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Chapter XI The Predicate Calculus
CHAPTER XI
THE PREDICATE CALCULUS 5 1. THEPROPOSITIONAL CALCULUS In Ch. IX, we have incidentally considered certain formulations of the propositional calculus and the predicate calculus. In this chapter, we shall describe a number of points in this area mor2 leisurely. It is well-known that the propositional calculus is essentially equivalent to Boolean algebra. For example, the following is an axiom system for Boolean algebra. T h e variables are x, y, z, etc., 0 is the only constant, a term is: (i) a constant or a variable; or (ii) a’ or ab when a, b are terms. A formula is a = b , if a, b are terms. One can then define 1 as 0’, a + b as ( a1b1 )’ And the axioms are:
.
(1) acr & a , (2) (3)
(4) (5) (6)
ab=ba, a(bc) = ( a b ) c , a0 = 0, if a b ‘ = O , then ab = a , if ab = a and ab‘ = a , then a = 0.
A well-known system of the propositional calculus is that of Whitehead-Russell, as modified by Bernays, and adopted by HilbertAckermann. The connectives “not” and “or” are used with the others defined. The axioms are: ’
(1)
(PVP)=%
(2)
P3(PVd,
(3)
( P v q )z l ( 4 v PI, = I ( ( r 3 PI 2 (7. =q)1.
(4) ( P = 4
The rules of inference are modus ponens and substitution if p , q, I, etc. are construed as proposition letters. If they are construed as schemata, then the rule of substitution is unnecessary.
If in my formula of the propositional calculus, we replace p , q, etc. by x, y, etc., complement by negation, intersection by conjunction, sum by disjunction, i t is possible to show that a formula A is a
30a
XI. THE PREDICATE CALCULUS
theorem in the propositional calculus if and only if A * = l theorem in Boolean algebra, A" being the transform of A .
is a
Another well-known system is that of Hibbert-Bernays in which the five common connectives are all taken as primitive. The axioms are:
The rules of inference are modus ponens with or without substitution as before. These systems are all known to be complete and consistent. The axioms are also independent. The usual method of proving independence is by choosing an interpretation in which the particular axiom is not true, but all theorems derivable from the other axioms are true. For example, if we take 1 as true, then the following interpretation gives the independence ( l a ) :
P
4
1 1 1 2 2 2
1 2
3 3 3
3
1 2 3
1 2 3
-p 3 3 3
2 2 2 1 1 1
P V 4 1 1 1 1
2 2 1 2 3
P&4
PE4
1 2
1
p = 4
'
1
3 3
3 3
2 2
3
1
1 1
3
3
3
3 3 3
3 3
,1 1 1
3
1
9 2. FORMULATIONS OF
’ r m PREDICATE
CALCULUS
309
Thus, every axiom except ( l a ) always gets the value 1, and if p and p I q always get the value 1, so does q . On the other hand, when p and q take the values 2 and 1 respectively, ( l a ) takes the value 3.
If one omits (5c) from the above system, one gets the “minimal calculus”. If one omits (5c) and replaces (5a) and (5b) by: ( p z - p ) 3 -p, (5b’)--p3 (pxq ) ; then the resulting system is the intuitionistic propositional calculus. It is also known that (5a) to (5c) can be replaced by: (5a’) ( P
-
=I -44) 3 (4 3 PI,
6’ ((P ) =
-
PI
2 4) 2
((P
(5b’),
3 4) 3 4).
This alternative formulation has, according to Lukasiewicz, the following property: drop (5c’), we get intuitionistic calculus, drop also (5b’), we get the minimal calculus.
$2. FORMULATIONS OF THE PREDICATE CALCULUS There are different formulations of the predicate calculus, variously known as quantification theory, the restricted or first order predicate calculus, the first order functional calculus. Often these diverse formulations are of interest for different purposes. We give here three alternative formulations. We may use proposition letters p , q, r, etc., variables x , y, z, etc., predicates F , G, etc. A proposition letter or a predicate followed by variables is an atomic proposition. If A and B are propositions and u is a variable, then -A, A V B, ( u ) A , ( E u ) B are propositions. A proposition containing n o quantifiers is an elementary proposition. Other connectives &, 3 , 5 may either be included initially or be introduced by definitions. A proposition is a tautology if i t as a truth function of its components which may or may not contain quantifiers is always true by the usual interpretation of truth-functional connectives. A given occurrence of a variable u is said to be bound i n a formula if i t is governed by a quantifier ( v ) or ( E u ) , otherwise free. If the result of writing general quantifiers for all free variables i n A before A is a theorem, we write FA.
2.1. System L, (see Quine, Matlzemnticnl Logic, 1951 ed.). *loo. If A is a tautology, F A *101. F ( v ) ( A I B )3 ( ( v ) A = )( u p ) *1O2. If ZI is not free in A , F A I ( v ) A .
XI. THE PREDICATE CALCCLUS
3 1y1
* 103.
If B is like A except for containing free occurrences of wherever A contains free occurrences of v, then t - ( v ) A 3 B . *104.
ZI
If A z B and A are theorems, so is B.
This system has *104, the modus ponens, as the only rule of inference, which is less general than the usual one i n so far as A and B contain n o free variables. Vacuous quantifiers are included so that ( v ) A is a proposition even when v is not free i n A . This enables one to dispense with a rule of generalization: if t-A, then t - ( v ) A , which is included in the usual systems. No rule for the particular quantifiers are included since (Ev) is to be defined as -(v)-. There is no rule of substitution since the axioms are already schemata in the manner of von Neumann. The principle *lo0 can be replaced by a finite number of specific schemata in the usual manner, and i t is sometimes convenient to do so. It may be noted (see 1. Symbolic Logic, 12 (1947), 130-132) that we can dispense with “102 if we replace “100, *101, and *lo4 respectively by: “100’.
If A is a tautology, t - ( v ) A .
*101’.
If v is not free in B, t - ( v ) ( A 3 B ) 3 ( A 3 ( v ) B ) .
*104’.
If t-A and t-AxB, then t-B.
The equivalence is quite direct. All the new principles are easy consequences of results in ML: *lo0 and *115; *159; “111. Conversely, *lo2 is proved by *loo’, *101’, *104’; *lo0 and “104 are special cases of *loo‘ and *104’; *lo1 can be proved by *101‘ and *lo3 with the help of *115 whose proof uses only the new principles.
As a second system, we present one in Herbrand’s thesis (Recherches sur la thkorie de la dtmonstration, Warsaw, 1930). There is no need to allow vacuous quantifiers (called “fictitious” quantifier i n his thesis, p. 29), and particular quantifiers are assumed to begin with, while truth functions besides and V are taken as defined. 2.2.
System L,.
L4 1. Rule of tautology. a tautology then t-A.
-
If A is an elementary proposition and
L, 2. Rules of inversion. Within a given proposition, if v is not free in B, we can replace each one by the other in any of the pairs: (1)
-
(Y)A, (EY) - A ;
(3) ( v ) ( A V
(2)
B), ( u ) A V B ;
L , 3. Rule of generalization.
-
(4)
(Ev)A,
( Y )
-
A;
(Ev)(A V B ) , (Ev)A V B.
If t-Av, then t-(v)Av.
4 2.
FORMULATIONS OF THE PREDICATE CALCULUS
311
Lb 4. Rule of particularization. If FAvv, then k(Eu)Avu, in other words, if t A and B is obtained from A by substituting u for v at some or all occurrences of u, then k ( E u ) B .
L, 5. Rule of simplification. If k A V A , then FA. L, 6.
Modus ponens.
If k A 3 B and P A , then t-B.
This remarkable system is specially useful for the study of proofs in the predicate calculus. Here are some concepts and results about the system:
2.21. Dreben’s lemma. If for all A, t-A follows from A V A , then t-(BV VB)zB.
v ..-
A proof of this which depends on 2.23 wi 1 be given at the end of this chapter.
-
2.22. Given any proposition H ( p , . -,q) built up from proposition letters by and V , the sign of each occurrence of each letter in H is defined thus: (1) p is positive in p ; (2) the sign of an occurrence of p in A is the opposite of its sign in A ; ( 3 ) the sign of an occurrence of p i n A is the same as its sign in A V B. (Thesis, p. 21.)
