Chapter III The Logical Constants

CHAPTER III THE LOGICAL CONSTANTS 3. If a and b are natural numbers and if [a, b] = 0 is a provable equation then we call the equation a = b a true proposition, and if the inequality Ia, b] > 0 is provable then the equation a = b is called a false proposition. Since the equations a = b, [a, b] = 0 may each be derived from the other it follows that a true proposition is a provable equation and conversely. Moreover, as lx, yl is a recursive function, for any natural numbers a, b there is a unique natural number c such that

[a, bl=c is provable; if this number c is zero then a = b is a true proposition, and if c is not zero then 1 -"-I a, b] = 0 is provable, so that a = b is a false proposition. Accordingly, any proposition is necessarily either true or false, and no proposition is both true and false. 3.1 We call tX(la, hi) the number of the proposition a=h, so that a true proposition has the number zero and the number of a false proposition is unity. Conversely, if tX(ja, bl) = 0 then [a, hi = 0 so that the proposition a = b is true, and if tX(la, hi) = 1 then [a, hi> 0 and a = b is false, and therefore the number of a proposition is zero if and only if the proposition is true and the number is unity if and only if the proposition is false. 3.11 The proposition l-"-la, hi = 0 is called the negation of the proposition a = h. Since tX(l-"-lx, YI)= l-"-lx,

yl

it follows that if the proposition a = h is true, so that [a, hi = 0, then the number of the negation of a = b is unity, and if a = b is false then the number of its negation is zero. Thus the negation 56

57

THE LOGICAL CONSTANTS

of a true proposition is a false one, and the negation of one that is false is a true proposition. 3.12 We denote propositions by single letters p, q, r with or without numerical subscripts. The negation of the proposition p is denoted by """ p, which is read 'not-p'. When the context renders the usage free from ambiguity we shall write simply 'p' for 'p is true' and <; p' for 'p is false'. 3.13 If p, q denote the propositions a = b, C = d respectively then the proposition

[a, bl+Ie, dl =0 is denoted by p & q which is read 'p and q', the proposition

[a, bl·le, d] =0 is denoted by p v q which is read 'p or q', and

(I

-'-I a, bl)·1 c, d] =

0

is denoted by p --+ q read 'p implies q', p --+ q is thus the same proposition as ,....., p v q. Finally, we denote the equation

(l-'-Ia, bj)·le, d] +(l-'-Ic, dj)·la, hi =0 by p ~ q which is read 'p is equivalent to q', so that p the same proposition as (p --+ q) & (q --+ pl. We observe that, since Ila, b], 01 = [a, b], therefore

~

q is

(a= b)~(la, b] =0)

and since and therefore

(l-'-x)(X(x)=x(I-'-(X(x))=O

(X((X(x))= (X(x) (x=O)~((X(x)=O)

(z »

O)~((X(x) =

I).

We shall justify the suggested readings of the signs and ~ in the next section.

&,

v, ,....." --+

58

THE LOGICAL CONSTANTS

3.14 From the equation [a, b] + [c, d] =0 it follows that [a, b] =0 and [c, dl =0, and conversely, and therefore p & q is true, if, and only if, p and q are both true. 3.141 The product [a, bj-]c, d] vanishes if either [a, b] = 0 or [e, dl =0; and if both [a, b] >0 and [c, d] >0 then [a, bl·le, d] >0, and so: p v q is true if and only if either p is true or q is true. 3.142 It follows that p -+ q is true if and only if either p is false or q is true, which we take to be the sense of the expression p implies q. 3.15 The definition of equivalence makes p and q equivalent if and only if p and q are both true or both false. From the proposition p +)- q and a proof of q we derive a proof of p, for if {1~la, blHe, dl+{I~le, dl}·la,

bl=O

[c, dl=O

and

are proved equations, then [a, b] =0 is a proved equation; accordingly, to prove some proposition p it suffices to prove a proposition equivalent to p. 3.16 The signs &, v, ,...." -+ and +)- are known as logical constants; their introduction effects a considerable economy in proof technique by revealing a variety of easily recognisable patterns of procedure. 3.2

