, X
Fr4
- Wv, K
- Qu, p
- Rs
KP-TYXF- Fr
F - Fr, T - Tr, 5 - Lv, K 1~17'15 s- Fr
Fr5
- Qu
Parameters [Pr] of Fr are inductively defined by means of the generating rules 9 - Ef, a - Vr, Q occurs in F Pr 1 SZ - Pr o f 9
9- Fr, 0 - Vr, 52 - Pr o f F , Q + 0 a - Pr of 3 e F
Pr3 Pr4
Pr5
-
S - Fr, X - Wv, 7'- Tr, K - Qu, p - Rs, ______52 - Pr of rcpTpXF
- Pr
ofF,Q
F - Fr, X - Wv, T - Tr, IC - Qu, p - Rs, Q occurs in T ~
52 - Pr of x p T p X 9
+X
286
Pr6
Pr7
A. A. MARKOV
F-Fr,t-Lv,
T-Tr,~-Qu,O-ProfF,f2+t 52 - Pro f K(TI&%=
F - Fr, 5 - Lv, T - Tr, K - Qu, 52 occurs in T __ 52 - Pr of K J T I(9
Fl and Lv are called Zetterterms [Lt]. Substituents [St] of Vr are defined as follows: Lt are St of Lv, Tr are St of wv. Let now 52 be a Vr, T a St of a. The (metalinguistic) operator of substitution of T for 52 is defined inductively by means of the following rules. A - At
Sbl
U - Tr, A - At
Sb2
F," .UA,
e F: U.
F," LA-J
-
~
'1cpF2LUAp X 9 , if 52 is not a Pr of S different from X F,".KPUPX~
--~
;*
K ~ F ?LU-IpXF," LFJ, if 52 is a Pr of 9different from X and X does not occur in T; KPF," LU, pYF," LFtL9A-J, if 52 is a Pr of different from X and X occurs in T.
CONSTRUCTIVE MATHEMATICAL LOGIC
287
IFF LUJ( (9, if D is not a Pr of 9different from 5 ; IF," LU,I 5F: L9J, if SZ is a Pr of 9different from 5 and t does not occur in T; F,RLKIUI t.FJ*< K IFF LUAI qF," LF: L9JJ, if Q is a Pr of 9different from and 5 occurs in T . 'K
K
288
A. A. MARKOV
the potential realizability of choosing a Fb [Fe] 9’of 3 LT_r,so that F i LSJ will be true. Sn7. A Cf 3 IT1 5 9 , where 5 is a Lv, T a Ct, F-[F, expresses the potential realizability of choosing a F1 2 occurring in j LT_r, so that F2 LF,is true. Sn8. A Cf V p T p X F , where X i s a Wv, p a Rs, T a Ct, F an XF, expresses that every Fr FZ LF_l, where B is a Fb [Fe] of 3 LT_I,is true. Sn9. A Cf VlTl 5 9 , where 5 is a Lv, T a Ct, F a 510, expresses that every Fr F$ L9_r, where 2 is a Fl occurring in j LT_r,is true. The applicability of a given normal algorithm to a given word can be expressed by a closed formula of L,, and therefore truth is undecidable for closed formulae of L,. At the same time a semantically complete calculus C, dealing with such formulae can be constructed. C, has the three axiomschemes:
A1
( T = U ) [T,U - Ct; jLTJ
A2
(TZU ) CT, u - Ct; jLTJ 3LUJ. v l ~tlF [t - L v ; ~ tF]. -
A3
7
+
3LUJ.
It uses the following 20 inference rules R1 R2 R3 R4 R5
F.2 L S J ~[Q - Vr;
352%
9 - Av of 52; 3Y
R7 R8
,
Fc” 2 1 [ X - WV;9 - X F ; p - Rs,9- FI] . 3p9pXS X
R6
- QF]
FSLTJ
LSJ[T - c t ]
3pTpXJ
.
