- Email: [email protected]
Truth-Functionality and the Ramified Theory of Types
TRUTH-FUNCTIONALITY AND THE RAMIFIED THEORY OF TYPES Hugues LEBLANC Temple University
George WEAVER B v n Mawr College
Take a calculus C with, say. ‘-’, ‘I)’ and , ‘V’ as its only logical operators. We shall say that C has a truth-functional interpretation if there is a family C of functions from the wffs (non-atomic as well as atomic) of C to {T,FI such that, informally speaking: (a) the theorems of C and the wffs of C evaluating to T under all members of Z coincide; and (b) the truth-values that the non-atomic wffs of C assume under members of Z hinge exclusively on the truth-values of their components or - - in
the case of quantifications - those of their substitution instances.
And we shall say that C has a strictly truth-functional interpretation if the family Z in question also obeys the following condition: (c) members of Z that agree on the atomic wffs of C agree on the nonatomic ones as well.’ The sentential calculus (SC, for short) has a strictly truth-functional interpretation; and - as follows from results of Beth, Schutte, and others2 - so does the first-order quantificational calculus (QC’ ). By contrast, the principal interpretation of QCP (the second-order quantificational calculus), though truth-functional, is not strictly truth-functional, a result due to Leblanc and Meyer [I9701? Passing to QC21m (the ramified second-order quantificational
1 For a formal account of the matter see section 11, where (b) is broken into three separate conditions: one dealing with negations, one with conditionals. and one with quantifications.
See Beth [ 19591 pp. 263-267, Schutte [ l!)67] pp. 5-14, Dunri a n d Belnap [ 19681, and Leblanc [ 19681 . Q C l , the first-order quantificational calculus with equality, also has a strictly truth-functional interpretation; see Leblanc [ 1969al on this matter.
2
That QCZ has a truth-functional interpretation follows in effect from results in Schutte [1960].
Ramified Theory of Types
149
calculus), we establish that:
(1) without Russell's Axiom of Reducibility, QC21m fares like SC and QC' (Theorems 1 and 3), but ( 2 ) upon adoption of that axiom, QC2im fares like QC2 (Theorems 8 and 11). Our results automatically extend to QC"1"
(the ramified theory of types):
(1 ') without Russell's axiom, QC"/" pretation, but
has a strictly truth-functional inter-
(2') upon adoption of the axiom, its principal interpretation - though still truth-functional - is no longer strictly truth-functional!
I The primitive signs of QC2/w are to be following parameters and variables:
'3', Y', '(),')),',',plus the
'5')
(i) N o individual parameters (among them 'u'), (ii) No individual variables (among them 'x'and 'y'), (iii) for each d from 0 on and each I from 1 on, No predicate parameters of degree d and level I (among them 'f', a predicate parameter of degree and level l), and
(iv) for each d from 0 on and each 1 from 1 on, Ho predicate variables of degree d and level 1 (among them I f ' , a predicate variable of degree and level l).5 The parameters in (i) and those in (iii) will be presumed to come in a definite order, known as their alphabetic order. The individual parameters and variables of QC2/" will be said to be of type 0; the predicate parameters and variables of QC2/" of degree 0, better known as sentence parameters and
Two questions remain open: I n the absence of Russcll's axiom does QCm:m (and, in particular, QC2:-) have any truth-functional interpretation that is not strictly truth-functional; and (and, in particular, QC?lm) have a truth-functional in the presence of that axiom does QPlm interpretation which unlike its principal one - is strictly truth-functional? Answers to these questions would shed a great deal of light on the ramified theory o f types.
4
~
5 Our Variables are what the literature would call
would Call free variables.
bound variables, and our parameters what it
I50
H. Leblanc, G. Weaver
variables, will be said to be of type -I; and the remaining predicate parameters and variables of QC2/" will be said to be of type 1. And, as usual, the formulas of QC2/" are to be all finite (but non-empty) strings of primitive signs of Q C ~ / - . Because we use both parameters and variables, our arsenal of syntactic variables will be rather large: 'P'to refer to parameters in general, 'V'to variables in general, ' X ' and 'Y'to individual parameters, 'X' to individual variables, ' I to 'individual signs (i.e., to individual parameters and variables), 'Fd/'', 'Gdll', and 'Hdl'' to predicate parameters of degree d and level I, 'Fdll' and -Gdll' to predicate variables of degreed and level I, and 'A', 'B', and 'C' to formulas. Three substitution conventions will be needed as we go along. Let A be a formula of QCU", V be a variable of QCU", and P be a parameter of QC2/" of the same type, same level, and (when V is a predicate variable) same degree as V: Under C I , the first of these conventions, (A)( V/P)will be the result of putting Veverywhere in A for P , and ( A ) ( P/ V ) that of putting P everywhere in A for V . The atomic well-fomed formulas (atomic wffs, for short) o f QC2/" are to be all formulas of QC2/" of the sort Fdh ( X I ,X 2 , . . . , X d h Where d 2 0.6 The well-formed formulas (wffs, for short) of QC2/" will be all atomic wffs of QC"", plus all formulas of QC2/" of the following three sorts: (i) -A, where A is a wff of QC2/", (ii) ( A 3 B), where A and B are wffs of QC2/", and (iii) (V V)A , where - for any parameter P of QC2/" of the same type, level, and degree as V - (A)(P/V)is a wff of QC2Iw.7 And the quasi-wffs(qwffs,for short) of QC2IW will be all formulas of QC2/" of the sort (A)( V/P),where A is a wff or qwff of QC21". P a parameter of QC2/" which occurs in A , and V a variable of QC2/" which is of the same type, level, and degree as P but does not occur in A . As usual, A counts as the immediate component of -A and of (VV)A (this for any variable V of QC'/"); and A and B count as the two immediate components of (A 3 R ) . And the components of a wff or qwff of QC2ioo are the wfi' or qwff itself, its immediate components, and the components of all its components.
6 When
d = 0, F d / I ( X , , X z , . . . , Xd) is t o be understood as just Fd//.
Under the present account of things formulas in which identical quantifiers overlap d o not count as well-formed. '&, 'V', 'E',and ' 3 ' , which we come to employ besides '-', *3', and 'Y', are presumed t o be defined in the usual manner. 7
Ramified Theory of n p e s
I51
For convenience's sake we assign levels to the wffs (as well as to the predicate parameters and variables) of QC2/". A being a wff of QC2/", let Ip(A) be 0 if no predicate parameter of QC2/= occurs in A ; let [,(A) be 1 if at least one predicate parameter of QCYooof level 1 occurs in A but none of higher level than 1 does; let Zp(A) be 2 if at least one predicate parameter of QC2/w of level 2 occurs in A but none of higher level than 2 does; etc. And let Zv(A)be similarly defined, but with the word 'variable' doing duty throughout for 'parameter'. We shall'then take the level Z(A) of A to be max(lp(A >, I+1). Now for our remaining two substitution conventions:
C2: Let A be a wff or qwff of QC2Irn;let X1,X2, . . . , and x d (d 2 0)be distinct individual parameters of QC*/=; and let I1,12, . . . ,and Id be (not necessarily distinct) individual signs of QC2/"".By (A)(Il,I2, . . . , Id/ X 1 , X 2 , . . . , x d ) we shall understand the result of simultaneously putting I1 everywhere in A for X1,12 for X2,. . . ,and Id for Xd.8 C3: Let A be a wff or qwff of QC2lw; let F d h be a predicate variable of QC2/" of degree d and level 1; in case d > 0, let X1,X2, . . . , and x d be distinct individual parameters of QC2/=; let B be a wff of QC2/"; and let F d 1 ' ( I l I J I 2 , ~,Id~),Fd1'(I12,I22,... -.. ,Id 2 ) , . . . ,andFdll(Il ,12k,. . . , I d k ) (k 2 0) be all the atomic components of A which contain FdTl. Case 1 :k = 0. Then ( A ) ( ~ / F d / ~ ( X2,. X i , .. ,Xd)) is to be A. Case 2:k > 0. (a) If F d h occurs in a component of A of the sort (VW'and B has a component of the sort (VOH (V the same variable in both cases), (A)(B/Fd(Xl,X2, . . . ,&)) is to be (A)(Fdll/Fd/l), where Fdll is in alphabetic order the first predicate parameter of QC2/= of degree d and level 1. (b) Otherwise, (A)@/ FdIl(Xl,X2, . . . ,x d ) ) is to be the result of putting (B)(Ilr12i, . . . , Idi/ x1 .X2, . . . , X d ) everywhere in A for Fdh(Ili, Izi, . . . , Idi), this for each i from 1 through k.9 The (substitution) instances of the quantifications of QC2/= will play a considerable role throughout. With C1 and C3 on hand, they can be identified as
8 Siinultaneous substitutions can of course be broken into sequences of single substitutions. See Church 119561, p. 8 2 , on this point.