PROPERTIES OF THE LOGICAL CONSTANTS

(1 ~x)Y= 0 (1 ~y)z=O

(1 ~x)z=O

and therefore, if p, q, r are any propositions ~ =~, bl = b2 , el =~, say, taking lal , ~I for x, IbI> b2 1for y and I~, <;1 for z respectively, we obtain the proof schema

p-+q q-+r ---_ .. p-+r

-

THE LOGICAL CONSTANTS

59

which is summed up by saying that '-)0' is a transitive relation. Similarly from the schema

x=o (l-"-x)y=O y=O

we derive the fundamental schema for implication (known as modus ponens)

3.22 Since addition and multiplication are commutative the following proof schemata are valid: p&q

pvq

»->«

q&p

qvp

q-
--, --, --. These show that '&', 'v' and '-
Since f=O and g=O follow from !+g=O, and conversely,

we have

p&q

--, P

and

p q p&q

It follows that '-
3.3 We consider next some of the important relations which hold between the logical constants. Since I .z: {I -"- (I -"- x)} = I .z: x it follows that 3.31 From the equation l-"-(x+y)=(l-"-x) (l-"-y) we deduce

60

THE LOGICAL CONSTANTS

3.32

"-' (p & q) ~ "-' P v "-' q

and from 1-'- xy = iX«1 -'- x) + (1-'- y)) follows 3.321 Furthermore, since x(y+z)=xy+xz, P v (q & r) is the same proposition as (p v q) 3.33

P v (q

& (p v

r) so that

& r) ~ (p v q) & (p v r)

and since {1-'-(x+yz)} (x+y) (x+z)={I-'-(x+y) (x+z)} (X+yz) = 0 therefore 3.331

P

&

(q v r)

~

(p

&

q) v (p

&

r)

For equivalent propositions we can prove interchangeability; this is given by following theorem: 3.34

If PI ~ P2 and ql ~ q2 then

We observe first that since the suffixes 1,2 are interchangeable in the given conditions PI ~ P2 , ql ~ q2 therefore it suffices to prove the foregoing propositions with implication in place of the equivalence relation. Since P -+ q is the same proposition as "-' P v q, the truth of the fourth proposition follows from that of the first and third, and the fifth then follows by means of the second. That "-' P2 -+ "-' PI follows from PI -+ P2 is shown by the equation

{I .s: (1 -'-x2 )} (1 -'-Xl) = (1 -'-Xl)-'-{(1 ..=... Xl) .z., x2 ( 1 .s: Xl)} by means of which {I -'-(1 -'-x 2 ) } (1 .z; Xl) = 0 follows from (I-'-X I )X 2=O.

Since "-' "-' r 3.341

+->-

r we deduce from 3.321 that Pvq

~

"-' ("-' P

& "-'

q)

so that the truth of the third proposition follows from that of the first and second.

THE LOGICAL CONSTANTS

61

It remains only to prove the schema PI -+ P2 gl -+ g2 PI & gl -+ P2 & g2

and this follows from the equation {I-'- (Xl + YI)}(X2+ Y2) = {(I-'-XI)X2 -'- YIX2}+ {(l-'- YI)Y2 -'-XIY2}

by means of which we derive {l-'-(xt +YI)}(X2+Y2)=O

from the equations (I-'-XI)X2=0, (I-'-Yl)Y2=0. 3.342 The importance of the results contained in theorem 3.34 lies in the fact that it allows us to replace any proposition, in an expression made up of logical constants, by an equivalent; the resulting expression will be equivalent to the original one and so by 3.15 a proof of the transformed expression suffices to prove the original. In particular, since the propositions Ia, hi = 0 and a = h are equivalent, we may without loss of generality suppose any proposition to have the form c=O. 3.343 As an example of the use of this principle we consider in detail a proof of the proposition

P:

(p -+ q) -+ {(1' V p) -+ (r v g)}

s', r' are equivalent to P, g, r respectively then p' -+ g' , r' v p' , r' v q' are equivalent to

If p',

p

-'>-

pi:

g ,r

v

P ,r

v

(pi

g respectively and so P is equivalent to -'>-

g/) -+ {(r' v pi)

-'>-

(r ' v q')}

Whatever the propositions p, g, r we may take pi, s'. r' to have the forms a = 0, b = 0, c = 0 respectively, and so P is equivalent to P*:

{I-'-(I-'-a)h} (l-'-ca)cb=O

where a, b, c are certain numerals, which may be derived from the provable equation (example 2.48)

G:

{I -'- (1 .s: x)y} (1 -'- zx)zy = 0

62

THE LOGICAL CONSTANTS

(where x, y, z are numeral variables) by substituting a, b, and c for z, y and z respectively. If we allow the letters p, q, r to have the dual roles of names for propositions and numeral variables then we may write G directly in the form {l--=-(l--=-p)q} (l--=-rp)rq=O; that is to say, we may formulate an equivalent of P simply by writing (l--=-p)q for p -7 q, rp for r vp and so on (though of course we change the significance of the letters when we make the transcription). 3.35

From the provable equations x(l--=-x)=O,l--=-{x+(l--=-x)}=O

we derive the truth of the propositions p v ~ p and ~ (p & ~ p)

which are known respectively as the principle of excluded middle (or tertium non datur) and the principle of noncontradiction. 3.4 PROPOSITIONAL FUNCTIONS If X is a variable and f(x), g(x) are two given recursive functions then the equation f(x) =;:g(x)

is called a propositionaljunction; if for some value a of x, f(a)=g(a) is provable then the propositional function is said to be true for the value a; if If(a), g(a)1 >0 is demonstrable then the propositional function is said to be false for the value a. More precisely, the equation f (x) = g(x) is called a one-variable propositional function, and f(x, y)=g(x, y) is a two-variable propositional function, and so on. Propositional functions are denoted by p(x), p(x, y) etc., according to the number of variables. 3.41 As in the case of propositions, if p(x), q(x) denote the propositional functions F(x)=f(x) , G(x)=g(x)

THE LOGICAL CONSTANTS

63

then we denote 1-'-IF(x), f(x)1 = 0 by '"'" p(x), IF(x), f(x)1

+ IG(y),

g(y)1 = 0 by p(x) .. q(y),

IF(x), f(x)j·IG(y), g(y)j = 0 by p(x) v q(y), r-..J

and

p(x)

V

q(y) by p(x) -+ q(y)

{p(x) -+ q(y)} .. {q(y) -+ p(x)} by p(x) ~ q(y).

3.42 The relations between the logical constants which we have established for propositions hold also for propositional functions, the proofs proceeding on the same lines; for instance we derive p(x) v'"'" p(x)

from y(l-'-y)=O by taking IF(x), !(x)1 for y. We use the term formula to cover both propositions and propositional functions. In the notation of this chapter the fundamental theorem 2.68 takes the form 3.43

(x=y) -)- {p(x) -+ p(y)}

3.5 The logical constants enable us to introduce the conditional equations of elementary algebra. Unlike the variable x which is characterised by the property that it may be replaced by zero or by S» wherever it occurs, the x in a conditional equation is a missing number sign which may only be replaced by some definite numeral. For instance, when we say that x = 3 is a solution of the equation x 2 = 9, we mean that a true equation results from replacing x by 3 in the second equation, but neither in x = 3 nor in x 2 = 9 may we replace x by zero or by Sx. In fact x = 3 and x 2 = 9 are not equations but propositional functions; and the fact that 3 is a value of x which satisfies x 2 = 9 is expressed by the implication F:

(x= 3) -+ (x 2 = 9) .

Formula F holds for any value of x as may readily be verified, for 1-'-13+Sr,31=0 and 1-'-13--'--Sr,31=0 (by example 2.42) and 32, 91 = 0, so that, by example 2.7303, {l-'-Ix, 31Hx 2 , 91=0

which completes the proof of formula F.

64

THE LOGICAL CONSTANTS

Similarly, the fact that the 'equation' x 2+6=5x has only the two solutions x = 2 and x = 3 is expressed by the implication (x 2+6=5x) --+ (x=2)

v

(x=3)

r.e. {1~lx2+6, 5xl}·lx, 21·jx, 31=0; denoting the left-hand side of this equation by f(x) we have f(3+r)= {1-=-r(r+ 1)}r(r+ 1)=0

and f(0)=f(I)=/(2)=0 which proves that f(x)=O. In addition to the logical constants we introduce the limited universal, existential and minimal operators A:, E: and L: as follows: A~U(x)=O) stands for the propositional function Lf(n)=O ; E:U(x)=O) for the propositional function Ilf(n)=O, and L:(!(x)=O) for the function f-lf(n). The operators 'A:', 'E:', 'L:' are read as 'for all x from 0 to n', 'for some x from 0 to n' and 'the least x from 0 to n'; we proceed to justify these suggested readings. (We use the term 'justify' to cover an informal discussion - there is no question of a formal proof, since only the sign itself and not the interpretation finds a place in the formal work.) 3.6