3
CONSTRUCTIVE MATHEMATlCAL LOGlC
289
R9 R10 R11 R12 R13
R15 R16
R17 Rl8 R19 R20 Implication applied to closed formulae of L, is then introduced as admissibility of a rule in C , . This implication is formalized in a language L,. which permits to build implications applied to formulae of L, and to form repeatedly conjunctions of formulae already obtained. The alphabet A2 of L2 is obtained from A, by adjoining the implication sign =I : A 2 e A , 2 . F1, Lv, Wv, Cn, Vr, At, Tr, Es, Qu, Ef, Lt, St, Ct, Fw, Fb, Fe, are defined as in L,. Fr of L, are now called formulae of first degree [FI d]. where F and 9 are Fld, are called implications. Strings of the form =I S9?,
290
A. A. MARKOV
Formulae [Fr] are now inductively defined by the three generating rules: Frl Fld are F r . Fr2
Im are F r .
Fr3
F - Fr, 9 - Fr &.%9 - Fr ~~~
Parameters [Pr] of Fr are inductively defined by means of the three rules:
9- F l d ; SZ - Pr of F in L, SZ-Prof@
Pr 1
Pr 3
@,F? - Fr; SZ - Pr o f 9
~-
~~~~
SZ - Prof &93,52 - Pr o f & 9 F
Let 52 be a Vr, T a St of a. The operator of substitution of T for Q[FF] is defined by means of nine rules Sbl-Sb9, where: Sbl-Sb7 are as in L1 with F l d instead of Fr; Sb8 is as Sb4 in L, with F l d instead of Fr and 2 in the role of a; Sb9 is as Sb4 in L, with & in the role of a. Cf and SZF are defined as in L, . V1 of Ct and Av of Vr are defined as in L,. Let 9 and 9 be Cf of L,. Let C' be the calculus, obtained from C, by adjoining to the list of inference rules of C, the rule F/9 of passing from F to 9'.We say that this rule is admissible in C, if there is an algorithm transforming every proof in C', into a proof of the same formula in C1. We define now the meaning of Cf in L, as follows. For Cf which are F 1d : as in L, . An Im 199[F,9-Cf of L,] expresses the admissibility of the rule F/9'in C,. A Cf &FF? [@, F? - Cf] expresses that both these Cf are true. No semantically complete calculus is possible for L,. A system of inference rules R'l-R'9 valid in L, can nevertheless be proposed. R'1 R'2
R'3
99' [F, 3 - Cf of L,] 9
@,3 ~~
3
~
.
F9,3 9 2 [ 2- Cf of L,] . 3 92
9
CONSTRUCTIVE MATHEMATICAL LOGIC 3
R’4
29 1
P 9 , 3 9%
R‘5
R’6
x39
&X9
[Z,9 - Cf of L,].
R’7 R’8
R‘9
3
F g LYJ9 for every AvP of Q ____ [a - Vr, Y - QFof L,]. 3 3QY9
-__
In this system the rule R 9 is of an essentially different nature in comparison with the other rules R‘l-R’8. Like Carnap’s rule for the generality quantifier this rule permits to pass to the Cf below the dash as soon as we have a unique general method of establishing every instance of the Cf above the dash. We intend now to introduce implication “+” applied to closed formulae f and X of L, as deducibility of .X from $ by means of the rules R’I-R‘9 with permission to use also arbitrary true closed formulae of L,. In the exact definition it will be appropriate to separate the applications of R’9. The definition will be inductive. % f X will mean that X is deducible from f and some true formula of L2 by means of the rules R l - R 8 . %$X will mean that there are 9, 9 and Q, satisfying the following conditions : 1. Y i s Q F o f L , ; 2. 3 is Cf of L,; 3. we possess a general method of establishing S f 3 F ; L9,9? for every Av B of 52;
4. %&$33QY9X. Here i is an arbitrary natural number. One can suppose that transfinite induction can also be necessary. Happily the facts are much more simple. Even the implication ‘‘A” gives nothing new in comparison with “%”. For an exact formulation of the corresponding result we need an algorithm 9, transforming every string of the form i $ X [$,Z - Cf of L,] into a closed formula of L,. This algorithm
292
A. A.MARKQV