9 It is easily verified that if (VFdl/)A and B are wffs of QC*lm, and in case d > 0 - X I , X z , . . . , and Xd are distinct individual parameters of QC2Im, then (A)(B/Fd(X, .&,. . . , X d ) ) is sure to be a wff of QC*/-. From this point on we drop a number of inessential parentheses, thus writing 'A 3 B'in lieu of '(A 3 B)', 'A(P/V)' in lieu of '(A)(P/V)', etc. ~
152
H. Leblanc, G. Weaver
follows: (i) the instances of a quantification of the sort (VX)A will be all wffs ofQCz/" of the sort A(X/X); (ii) the instances of a quantification of the sort (VFdIl)A will be all wffs of QC2/" of the sort A(B/Fd/l(Xl,X2, . . . ,&)), where B is any wff of QC2/"" and - in case d > 0 - X I , X2, . . . , and Xd are distinct individual parameters of QC2/w; and (iii) the predicutive instances of (VFdl1)A will be all wffs of QC2/" of the sort A(B/Fdh(X, , X 2 , . . . ,&)), where B is wff of QC21w of level 1 or less (and - in case d > 0 - X1 ,X2, . . . , and X d are distinct individual parameters of QC2/"). Turning at last to the deductive apparatus of QCZI", we take the axioms of QC2/O0 to be all wffs of QC2/"" of any of the following seven sorts:
Al. A > @ > A ) , A2. (A 3 (B 3 C)) 3 ((A 3 B ) 3 (A 3 C)), A3. (-A 3 -B) 3 (B 3 A ) , A4. (VV)(A 3 B ) 3 ((VV)A 3 ( V V B ) , A5. A 3 ( V W , A6. ( V X ) A 3 A *, where A * is an instance of (VX)A,
A7. (VFdll)A 3 A *, where A * is a predicative instance of (VF'd/l)A, plus all wffs of QC2/"" of the sort (VV)A, where - for any parameter P of QC2lrn of the same type, level, and degree as Y that is foreign to (VQ.4 - A(P/V) is an axiom of QC2/"".10 We understand by the ponential of two wffs A and A 3 B of QC"" the wff B. Where A is a wff and S a set of wffs of QC2/", we understand by a proof in QC2/" of A from S any column 81
B2
BP of wffs of QC2'"" such that Bp is A and, for each i from 1 through p , Bi belongs to S, is an axiom of QC2/", or is the ponential of two 10 The trick of counting ( V V ) A an axiom if for suitable P A ( P / V ) is an axiom, stems from Fitch [ 19481. Note in connection with AS that with A 3 (V V)A - and, hence, A presumed here to be a wff o f Q C I P , V is sure to be foreign to A . ~
Ramified Theory of Types
I53
earlier wffs in the column; we say that A is provable in QC2/- from S if there is in QC2i"O a proof of A from S; we say that S is syntacti-
cally consistent in QC2i" if there is no wff of QC21" such that both it and its negation are provable in QC2/"0 from S; and we say that A is a theorem of QC2lm ( FA, for short) if there is in QC2/"O a proof of A from 8. Russell's Axiom of Reducibility, as tailored to suit QC2/"O, runs: A8. (gFd/1)(vX,)(vX2). . . (vXd)(Fd!l(X1, X2, . , . , X d ) f Fdh(Xl,X2, . . . , Xd)), where d > 0 and X I , X2, . . . , and Xd are distinct individual variables of QC2l"O. It will count as an axiom of QC2l"O in sections 111 and IV below, where
(VF~/A [ ) 3A * will thereby become provable in QC2/" for every instance A * of (V@/l)A.11
I1 Towards sharpening our accounts of truth-functionality and strict truthfunctionality on p. 148, understand by a truth-value function for QC2Im any function from the wffs of QC2/"O to {T, F} ; understand by an instantial function for (the quantifications of) QC2/"Oany function that pairs with each quantification ( V Q 4 of QC2/O0a set of instances of (VQ4 ;and b y a permissible instantial function for QC2/"O understand any instantial function1 for QC2/* such that, for any quantification ( V I / ) A of Q C 2 P a n d any member A*ofl((VV)A), k(VV)A > A * . We shall say that QC21°0has a truth-functional interpretation if there is a family Z of truth-value functions for QC2/" such that:
(1) for any wff A of QC2/m, FA if and only if 4 A ) = T for every member a o f c, (2.1) for any member a of 2 and any two negations -A and -A' of QC2/"O,if &I) = a@'), then &(-A) = a(-A'), 1 1 In the absence of A8, ( V F d / / ) A 3 A is provable in Q C 2 / - for an occasional non-predicative instance A * of (VFd//)A. For example, '(Vf')f(u) 3 ( ( f ( u ) )& (Vf)f(u)) V ( f l u ) & -(Vf)f(u)))' is provable in QC2/'.", even though the wff ' ( f l u )& (Vf)f(u)) V ( f l u )& -(Vnf(u)Y is one level higher than the predicate variable 'f'. But only upon adoption of A8 does (VFd//)A 3 A *become provable in QC2/'." for every non-predicative instance A * of
( VFd/l)A,
IS4
H.Leblanc, G. Weaver
(2.2) for any negation -A of QC2/= and any two members a and a' of C, if .(A) = a'@), then a(-A) = &'(-A), (3.1) for any member a of C and any two conditionals A 3 B and A' 3 E' of Q C ~ / = ,if &(A)= a(A') and a(B) = a(B'), then a(A 3 B ) = a(A' 3 B'), (3.2) for any conditional A 3 B of QC21°0 and any two members a and a' of C, if &(A)= @'(A)and a(B)= a'(B), then or(A 3 B) = a'(A 3 B), and (4) there is a permissible instantiation function I for QC21rn such that:
(4.1) for any member a of C, any two quantifications (VI/)A and (VV')A' of QC2/"" ( V and V' here variables of the same type, level, and degree) such that I(( V W) is at least as large as I((VV')A'), and any functionf from I(( VV)A) onto I((VV)A'), if a(A *) = .(f(A *)) for every member A * of I(( VW), then a(( VW) = a(( VV)A'), (4.2) for any member a of C , any two quantifications ( V V A and (VV')A' of QC2Im (V and V' as in (4.1)) such that I((VV)A) is smaller than I((VV)A'),and any functionffrom I(( VV')A') onto I(( VW),if .(A * ') = a(f(A for every member A * ' of Z((VV')A'), then a(( V V A ) = a(( VV')A'), and * I ) )
(4.3) for any quantification (VVA of QC2Im and any two members (Y and a ' of Z, if &(A*) = a'(A *) for every member A * of N V W)),then a((v VA) = NV W).12
And we shall say that QC2/O0 has a strictly truth-functional interpretation if there is a family L: of truth-value functions for QCZi" which satisfies conditions (1)-(4) plus the following: ( 5 ) for any two members (Y and a' of C,if .(A) = a'@) for every atomic wff A of QCz/OO, then &(A) = a'@) for every wff A of QC2/O0.
1 2 The reader may wish to verify that conditions (2)-(4) are equivalent t o tlic following:
(2') for any member a Jf Z and any negation -A of QC2'w. a(-A ) = T if and only if &(A)= F. (3') for any member a of Z and any conditi(inal A 2 /I of Q c " i " % dA 3 / I ) = T if and only if a(A) = F or a(B)= T, and (4') for any member a of 2: and any quantification (VV)A of QCZ/-, a(( V V)A) = T if and only if44*) = T for every instance A * o f ( V V)A such that t ( V V)A 3 A *.
Ramified Theory of Types
15s
Now consider the family &consisting of every truth-value function 01 for
QC21W such that:
(i) for any negation -A of QC21W, a(-A) = T if and only if a(A) = F, (ii) for any conditional A 3 B of QCZl", a(A 3 B) = T if and only if &(A) = F or a(B) = T. (iii) for any quantification of QC2/W40fthe sort (VX)A, a((VX)A) = T if and only if &(A*) = T for every instance A * of ( V q A , and
(iv) for any quantification of Q C 2 P of the sort ( V F d h ) A , a((VFd/l)A) = T if and only if &(A*) = T for every predicative instance A * of (\clFd/l)A. It is clear that C1 - the intended or principal interpretation of QC2/- satisfies conditions (2)-(4). Towards showing that C1 satisfies condition (1) as well, let S be any finite set of wffs of QC2/" that is syntactically consistent in QC2/O'. Following the instructions in Leblanc [1969j , 1 3 one can extend S into a set SW such that: (iw) for any negation -A of QC2/O0,-A belongs to Sm if and only if A does not belong to Sw, (ii-) for any conditional A 3 B of QC2/w, A 3 B belongs to Sw if and only if A does not belong to Sm o r B does, (iii,) for any quantification of QC21Wof the sort (VX)A, if (VX)A does not belong to Sm, then there is an individual parameter X of QC2lw such that A ( X / X ) does not belong to SW, and (iv-) for any quantification of QC2/- of the sort (VFdb)A, (VFd/l)A does not belong to Sm, then there is a predicate parameter Fdll of QC21rn of the same degree and level as Fdll such that A ( F d h / F d h ) does not belong to Srn either. But, if a quantification ( V X ) A of QC2/OCbelongs to SOO,then in view of A6 so does every instance of (VX)A; and, if a quantification ( V F d h ) A of QC2/- belongs to SW,then in view of A7 so does every predicative instance of (VFdl1)A. Hence each member of SOO- and, by rebound, of S - is sure to evaluate to T under the member a of Z1 that assigns T to every wff of Q C 2 P in Soo and F to everyone not in SW. Hence, in particular, if { - A } is syntactically consistent in QC2/w, then a(A) = F, this for any wff A of QC2/-. Hence any wff of QC2jW evaluating to T under all members of Z1 is a theorem of Q C 2 P But any theorem of QC2/= can be shown to evaluate to T under all members of X i . Hence El meets all four of conditions (1)-(4). Hence: Theorem 1. In the absence of the Axiom of Reducibility, QC21m has a truth-functional interpretation. 13
The instructions stern largely from Henkin [ 1949)
156
H. Leblanc, G. Weaver
Proof that C1 meets condition (5) as well (and, hence, that in the absence of the Axiom of Reducibility QC2/- has a strictly truth-functional interpretation) is by an induction on what we call the rank r(A) of a wff A of QC2IW, and proofs of the basis and the inductive step of tlus induction are by subsidiary inductions on the length G ( A ) of A . Asusual, { ( A ) = 1 whenA is atornic;,l(-A)=d(A)+l;d(A> B ) = G(A)td(B)tl; andd((VV)A) = G(A(P/V))+l,where P is the alphabetically earliest parameter of QC2/- of the same type, level, and degree as V . Definition of the rank of A is more elaborate. By a term'naring sequence n 2 , . . , ni, . . . ), o f natural numbers understand any sequence of the sort hl, where (1) for each 1 from 1 through j , nl > 0 and (2) for each 1 from j+l on, nz = 0. 011, n 2 , . . . , nj, . . .) and h i , n 2 , . . . , ni', . . .) being terminating sequences ofnaturalnumbers,take(nl,n2,... ,n,,. . . )toalphabeticallyprecede , , ( n l ,n2, . . . ,n)', , . .) if there is an 1 (1 2 1) such that nm = nh for any m from 1t1 on, but nl < ni. Take a terminating sequence of natural numbers to be of rank I if none alphabetically precedes it, take the sequence to be of rank 2 if exactly one alphabetically precedes it, etc. Take the associated terminating sequence of a wff A of Q C 2 P to be ( n l ,n2, . . , ,ni, , . .), where - for each 1 from 1 on - n is the number of times 'V' occurs in A flanked on the right by a predicate variable of level 1. And take the rank r(A) of A to be that of its associated terminating sequence. It is easily verified that: (a) r(A) = 1 if and only if A contains no predicate variable, (b) r(-A) = r(A),(c) r(A(B/Fd/l(X1, X2,. . . , Xd)))so long as B is of level 1 or less. ,
I
t
Theorem 2. Let aand a' be members of XI such that &(A) = &'@)for every atomic wff A of QG'f-. Then a(A) = &'(A)for every wff A of QC?/*. hoof by mathematical induction on the rank r(A) of an arbitrary wffA of QC2IW. Basis: r(A) = 1, in which case A contains n o predicate variable. Proof that a(A) = a'@) here is by a first induction on the length d ( A ) of A . (i) Suppose / ( A ) = 1, in which case A is atomic. Then a(A) = a'@) by the hypothesis on a and a'. (ii) Suppose L ( A ) > 1, and A is of the sort -B. Since d(B) < d(-B), a(B)= a'@) by the hypothesis of the first induction on length. Hence a(++) = a'(-B). Or suppose A is of the sort B 3 C. Since G(B)< G(B 3 C ) and C (C) < d(B 3 C), a(B) = a'(B) and a(C) = a'(Q by the hypothesis of the first induction on length. Hence a(B 3 C ) = a'@ 3 C). Or suppose A is of the sort (VX)B, and let X be an arbitrary individual parameter of QC2/m. Since
Ramified Theory of Types
157
G (B(X/X))< G ((VX)B), a(B(X/X)) = a'(B(X/X)) by the hypothesis of the first induction on length. Hence a((VX)B)= a'((VX)B).