-
T h e rules of inversion i n L, are specially convenient for bringing a proposition to the prenex normal form: i.e., a proposition with all quantifiers standing at the beginning. Given an occurrence of a quantifier in a proposition, when brought to the beginning, i t may remain the same, or change from general to particular or vice versa. T h e definition 2.22 can be extended in obvious manner to occurrences of any propositions in a given proposition, we need only add that the sign of an occurrence of A in B is the same as its sign i n ( v ) B or ( E v ) B . It then happens that a quantifier remains the same when brought to the beginning, if and only if the component proposition beginning with the quantifier has a positive occurrence in the whole proposition. One peculiar feature of Lb is that the rule of tautology only uses elementary propositions. For the system, we can actually derive:
FA.
Lh 1’. The strengthened rule of tautology.
If A is a tautology,
Given a tautologous elementary proposition, we wish to show that
i f we substitute more complex propositions for the atomic propositions i n it, the results are always theorems of Ls. For this purpose, we may as well replace all atomic propositions by proposition letters and consider some given tautology H ( p , -,q). In general, each letter p,. -,q in H may have a number of positive occurrences and a number of negative Qccurrences. By the rules of
-
-
3 12
XI.
THE PREDICATE CALCULUS
inversion, we may assume that the propositions to substitute the letters
p,..., q are in the prenex normal form. We prove first a special case when each letter p , - -,q has at most one positive occurrence and one negative occurrence in H : 2.23. For every elementary tautology H ( p , - - ., q ) of this restricted type, all substitution results from H are theorems of L,.
-
While each result is obtained when we make all substitutions on the different proposition letters at the same time, we imagine that we break the process into different steps, one step for each proposition letter. Suppose we have shown that H ( B y , -,A ) is shown to be a theorem, we wish to show that if instead of By we substitute (Ey)By or (y)By for p , we again get a theorem. Suppose p has one positive occurrence and one negative occurrence and suppose, for brevity, ( E y ) B y is to replace p . Hence, we wish to derive, from
-
H(p,p,- - -,A ) :
H((Ex)Bx, (Ey)By,
-
* *
,A )
or b ) ( E x ) H ( B x , B Y ,
-
* *,
A).
By hypothesis, H ( B y , By,. -,A ) is a theorem. Hence, by Lb 4 and L6 3, we get the desired theorem. From 2.23, i t follows that for any A,C,B,D: t - B x B , and (1)
I- ( ( A 3 B )
82
( c 3 D)) 3 ( ( A v c ) 3 ( B v u ) ) .
If we substitute B for C and modus ponens: I- ( ( B
v
D, B V B for A i n (l), then we get by
B) 3 B) 3 ((B
v
B
v
B)
= l(
B
v B)).
By 2.21 and the rule of simplification, we have also ( B V B ) z , B . Hence, if I - B V B V B , then I-BVB. Therefore, by the rule of simplification, if t-B V B V B, then t-B. If we repeat the process by substituting B V B V B , etc. for A i n (l), we get:
2.24.
If t-B V
0 .
- V B, then
I-B.
With the help of this, we can now prove the general theorem:
Theorem 1. The strengthened rule o f tautology L4 I' is derivable in Ls; there is an effective (in fact, primitive recursive) method, szlch that for every tautology A, we can find a proof for it in L,. We
illustrate
the procedure by a simple example.
Suppose
H(p;p,p,p;q,q;q) is an elementary tautology with one positive occurrence of p , two of q, three negative occurrences of p , one of q. And we wish to prove the result of substituting (x)(Ey)Fxy for p, (x)Gx for q.
S 2. FORMULATIONS
OF THE PREDICATE CALCGZUS
3 13
By Lh 2, Lh 3, and L, 4, it is sufficient to prove: (2)
H((Ey)Fxy; (EY)FXY (2)
,
G (2)
,( E y ) F x y ,
( 2 ) G*
(EY)FxY;
; (2) Gz).
To prove this, by 2.24, Lh 3, and L, 4, i t is sufficient to prove: (3)
H ( F x u ; F x u , F x v , F x w ; ( z ) G z , (z)Gz; (2)G.z) V H ( F x v ; * - . )
V H(Fxw; **-).
For the same reason, this i n turn follows from (4)
I-I( F x u ;
--
*
; G z , G y ; Cz) V El( F x v ;
V H(Fxw;
V H(Fxv;
* * - ; *
Gz)
v
H(Fxu;
*
-
* - * ;
*
;G z )
Gy)
- ; G y ) V H ( Fxcu; - ; G y ) . * *
Now we wish to show that, since H(p;p,p,p;q,q;q)is a tautology, (4) is also one and falls under L h 1. Instead of the two atomic propositions p and q, we now have five: FXZI,Fxv, Fxw, Gx,Gy. If we put H(p;n,b,c;d,e;q) into a conjunctive normal form, t h i n since signs
of the atomic propositions are preserved, each conjunctive term must be a part of (5)
p V
- - a V
6 V
Moreover, since the original H term must contain either p and at and at least one of d and e. Now of (4) into the corresponding form (4')
H ( a ; a , 6 , c; d , e ; d )
v*
c
-V
V d V e V
-
q.
is tau tologous, each conjunctive least one of -a, -6, -c, or -q if we turn each disjunctive term and get H ( c ; a, 6 , c ; d , e; e),
then we can show that (4') must be true for all possible truth values of a,b,c,d,e. First, consider the corresponding form for (3): (3')
H ( a ; a , 6 , c ; q , q ; q) V H ( 6 ;
a * - )
v H(c;
-.*).
This must he a tautology. If a,b,c are all false, every conjunctive term (a disjunction) i n every disjunctive term of (3') must be true since it must contain either q and -q, or at least one of -n, 4, -c. * If at least one of a,b,c is true, say b, then, H(b;n,b,c;q,q;q) must be always true. Hence, if we take the conjunctive normal form of (3")
N(a;
a,
b,
c;
d , e ; 4) V H ( 6 ;
-
* *)
v
H(c;
*),
XI. THE PREDICATE CALCULUS
314
then each conjunctive term must contain either d and -q or e and -q, provided we drop all terms aV-a, bV-b, cV-c. But (4’) is equivalent to a disjunction of the two propositions obtained from (3”) by substituting d and e for q . For any possible truth values of n,b, c,d,e, if d, e are both false, then every one is true since -q occurs. If at least one is true, take that one. This completes the proof of Theorem 1, since the example includes all the essential features for the general case. The proof is based on the method i n Herbrand’s thesis. But the argument put forward in the example above is more complicated than Herbrand’s, because we feel there are some gaps in his simpler proof. In Whitehead-Russell, the whole * 9 is devoted to a sketch of a proof of Theorem 1 along quite a different line. Since they do not establish the completeness of their axioms for the propositional calculus, they cannot use arguments of the type as exemplified i n the above proof. In his thesis, Herbrand continues to exhibit a procedure whereby he can transform any given proof i n Lh into one which does not use the rule of modus ponens. This is the famous:
2.25.
Herbrand’s Theorem. There is an effective (in fact, primitive recursive) procedure whereby, given any proof i n an ordinary formulation of the predicate calculus, we can turn i t into a proof of Lb with no appeal to the rule of modus ponens. There is an extensive literature on this theorem mostly following the alternative treatment by Gentzen. We shall not attempt an exposition of Herbrand’s original proof which, however, we believe, contains many interesting distinctive features well worth exploring. Instead, we shall consider some related results and some different formulations. In mathematics, we constantly use the following mode of reasoning: “There exist y such that Ay. Let x be one such and consider whether this x has such and such other properties.” In the symbolism of the predicate calculus, one might wish to formalize this by using a rule “from ( E y ) A y , infer Ax”. But since the x chosen is not an arbitrary object but an arbitrary object which happens to have the property A, the variable x i n such a rule cannot be expected to behave like ordinary variables i n mathematical logic. T o overcome this difficulty, Hilbert introduced the 6-symbol and infers A(e,Ax) f r o m ( E x ) A x . This symbol is treated thoroughly in Hilbert-Bernays’ book, volume 11. The predicate calculus is then formulated a little differently. In addition to the variables, s-expressions ~ J x , etc. also become terms
$2.