can be inductively defined by the conditions :
L%LLF9,+3F9, &F329p3
&F29,
.%LA3 9F9J+& V $9 3 9-9, 9%
2
29,e & 2 .# v f 9
9LA 9& X S Je &B+
3
&F#9,
&X, gL>&LfJ,
*
2 LL&X5?AA 9Lf 2-9 LA
Here 9, 9, S, 3 can be arbitrary Cf of L, ; # can either be an arbitrary 9 - C f of L,]; X , 9, A can Cf of L, or a Fr of L, of the form 399[2F, be arbitrary Cf of L2 such that & X 9 is not a formula of L,. We see easily that the result of application of W to a string of the form k 4 ? 9 [ c . X , 8 - C f of L,] is everytimes a CF of L, independent of i. Now we can state the THEOREM 1. Whatever be the closedf o r m u l a e X a n d 9 of L, and the natural number i, we have L X 9 if and only i f W ‘ , s % 9 J holds. “The deducibility of 9from 3? of rank i” is thus expressible by a formula of L, independent of i. In the sequel we omit the rank index i over the arrow. It is natural to define the deductional negation of a closed formula X of L, by means of 7XS-r .X(F # 5). It follows from Theorem 1 that we have 7 -19-if and only if F is true. Here F is an arbitrary Cf of L, and the meaning of the ordinary negation sign i is defined in terms of implication 3 : 19-=+39.7 # 7). Since the applicability of a normal algorithm to a word can be expressed by a closed formula of L,, this may be considered as the proof of a form of the principle of constructive selection: if the hypothesis of the nonapplicability of a normal algorithm to a word is reduced ad absurdum by means of certain inference rules then the algorithm does apply to this word. Another consequence of Theorem 1 is the deduction theorem which states that we have +99,if and only if zS9, where S and 9 are Cf of L,.
CONSTRUCTIVE MATHEMATICAL LOGIC
293
The deductive implication can now be formalized in a language L, permitting to form implications, applied to formulae of L,, and conjunctions of formulae already obtained. The construction of consecutive languages L, can be so continued that implication in every language L,(n 2 3) will have a deductive meaning. The sequence of languages L, so obtained can be incorporated into one language L,, in which implications and conjunctions can be formed without restrictions. The operator W can be extended to L, by means of the stipulations B L+%LYj $9 L+ W L X j W L L Y A j ,
W L&X-Yj
&9LX-JW L 2 - J .
Now this operator converts every closed formula of L, into a closed formula of L,. The language L, is thus reduced to L,. Let us now introduce the generality quantifier V. In language formulae are built from formulae of L, by successive left adjoinings of strings of the form VQ [Q-variable]. V Q X where sf? is an Q-formula means that we possess a method for proving every formula F: LX,,where 9is an arbitrary admissible value of Q. A natural system of inference rules R"I-R"I3 can be proposed for L,+, : R"l-R"5 identical with R'l-R'5. R"6-R"8 as Rr6-R'8 with the difference that X and 9 are now Cf of L,.
R"9
VO
96 [Q - V r , Y - Q F o f L , , 9 - Cf of L,]. 3 3896 3
R"10 R"11 R"12
R"13
Fg LJVJ for every Av P of 8 __ __ . ~ _ _ _
VQJV
Here R"13 is Carnap's rule, corresponding to the semantics of the generality quantifier. In virtue of R"13 we could replace R'9 by the simpler rule R"9. Now we can introduce a hierarchy of deductive implications for Cf of
294
A. A. MARKOV
7 f Y will mean that .X is deducible from 2 and a list of true formulae of L,+
by means of R”l-R”12.
2 f . X will mean that we have an Q-formula Jlr of L,+l such that bfF$ is provable for every admissible value B of SZ, and that X is by means of deducible from f , V O N and a list of true formulae of L,,
R’ 1-R’ 12. 7 $ X , where p is a limit ordinal, will mean that we have 2 f . X for some ci