Inductive Step: r(A) > 1. Proof that .(A) = a'@) here is by a second induction (this one without Basis) on the length! ( A ) of A . (i) Suppose A is of the sort -B. Since r(B) = r(-B) and G (B) < G (-B), a(B) = a'(B) by the hypothesis of the second induction on length. Hence a(-@ = a'(-B). (ii) Suppose A is of the sort B 3 C. Since r(B) < r(B 3 C) and " (B) < t ( B 3 C), a(B) = a'(B) by the hypothesis of the second induction on length when 4 B ) = r(B 3 C), otherwise by the hypothesis of the induction on rank. Similarly, a(C) = a'(C). Hence a(B 3 C) = a'(B 3 C). (iii) Suppose A is of the sort ( V D . Since r(B(X/X)) = r((VX)B) and t (B(X/X)) < G ((VX)B), a(B(X/X)) = a'(B(X/X)) by the hypothesis of the second induction on length. Hence a((VX)B) = a'((VX)B). (iv) Suppose A is of the sort (V'Fd/l)B, let C be an arbitrary wff of QC2/O0of level I or less, and - in case d > 0 let X I ,X2, . . . , and X d be distinct individual parameters of QC2/=. Since r l B ( c / F / l ( X ~X2, , . . . ,Xd)))< r((VFd/z)B),(u(B(cIFd/l(x~ J 2 . . . . , Xd)))= aV(B(C/Fd/l(X1,X2, . . . ,Xd)))by the hypothesis of the induction on rank. Hence a(( tlFd/l)B) = a'((VN/l)B). Hence :
Theorem 3. In the absence of the Axiom of Reducibility, QC21w has a strictly truth-functional interpretation. I1I Now count as an axiom,of Q C 2 / - any wff of QC21m of any of the seven sorts A1 -A7 on p. 152, of the sort A8 on p. 153 (= the Axiom of Reducibility), or o f the sort ( V W , where - for any parameter P of QC2Im of the same type, level, and degree as Vthat is foreign to ( V V ) - A ( P / V ) is an axiom, of QC21°0. Where A is a wff and S a set of wffs of QC2/-, we shall understand by a proofr in QC21m of A from S any finite column of wffs of QC2/"" such that: (1) the last entry in the column is A and (2) every entry in the column belongs to S, is an axiom, of QC2/"", or is the ponential of two earlier wffs in the column; we shall say that A is provable, in QC2:- from S if there is in QC2/- a proof of A from S; we shall say that S is synratically consistent, in QC2Im if there is no wff of QCIl- such that both it and its negation are provable, in QC21°0 from S ;and we shall say that A is a theorem, of QC21°0 ( krA, for short) if there is in QC21m a proof, of A from S.
1.58
H.Leblanc. G. Weaver
We first obtain a generalization of A8 with ‘Fdll’ in place of ‘Fdl”, and ‘Fdh’ (Theorem 5 ) ;and, given this result, we obtain the counterpart in QC2/“Oof the familiar Axiom o f Comprehension (Theorem 6 ) . Proof that, for any quantification of QC2/w of the sort (VFdll)A and any instance A * of (VFd/l)A, ‘@/l’’ in place of
( V F d / l ) A 3 A*
is a theorem, of QC2lm, can then be retrieved from Henkin [ 1 9 5 3 ] , a task we leave to the reader.14 In the course of proving Theorems 5 and 6 we use some thirteen lemmas, listed without proof under the common heading Lemma 4. These, as the reader may verify, do not call for A8; so throughout we write ‘t-’ in place of ‘tr’. Lemma 4. (a) Zf I- A and t A 3 B, then k B.
(b) FA 3 ((B 3 B ) = A ) . (c) 1-A 3 (-(B 3 B ) A ) . (d)If / - A >Band I--A 3 B , t h e n FB.
-
(e) I- (VXl)(VX2) . . @&)(A = B) 3 ((VX,)(VX2) . . . (VXd)(A C ) 3 ( V X , ) ( V X 2 ) . . . ( V X d ) ( C r B)). (0 k ( v x l ) ( V x 2 ) . * ( v x d ) ( A B) 3 ( ( v X i ) ( V x 2 ) . . . (vXd)(B c) 3 ( v X l ) ( v x 2 ) . . ( v X d ) ( A c)). (g) I f I-A(P/V) 3 B(P/V), then I- ( 3 VpI 3 ( 3 V)B, where P is foreign to (3 V)A and (3 V)B.
(h) I f I-A(P/V)3 B, then I- ( 3 V)A ( 3 V)A and To B.
3
B, where P is foreign to
(i) I f f - A3(B(P/V)>C(P/V)),then I - A 3 ( ( 3 V ) B 3 ( 3 V ) C ) , where P is foreign to A , ( 3 V)B, and ( 3 V)C. (j) I f t ( g V ) ( A >@,then I-(VV)A 3 B . 1 5
(k)Ifl-(3yXA 3 B),th en l - A 3 ( 3 V ) B . 1 6
14
A detailcd proof of t h e result will appcar
15
With (VV)A 3 8 presumcd herc
16
W i t h A >(3V)Eprcsunied h e r e i o h e a wlTofQC?:m. C’issurc tohcforcign t o A .
to
in
Lehlunc I19721.
hc a w f f of QC?lm, V is surc
to
he folcign
to
R
Ramified Theory of Types
159
H. Leblanc. G. Weaver
I60
Hence by Lemma 4Q) and the hypothesis on @ / 1
I-I(VGd/l)(VFd/l)(3X1)(VX2) * . . ( V X d ) ( G d / l ( X l , X 2 , .. . , X d ) s Fdl'(X1 , X 2 , . . . ,&)) 3 ( 3 F d / ' ) ( v x 1 ) ( V X 2 ) .. . ( v x d ) (FCIIl(X1, X 2 , . . . ,X d ) fF d / l ' ( X , , X 2 , . . . ,X d ) ) . Hence by Lemmas 4(m) and 4(a)
I- , ( 3 F d / l ) ( V X l ) ( V X 2 ) .. . ( V X d ) ( F d / l ( X , , X 2 , .. . , X d ) Fd/"(Xl,X2, .
. . ,X d ) ) .
Theorem 6 . I- I ( 3 F d / l ) ( V X l ) ( V X 2 ) . . . ( V X d ) ( F d / l ( X l ,X 2 , . . . , X d ) A ) , where F d h is foreign to A and in case d > 0 - X I , X 2 , . . . , and Xd are distinct individual variables of QC2iW. ~
Proof: Let Fd/l be a predicate parameter of QC2;"O of degree d and level 1 foreign t o A . Case 1: d
= 0.
I- ,A
By Lemmas 4(b) and 4(c)
3 ((@/I
3 Fdh) = A )
and
1I-A
3 (-(Fd/' 3 F d h )
A).
Hence by Lemma 4(1)
I- (3Fdh)(A 3 (Fdh = A ) ) and
I- r( 3 Fd/l)(-A 3 (Fd/l = A ) ) . Hence by Lemma 4(k) and the hypothesis on Fd/l
I-,A
3
(3FCI/')(Fd/' = A )
and
kr-A 3 ( 3 F d / l ) ( F d / l Z A ) . Hence by Lemma 4(d)
I-r ( 3 ~ d / l ) ( ~ d=/Al ) . Case 2: d > 0.1' being the level of A , let Cd/l' be a predicate parameter of Q C 2 / - of degree d and level I' foreign t o A and distinct from F d h , atid let be a predicate variable of QC21m of degree d and level I' foreign to A .
Ramified Theory of Types
Theorem 7.
161
k r (VFd11)A 3 A*, for any instance A * of (VFd/l)A.
To show that in the presence of A 8 QC21m still has a truth-functional interpretation, read 'I-rA' for 'I-A'in condition (1) o n p . 153, drop the (now idle) qualifier 'permissible' in condition (4) on p. 154, and define C2 as C1 was defined on p. 155, but with (iv) now amended t o read: (iv') for any quantification of Q C 2 / m o f the sort (VFd/l)A, a((VFd/l)A) = T if and only if a(A*) = T for every instance A * of (VFd/l)A. Supposing a finite set S of wffs of Q C 2 / m t o be syntactically consistent, in QC2lm, one can again extend S into a set Sm boasting features (im)- (iv-) . But, if a quantification (VX)A of QC2/03 belongs to Sm, then on pp.
162
H. Leblanc, G. Weaver
in view again of A6 so does every instance of (VX)A; and if a quantification ( V f l / l ) A of QC21m, then in view of Theorem 7 so does every instance of ( V f l / l ) A . Hence each member o f S is sure to evaluate to T under the member a of Z2 that assigns T to every wff of QC2Im in S m and F to everyone not in SOO.Hence any wff of QC2/m evaluating to T under all the members of C2 is a theorem, of QC2lm. But any theorem, of QC2/OO can be shown to evaluate to T under all members of & . Hence C2 meets all four conditions (1)--(4). Hence :
Theorem 8. In the presence (as well as in the absence) of the Axiom of Reducibility, QC-'lm has a truth-functional interpretation. IV Proof that 2 2 - now the principal interpretation of QC21m fails to meet condition (5) on p. 157 calls for some model-theoretic notions (one of them adapted to serve the purpose at hand). Take a model to be any non-empty set. Where D is a domain, take a D-interpretation of the parameters of QC2 im to be any function f D from the predicate parameters of QC2/m of degree 0 to {T, F}, from the individual parameters of QC2lm to D, and from the predicate parameters of QC2/m of non-zero degree d to the power set of @(=D x D x . . . x D).17 d times Where D is a domain, fD a D-interpretation of the parameters of QC2lrn , and P a parameter of QC2Im, take aP-variant of ID to be any D-interpretation of the parameters of QC2i" that agrees with I D on all the parameters of QC21m other than P (and possibly on P as well). And, where A is a wff of QC21m, D a domain, and I D a D-interpretation of the parameters of QCZi", take A to be true on I D if: ~
(a) in case A is a predicate parameter of QC2/m of degree 0, ID(A)= T, (b) in case A is of the sort Fdh(X1 ,X2, . . . , X d ) for some non-zero d , ( I D ( x ~ )1, 0 ( ~ 2 ) ,. . . ,ID(&)) belongs to 10 (Fdll),
(c) in caseA is of the sort -B, B is not true on I D , (d) in case A is of the sort B 3 C, B is not true on ID or Cis, and 1 ' In model-tlicorctic accounts 01' O('2 '-. / , ) ( F d / / ) . where d > 0. would be required to be subset of nl level 1. But tlic r c s ~ r i c t i t ~would n p h y 1 1 0 colt Iicrc. a n d 15 i i i l c i i t i o n a l l y
dropped.