315
FORMULATIONS OF ?WE PREDICATE CALCULUS
and Fv, Gvu, etc. are propositions for arbitrary terms v, u, etc. If, e.g., we take the system Lh, we can obtain a new system Lb by some changes. 2.3. System La. The system includes L,, 1 to Lh7 with the propositions reconstrued in the extended sense, and a new rule: (6)
The 6-rule.
t- ( E v ) A Y 3 A ( e y A v ) .
Incidentally, given the e-symbol, i t is possible to omit quantifiers in the primitive apparatus and introduce them by definitions.
( E v ) A v if and only if A(e,Av). ( v ) A v if and only if A(e,-Av). For the system La, two fundamental theorems are proved in Hilbert-Bernays I1 (stated on p. 18). If F is a system formalized in the framework of Lb and, i n general, contains additional symbols for in-
2.31. 2.32.
dividuals and predicates, as well as a finite number of new axioms A,, * A, which do not contain the 6-symbol, but may contain quantifiers, then:
-.,
2.33. The first e-theorem. If the axioms A,, * A, contain no bound variables and B is a theorem of F which contains no bound variables, then B can also be derived from A,, A, without using bound variables. a,
a * - ,
2.34. The second e-theorem. If B does not contain the e-symbol but may contain quantifiers, i t can be derived from A,, A , without using the e-symbol (hence, can be derived in L,). - - a ,
In Hilbert-Bernays, Herbrand’s theorem is deduced from these, and the two 6-theorems are extended to imbed also the axioms for equality. We shall not enter upon proofs of these results. If we combine the use of the 6-symbol with the method of natural deduction as developed by Quine (Methods of Logic, 1950), we obtain a rather symmetric system L,, which has the same terms and propositions as La but uses different rules. In particular, the deduction theorem is now taken as a primitive rule, and we are allowed to write down any new axiom as a hypothesis. The final result is a theorem of L, only if i t is freed from all hypotheses. Formally, a hypothesis may be marked by a * at the beginning which stands also i n front of all consequences of the hypothesis. A line in a proof of L, is a theorem of L, only when i t does not begin with a *
.
L, 1. Rule of assumption. We may set down any proposition A as a line at any stage in the course of a deduction, provided we prefix a new * on the line and all its consequences.
XIiI. TlIE PREDICATE CALCCLGS
3 16
L,2. Rule of deduction. If under the assumption of * A we arrive at *B, then we may write a new line A x R , with one * less.
L,3. Rule of tautology. We may write a line which is implied truth-functionally by one or more previous lines taken together.
L, 5. Rule of instantiation.
t-Au. I-
If v is a variable and kAv, then
Rule of generalization.
L,4.
t- ( v ) A s .
If zi is any term and k ( v ) A v , then
L,6. Rule of particularization. If ~l is any term and t-Azt, then ( E v )Av. L,7. Rule of specification. If t-(Ev)Av, then ~ - A ( E , A u ) .
For the system L,, it is rather direct to prove something like the second &-theorem. For this purpose, we need an additional rule which is no longer derivable if we drop L,7:
If ~ A u and ~ Bv is not
L,7’. Weakened rule of specification. free in B, then t-(Ev)AuI>B. We have then:
--
2.4. If B is derivable in L, from A,, -,A, and the &-symbol does not occur in A,, A,”,B, then B can be derived from A,, * A, in L, with the help of L,7’ but without using the s-symbol.
-
- -,
0 ,
Suppose a derivation of B in L, from A,, E,ClX,
- - .,A ,
is given and
.. , E , C n X
are all the &--termsintroduced in the proof by L,7, arranged in the order in which they appear in the proof. It follows that if instead of using L,7, we just write down each of Cl(e,Clx), C,,(ezC.x), we get again a proof of B, but this time from A,, -,A,, together with Cl(ESlx), C,,(s,C,x). T h e problem now is to find another proof i n which the additional premises are dispensed with.
--
- - .,
-
-a,
Consider the last of these: C,(B,C,X). By the rule L, 2, we can prove “c,(6,cnx) 3 B” from A,, -,A,, c,(E,c~x), Cn-i(&$n-,X) without using Lg7. Throughout the proof, we can replace c,C,x everywhere by a new variable ‘Y’ (say) and obtain a proof of “C.vx B’7. By the additional rule L, 7’ assumed especially for this purpose, we can then derive “ ( E v ) C , v 13 B” without using L, 7.
--
- - -,
But “C,(s,C,x)” was originally introduced by L, 7. Its premise “(Ex)C,x” is, therefore, derivable from A,, * -,A?, C,(exClx), * -, Cn-,( 6,C.-1x) already. Hence, since “ ( E u ) C g 2 B” IS also derivable
-
-
.
9
____
3. COMI’LETENESS OF THE PREDICATE CALCULUS
3i4
-
from the same, B is already derivable from 141, Cn-I(~,C,-l~) without using L, 7.
-
* *,
- - -,
A,, CI(&,C,x),
Hence, we have succeeded in getting rid of the premise C,(E,C,X). Repeating the same process, we can get rid of the other premises Cn-I(E,C,-lx),-,C,(E,C,X) one by one and obtain a proof of B from A , , - - - , A , in which L,7 is not applied. In the resulting proof, there may still appear s-terms by L, 1, L,3, and L, 5. Suppose that
introduced
. ,E , H ~ x
E,H~x,
*
are all the &-terms i n the resulting proof, arranged in the order in which their last occurrences appear. Replace 6,Hkx throughout the proof by a new free variable, and we get again a proof of B from A,, . -,A , in which L, 7 is not used. Repeat the same with E , H ~ - ~ x , and so on. We finally get a proof B from A , ,-* -,A , in which the &-symbol does not occur. Sometimes the predicate calculus is taken as to include also the theory of equality. Hence, one particular predicate R x y is chosen and rewritten as x=y, together with special axioms for them. In this way, we obtain a system:
2.5. L , l and L, 1. L,2.
System L,: e.g., system L, plus the axioms for equality
L,2.
tv=v.
If B is like A except for containing free occurrences of u in place of some free occurrences of D, and t v = u , then I-A-B.
It is also often convenient to include constant names and functors. Then variables give way to the more general notion of terms which include constants, variables, and functors followed by terms. As these additional symbols usually do not affect general considerations about the predicate calculus, we shall often neglect them, though taking them for granted when necessary.
5 3. COMPLETENESS OF THE PREDICATE CALCULUS Using the notion of maximal consistent extensions (Godel, “Eine Eigenshaf t der Realisieung des Aussagenkalkiils”, Ergebnisse eines Math. Kolloquiums, 4 (1933), 20-21; L. Henkin, J. Symbolic Logic, 14 (1949), 159-la), i t is possible to give a short proof of the completeness of the systems L,, L,, La, L,.