3.61 The logical constants and the operators may be regarded simply as abbreviations for the expressions by means of which they were introduced, in which case in any formal proof the logical constants and operators must be eliminated and replaced by the expressions which they denote. Alternatively we may regard these signs as an additional part of the formal system satisfying, by definition, the equivalences

{(a=b) v (e=d)} ** {la, bl·le, dl =O} {(a=b) & (e=d)} ** {la, bl + [c, dl =O} {(a=b) -)- (e=d)} ** {(l~la, b\)le, dl =O} {(a=b) ** (c=d)} ** {(l .z: la, b\)lc, dl + (1 ~ [c, d\)la, bl =O} A:U(x)=O) E:U(x)=O)

** (Lf(n) = 0) ** (Ilf(n) =0)

L:U(x) =0) = flJ(n)

THE LOGICAL CONSTANTS

65

which must be added to the list of permitted formulae in a proof; in this case the new signs cannot be totally eliminated but expressions containing them may be transformed into equivalent equations in which they do not appear. The sign x in the operators A~(/(x)=O), E:(/(x) = 0) and L:(f(x)=O) is not a true variable but an auxiliary sign known as a bound variable. We could readily reserve a special class of signs for bound variables but since there is very little risk of confusion it is customary to use the same signs as for variables. The fact that the x in A:(f(x)=O) is a bound variable may be expressed formally by the rule that A:(f(x)=O) may be replaced by A~(f(y)=O), or by the expression obtained by replacing x by any other variable, but substitution for the bound variable is not allowed, with a corresponding rule for the other operators. Alternatively we may formulate the rule as an equivalence

in which x and y may be replaced by any other variables, with corresponding equivalences for the other operators. 3.62

A~(f(x)=O)

is the proposition 1:'j(O)=O, that is 1(0)=0, and is equivalent to

A~(f(x)=O)

if for some P,

1(0)=0 & 1(1)=0 & ... &l(p)=O,

then since 1:'j(n+ 1)=1:'j(n)+/(n+ 1), so that A~+l(f(x)=O) is the propositionA~(f(x)=O)&/(p+I)=O, it follows that A:+l(f(x)=O) is equivalent to 1(0)=0 & 1(1)=0 & ... & I(p+ 1)=0,

so that for any assigned p, A:(f(x)=O) is equivalent to 1(0)=0 & 1(1)=0 & ... &l(p)=O.

Similarly, since IIj(n+ 1) = IIf(n)· I(n + 1), E:(f(x)=O) is equivalent to 1(0)=0 v 1(1)=0 v ... v l(p)=O.

66

THE LOGICAL CONSTANTS

L:

For the interpretation of the operator we recall the characteristic properties of the function pt(n) proved in the last chapter. From the proved equations (2.942 and 2.949) we have pt(n),;;;;;,n,

and whence we obtain the formula E~(f(x)=O) ~ {f(pt(n))=O &pt(n) ,;;;;;, n}

which says that if there is a value of x between 0 and n for which f(x) vanishes, then pt(n) is one of these values; and from the

equation (2.9493) we have j(n)=O ~ Pt(x) ,;;;;;, n

which is equivalent to n
which says that f(n) does not vanish for an n less than Pt(x), so that if f(x) vanishes for some value of x from 0 to n, then pt(n) is the least of these values. If f (x) > 0 for all x from 0 to n, it follows from the equation (2.945) IIj(n)pt(n) = 0

that 3.7

L~(f(x) = 0) = pj(n) = 0 MATHEMATICAL INDUCTION

It follows from theorem 2.8 that the proof schema p(x)

p(O) p(x+ 1)

~

p(x)

is valid. This schema is known as the schema of mathematical induction:

67

THE LOGICAL CONSTANTS

The companion formula 3.8

[p(O)

& A~{p(x) ~

p(x+

1m

~ p(n)

may be called the principle of mathematical induction. We derive formula 3.8 from 3.81

[p(O)

& A~{p(x) ~

p(x+ I)}] ~ p(n+ 1) ;

any p(x) has an equivalent propositional function, f (x) = 0, say; by examples 3.01, 3.02 and 3.322 we see that formula 3.81 is equivalent to r(n)=O where r(n) = {I--'- f(O)}l1o(n)f(n + 1) , O(x) = f(x) + {I--'- f(x+ I)}.