3
Ramified Theory of Types
163
(e) in case A is of the sort (VT/)B, B(P/y) (P here the alphabetically earliest parameter of Q C 2 / - of the same type, level, and degree as V that is foreign to (V u)B) is true on every P-variant Of ID. Crucial among D-interpretations are those - to be called Henkin D-interpretations - where each member of the domain D is assigned to an individual parameter of QC21m (more formally, where for each member d of the domain D there is an individual parameter X of QC2/- such that I D Q = d). Extending a familiar result for QCl , one can indeed show that: Theorem 9. Let D be a domain, and ID be a Henkin D-interpretation of the parameters of QC-'l*. Then a quantification ( V X ) A of QC2/" is true on ID ifand only if every instance of ( V X ) A is true on ID . I * And crucial among Henkin D-interpretations are those where the domain D is finite. Take indeed an arbitrary quantification (VFdf1)A of QC2/m; let Fdll be the alphabetically earliest predicate parameter of Q C 2 / - of the same level and degree as Fdll that is foreign to (VFdfl)A;in case d > 0, let X I , X 2 , . . . ,and x d be in alphabetic order the first d individual parameters of Q C 2 P that are foreign t o (VFdlrJA;let D be a finite domain; let I D be a Henkin D-interpretation of the parameters of QC21m; and let Ib be an Fdh. variant of 10.There exists then a wff B of Q C 2 / - , to be known as the wffassociate of I D , such that A(Fd/l/Fd/l)is true on Ib if and only if A ( B / P / ! ( X I , ~ 2. .,. ,&)) is true on I D ; and, as a result, ( v F ~ / ' ) Apans out true on ID if and only if all its instances do. B is as in the following table, where dl , d2, . . . , and d,, ( n > 0) serve as the various members of D, Yi is for each i from 1 through n the alphabetically earliest individual parameter of Q C 2 / - to which member di of D is assigned in I D , and I1 = 12 is short for (Vf)(f(ll) 3 f(I2)): And proof that A(Fdll/Fdf') is true on Ib if and only if A(B/@/I(X1 , X 2 , . . . , &)) is true on ID is by mathematical induction on the length of A . Now suppose that (VFdf1)A is true on I D . Then A ( B / F d f l ( X l ,X 2 , . . . , x d ) ) is sure to be true on ID for any wff B of QC21m and - in case d > 0 -any distinct individual parameters X i , X2, . . , and X&of QC*/m. Suppose, on the other hand, that (VFdl1)A is not true on ID. Then there is an Fdfrvarient Ib of ID such that A(Fd/l/Fd/l)is not true on Ib, and hence by the
.
1s A detailed proof of Theorem 9 will appear it1 Lehlanc [ 1 9 7 1 ] . TheD-interpretations under consideration are called Ifc,rkirr L)-irrlcr/~rc,tulir,rlsbecause of t h e rolc their counterparts for QCl play in Henkin [ 19491.
I64
H. Leblanc, G. Weaver
etc.
foregoing result there is a wff B of QC2/" (to wit: the wff-associate of I D ) , and - in case d > 0 -- there are distinct individual parameters X 1 , X 2 , . . . , and X d of @?la", such that A(B/@h(X,, X2,. ,Xd)) is not true on ID itself. Hence:
..
Theorem 10. Let D be a finite domain, and ID be a Henkin D-interpretation of the parameters of QC2/". Then a quantification (tlFd/l)A of Qbl" is true on I D if and only if every instance of ( V P h ) A is true on ID. Proof of our last theorem is within reach. Indeed, let D and D' respectively be { 1) and {I, 2) ;let ID assign T to every predicate parameter of QC2/0° of degree 0 (1 t o every individual parameter of QC21m), and od to every predicate parameter of QC2/0° of non-zero degree d; let ID' assign T to every predicate parameter of QC2/" of degree 0, 1 to 'x', 2 to every other individual
Ramified Theory of Types
165
parameter of QC2/", and D'd to every predicate parameter of Q C 2 P of nonzero degree d; let a be the truth-value function for QC2l0" such that, for every wff A of QC21°0, a(A) = T if and only if A is true on ID;and let a' be the one such that, for any such A , &'(A)= T if and only if A is true on ID'. It is easily verified with the aid of Theorems 9 and 10 that a and 01' both belong to the family Z2 of p. 161, that a and 01' agree on all atomic wffs of QC2/m (indeed, on all wffs of QC2/" of rank I), and yet that 01 and a' disagree on '( 3f)( 3x)( 3y)cf(x) & -f(y))',which evaluates to F on 01 but to T on a'. (For proof that the wff in question evaluates to F on a,note that '(3 x)( 3y)(f(x) & -f(y))' cannot be true on either of the two f-variants of ID; for proof that it evaluates to T on a',note that '(3x)( 3y)(f(x) & -f(y))' is true on thef-variant of ID' that assigns (1) to 'f'.)19 Hence C2 does not meet condition (5) on p. 154. Hence:
.Theorem 11.
In the presence of the Axiom of Reducibility, QCZ/" has a truth-functions[ interpretation which is not strictly truth-functional.2 0 V
Church [1956] sketches a semantic account of QC21°" with two truthvalues per level.21 The account uses models. To mimic it here, acknowledge with Church two truth-values TZ and FI for each level 1. Understand by a system of truth-value functions for QC2/" any sequence of the sort (a1, a2,a3,. . ), where - for each I from 1 on - al is a function from the wffs of Q C 2 P o f level 1 to (TI, FI} such that: ,
(i) al(-A) = TI if and only if q ( A ) = Fl, (ii) q ( A 3 B) = TI if and only if q ( A ) = n' or a@) = T k ( j here the
level of A and k that of B),
19
The example comes from Leblanc and Mcyer [ 19701.
The above proof of Theorem I 1 borrows from Leblanc and Mcyer [ 1970], where Q C I is shown t o have a truth-functional interprctation which is not strictly truth-functional. Another proof of Theorem 1 1 can be had via that result in Leblanc and Meyer [ 19701. Indeed, with Theorem 7 31 hand, one can construct for each wff A of Q C Z P a wff E of QCI such that A is provable, in QCZim if and only i f R IS provable in QC!. B is gotten froni A by systematically lowering the level of each predicate variable and parameter in A t o I , and then writing f:d and Fd everywhere for F d / l and Fdll,respectively. See the article Semantic Deviations in this volume. 20
? I See pp, 347-348, footnote 577
166
H, Leblanc, G. Weaver
(iii) a/((V X ) A ) = TI if and only if cq(A(X/X)) = TI for every individual parameter X of QC2/=, and
(iv) a [ ( ( V F d l i ) A ) = TI (j< I ) if and only if a&l(B/Fdh(X1.X2, . . . ,Xd)))= Tk (k here the level ofA(B/@h(X1.X2,. . . ,x d ) ) ) for every wff B of QCZ/" of level j or less, and - in case d > 0 - any distinct individual
parameters XI, X 2 . .
. . , and Xd of QC2/OO.
And take a wff A of QC2/m of level I to be true on a system (a1,a2, a3, . . . ) of truth-value functions for QC2Irn if q ( A ) = TI. It can be verified that: (1) t A if and only if A is true on every system of truth-value functions
for QC2/", and
(2) two systems of truth-value functions for QC2/" that agree on all the atomic wffs of QC2/OO are sure t o agree on all the non-atomic ones. Indeed, the induction that saw Theorem 2 through will - mutatis mutandis see (2) through. Suppose, however, that B in condition (iv) is allowed t o be of any level whatever. It then follows that: (3) I- ,A if and only if A is true on every system of truth-value functions for QC2/m,
but ( 2 ) fails, as a straightforward generalization of the argument in 1V will show. So analogues of Theorems 1 , 3 , 8 , and 1 1 hold for Church's account. References Beth, E. W., 1959, The Foundations of Mathematics (North-Holland, Amsterdam). Church, A., 1956, Introduction to Mathematical Logic, Volume 1 (Princeton). Dunn, M. and Belnap, N. D., 1968, The Substitution Interpretation of the Quantifiers, N o ~ sV , O ~ .2, pp. 177-185. Fitch, F. B . , 1948, Intuitionistic Modal Logic with Quantifiers, Portuguliae Mathemntica, VOI. 7 , p p . 113-118. Henkin, L., 1949, The Completeness of the First-order Functional Calculus, The Journalof Symbolic Logic, vol. 14, pp. 159-166. Henkin, L., 1953, Banishing the Rule of Substitution for Functional Variables, The Journal of Symbolic Logic, vol. 18, pp. 201-208. Leblanc, H., 1968, A Simplified Account of Validity and Implication for Quantificational Logic, The Journalof Symbolic Logic, vol. 3 3 , pp. 231 -235.
167
Leblanc, H., 1969a, A Simplified Strong Completeness Proof for Q C = , A k t e n des XIV. Internationalen Kongresses f i r Philosophie, Volume III, (Vienna) pp. 83-96. Leblanc, H., 1969, Three Generalizations of a Theorem of Beth’s, Logique et Analyse. VOI. 12, pp. 205-220. Leblanc, H., 1972, Truth-Value Semantics (Amsterdam) forthcoming. Leblanc, H., and Meyer, R. K.,1970, Truth-value Semantics for the Theory of Types, in: Philosophical Problems in Logic: Some Recent Developments, ed. K. Larnbert (Reidel, Dordrecht), pp. 77-101. Schutte, K., 1960, Syntactical and Semahtical Properties of Simple Type Theory, The Journal of Symbolic Logic, vol. 25, pp. 305-326. Schutte, K., 1962, Lecture Notes in Mathematical Logic, Vol. I (The Pennsylvania State University) .