Theorem 2. The systems L,, L,, L,, L, are all complete; in fact, not only a single formula, but any enumerable set of formulae, if not
318
XI. THE PREDICATE CALCULUS
containing the 6-symbol and consistent with these systems (i.e., no contradictions arising when they are taken as axioms) has an enzlmernble model. The proof can be obtained for Lb and L, first, then, by 2.41, the theorem also holds for L, and Lb. We assume all the quantificational formulae, i.e., say, all formulae of L,, are enumerated in a definite manner so that each is correlated with a unique positive integer in the standard ordering: ql, q2, etc. If a consistent set Soof quantificational formulae is given, its maximum consistent extension is constructed in the following manner. If So plus q1 is consistent, then S, is So plus q l ; otherwise S, is the same as So. In general, if Sn plus q,+] is consistent, then Sn+, is S,, plus qn+,i Otherwise, Sn+,is the same as S,. The maximum consistent extension is the union of the sets S;. Let So be a consistent set of formulae in L,. We may assume all free variables as bound at the beginning with general quantifiers (hence, only “closed” formulae) and then eliminate all general quantifiers and all connectives besides “neither-nor” i n familiar manner. Construct the maximum extension (call it S) of So within the system L, or Lb. Call a closed formula true or false according as whether it belongs to S or not. We assert that hereby we get a model of So in the enumerable domain D of all constant 6-terms (i.e., those containing no free variables). In the first place, if P is an arbitrary predicate with n arguments and a,, *, an are n constant 6-terms, then either P(a,, a,) or -P(a,, a,), but not both, is true, because S is maximal consistent. In the second place, “neither p nor q” is true if and only if neither p is true nor q is true, again because S is maximal consistent. In the third place, ( E x ) F ( x ) is true if and only if there is some constant s-term a such that F ( a ) is true, because “ ( E x ) F ( x ) ~ F ( s , F x ) ” belongs to S. Moreover, since So is a subset of S , all formulae in So are true. 0 ,
0 ,
This completes the proof of Theorem 2, completeness being the special case when So contains a single formula. There is a long history behind this theorem. In 1915, Lowenheim (Math. Annalen, 76, 4 4 7 4 7 0 ) proved:
3.1. Every quantificational formula, if satisfiable in any (nonempty) domain at all, is satisfiable in an enumerable domain. In 1920, Skolem (Vidensk. Skrifter I, Mat. Naturw. Klasse, Oslo, no. 4) improved the proof and extended i t to any enumerable set of formulae. These proofs all use the axiom of choice. T h e idea is
5 3. COMPLETENESS OF THE PREDICATE CALCULUS
particularly simple and illuminating. given in the prenex normal form:
319
Suppose a single formula is
p : ( x ) ( E y > ( E z ) ( u l ) ( u > ( E v ) H ( x , YY
ZY WY U Y V
).
Following Godel, let us call "Skolem functions for p and M" any functions f ( x ) , g ( x ) , h ( x , t v , z ~ )in M such that for any elements x, w, u of M the following is true:
P": m x , fb),& > Y
W , U Y
h(XY
WY
#I>.
By the axiom of choice, there exist such functions f , g , and h. These functions can be applied fruitfully in many considerations. Hence if a quantificational formula p is satisfiable at all, it is satisfiable i n a denumerable domain. Thus, if 0 is an arbitrary object, the domain is the union of all sets Mi, such that M,=O, and Mi+1 contains all and only those objects which either belong to M ior are f ( n ) , g(a) or h(a,b,c) for some a,b, and c that belong to Mi. In 1923 and 1929 (cf. ibid, no. 4 for 1929), Skolem proved 3.1 and its generalization to infinitely many formulae without using the axiom of choice. Skolem merely assumes that the quanti€icational formulae be given i n prenex normal form. But, for brevity, let us conside? a single formula i n the Skolem normal form; Skolem takes H(l, *, 1; 2, *, n + 1) as H,, and then considers all the m-tuples of the positive integers n o greater than n + 1, ordering them i n a n arbitrary manner with the m-tuple (1, * * *, 1) as the first. If (G,, ,tim) is the i-th m-tuple, then Hi is: 0 ,
He then considers all the m-tuples of the positive integers used so far, and again couples the i-th m-tuple with the n-tuple consisting of the n consecutive integers starting with n ( i - 1) +2. In this way he defines a sequence of quantifier-free conjunctions HI,H,, all gotten from the formula A. 0 .
His proof of 3.1 contains two parts:
3.2. If a quantificational formula A is satisfiable at all, then none of the formulas -HI, -H2, is a tautology.
-
is a tautology, then 3.3. If none of the formulas -HI, -H2,. the formula A itself is satisfiable in the domain of positive integers. 0 -
XI. THE PREDICATE CALCULUS
320
At about this time, Herbrand (Comptes YeBdas, Paris, 188 (1929), 1076) proved as part of his famous theorem: 3.4. If --A is not a theorem of the predicate calculus, then none of -HI, -H2, *, is a tautology. From 3.3 and 3.4, the completeness of the predicate calculus fol-
-
lows as a corollary.
Soon after, Godel independently proved these theorems (Monntsh. His proof is widely known through Hilbert-Ackermann.
Math. Phys. 37 (1930), 349-360).
While a proof of Herbrand’s theorem (see 2.25 above) requires metamathematical considerations about the structure of proofs, i t has been observed by Beth and Dreben that one can derive from 3.3 and 3.4 a weaker Herbrand theorem which uses a general recursive rather than a primitive recursive procedure to get a proof not using modus must ponens. Thus, by 3.3, if A is a theorem, some of HI,H,, be a tautology. Since we can effectively construct H I , H,, from A and test for tautology, we can effectively find some tautologous Hi, provided only A is a theorem. By the proof of 3.4, we can actually write out a proof of A using only L, 1 to L , 5, i.e., not using the rule of modus ponens.
-----
We append a remark on the relation between the axiom of choice and the s-symbol. Clearly there is some connection between the axiom of choice and rules governing Hilbert’s s-operator. Occasionally, the s-rule is referred to as a generalized principle of choice. This is misleading. For example, if the axiom of choice were a special case of the srule, why does the consistency of the axiom of choice not follow from the s-theorems according to which application of the s-rules can be dispensed with if the 6-operator occurs in neither the axioms nor the conclusions? Indeed, if the axiom of choice were derivable from the s-rule, we would, by the s-theorems, be able to derive the axiom of choice from the other axioms of set theory. But, as we know, the independence of the axiom of choice is an unsolved problem. Of course, there is at least one difference between the 6-rule and the axiom of choice. The. former makes a single selection, while the latter requires that a simultaneous choice from each member of a given set be made and that all these selected items be put together to generate a new set. Hence, there is no reason to suppose that, in general, the axiom of choice follows from the s-rule. If the following (x)(Ey)(a)(xsy
( E W ) ( W S& Xx
~,(usw)))
4 3 . COMPLETEhiSS OF T€IE PREDICATE CALCULUS .-
_-
32 1
happens to be a theorem in a certain system of set theory, then the axiom of choice does follow from the 6-rule i n that system. But then i t would be highly unlikely that the e-operator did not appear in the axioms of the system. There are also cases where, although the s-rule would yield the desired result, the axiom of choice would not. For example, in the Zermelo theory we can infer “ ( x ) R ( x ,syRxy)” from “(x) (Ey)Rxy” by the s-rule, but we cannot infer “there exists f , ( x ) R ( x , f x ) ” from “(x)(Ey)Rxy” by the axiom of choice, on account of the absence of a universal set in Zermelo’s theory. Most of the remarks in this section and the preceding section appeared i n a symposium paper in Mathematical Interpretation of Formal Systems, Amsterdam, 1955, pp. 57-84. The completeness theorem of the predicate calculus has been applied to prove results in algebra by Malcev, Henkin, A. Robinson, Tarski. There is also a complementary trend of proving results of interest to logic from the algebraic approach. The completeness theorem is applied in the following form: if A is an enumerable set and B a finite set of formulae in the predicate calculus, and if there is, for every finite subset A’ of A , a model satisfying both B and A’, then there is a model satisfying A and B. For example, this can be applied to prove that every partial ordering can be extended to a total ordering. That a relation R is a total ordering can be expressed in the predicate calculus saying that
R is transitive and that Rxy or Ryx for all distinct x and y . These propositions on R form the finite set B. If R is given as a partial ordering, then A is the set of all propositions R(a, b ) which are true, i.e., i n which the objects a and b stand i n the partial ordering relation R. Since any finite subset of A can be made into a total ordering, a model can he found to satisfy both B and any finite set of A . Hence, a model can be found to satisfy both A and B. That is to say, R can be extended to a total ordering.
Additional Note. B. S . Dreben gives the following proof of 2.21 stated on p. 311 above. Consider the case when B is (x)(Ey)(x)Gxyx and there are three occurrences of B in the premiss. Let Gi be G x ~ ~ - ~ ~ 3 i - 1x3i, then, by 2.23, ( ( G V G V G) =’GI) V ((GIV G V G 6 ) 2 Gd V
((G V GBV GI)=’ G)
is a theorem. Hence, by repeated applications of LA 3 and Ld 4, we get B in place of each Gi(i= 1, * 9). Hence, the conclusion of 2.21 follows from its hypothesis.