Since r(O)={I--'-f(O)} {/(O)+(I--'-f(I))}f(I)=O and r(n + 1) = {I--'- f(O)}l1o(n) {/(n + 1) + (1--'- f(n+ 2))} f(n + 2) =r(n)f(n+2)

so that {I--'-r(n)}r(n+I)=O, whence r(n)=O follows by the induction schema. If we denote the propositional function 3.8 by P(n) then P(O) is p(O) & {p(O) ~ p(I)} ~ p(O)

which holds by example 3.031, and P(n+ 1) is equivalent to p(O) & A:{p(x) ~ p(x+ I)} & {p(n+ 1) ~ p(n+2)} ~ p(n+ 1) which follows from 3.81 (and example 3.031). Since P(O) and Pin-i- 1) are proved then P(n) follows by 2.7. 3.9 We collect here for reference the principle properties of the operators A, E and L; the fractional part of the numbers of the following formulae are the same as in the numbers of the theorems in the previous chapter of which these formulae are transcriptions. 3.91 3.92

q(n)->-{a<:n~p(a)} q(n)~ A:p(a) {a<:n&p(a)}~q(n) E:p(a)~q(n)

~

3.921

p(n)

3.942

L~p(x)

E:p(a) ;

<: n ;

68

THE LOGICAL CONSTANTS

3.945 3.949 3.9493

r-o-IE".,p(x) -+ {L:p(x)=O} ; E".,p(x) -+ p(L:P(x» pea) -+ {L:p(x)
3.95 We consider next some further theorems on these operators which are needed in the sequel. In the following proofs we systematically employ the corresponding lower case letter as the representing function of a predicate expressed by a capital letter. The product IIj(n) and the sum Ej(n) will also be denoted by II I (x) and E I(x) respectively.

lt~n

:r:~"

H~{F(x)~G(x)}

3.951

H~{~F(x)-+~G(x)}

In terms of representing functions we have to derive the equation ~ h)

IIj(n»IIIl(n) = 0

(i)

(I

from

(I~h) (l~f(x»g(x)=O.

(I

.i:

For n=O the derivation is obvious. By 2.7, (l~p)..:...pq=I":"'p, and thence by 2.7 (over q) l..:...pq=(I..:...p)

{1":"'(I~q)}+(I..:...q).

It follows that [l~{(I~h} (I~IIj(n})IIIl(n)}](1~h) (I~IIj(n+I})IIIl(n+I)=

= [I ~ {(I ~h) (I ~IIj(n»IIIl(n)}] (I ~h)g(n+ I)IIIl(n) {(I":'" IIj(n}) (I..:... (I ~ f(n+ I») + (I ~ f(n + I»}=O,

which completes a proof of (i) by 2.7. 3.9511

For

H~{ F(x)-+G(x)}

H

-+{A~F(x)-+ A:G(x)}

[1~{(I~h) (I~Ej(n})EIl(n)}](I~h) {(1~Ej(n})~f(n+l)}

(EIl(n)+g(n+ I»

=

=[I":"'{(I~h) (I":'" Ej(n})EIl(n)}] (I~h) {(I~f(n+I»~Ej(n)}

g(n+ 1)=0,

using the hypothesis (1 ~h) (I

~f(x})g(x)=O.

THE LOGICAL CONSTANTS

69

A similar proof shows also that H~(F(x)~G)

3.952

H~(~F(x)~G)

is valid. In either of these schemata, by taking 0 = 0 for H we see that the schemata are valid also if H is suppressed. E';(x.;;;n" F(x» ~ E:F(x)

3.953

x.;;;n " F(x) ~ F(x)

For

whence the result follows by 3.951. 3.954

By example 3.831. E:F(x) "A:G(x) ~ E:{F(x) "G(x)} .

3.955

The result is evident for n = O. Now [l--'-(l--'-a) (l-=-b)c] (1--'- (a+q» (l-=-bp)c(p+q) =

[1--'-(I-=-a) (l-=-b)c] (l-=-(a+q» (l-=-b)cp, since (l--'-q)q=O and (l-=-bp)p= (l-'-b)p,

= [l--'-(I-=-a) (l-=-b)c] {(l-=-a) (l-=-b)cp--'-q(l-'-b)cp}=O,

and therefore [1-'-(1-=-

.2 g(x» (I-=- II j(x» II (f(x)+g(x»]

z~n

(I-=-

a:~n

L

x~n

g(x» (I-=-

x~n+l

II /(x» II x:S;;;n+l

x~n+l

(f(x)+g(x»=O,

whence 3.955 follows by 2.8. 3.956

For 3.957

E:F(x) A~(x .;;; n)

-7>

-7>

E: (x .;;; n " F(x» .