George WEAVER B v n Mawr College
Take a calculus C with, say. ‘-’, ‘I)’ and , ‘V’ as its only logical operators. We shall say that C has a truth-functional interpretation if there is a family C of functions from the wffs (non-atomic as well as atomic) of C to {T,FI such that, informally speaking: (a) the theorems of C and the wffs of C evaluating to T under all members of Z coincide; and (b) the truth-values that the non-atomic wffs of C assume under members of Z hinge exclusively on the truth-values of their components or - - in
the case of quantifications - those of their substitution instances.
And we shall say that C has a strictly truth-functional interpretation if the family Z in question also obeys the following condition: (c) members of Z that agree on the atomic wffs of C agree on the nonatomic ones as well.’ The sentential calculus (SC, for short) has a strictly truth-functional interpretation; and - as follows from results of Beth, Schutte, and others2 - so does the first-order quantificational calculus (QC’ ). By contrast, the principal interpretation of QCP (the second-order quantificational calculus), though truth-functional, is not strictly truth-functional, a result due to Leblanc and Meyer [I9701? Passing to QC21m (the ramified second-order quantificational
1 For a formal account of the matter see section 11, where (b) is broken into three separate conditions: one dealing with negations, one with conditionals. and one with quantifications.
See Beth [ 19591 pp. 263-267, Schutte [ l!)67] pp. 5-14, Dunri a n d Belnap [ 19681, and Leblanc [ 19681 . Q C l , the first-order quantificational calculus with equality, also has a strictly truth-functional interpretation; see Leblanc [ 1969al on this matter.
2
That QCZ has a truth-functional interpretation follows in effect from results in Schutte [1960].
Ramified Theory of Types
149
calculus), we establish that:
(1) without Russell's Axiom of Reducibility, QC21m fares like SC and QC' (Theorems 1 and 3), but ( 2 ) upon adoption of that axiom, QC2im fares like QC2 (Theorems 8 and 11). Our results automatically extend to QC"1"
(the ramified theory of types):
(1 ') without Russell's axiom, QC"/" pretation, but
has a strictly truth-functional inter-
(2') upon adoption of the axiom, its principal interpretation - though still truth-functional - is no longer strictly truth-functional!
I The primitive signs of QC2/w are to be following parameters and variables:
'3', Y', '(),')),',',plus the
'5')
(i) N o individual parameters (among them 'u'), (ii) No individual variables (among them 'x'and 'y'), (iii) for each d from 0 on and each I from 1 on, No predicate parameters of degree d and level I (among them 'f', a predicate parameter of degree and level l), and
(iv) for each d from 0 on and each 1 from 1 on, Ho predicate variables of degree d and level 1 (among them I f ' , a predicate variable of degree and level l).5 The parameters in (i) and those in (iii) will be presumed to come in a definite order, known as their alphabetic order. The individual parameters and variables of QC2/" will be said to be of type 0; the predicate parameters and variables of QC2/" of degree 0, better known as sentence parameters and
Two questions remain open: I n the absence of Russcll's axiom does QCm:m (and, in particular, QC2:-) have any truth-functional interpretation that is not strictly truth-functional; and (and, in particular, QC?lm) have a truth-functional in the presence of that axiom does QPlm interpretation which unlike its principal one - is strictly truth-functional? Answers to these questions would shed a great deal of light on the ramified theory o f types.
4
~
5 Our Variables are what the literature would call
would Call free variables.
bound variables, and our parameters what it
I50
H. Leblanc, G. Weaver
variables, will be said to be of type -I; and the remaining predicate parameters and variables of QC2/" will be said to be of type 1. And, as usual, the formulas of QC2/" are to be all finite (but non-empty) strings of primitive signs of Q C ~ / - . Because we use both parameters and variables, our arsenal of syntactic variables will be rather large: 'P'to refer to parameters in general, 'V'to variables in general, ' X ' and 'Y'to individual parameters, 'X' to individual variables, ' I to 'individual signs (i.e., to individual parameters and variables), 'Fd/'', 'Gdll', and 'Hdl'' to predicate parameters of degree d and level I, 'Fdll' and -Gdll' to predicate variables of degreed and level I, and 'A', 'B', and 'C' to formulas. Three substitution conventions will be needed as we go along. Let A be a formula of QCU", V be a variable of QCU", and P be a parameter of QC2/" of the same type, same level, and (when V is a predicate variable) same degree as V: Under C I , the first of these conventions, (A)( V/P)will be the result of putting Veverywhere in A for P , and ( A ) ( P/ V ) that of putting P everywhere in A for V . The atomic well-fomed formulas (atomic wffs, for short) o f QC2/" are to be all formulas of QC2/" of the sort Fdh ( X I ,X 2 , . . . , X d h Where d 2 0.6 The well-formed formulas (wffs, for short) of QC2/" will be all atomic wffs of QC"", plus all formulas of QC2/" of the following three sorts: (i) -A, where A is a wff of QC2/", (ii) ( A 3 B), where A and B are wffs of QC2/", and (iii) (V V)A , where - for any parameter P of QC2/" of the same type, level, and degree as V - (A)(P/V)is a wff of QC2Iw.7 And the quasi-wffs(qwffs,for short) of QC2IW will be all formulas of QC2/" of the sort (A)( V/P),where A is a wff or qwff of QC21". P a parameter of QC2/" which occurs in A , and V a variable of QC2/" which is of the same type, level, and degree as P but does not occur in A . As usual, A counts as the immediate component of -A and of (VV)A (this for any variable V of QC'/"); and A and B count as the two immediate components of (A 3 R ) . And the components of a wff or qwff of QC2ioo are the wfi' or qwff itself, its immediate components, and the components of all its components.
6 When
d = 0, F d / I ( X , , X z , . . . , Xd) is t o be understood as just Fd//.
Under the present account of things formulas in which identical quantifiers overlap d o not count as well-formed. '&, 'V', 'E',and ' 3 ' , which we come to employ besides '-', *3', and 'Y', are presumed t o be defined in the usual manner. 7
Ramified Theory of n p e s
I51
For convenience's sake we assign levels to the wffs (as well as to the predicate parameters and variables) of QC2/". A being a wff of QC2/", let Ip(A) be 0 if no predicate parameter of QC2/= occurs in A ; let [,(A) be 1 if at least one predicate parameter of QCYooof level 1 occurs in A but none of higher level than 1 does; let Zp(A) be 2 if at least one predicate parameter of QC2/w of level 2 occurs in A but none of higher level than 2 does; etc. And let Zv(A)be similarly defined, but with the word 'variable' doing duty throughout for 'parameter'. We shall'then take the level Z(A) of A to be max(lp(A >, I+1). Now for our remaining two substitution conventions:
C2: Let A be a wff or qwff of QC2Irn;let X1,X2, . . . , and x d (d 2 0)be distinct individual parameters of QC*/=; and let I1,12, . . . ,and Id be (not necessarily distinct) individual signs of QC2/"".By (A)(Il,I2, . . . , Id/ X 1 , X 2 , . . . , x d ) we shall understand the result of simultaneously putting I1 everywhere in A for X1,12 for X2,. . . ,and Id for Xd.8 C3: Let A be a wff or qwff of QC2lw; let F d h be a predicate variable of QC2/" of degree d and level 1; in case d > 0, let X1,X2, . . . , and x d be distinct individual parameters of QC2/=; let B be a wff of QC2/"; and let F d 1 ' ( I l I J I 2 , ~,Id~),Fd1'(I12,I22,... -.. ,Id 2 ) , . . . ,andFdll(Il ,12k,. . . , I d k ) (k 2 0) be all the atomic components of A which contain FdTl. Case 1 :k = 0. Then ( A ) ( ~ / F d / ~ ( X2,. X i , .. ,Xd)) is to be A. Case 2:k > 0. (a) If F d h occurs in a component of A of the sort (VW'and B has a component of the sort (VOH (V the same variable in both cases), (A)(B/Fd(Xl,X2, . . . ,&)) is to be (A)(Fdll/Fd/l), where Fdll is in alphabetic order the first predicate parameter of QC2/= of degree d and level 1. (b) Otherwise, (A)@/ FdIl(Xl,X2, . . . ,x d ) ) is to be the result of putting (B)(Ilr12i, . . . , Idi/ x1 .X2, . . . , X d ) everywhere in A for Fdh(Ili, Izi, . . . , Idi), this for each i from 1 through k.9 The (substitution) instances of the quantifications of QC2/= will play a considerable role throughout. With C1 and C3 on hand, they can be identified as
8 Siinultaneous substitutions can of course be broken into sequences of single substitutions. See Church 119561, p. 8 2 , on this point.