-
a,
THE PREDICATE CALCULUS 5 1. THEPROPOSITIONAL CALCULUS In Ch. IX, we have incidentally considered certain formulations of the propositional calculus and the predicate calculus. In this chapter, we shall describe a number of points in this area mor2 leisurely. It is well-known that the propositional calculus is essentially equivalent to Boolean algebra. For example, the following is an axiom system for Boolean algebra. T h e variables are x, y, z, etc., 0 is the only constant, a term is: (i) a constant or a variable; or (ii) a’ or ab when a, b are terms. A formula is a = b , if a, b are terms. One can then define 1 as 0’, a + b as ( a1b1 )’ And the axioms are:
.
(1) acr & a , (2) (3)
(4) (5) (6)
ab=ba, a(bc) = ( a b ) c , a0 = 0, if a b ‘ = O , then ab = a , if ab = a and ab‘ = a , then a = 0.
A well-known system of the propositional calculus is that of Whitehead-Russell, as modified by Bernays, and adopted by HilbertAckermann. The connectives “not” and “or” are used with the others defined. The axioms are: ’
(1)
(PVP)=%
(2)
P3(PVd,
(3)
( P v q )z l ( 4 v PI, = I ( ( r 3 PI 2 (7. =q)1.
(4) ( P = 4
The rules of inference are modus ponens and substitution if p , q, I, etc. are construed as proposition letters. If they are construed as schemata, then the rule of substitution is unnecessary.
If in my formula of the propositional calculus, we replace p , q, etc. by x, y, etc., complement by negation, intersection by conjunction, sum by disjunction, i t is possible to show that a formula A is a
30a
XI. THE PREDICATE CALCULUS
theorem in the propositional calculus if and only if A * = l theorem in Boolean algebra, A" being the transform of A .
is a
Another well-known system is that of Hibbert-Bernays in which the five common connectives are all taken as primitive. The axioms are:
The rules of inference are modus ponens with or without substitution as before. These systems are all known to be complete and consistent. The axioms are also independent. The usual method of proving independence is by choosing an interpretation in which the particular axiom is not true, but all theorems derivable from the other axioms are true. For example, if we take 1 as true, then the following interpretation gives the independence ( l a ) :
P
4
1 1 1 2 2 2
1 2
3 3 3
3
1 2 3
1 2 3
-p 3 3 3
2 2 2 1 1 1
P V 4 1 1 1 1
2 2 1 2 3
P&4
PE4
1 2
1
p = 4
'
1
3 3
3 3
2 2
3
1
1 1
3
3
3
3 3 3
3 3
,1 1 1
3
1
9 2. FORMULATIONS OF
’ r m PREDICATE
CALCULUS
309
Thus, every axiom except ( l a ) always gets the value 1, and if p and p I q always get the value 1, so does q . On the other hand, when p and q take the values 2 and 1 respectively, ( l a ) takes the value 3.
If one omits (5c) from the above system, one gets the “minimal calculus”. If one omits (5c) and replaces (5a) and (5b) by: ( p z - p ) 3 -p, (5b’)--p3 (pxq ) ; then the resulting system is the intuitionistic propositional calculus. It is also known that (5a) to (5c) can be replaced by: (5a’) ( P
-
=I -44) 3 (4 3 PI,
6’ ((P ) =
-
PI
2 4) 2
((P
(5b’),
3 4) 3 4).
This alternative formulation has, according to Lukasiewicz, the following property: drop (5c’), we get intuitionistic calculus, drop also (5b’), we get the minimal calculus.
$2. FORMULATIONS OF THE PREDICATE CALCULUS There are different formulations of the predicate calculus, variously known as quantification theory, the restricted or first order predicate calculus, the first order functional calculus. Often these diverse formulations are of interest for different purposes. We give here three alternative formulations. We may use proposition letters p , q, r, etc., variables x , y, z, etc., predicates F , G, etc. A proposition letter or a predicate followed by variables is an atomic proposition. If A and B are propositions and u is a variable, then -A, A V B, ( u ) A , ( E u ) B are propositions. A proposition containing n o quantifiers is an elementary proposition. Other connectives &, 3 , 5 may either be included initially or be introduced by definitions. A proposition is a tautology if i t as a truth function of its components which may or may not contain quantifiers is always true by the usual interpretation of truth-functional connectives. A given occurrence of a variable u is said to be bound i n a formula if i t is governed by a quantifier ( v ) or ( E u ) , otherwise free. If the result of writing general quantifiers for all free variables i n A before A is a theorem, we write FA.
2.1. System L, (see Quine, Matlzemnticnl Logic, 1951 ed.). *loo. If A is a tautology, F A *101. F ( v ) ( A I B )3 ( ( v ) A = )( u p ) *1O2. If ZI is not free in A , F A I ( v ) A .
XI. THE PREDICATE CALCCLUS
3 1y1
* 103.
If B is like A except for containing free occurrences of wherever A contains free occurrences of v, then t - ( v ) A 3 B . *104.
ZI
If A z B and A are theorems, so is B.
This system has *104, the modus ponens, as the only rule of inference, which is less general than the usual one i n so far as A and B contain n o free variables. Vacuous quantifiers are included so that ( v ) A is a proposition even when v is not free i n A . This enables one to dispense with a rule of generalization: if t-A, then t - ( v ) A , which is included in the usual systems. No rule for the particular quantifiers are included since (Ev) is to be defined as -(v)-. There is no rule of substitution since the axioms are already schemata in the manner of von Neumann. The principle *lo0 can be replaced by a finite number of specific schemata in the usual manner, and i t is sometimes convenient to do so. It may be noted (see 1. Symbolic Logic, 12 (1947), 130-132) that we can dispense with “102 if we replace “100, *101, and *lo4 respectively by: “100’.
If A is a tautology, t - ( v ) A .
*101’.
If v is not free in B, t - ( v ) ( A 3 B ) 3 ( A 3 ( v ) B ) .
*104’.
If t-A and t-AxB, then t-B.
The equivalence is quite direct. All the new principles are easy consequences of results in ML: *lo0 and *115; *159; “111. Conversely, *lo2 is proved by *loo’, *101’, *104’; *lo0 and “104 are special cases of *loo‘ and *104’; *lo1 can be proved by *101‘ and *lo3 with the help of *115 whose proof uses only the new principles.
As a second system, we present one in Herbrand’s thesis (Recherches sur la thkorie de la dtmonstration, Warsaw, 1930). There is no need to allow vacuous quantifiers (called “fictitious” quantifier i n his thesis, p. 29), and particular quantifiers are assumed to begin with, while truth functions besides and V are taken as defined. 2.2.
System L,.
L4 1. Rule of tautology. a tautology then t-A.
-
If A is an elementary proposition and
L, 2. Rules of inversion. Within a given proposition, if v is not free in B, we can replace each one by the other in any of the pairs: (1)
-
(Y)A, (EY) - A ;
(3) ( v ) ( A V
(2)
B), ( u ) A V B ;
L , 3. Rule of generalization.
-
(4)
(Ev)A,
( Y )
-
A;
(Ev)(A V B ) , (Ev)A V B.
If t-Av, then t-(v)Av.
4 2.
FORMULATIONS OF THE PREDICATE CALCULUS
311
Lb 4. Rule of particularization. If FAvv, then k(Eu)Avu, in other words, if t A and B is obtained from A by substituting u for v at some or all occurrences of u, then k ( E u ) B .
L, 5. Rule of simplification. If k A V A , then FA. L, 6.
Modus ponens.
If k A 3 B and P A , then t-B.
This remarkable system is specially useful for the study of proofs in the predicate calculus. Here are some concepts and results about the system:
2.21. Dreben’s lemma. If for all A, t-A follows from A V A , then t-(BV VB)zB.
v ..-
A proof of this which depends on 2.23 wi 1 be given at the end of this chapter.
-
2.22. Given any proposition H ( p , . -,q) built up from proposition letters by and V , the sign of each occurrence of each letter in H is defined thus: (1) p is positive in p ; (2) the sign of an occurrence of p in A is the opposite of its sign in A ; ( 3 ) the sign of an occurrence of p i n A is the same as its sign in A V B. (Thesis, p. 21.)