{E:F(x)

E:F(x)

-<8-

-7>

E: (x .;;; n " F(x»)} .

E: (x .;;; n " F(x» •

By 3.952 and 3.956. 3.958

F(n)

-7>

E:F(x)

For (1-'-/(0»/(0)=0 and {1-=-/(n+l)}

{II l(x)}/(n+l)=O. x~n

70

THE LOGICAL CONSTA1'i'l'S

3.959 If G does not contain the variable x and n is a variable not contained in F or G then F(x)~G

EJ7;,F{x)~G

The case n = 0 is obvious. Since (p -> q) & (r -+ q) -+ (p fore (E:F(x) -+ G)

Y

r -+ q), as is readily proved, there-

(F(n + I) -+ G) ->- (E~+l F(x) -+ G)

&

and so, from F(n+ I) -+ G follows (E:F(x) -+ G) -+ (E:+l F(x) -+ G)

and the proof is completed by 3.7. It follows that the schema x
is valid, and so in particular is the schema x
3.96

F---+ A~G(x)

The validity of the schema for n = 0 is obvious. Since {I-'- (1-'- f)

L

x~n

g(x)} (1-'- f) {

={l-'-(I-'-f) =

L

x~n

g(x)+g(n+ In

L g(x)}(I-'-f)g(n+l)

:z:':::;;n

0, using the hypothesis (1-'- f)g(x) = 0,

the proof is completed by an application of 3.7. 3.961

For G(O) -+ G(O) and

{I-'-

L

"',,;;,,+ 1

g(x)}

L

,,"',,+ 1

g(x)={(I-'-g(n+I»-'-

L g(x)}g(n+ 1) II g(x)=O .

x~n

x~n

71

THE LOGICAL CONSTANTS

3.962 By 3.96 and 3.961. F(x)~(1(x)

F(x)-G(x)

3.963

~F(x)~~(1(x)

By 3.951, 3.9511. 3.964 The case n=O is obvious. To prove the equation

",-,A~G(x) --0>-

E: ",-,G(x) we use

[1 ~ {I ~ (1 ~ a)} b] {1-'-(1-'- (a + c»} b(1-'- c) = 0 which is proved by applying 2.8 to the variable c; taking ~ g(x) for a, II (1 ~g(x» for band g(n+ 1) for c, the z~n

z~n

proof is completed by an application of schema 2.8. Similarly to prove E~

"'-' G(x)

--0>- "'-'

A: G(x)

we use the equation {1-'-(I~a) (l-'-b)}

{(I-'-b)-'-bc} {(I-'-a)-'-c}

(which is also proved by applying 2.8 to the variable c) with the same substitution for a, band c. 3.9641 have

Taking "'-' G(x) for F(x) A:(x <; n

--0>-

III

3.957 and using 3.964 we

G(x»**A:G(x) .

From the propositional schema p-+-q --q-+-r-p

and 3.964 we derive immediately 3.965

F(n)-+ A~(1(x) E~-.-G(x) -+-.-F(n)

72

THE LOGICAL CONSTANTS

x
3.966

Denote this implication by P(n). P(O) is evident, and {x
But

E: G(y) -+ E:+l G(y)

so that

{P(n) -+ E:G(y)} -+ {P(n) -+ E:+lG(y)}

whence, taking y=x" f(y)=O for G(y), {x
But {x=n+ 1 "f(x)=O} -+ (x=n+ 1) -+ E:+l{x

=

{f(n+ 1)=O}

&

y

f(y)=O} , by 3.958,

&

so that x=n+ 1 & f(x)=O -+ [P(n) -+ E:+l{y=x & f(y)=O}].