9 It is easily verified that if (VFdl/)A and B are wffs of QC*lm, and in case d > 0 - X I , X z , . . . , and Xd are distinct individual parameters of QC2Im, then (A)(B/Fd(X, .&,. . . , X d ) ) is sure to be a wff of QC*/-. From this point on we drop a number of inessential parentheses, thus writing 'A 3 B'in lieu of '(A 3 B)', 'A(P/V)' in lieu of '(A)(P/V)', etc. ~
152
H. Leblanc, G. Weaver
follows: (i) the instances of a quantification of the sort (VX)A will be all wffs ofQCz/" of the sort A(X/X); (ii) the instances of a quantification of the sort (VFdIl)A will be all wffs of QC2/" of the sort A(B/Fd/l(Xl,X2, . . . ,&)), where B is any wff of QC2/"" and - in case d > 0 - X I , X2, . . . , and Xd are distinct individual parameters of QC2/w; and (iii) the predicutive instances of (VFdl1)A will be all wffs of QC2/" of the sort A(B/Fdh(X, , X 2 , . . . ,&)), where B is wff of QC21w of level 1 or less (and - in case d > 0 - X1 ,X2, . . . , and X d are distinct individual parameters of QC2/"). Turning at last to the deductive apparatus of QCZI", we take the axioms of QC2/O0 to be all wffs of QC2/"" of any of the following seven sorts:
Al. A > @ > A ) , A2. (A 3 (B 3 C)) 3 ((A 3 B ) 3 (A 3 C)), A3. (-A 3 -B) 3 (B 3 A ) , A4. (VV)(A 3 B ) 3 ((VV)A 3 ( V V B ) , A5. A 3 ( V W , A6. ( V X ) A 3 A *, where A * is an instance of (VX)A,
A7. (VFdll)A 3 A *, where A * is a predicative instance of (VF'd/l)A, plus all wffs of QC2/"" of the sort (VV)A, where - for any parameter P of QC2lrn of the same type, level, and degree as Y that is foreign to (VQ.4 - A(P/V) is an axiom of QC2/"".10 We understand by the ponential of two wffs A and A 3 B of QC"" the wff B. Where A is a wff and S a set of wffs of QC2/", we understand by a proof in QC2/" of A from S any column 81
B2
BP of wffs of QC2'"" such that Bp is A and, for each i from 1 through p , Bi belongs to S, is an axiom of QC2/", or is the ponential of two 10 The trick of counting ( V V ) A an axiom if for suitable P A ( P / V ) is an axiom, stems from Fitch [ 19481. Note in connection with AS that with A 3 (V V)A - and, hence, A presumed here to be a wff o f Q C I P , V is sure to be foreign to A . ~
Ramified Theory of Types
I53
earlier wffs in the column; we say that A is provable in QC2/- from S if there is in QC2i"O a proof of A from S; we say that S is syntacti-
cally consistent in QC2i" if there is no wff of QC21" such that both it and its negation are provable in QC2/"0 from S; and we say that A is a theorem of QC2lm ( FA, for short) if there is in QC2/"O a proof of A from 8. Russell's Axiom of Reducibility, as tailored to suit QC2/"O, runs: A8. (gFd/1)(vX,)(vX2). . . (vXd)(Fd!l(X1, X2, . , . , X d ) f Fdh(Xl,X2, . . . , Xd)), where d > 0 and X I , X2, . . . , and Xd are distinct individual variables of QC2l"O. It will count as an axiom of QC2l"O in sections 111 and IV below, where
(VF~/A [ ) 3A * will thereby become provable in QC2/" for every instance A * of (V@/l)A.11
I1 Towards sharpening our accounts of truth-functionality and strict truthfunctionality on p. 148, understand by a truth-value function for QC2Im any function from the wffs of QC2/"O to {T, F} ; understand by an instantial function for (the quantifications of) QC2/"Oany function that pairs with each quantification ( V Q 4 of QC2/O0a set of instances of (VQ4 ;and b y a permissible instantial function for QC2/"O understand any instantial function1 for QC2/* such that, for any quantification ( V I / ) A of Q C 2 P a n d any member A*ofl((VV)A), k(VV)A > A * . We shall say that QC21°0has a truth-functional interpretation if there is a family Z of truth-value functions for QC2/" such that:
(1) for any wff A of QC2/m, FA if and only if 4 A ) = T for every member a o f c, (2.1) for any member a of 2 and any two negations -A and -A' of QC2/"O,if &I) = a@'), then &(-A) = a(-A'), 1 1 In the absence of A8, ( V F d / / ) A 3 A is provable in Q C 2 / - for an occasional non-predicative instance A * of (VFd//)A. For example, '(Vf')f(u) 3 ( ( f ( u ) )& (Vf)f(u)) V ( f l u ) & -(Vf)f(u)))' is provable in QC2/'.", even though the wff ' ( f l u )& (Vf)f(u)) V ( f l u )& -(Vnf(u)Y is one level higher than the predicate variable 'f'. But only upon adoption of A8 does (VFd//)A 3 A *become provable in QC2/'." for every non-predicative instance A * of
( VFd/l)A,
IS4
H.Leblanc, G. Weaver
(2.2) for any negation -A of QC2/= and any two members a and a' of C, if .(A) = a'@), then a(-A) = &'(-A), (3.1) for any member a of C and any two conditionals A 3 B and A' 3 E' of Q C ~ / = ,if &(A)= a(A') and a(B) = a(B'), then a(A 3 B ) = a(A' 3 B'), (3.2) for any conditional A 3 B of QC21°0 and any two members a and a' of C, if &(A)= @'(A)and a(B)= a'(B), then or(A 3 B) = a'(A 3 B), and (4) there is a permissible instantiation function I for QC21rn such that:
(4.1) for any member a of C, any two quantifications (VI/)A and (VV')A' of QC2/"" ( V and V' here variables of the same type, level, and degree) such that I(( V W) is at least as large as I((VV')A'), and any functionf from I(( VV)A) onto I((VV)A'), if a(A *) = .(f(A *)) for every member A * of I(( VW), then a(( VW) = a(( VV)A'), (4.2) for any member a of C , any two quantifications ( V V A and (VV')A' of QC2Im (V and V' as in (4.1)) such that I((VV)A) is smaller than I((VV)A'),and any functionffrom I(( VV')A') onto I(( VW),if .(A * ') = a(f(A for every member A * ' of Z((VV')A'), then a(( V V A ) = a(( VV')A'), and * I ) )
(4.3) for any quantification (VVA of QC2Im and any two members (Y and a ' of Z, if &(A*) = a'(A *) for every member A * of N V W)),then a((v VA) = NV W).12
And we shall say that QC2/O0 has a strictly truth-functional interpretation if there is a family L: of truth-value functions for QCZi" which satisfies conditions (1)-(4) plus the following: ( 5 ) for any two members (Y and a' of C,if .(A) = a'@) for every atomic wff A of QCz/OO, then &(A) = a'@) for every wff A of QC2/O0.
1 2 The reader may wish to verify that conditions (2)-(4) are equivalent t o tlic following:
(2') for any member a Jf Z and any negation -A of QC2'w. a(-A ) = T if and only if &(A)= F. (3') for any member a of Z and any conditi(inal A 2 /I of Q c " i " % dA 3 / I ) = T if and only if a(A) = F or a(B)= T, and (4') for any member a of 2: and any quantification (VV)A of QCZ/-, a(( V V)A) = T if and only if44*) = T for every instance A * o f ( V V)A such that t ( V V)A 3 A *.
Ramified Theory of Types
15s
Now consider the family &consisting of every truth-value function 01 for
QC21W such that:
(i) for any negation -A of QC21W, a(-A) = T if and only if a(A) = F, (ii) for any conditional A 3 B of QCZl", a(A 3 B) = T if and only if &(A) = F or a(B) = T. (iii) for any quantification of QC2/W40fthe sort (VX)A, a((VX)A) = T if and only if &(A*) = T for every instance A * of ( V q A , and
(iv) for any quantification of Q C 2 P of the sort ( V F d h ) A , a((VFd/l)A) = T if and only if &(A*) = T for every predicative instance A * of (\clFd/l)A. It is clear that C1 - the intended or principal interpretation of QC2/- satisfies conditions (2)-(4). Towards showing that C1 satisfies condition (1) as well, let S be any finite set of wffs of QC2/" that is syntactically consistent in QC2/O'. Following the instructions in Leblanc [1969j , 1 3 one can extend S into a set SW such that: (iw) for any negation -A of QC2/O0,-A belongs to Sm if and only if A does not belong to Sw, (ii-) for any conditional A 3 B of QC2/w, A 3 B belongs to Sw if and only if A does not belong to Sm o r B does, (iii,) for any quantification of QC21Wof the sort (VX)A, if (VX)A does not belong to Sm, then there is an individual parameter X of QC2lw such that A ( X / X ) does not belong to SW, and (iv-) for any quantification of QC2/- of the sort (VFdb)A, (VFd/l)A does not belong to Sm, then there is a predicate parameter Fdll of QC21rn of the same degree and level as Fdll such that A ( F d h / F d h ) does not belong to Srn either. But, if a quantification ( V X ) A of QC2/OCbelongs to SOO,then in view of A6 so does every instance of (VX)A; and, if a quantification ( V F d h ) A of QC2/- belongs to SW,then in view of A7 so does every predicative instance of (VFdl1)A. Hence each member of SOO- and, by rebound, of S - is sure to evaluate to T under the member a of Z1 that assigns T to every wff of Q C 2 P in Soo and F to everyone not in SW. Hence, in particular, if { - A } is syntactically consistent in QC2/w, then a(A) = F, this for any wff A of QC2/-. Hence any wff of QC2jW evaluating to T under all members of Z1 is a theorem of Q C 2 P But any theorem of QC2/= can be shown to evaluate to T under all members of X i . Hence El meets all four of conditions (1)-(4). Hence: Theorem 1. In the absence of the Axiom of Reducibility, QC21m has a truth-functional interpretation. 13
The instructions stern largely from Henkin [ 1949)
156
H. Leblanc, G. Weaver
Proof that C1 meets condition (5) as well (and, hence, that in the absence of the Axiom of Reducibility QC2/- has a strictly truth-functional interpretation) is by an induction on what we call the rank r(A) of a wff A of QC2IW, and proofs of the basis and the inductive step of tlus induction are by subsidiary inductions on the length G ( A ) of A . Asusual, { ( A ) = 1 whenA is atornic;,l(-A)=d(A)+l;d(A> B ) = G(A)td(B)tl; andd((VV)A) = G(A(P/V))+l,where P is the alphabetically earliest parameter of QC2/- of the same type, level, and degree as V . Definition of the rank of A is more elaborate. By a term'naring sequence n 2 , . . , ni, . . . ), o f natural numbers understand any sequence of the sort hl, where (1) for each 1 from 1 through j , nl > 0 and (2) for each 1 from j+l on, nz = 0. 011, n 2 , . . . , nj, . . .) and h i , n 2 , . . . , ni', . . .) being terminating sequences ofnaturalnumbers,take(nl,n2,... ,n,,. . . )toalphabeticallyprecede , , ( n l ,n2, . . . ,n)', , . .) if there is an 1 (1 2 1) such that nm = nh for any m from 1t1 on, but nl < ni. Take a terminating sequence of natural numbers to be of rank I if none alphabetically precedes it, take the sequence to be of rank 2 if exactly one alphabetically precedes it, etc. Take the associated terminating sequence of a wff A of Q C 2 P to be ( n l ,n2, . . , ,ni, , . .), where - for each 1 from 1 on - n is the number of times 'V' occurs in A flanked on the right by a predicate variable of level 1. And take the rank r(A) of A to be that of its associated terminating sequence. It is easily verified that: (a) r(A) = 1 if and only if A contains no predicate variable, (b) r(-A) = r(A),(c) r(A(B/Fd/l(X1, X2,. . . , Xd)))so long as B is of level 1 or less. ,
I
t
Theorem 2. Let aand a' be members of XI such that &(A) = &'@)for every atomic wff A of QG'f-. Then a(A) = &'(A)for every wff A of QC?/*. hoof by mathematical induction on the rank r(A) of an arbitrary wffA of QC2IW. Basis: r(A) = 1, in which case A contains n o predicate variable. Proof that a(A) = a'@) here is by a first induction on the length d ( A ) of A . (i) Suppose / ( A ) = 1, in which case A is atomic. Then a(A) = a'@) by the hypothesis on a and a'. (ii) Suppose L ( A ) > 1, and A is of the sort -B. Since d(B) < d(-B), a(B)= a'@) by the hypothesis of the first induction on length. Hence a(++) = a'(-B). Or suppose A is of the sort B 3 C. Since G(B)< G(B 3 C ) and C (C) < d(B 3 C), a(B) = a'(B) and a(C) = a'(Q by the hypothesis of the first induction on length. Hence a(B 3 C ) = a'@ 3 C). Or suppose A is of the sort (VX)B, and let X be an arbitrary individual parameter of QC2/m. Since
Ramified Theory of Types
157
G (B(X/X))< G ((VX)B), a(B(X/X)) = a'(B(X/X)) by the hypothesis of the first induction on length. Hence a((VX)B)= a'((VX)B).