-
T h e rules of inversion i n L, are specially convenient for bringing a proposition to the prenex normal form: i.e., a proposition with all quantifiers standing at the beginning. Given an occurrence of a quantifier in a proposition, when brought to the beginning, i t may remain the same, or change from general to particular or vice versa. T h e definition 2.22 can be extended in obvious manner to occurrences of any propositions in a given proposition, we need only add that the sign of an occurrence of A in B is the same as its sign i n ( v ) B or ( E v ) B . It then happens that a quantifier remains the same when brought to the beginning, if and only if the component proposition beginning with the quantifier has a positive occurrence in the whole proposition. One peculiar feature of Lb is that the rule of tautology only uses elementary propositions. For the system, we can actually derive:
FA.
Lh 1’. The strengthened rule of tautology.
If A is a tautology,
Given a tautologous elementary proposition, we wish to show that
i f we substitute more complex propositions for the atomic propositions i n it, the results are always theorems of Ls. For this purpose, we may as well replace all atomic propositions by proposition letters and consider some given tautology H ( p , -,q). In general, each letter p,. -,q in H may have a number of positive occurrences and a number of negative Qccurrences. By the rules of
-
-
3 12
XI.
THE PREDICATE CALCULUS
inversion, we may assume that the propositions to substitute the letters
p,..., q are in the prenex normal form. We prove first a special case when each letter p , - -,q has at most one positive occurrence and one negative occurrence in H : 2.23. For every elementary tautology H ( p , - - ., q ) of this restricted type, all substitution results from H are theorems of L,.
-
While each result is obtained when we make all substitutions on the different proposition letters at the same time, we imagine that we break the process into different steps, one step for each proposition letter. Suppose we have shown that H ( B y , -,A ) is shown to be a theorem, we wish to show that if instead of By we substitute (Ey)By or (y)By for p , we again get a theorem. Suppose p has one positive occurrence and one negative occurrence and suppose, for brevity, ( E y ) B y is to replace p . Hence, we wish to derive, from
-
H(p,p,- - -,A ) :
H((Ex)Bx, (Ey)By,
-
* *
,A )
or b ) ( E x ) H ( B x , B Y ,
-
* *,
A).
By hypothesis, H ( B y , By,. -,A ) is a theorem. Hence, by Lb 4 and L6 3, we get the desired theorem. From 2.23, i t follows that for any A,C,B,D: t - B x B , and (1)
I- ( ( A 3 B )
82
( c 3 D)) 3 ( ( A v c ) 3 ( B v u ) ) .
If we substitute B for C and modus ponens: I- ( ( B
v
D, B V B for A i n (l), then we get by
B) 3 B) 3 ((B
v
B
v
B)
= l(
B
v B)).
By 2.21 and the rule of simplification, we have also ( B V B ) z , B . Hence, if I - B V B V B , then I-BVB. Therefore, by the rule of simplification, if t-B V B V B, then t-B. If we repeat the process by substituting B V B V B , etc. for A i n (l), we get:
2.24.
If t-B V
0 .
- V B, then
I-B.
With the help of this, we can now prove the general theorem:
Theorem 1. The strengthened rule o f tautology L4 I' is derivable in Ls; there is an effective (in fact, primitive recursive) method, szlch that for every tautology A, we can find a proof for it in L,. We
illustrate
the procedure by a simple example.
Suppose
H(p;p,p,p;q,q;q) is an elementary tautology with one positive occurrence of p , two of q, three negative occurrences of p , one of q. And we wish to prove the result of substituting (x)(Ey)Fxy for p, (x)Gx for q.
S 2. FORMULATIONS
OF THE PREDICATE CALCGZUS
3 13
By Lh 2, Lh 3, and L, 4, it is sufficient to prove: (2)
H((Ey)Fxy; (EY)FXY (2)
,
G (2)
,( E y ) F x y ,
( 2 ) G*
(EY)FxY;
; (2) Gz).
To prove this, by 2.24, Lh 3, and L, 4, i t is sufficient to prove: (3)
H ( F x u ; F x u , F x v , F x w ; ( z ) G z , (z)Gz; (2)G.z) V H ( F x v ; * - . )
V H(Fxw; **-).
For the same reason, this i n turn follows from (4)
I-I( F x u ;
--
*
; G z , G y ; Cz) V El( F x v ;
V H(Fxw;
V H(Fxv;
* * - ; *
Gz)
v
H(Fxu;
*
-
* - * ;
*
;G z )
Gy)
- ; G y ) V H ( Fxcu; - ; G y ) . * *
Now we wish to show that, since H(p;p,p,p;q,q;q)is a tautology, (4) is also one and falls under L h 1. Instead of the two atomic propositions p and q, we now have five: FXZI,Fxv, Fxw, Gx,Gy. If we put H(p;n,b,c;d,e;q) into a conjunctive normal form, t h i n since signs
of the atomic propositions are preserved, each conjunctive term must be a part of (5)
p V
- - a V
6 V
Moreover, since the original H term must contain either p and at and at least one of d and e. Now of (4) into the corresponding form (4')
H ( a ; a , 6 , c; d , e ; d )
v*
c
-V
V d V e V
-
q.
is tau tologous, each conjunctive least one of -a, -6, -c, or -q if we turn each disjunctive term and get H ( c ; a, 6 , c ; d , e; e),
then we can show that (4') must be true for all possible truth values of a,b,c,d,e. First, consider the corresponding form for (3): (3')
H ( a ; a , 6 , c ; q , q ; q) V H ( 6 ;
a * - )
v H(c;
-.*).
This must he a tautology. If a,b,c are all false, every conjunctive term (a disjunction) i n every disjunctive term of (3') must be true since it must contain either q and -q, or at least one of -n, 4, -c. * If at least one of a,b,c is true, say b, then, H(b;n,b,c;q,q;q) must be always true. Hence, if we take the conjunctive normal form of (3")
N(a;
a,
b,
c;
d , e ; 4) V H ( 6 ;
-
* *)
v
H(c;
*),
XI. THE PREDICATE CALCULUS
314
then each conjunctive term must contain either d and -q or e and -q, provided we drop all terms aV-a, bV-b, cV-c. But (4’) is equivalent to a disjunction of the two propositions obtained from (3”) by substituting d and e for q . For any possible truth values of n,b, c,d,e, if d, e are both false, then every one is true since -q occurs. If at least one is true, take that one. This completes the proof of Theorem 1, since the example includes all the essential features for the general case. The proof is based on the method i n Herbrand’s thesis. But the argument put forward in the example above is more complicated than Herbrand’s, because we feel there are some gaps in his simpler proof. In Whitehead-Russell, the whole * 9 is devoted to a sketch of a proof of Theorem 1 along quite a different line. Since they do not establish the completeness of their axioms for the propositional calculus, they cannot use arguments of the type as exemplified i n the above proof. In his thesis, Herbrand continues to exhibit a procedure whereby he can transform any given proof i n Lh into one which does not use the rule of modus ponens. This is the famous:
2.25.
Herbrand’s Theorem. There is an effective (in fact, primitive recursive) procedure whereby, given any proof i n an ordinary formulation of the predicate calculus, we can turn i t into a proof of Lb with no appeal to the rule of modus ponens. There is an extensive literature on this theorem mostly following the alternative treatment by Gentzen. We shall not attempt an exposition of Herbrand’s original proof which, however, we believe, contains many interesting distinctive features well worth exploring. Instead, we shall consider some related results and some different formulations. In mathematics, we constantly use the following mode of reasoning: “There exist y such that Ay. Let x be one such and consider whether this x has such and such other properties.” In the symbolism of the predicate calculus, one might wish to formalize this by using a rule “from ( E y ) A y , infer Ax”. But since the x chosen is not an arbitrary object but an arbitrary object which happens to have the property A, the variable x i n such a rule cannot be expected to behave like ordinary variables i n mathematical logic. T o overcome this difficulty, Hilbert introduced the 6-symbol and infers A(e,Ax) f r o m ( E x ) A x . This symbol is treated thoroughly in Hilbert-Bernays’ book, volume 11. The predicate calculus is then formulated a little differently. In addition to the variables, s-expressions ~ J x , etc. also become terms
$2.