Inductive Step: r(A) > 1. Proof that .(A) = a'@) here is by a second induction (this one without Basis) on the length! ( A ) of A . (i) Suppose A is of the sort -B. Since r(B) = r(-B) and G (B) < G (-B), a(B) = a'(B) by the hypothesis of the second induction on length. Hence a(-@ = a'(-B). (ii) Suppose A is of the sort B 3 C. Since r(B) < r(B 3 C) and " (B) < t ( B 3 C), a(B) = a'(B) by the hypothesis of the second induction on length when 4 B ) = r(B 3 C), otherwise by the hypothesis of the induction on rank. Similarly, a(C) = a'(C). Hence a(B 3 C) = a'(B 3 C). (iii) Suppose A is of the sort ( V D . Since r(B(X/X)) = r((VX)B) and t (B(X/X)) < G ((VX)B), a(B(X/X)) = a'(B(X/X)) by the hypothesis of the second induction on length. Hence a((VX)B) = a'((VX)B). (iv) Suppose A is of the sort (V'Fd/l)B, let C be an arbitrary wff of QC2/O0of level I or less, and - in case d > 0 let X I ,X2, . . . , and X d be distinct individual parameters of QC2/=. Since r l B ( c / F / l ( X ~X2, , . . . ,Xd)))< r((VFd/z)B),(u(B(cIFd/l(x~ J 2 . . . . , Xd)))= aV(B(C/Fd/l(X1,X2, . . . ,Xd)))by the hypothesis of the induction on rank. Hence a(( tlFd/l)B) = a'((VN/l)B). Hence :
Theorem 3. In the absence of the Axiom of Reducibility, QC21w has a strictly truth-functional interpretation. I1I Now count as an axiom,of Q C 2 / - any wff of QC21m of any of the seven sorts A1 -A7 on p. 152, of the sort A8 on p. 153 (= the Axiom of Reducibility), or o f the sort ( V W , where - for any parameter P of QC2Im of the same type, level, and degree as Vthat is foreign to ( V V ) - A ( P / V ) is an axiom, of QC21°0. Where A is a wff and S a set of wffs of QC2/-, we shall understand by a proofr in QC21m of A from S any finite column of wffs of QC2/"" such that: (1) the last entry in the column is A and (2) every entry in the column belongs to S, is an axiom, of QC2/"", or is the ponential of two earlier wffs in the column; we shall say that A is provable, in QC2:- from S if there is in QC2/- a proof of A from S; we shall say that S is synratically consistent, in QC2Im if there is no wff of QCIl- such that both it and its negation are provable, in QC21°0 from S ;and we shall say that A is a theorem, of QC21°0 ( krA, for short) if there is in QC21m a proof, of A from S.
1.58
H.Leblanc. G. Weaver
We first obtain a generalization of A8 with ‘Fdll’ in place of ‘Fdl”, and ‘Fdh’ (Theorem 5 ) ;and, given this result, we obtain the counterpart in QC2/“Oof the familiar Axiom o f Comprehension (Theorem 6 ) . Proof that, for any quantification of QC2/w of the sort (VFdll)A and any instance A * of (VFd/l)A, ‘@/l’’ in place of
( V F d / l ) A 3 A*
is a theorem, of QC2lm, can then be retrieved from Henkin [ 1 9 5 3 ] , a task we leave to the reader.14 In the course of proving Theorems 5 and 6 we use some thirteen lemmas, listed without proof under the common heading Lemma 4. These, as the reader may verify, do not call for A8; so throughout we write ‘t-’ in place of ‘tr’. Lemma 4. (a) Zf I- A and t A 3 B, then k B.
(b) FA 3 ((B 3 B ) = A ) . (c) 1-A 3 (-(B 3 B ) A ) . (d)If / - A >Band I--A 3 B , t h e n FB.
-
(e) I- (VXl)(VX2) . . @&)(A = B) 3 ((VX,)(VX2) . . . (VXd)(A C ) 3 ( V X , ) ( V X 2 ) . . . ( V X d ) ( C r B)). (0 k ( v x l ) ( V x 2 ) . * ( v x d ) ( A B) 3 ( ( v X i ) ( V x 2 ) . . . (vXd)(B c) 3 ( v X l ) ( v x 2 ) . . ( v X d ) ( A c)). (g) I f I-A(P/V) 3 B(P/V), then I- ( 3 VpI 3 ( 3 V)B, where P is foreign to (3 V)A and (3 V)B.
(h) I f I-A(P/V)3 B, then I- ( 3 V)A ( 3 V)A and To B.
3
B, where P is foreign to
(i) I f f - A3(B(P/V)>C(P/V)),then I - A 3 ( ( 3 V ) B 3 ( 3 V ) C ) , where P is foreign to A , ( 3 V)B, and ( 3 V)C. (j) I f t ( g V ) ( A >@,then I-(VV)A 3 B . 1 5
(k)Ifl-(3yXA 3 B),th en l - A 3 ( 3 V ) B . 1 6
14
A detailcd proof of t h e result will appcar
15
With (VV)A 3 8 presumcd herc
16
W i t h A >(3V)Eprcsunied h e r e i o h e a wlTofQC?:m. C’issurc tohcforcign t o A .
to
in
Lehlunc I19721.
hc a w f f of QC?lm, V is surc
to
he folcign
to
R
Ramified Theory of Types
159
H. Leblanc. G. Weaver
I60
Hence by Lemma 4Q) and the hypothesis on @ / 1
I-I(VGd/l)(VFd/l)(3X1)(VX2) * . . ( V X d ) ( G d / l ( X l , X 2 , .. . , X d ) s Fdl'(X1 , X 2 , . . . ,&)) 3 ( 3 F d / ' ) ( v x 1 ) ( V X 2 ) .. . ( v x d ) (FCIIl(X1, X 2 , . . . ,X d ) fF d / l ' ( X , , X 2 , . . . ,X d ) ) . Hence by Lemmas 4(m) and 4(a)
I- , ( 3 F d / l ) ( V X l ) ( V X 2 ) .. . ( V X d ) ( F d / l ( X , , X 2 , .. . , X d ) Fd/"(Xl,X2, .
. . ,X d ) ) .
Theorem 6 . I- I ( 3 F d / l ) ( V X l ) ( V X 2 ) . . . ( V X d ) ( F d / l ( X l ,X 2 , . . . , X d ) A ) , where F d h is foreign to A and in case d > 0 - X I , X 2 , . . . , and Xd are distinct individual variables of QC2iW. ~
Proof: Let Fd/l be a predicate parameter of QC2;"O of degree d and level 1 foreign t o A . Case 1: d
= 0.
I- ,A
By Lemmas 4(b) and 4(c)
3 ((@/I
3 Fdh) = A )
and
1I-A
3 (-(Fd/' 3 F d h )
A).
Hence by Lemma 4(1)
I- (3Fdh)(A 3 (Fdh = A ) ) and
I- r( 3 Fd/l)(-A 3 (Fd/l = A ) ) . Hence by Lemma 4(k) and the hypothesis on Fd/l
I-,A
3
(3FCI/')(Fd/' = A )
and
kr-A 3 ( 3 F d / l ) ( F d / l Z A ) . Hence by Lemma 4(d)
I-r ( 3 ~ d / l ) ( ~ d=/Al ) . Case 2: d > 0.1' being the level of A , let Cd/l' be a predicate parameter of Q C 2 / - of degree d and level I' foreign t o A and distinct from F d h , atid let be a predicate variable of QC21m of degree d and level I' foreign to A .
Ramified Theory of Types
Theorem 7.
161
k r (VFd11)A 3 A*, for any instance A * of (VFd/l)A.
To show that in the presence of A 8 QC21m still has a truth-functional interpretation, read 'I-rA' for 'I-A'in condition (1) o n p . 153, drop the (now idle) qualifier 'permissible' in condition (4) on p. 154, and define C2 as C1 was defined on p. 155, but with (iv) now amended t o read: (iv') for any quantification of Q C 2 / m o f the sort (VFd/l)A, a((VFd/l)A) = T if and only if a(A*) = T for every instance A * of (VFd/l)A. Supposing a finite set S of wffs of Q C 2 / m t o be syntactically consistent, in QC2lm, one can again extend S into a set Sm boasting features (im)- (iv-) . But, if a quantification (VX)A of QC2/03 belongs to Sm, then on pp.
162
H. Leblanc, G. Weaver
in view again of A6 so does every instance of (VX)A; and if a quantification ( V f l / l ) A of QC21m, then in view of Theorem 7 so does every instance of ( V f l / l ) A . Hence each member o f S is sure to evaluate to T under the member a of Z2 that assigns T to every wff of QC2Im in S m and F to everyone not in SOO.Hence any wff of QC2/m evaluating to T under all the members of C2 is a theorem, of QC2lm. But any theorem, of QC2/OO can be shown to evaluate to T under all members of & . Hence C2 meets all four conditions (1)--(4). Hence :
Theorem 8. In the presence (as well as in the absence) of the Axiom of Reducibility, QC-'lm has a truth-functional interpretation. IV Proof that 2 2 - now the principal interpretation of QC21m fails to meet condition (5) on p. 157 calls for some model-theoretic notions (one of them adapted to serve the purpose at hand). Take a model to be any non-empty set. Where D is a domain, take a D-interpretation of the parameters of QC2 im to be any function f D from the predicate parameters of QC2/m of degree 0 to {T, F}, from the individual parameters of QC2lm to D, and from the predicate parameters of QC2/m of non-zero degree d to the power set of @(=D x D x . . . x D).17 d times Where D is a domain, fD a D-interpretation of the parameters of QC2lrn , and P a parameter of QC2Im, take aP-variant of ID to be any D-interpretation of the parameters of QC2i" that agrees with I D on all the parameters of QC21m other than P (and possibly on P as well). And, where A is a wff of QC21m, D a domain, and I D a D-interpretation of the parameters of QCZi", take A to be true on I D if: ~
(a) in case A is a predicate parameter of QC2/m of degree 0, ID(A)= T, (b) in case A is of the sort Fdh(X1 ,X2, . . . , X d ) for some non-zero d , ( I D ( x ~ )1, 0 ( ~ 2 ) ,. . . ,ID(&)) belongs to 10 (Fdll),
(c) in caseA is of the sort -B, B is not true on I D , (d) in case A is of the sort B 3 C, B is not true on ID or Cis, and 1 ' In model-tlicorctic accounts 01' O('2 '-. / , ) ( F d / / ) . where d > 0. would be required to be subset of nl level 1. But tlic r c s ~ r i c t i t ~would n p h y 1 1 0 colt Iicrc. a n d 15 i i i l c i i t i o n a l l y
dropped.
3
Ramified Theory of Types
163
(e) in case A is of the sort (VT/)B, B(P/y) (P here the alphabetically earliest parameter of Q C 2 / - of the same type, level, and degree as V that is foreign to (V u)B) is true on every P-variant Of ID. Crucial among D-interpretations are those - to be called Henkin D-interpretations - where each member of the domain D is assigned to an individual parameter of QC21m (more formally, where for each member d of the domain D there is an individual parameter X of QC2/- such that I D Q = d). Extending a familiar result for QCl , one can indeed show that: Theorem 9. Let D be a domain, and ID be a Henkin D-interpretation of the parameters of QC-'l*. Then a quantification ( V X ) A of QC2/" is true on ID ifand only if every instance of ( V X ) A is true on ID . I * And crucial among Henkin D-interpretations are those where the domain D is finite. Take indeed an arbitrary quantification (VFdf1)A of QC2/m; let Fdll be the alphabetically earliest predicate parameter of Q C 2 / - of the same level and degree as Fdll that is foreign to (VFdfl)A;in case d > 0, let X I , X 2 , . . . ,and x d be in alphabetic order the first d individual parameters of Q C 2 P that are foreign t o (VFdlrJA;let D be a finite domain; let I D be a Henkin D-interpretation of the parameters of QC21m; and let Ib be an Fdh. variant of 10.There exists then a wff B of Q C 2 / - , to be known as the wffassociate of I D , such that A(Fd/l/Fd/l)is true on Ib if and only if A ( B / P / ! ( X I , ~ 2. .,. ,&)) is true on I D ; and, as a result, ( v F ~ / ' ) Apans out true on ID if and only if all its instances do. B is as in the following table, where dl , d2, . . . , and d,, ( n > 0) serve as the various members of D, Yi is for each i from 1 through n the alphabetically earliest individual parameter of Q C 2 / - to which member di of D is assigned in I D , and I1 = 12 is short for (Vf)(f(ll) 3 f(I2)): And proof that A(Fdll/Fdf') is true on Ib if and only if A(B/@/I(X1 , X 2 , . . . , &)) is true on ID is by mathematical induction on the length of A . Now suppose that (VFdf1)A is true on I D . Then A ( B / F d f l ( X l ,X 2 , . . . , x d ) ) is sure to be true on ID for any wff B of QC21m and - in case d > 0 -any distinct individual parameters X i , X2, . . , and X&of QC*/m. Suppose, on the other hand, that (VFdl1)A is not true on ID. Then there is an Fdfrvarient Ib of ID such that A(Fd/l/Fd/l)is not true on Ib, and hence by the
.
1s A detailed proof of Theorem 9 will appear it1 Lehlanc [ 1 9 7 1 ] . TheD-interpretations under consideration are called Ifc,rkirr L)-irrlcr/~rc,tulir,rlsbecause of t h e rolc their counterparts for QCl play in Henkin [ 19491.
I64
H. Leblanc, G. Weaver
etc.
foregoing result there is a wff B of QC2/" (to wit: the wff-associate of I D ) , and - in case d > 0 -- there are distinct individual parameters X 1 , X 2 , . . . , and X d of @?la", such that A(B/@h(X,, X2,. ,Xd)) is not true on ID itself. Hence:
..
Theorem 10. Let D be a finite domain, and ID be a Henkin D-interpretation of the parameters of QC2/". Then a quantification (tlFd/l)A of Qbl" is true on I D if and only if every instance of ( V P h ) A is true on ID. Proof of our last theorem is within reach. Indeed, let D and D' respectively be { 1) and {I, 2) ;let ID assign T to every predicate parameter of QC2/0° of degree 0 (1 t o every individual parameter of QC21m), and od to every predicate parameter of QC2/0° of non-zero degree d; let ID' assign T to every predicate parameter of QC2/" of degree 0, 1 to 'x', 2 to every other individual
Ramified Theory of Types
165
parameter of QC2/", and D'd to every predicate parameter of Q C 2 P of nonzero degree d; let a be the truth-value function for QC2l0" such that, for every wff A of QC21°0, a(A) = T if and only if A is true on ID;and let a' be the one such that, for any such A , &'(A)= T if and only if A is true on ID'. It is easily verified with the aid of Theorems 9 and 10 that a and 01' both belong to the family Z2 of p. 161, that a and 01' agree on all atomic wffs of QC2/m (indeed, on all wffs of QC2/" of rank I), and yet that 01 and a' disagree on '( 3f)( 3x)( 3y)cf(x) & -f(y))',which evaluates to F on 01 but to T on a'. (For proof that the wff in question evaluates to F on a,note that '(3 x)( 3y)(f(x) & -f(y))' cannot be true on either of the two f-variants of ID; for proof that it evaluates to T on a',note that '(3x)( 3y)(f(x) & -f(y))' is true on thef-variant of ID' that assigns (1) to 'f'.)19 Hence C2 does not meet condition (5) on p. 154. Hence:
.Theorem 11.
In the presence of the Axiom of Reducibility, QCZ/" has a truth-functions[ interpretation which is not strictly truth-functional.2 0 V
Church [1956] sketches a semantic account of QC21°" with two truthvalues per level.21 The account uses models. To mimic it here, acknowledge with Church two truth-values TZ and FI for each level 1. Understand by a system of truth-value functions for QC2/" any sequence of the sort (a1, a2,a3,. . ), where - for each I from 1 on - al is a function from the wffs of Q C 2 P o f level 1 to (TI, FI} such that: ,
(i) al(-A) = TI if and only if q ( A ) = Fl, (ii) q ( A 3 B) = TI if and only if q ( A ) = n' or a@) = T k ( j here the
level of A and k that of B),
19
The example comes from Leblanc and Mcyer [ 19701.
The above proof of Theorem I 1 borrows from Leblanc and Mcyer [ 1970], where Q C I is shown t o have a truth-functional interprctation which is not strictly truth-functional. Another proof of Theorem 1 1 can be had via that result in Leblanc and Meyer [ 19701. Indeed, with Theorem 7 31 hand, one can construct for each wff A of Q C Z P a wff E of QCI such that A is provable, in QCZim if and only i f R IS provable in QC!. B is gotten froni A by systematically lowering the level of each predicate variable and parameter in A t o I , and then writing f:d and Fd everywhere for F d / l and Fdll,respectively. See the article Semantic Deviations in this volume. 20
? I See pp, 347-348, footnote 577
166
H, Leblanc, G. Weaver
(iii) a/((V X ) A ) = TI if and only if cq(A(X/X)) = TI for every individual parameter X of QC2/=, and
(iv) a [ ( ( V F d l i ) A ) = TI (j< I ) if and only if a&l(B/Fdh(X1.X2, . . . ,Xd)))= Tk (k here the level ofA(B/@h(X1.X2,. . . ,x d ) ) ) for every wff B of QCZ/" of level j or less, and - in case d > 0 - any distinct individual
parameters XI, X 2 . .
. . , and Xd of QC2/OO.
And take a wff A of QC2/m of level I to be true on a system (a1,a2, a3, . . . ) of truth-value functions for QC2Irn if q ( A ) = TI. It can be verified that: (1) t A if and only if A is true on every system of truth-value functions
for QC2/", and
(2) two systems of truth-value functions for QC2/" that agree on all the atomic wffs of QC2/OO are sure t o agree on all the non-atomic ones. Indeed, the induction that saw Theorem 2 through will - mutatis mutandis see (2) through. Suppose, however, that B in condition (iv) is allowed t o be of any level whatever. It then follows that: (3) I- ,A if and only if A is true on every system of truth-value functions for QC2/m,
but ( 2 ) fails, as a straightforward generalization of the argument in 1V will show. So analogues of Theorems 1 , 3 , 8 , and 1 1 hold for Church's account. References Beth, E. W., 1959, The Foundations of Mathematics (North-Holland, Amsterdam). Church, A., 1956, Introduction to Mathematical Logic, Volume 1 (Princeton). Dunn, M. and Belnap, N. D., 1968, The Substitution Interpretation of the Quantifiers, N o ~ sV , O ~ .2, pp. 177-185. Fitch, F. B . , 1948, Intuitionistic Modal Logic with Quantifiers, Portuguliae Mathemntica, VOI. 7 , p p . 113-118. Henkin, L., 1949, The Completeness of the First-order Functional Calculus, The Journalof Symbolic Logic, vol. 14, pp. 159-166. Henkin, L., 1953, Banishing the Rule of Substitution for Functional Variables, The Journal of Symbolic Logic, vol. 18, pp. 201-208. Leblanc, H., 1968, A Simplified Account of Validity and Implication for Quantificational Logic, The Journalof Symbolic Logic, vol. 3 3 , pp. 231 -235.
167
Leblanc, H., 1969a, A Simplified Strong Completeness Proof for Q C = , A k t e n des XIV. Internationalen Kongresses f i r Philosophie, Volume III, (Vienna) pp. 83-96. Leblanc, H., 1969, Three Generalizations of a Theorem of Beth’s, Logique et Analyse. VOI. 12, pp. 205-220. Leblanc, H., 1972, Truth-Value Semantics (Amsterdam) forthcoming. Leblanc, H., and Meyer, R. K.,1970, Truth-value Semantics for the Theory of Types, in: Philosophical Problems in Logic: Some Recent Developments, ed. K. Larnbert (Reidel, Dordrecht), pp. 77-101. Schutte, K., 1960, Syntactical and Semahtical Properties of Simple Type Theory, The Journal of Symbolic Logic, vol. 25, pp. 305-326. Schutte, K., 1962, Lecture Notes in Mathematical Logic, Vol. I (The Pennsylvania State University) .