315
FORMULATIONS OF ?WE PREDICATE CALCULUS
and Fv, Gvu, etc. are propositions for arbitrary terms v, u, etc. If, e.g., we take the system Lh, we can obtain a new system Lb by some changes. 2.3. System La. The system includes L,, 1 to Lh7 with the propositions reconstrued in the extended sense, and a new rule: (6)
The 6-rule.
t- ( E v ) A Y 3 A ( e y A v ) .
Incidentally, given the e-symbol, i t is possible to omit quantifiers in the primitive apparatus and introduce them by definitions.
( E v ) A v if and only if A(e,Av). ( v ) A v if and only if A(e,-Av). For the system La, two fundamental theorems are proved in Hilbert-Bernays I1 (stated on p. 18). If F is a system formalized in the framework of Lb and, i n general, contains additional symbols for in-
2.31. 2.32.
dividuals and predicates, as well as a finite number of new axioms A,, * A, which do not contain the 6-symbol, but may contain quantifiers, then:
-.,
2.33. The first e-theorem. If the axioms A,, * A, contain no bound variables and B is a theorem of F which contains no bound variables, then B can also be derived from A,, A, without using bound variables. a,
a * - ,
2.34. The second e-theorem. If B does not contain the e-symbol but may contain quantifiers, i t can be derived from A,, A , without using the e-symbol (hence, can be derived in L,). - - a ,
In Hilbert-Bernays, Herbrand’s theorem is deduced from these, and the two 6-theorems are extended to imbed also the axioms for equality. We shall not enter upon proofs of these results. If we combine the use of the 6-symbol with the method of natural deduction as developed by Quine (Methods of Logic, 1950), we obtain a rather symmetric system L,, which has the same terms and propositions as La but uses different rules. In particular, the deduction theorem is now taken as a primitive rule, and we are allowed to write down any new axiom as a hypothesis. The final result is a theorem of L, only if i t is freed from all hypotheses. Formally, a hypothesis may be marked by a * at the beginning which stands also i n front of all consequences of the hypothesis. A line in a proof of L, is a theorem of L, only when i t does not begin with a *
.
L, 1. Rule of assumption. We may set down any proposition A as a line at any stage in the course of a deduction, provided we prefix a new * on the line and all its consequences.
XIiI. TlIE PREDICATE CALCCLGS
3 16
L,2. Rule of deduction. If under the assumption of * A we arrive at *B, then we may write a new line A x R , with one * less.
L,3. Rule of tautology. We may write a line which is implied truth-functionally by one or more previous lines taken together.
L, 5. Rule of instantiation.
t-Au. I-
If v is a variable and kAv, then
Rule of generalization.
L,4.
t- ( v ) A s .
If zi is any term and k ( v ) A v , then
L,6. Rule of particularization. If ~l is any term and t-Azt, then ( E v )Av. L,7. Rule of specification. If t-(Ev)Av, then ~ - A ( E , A u ) .
For the system L,, it is rather direct to prove something like the second &-theorem. For this purpose, we need an additional rule which is no longer derivable if we drop L,7:
If ~ A u and ~ Bv is not
L,7’. Weakened rule of specification. free in B, then t-(Ev)AuI>B. We have then:
--
2.4. If B is derivable in L, from A,, -,A, and the &-symbol does not occur in A,, A,”,B, then B can be derived from A,, * A, in L, with the help of L,7’ but without using the s-symbol.
-
- -,
0 ,
Suppose a derivation of B in L, from A,, E,ClX,
- - .,A ,
is given and
.. , E , C n X
are all the &--termsintroduced in the proof by L,7, arranged in the order in which they appear in the proof. It follows that if instead of using L,7, we just write down each of Cl(e,Clx), C,,(ezC.x), we get again a proof of B, but this time from A,, -,A,, together with Cl(ESlx), C,,(s,C,x). T h e problem now is to find another proof i n which the additional premises are dispensed with.
--
- - .,
-
-a,
Consider the last of these: C,(B,C,X). By the rule L, 2, we can prove “c,(6,cnx) 3 B” from A,, -,A,, c,(E,c~x), Cn-i(&$n-,X) without using Lg7. Throughout the proof, we can replace c,C,x everywhere by a new variable ‘Y’ (say) and obtain a proof of “C.vx B’7. By the additional rule L, 7’ assumed especially for this purpose, we can then derive “ ( E v ) C , v 13 B” without using L, 7.
--
- - -,
But “C,(s,C,x)” was originally introduced by L, 7. Its premise “(Ex)C,x” is, therefore, derivable from A,, * -,A?, C,(exClx), * -, Cn-,( 6,C.-1x) already. Hence, since “ ( E u ) C g 2 B” IS also derivable
-
-
.
9
____
3. COMI’LETENESS OF THE PREDICATE CALCULUS
3i4
-
from the same, B is already derivable from 141, Cn-I(~,C,-l~) without using L, 7.
-
* *,
- - -,
A,, CI(&,C,x),
Hence, we have succeeded in getting rid of the premise C,(E,C,X). Repeating the same process, we can get rid of the other premises Cn-I(E,C,-lx),-,C,(E,C,X) one by one and obtain a proof of B from A , , - - - , A , in which L,7 is not applied. In the resulting proof, there may still appear s-terms by L, 1, L,3, and L, 5. Suppose that
introduced
. ,E , H ~ x
E,H~x,
*
are all the &-terms i n the resulting proof, arranged in the order in which their last occurrences appear. Replace 6,Hkx throughout the proof by a new free variable, and we get again a proof of B from A,, . -,A , in which L, 7 is not used. Repeat the same with E , H ~ - ~ x , and so on. We finally get a proof B from A , ,-* -,A , in which the &-symbol does not occur. Sometimes the predicate calculus is taken as to include also the theory of equality. Hence, one particular predicate R x y is chosen and rewritten as x=y, together with special axioms for them. In this way, we obtain a system:
2.5. L , l and L, 1. L,2.
System L,: e.g., system L, plus the axioms for equality
L,2.
tv=v.
If B is like A except for containing free occurrences of u in place of some free occurrences of D, and t v = u , then I-A-B.
It is also often convenient to include constant names and functors. Then variables give way to the more general notion of terms which include constants, variables, and functors followed by terms. As these additional symbols usually do not affect general considerations about the predicate calculus, we shall often neglect them, though taking them for granted when necessary.
5 3. COMPLETENESS OF THE PREDICATE CALCULUS Using the notion of maximal consistent extensions (Godel, “Eine Eigenshaf t der Realisieung des Aussagenkalkiils”, Ergebnisse eines Math. Kolloquiums, 4 (1933), 20-21; L. Henkin, J. Symbolic Logic, 14 (1949), 159-la), i t is possible to give a short proof of the completeness of the systems L,, L,, La, L,.
Theorem 2. The systems L,, L,, L,, L, are all complete; in fact, not only a single formula, but any enumerable set of formulae, if not
318
XI. THE PREDICATE CALCULUS
containing the 6-symbol and consistent with these systems (i.e., no contradictions arising when they are taken as axioms) has an enzlmernble model. The proof can be obtained for Lb and L, first, then, by 2.41, the theorem also holds for L, and Lb. We assume all the quantificational formulae, i.e., say, all formulae of L,, are enumerated in a definite manner so that each is correlated with a unique positive integer in the standard ordering: ql, q2, etc. If a consistent set Soof quantificational formulae is given, its maximum consistent extension is constructed in the following manner. If So plus q1 is consistent, then S, is So plus q l ; otherwise S, is the same as So. In general, if Sn plus q,+] is consistent, then Sn+, is S,, plus qn+,i Otherwise, Sn+,is the same as S,. The maximum consistent extension is the union of the sets S;. Let So be a consistent set of formulae in L,. We may assume all free variables as bound at the beginning with general quantifiers (hence, only “closed” formulae) and then eliminate all general quantifiers and all connectives besides “neither-nor” i n familiar manner. Construct the maximum extension (call it S) of So within the system L, or Lb. Call a closed formula true or false according as whether it belongs to S or not. We assert that hereby we get a model of So in the enumerable domain D of all constant 6-terms (i.e., those containing no free variables). In the first place, if P is an arbitrary predicate with n arguments and a,, *, an are n constant 6-terms, then either P(a,, a,) or -P(a,, a,), but not both, is true, because S is maximal consistent. In the second place, “neither p nor q” is true if and only if neither p is true nor q is true, again because S is maximal consistent. In the third place, ( E x ) F ( x ) is true if and only if there is some constant s-term a such that F ( a ) is true, because “ ( E x ) F ( x ) ~ F ( s , F x ) ” belongs to S. Moreover, since So is a subset of S , all formulae in So are true. 0 ,
0 ,
This completes the proof of Theorem 2, completeness being the special case when So contains a single formula. There is a long history behind this theorem. In 1915, Lowenheim (Math. Annalen, 76, 4 4 7 4 7 0 ) proved:
3.1. Every quantificational formula, if satisfiable in any (nonempty) domain at all, is satisfiable in an enumerable domain. In 1920, Skolem (Vidensk. Skrifter I, Mat. Naturw. Klasse, Oslo, no. 4) improved the proof and extended i t to any enumerable set of formulae. These proofs all use the axiom of choice. T h e idea is
5 3. COMPLETENESS OF THE PREDICATE CALCULUS
particularly simple and illuminating. given in the prenex normal form:
319
Suppose a single formula is
p : ( x ) ( E y > ( E z ) ( u l ) ( u > ( E v ) H ( x , YY
ZY WY U Y V
).
Following Godel, let us call "Skolem functions for p and M" any functions f ( x ) , g ( x ) , h ( x , t v , z ~ )in M such that for any elements x, w, u of M the following is true:
P": m x , fb),& > Y
W , U Y
h(XY
WY
#I>.
By the axiom of choice, there exist such functions f , g , and h. These functions can be applied fruitfully in many considerations. Hence if a quantificational formula p is satisfiable at all, it is satisfiable i n a denumerable domain. Thus, if 0 is an arbitrary object, the domain is the union of all sets Mi, such that M,=O, and Mi+1 contains all and only those objects which either belong to M ior are f ( n ) , g(a) or h(a,b,c) for some a,b, and c that belong to Mi. In 1923 and 1929 (cf. ibid, no. 4 for 1929), Skolem proved 3.1 and its generalization to infinitely many formulae without using the axiom of choice. Skolem merely assumes that the quanti€icational formulae be given i n prenex normal form. But, for brevity, let us conside? a single formula i n the Skolem normal form; Skolem takes H(l, *, 1; 2, *, n + 1) as H,, and then considers all the m-tuples of the positive integers n o greater than n + 1, ordering them i n a n arbitrary manner with the m-tuple (1, * * *, 1) as the first. If (G,, ,tim) is the i-th m-tuple, then Hi is: 0 ,
He then considers all the m-tuples of the positive integers used so far, and again couples the i-th m-tuple with the n-tuple consisting of the n consecutive integers starting with n ( i - 1) +2. In this way he defines a sequence of quantifier-free conjunctions HI,H,, all gotten from the formula A. 0 .
His proof of 3.1 contains two parts:
3.2. If a quantificational formula A is satisfiable at all, then none of the formulas -HI, -H2, is a tautology.
-
is a tautology, then 3.3. If none of the formulas -HI, -H2,. the formula A itself is satisfiable in the domain of positive integers. 0 -
XI. THE PREDICATE CALCULUS
320
At about this time, Herbrand (Comptes YeBdas, Paris, 188 (1929), 1076) proved as part of his famous theorem: 3.4. If --A is not a theorem of the predicate calculus, then none of -HI, -H2, *, is a tautology. From 3.3 and 3.4, the completeness of the predicate calculus fol-
-
lows as a corollary.
Soon after, Godel independently proved these theorems (Monntsh. His proof is widely known through Hilbert-Ackermann.
Math. Phys. 37 (1930), 349-360).
While a proof of Herbrand’s theorem (see 2.25 above) requires metamathematical considerations about the structure of proofs, i t has been observed by Beth and Dreben that one can derive from 3.3 and 3.4 a weaker Herbrand theorem which uses a general recursive rather than a primitive recursive procedure to get a proof not using modus must ponens. Thus, by 3.3, if A is a theorem, some of HI,H,, be a tautology. Since we can effectively construct H I , H,, from A and test for tautology, we can effectively find some tautologous Hi, provided only A is a theorem. By the proof of 3.4, we can actually write out a proof of A using only L, 1 to L , 5, i.e., not using the rule of modus ponens.
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We append a remark on the relation between the axiom of choice and the s-symbol. Clearly there is some connection between the axiom of choice and rules governing Hilbert’s s-operator. Occasionally, the s-rule is referred to as a generalized principle of choice. This is misleading. For example, if the axiom of choice were a special case of the srule, why does the consistency of the axiom of choice not follow from the s-theorems according to which application of the s-rules can be dispensed with if the 6-operator occurs in neither the axioms nor the conclusions? Indeed, if the axiom of choice were derivable from the s-rule, we would, by the s-theorems, be able to derive the axiom of choice from the other axioms of set theory. But, as we know, the independence of the axiom of choice is an unsolved problem. Of course, there is at least one difference between the 6-rule and the axiom of choice. The. former makes a single selection, while the latter requires that a simultaneous choice from each member of a given set be made and that all these selected items be put together to generate a new set. Hence, there is no reason to suppose that, in general, the axiom of choice follows from the s-rule. If the following (x)(Ey)(a)(xsy
( E W ) ( W S& Xx
~,(usw)))
4 3 . COMPLETEhiSS OF T€IE PREDICATE CALCULUS .-
_-
32 1
happens to be a theorem in a certain system of set theory, then the axiom of choice does follow from the 6-rule i n that system. But then i t would be highly unlikely that the e-operator did not appear in the axioms of the system. There are also cases where, although the s-rule would yield the desired result, the axiom of choice would not. For example, in the Zermelo theory we can infer “ ( x ) R ( x ,syRxy)” from “(x) (Ey)Rxy” by the s-rule, but we cannot infer “there exists f , ( x ) R ( x , f x ) ” from “(x)(Ey)Rxy” by the axiom of choice, on account of the absence of a universal set in Zermelo’s theory. Most of the remarks in this section and the preceding section appeared i n a symposium paper in Mathematical Interpretation of Formal Systems, Amsterdam, 1955, pp. 57-84. The completeness theorem of the predicate calculus has been applied to prove results in algebra by Malcev, Henkin, A. Robinson, Tarski. There is also a complementary trend of proving results of interest to logic from the algebraic approach. The completeness theorem is applied in the following form: if A is an enumerable set and B a finite set of formulae in the predicate calculus, and if there is, for every finite subset A’ of A , a model satisfying both B and A’, then there is a model satisfying A and B. For example, this can be applied to prove that every partial ordering can be extended to a total ordering. That a relation R is a total ordering can be expressed in the predicate calculus saying that
R is transitive and that Rxy or Ryx for all distinct x and y . These propositions on R form the finite set B. If R is given as a partial ordering, then A is the set of all propositions R(a, b ) which are true, i.e., i n which the objects a and b stand i n the partial ordering relation R. Since any finite subset of A can be made into a total ordering, a model can he found to satisfy both B and any finite set of A . Hence, a model can be found to satisfy both A and B. That is to say, R can be extended to a total ordering.
Additional Note. B. S . Dreben gives the following proof of 2.21 stated on p. 311 above. Consider the case when B is (x)(Ey)(x)Gxyx and there are three occurrences of B in the premiss. Let Gi be G x ~ ~ - ~ ~ 3 i - 1x3i, then, by 2.23, ( ( G V G V G) =’GI) V ((GIV G V G 6 ) 2 Gd V
((G V GBV GI)=’ G)
is a theorem. Hence, by repeated applications of LA 3 and Ld 4, we get B in place of each Gi(i= 1, * 9). Hence, the conclusion of 2.21 follows from its hypothesis.
-
a,