Since x
&

f(x)=O -+ {x
&

f(x) =O}

Y

{x=n+ 1

&

f(x) =O} ,

it follows that {x
&

f(x)=O} -+ [P(n) -+ E:+ 1{y=x

&

f(y)=O}]

whence P(n) -+ [{x
&

f(x)=O} -)- E:+l{y=x

&

i.e. P(n) -+ P(n+ 1), which completes the proof. If F does not depend upon x then: 3.9661

A~(F "G(x» ~ F " A:G(x)

3.9662

A:(F

3.9663

E:(F & G(x» ~ F & E:G(x)

3.9664

E~(F

Y

Y

G(x» ~ F

G(x»

~

F

Y

Y

A:G(x)

E:G(x) .

f(y)=O}]

73

THE LOGICAL CONSTANTS

F .. G(x) ~ G(x)

From

A~(F" G(x» ~ A~G(x)

by 3.9511 we derive

F .. G(x)

and from

by 3.959, 3.961 we derive

~

F

A~(F" G(x» ~

F

A=(F .. G(x» ~ F .. A:G(x) follows.

whence Conversely from

A=G(y) ~ {x
follows

F .. A=G(y) ~ F .. {x <; n ~ G(x)} ,

whence

F .. A:G(y) ~ {x,;;;;n ~ (F .. G(x»} ,

and so, by 3.96, 3.9641 F .. A:G(y) ~ A~(F .. G(x»

which completes the proof of 3.9661. Again, by 3.9641, A:G(y) ~ {x,;;;;n ~G(x)}

and

80

F v A:G(y) ~ F v {x
F v A:G(y)

and so

F v A:G(y) ~ A:(F v G(x»

Conversely

A:(F v G(y» ~ {x,;;;;n ~ (F v G(x»}

and so

A:(F

and therefore

A:(F v G(y» .. ,......, F ~ A:G(y) ,

v

{(x,;;;;n)

~

whence

(F v G(x»} ,aB

above.

G(y» .. ,......, F ~ {x
whence 3.9662 follows. Theorems 3.9663 and 3.9664 follow from 3.9662, 3.9661 by taking ,......, F for F and '" G(x) for G(x).

3.97 The Oounting Operator N:F(x) '1'0 express the number of roots of an equation f (x) = 0 in the

74

THE LOGICAL CONSTANTS

range O.;;;;x.;;;;n, or what is the same thing, the number of values of x in this range for which the propositional function f (x) = 0 {I(x) = O} defined by the is true we introduce the function recursion

N:

N~{t(x) = O}= 1..:.. 1(0)

N:+1{/(x)=O}=N:{t(x)= O}+ {I..:.. I(n+

I)} .

If I(x) is the representing function of the propositional function F(x) then we write N:F(x)=N:{t(x)=O}. The first theorem we prove for the counting operator is F(x)_G(x)

3.971

~F(x) = N~G(x)

If D(a, b) denotes the positive difference between II..:..a, I..:..bl and I..:..{1..:..!(l..:..a)b, (I":"b)al}

then

Dia, 0)=0, D(O, Sb)=D(Sa, Sb)=O,

so that D(a, Sb) = 0 and therefore D(a, b) = 0 and finally II..:..a, I..:..bl=l..:..{I..:..I(I..:..a)b, (I..:..b)al}.

Thus from {l..:.. f(x)}g(x) = {l..:..g(x)}f(x) = 0 we derive I..:..f(x)=1..:..g(x). N~F(x) = N~G(x)

It follows that and

IN~+1F(x), N:+1G(x)1 = IN:F(x), N:G(x)1

whence, by 2.8, 3.972

I

IN~F(x), N:G(x) =

{N:F(x) = O}

+r

0 , which completes the proof. A~ '"'"' F(x) .

For n=O the equivalence becomes simply (1..:../(0»=0 +r(I..:..f(O»=O.

Since N:+ 1(f(x) = 0) =N:(f(x) = 0) + (1..:.. I(n +-1» therefore {N;+ I F(x) = O}

+r

{N~F(x) =

O}

& '"'"'

F(n+ 1) ;

75

TH}, LOGICAl. CONSTANTS

moreover

A~+I

[{N:F(x)=O}

,...., F(x)

--3>-

** {A~

A:,...., F(x)]

,...., F'(x)} &: ,...., F(n+ I), and so

--3>-

{[N:+IF(x)=O}

--3>-

{A:+I,...., F(x)}]

whence, by 3.7, Conversely [A: ,...., F(x)

--3>-

{N:F(x) = O}] A~,....,

whence

F(x)

--3>-

[A:+ 1

--3>-

,.....,

F(x)

--3>-

{N:-H F(x) = On

{N:F(x)=O}.

The equivalence 3.972 may also be expressed in the form {N:F(x) > O} 3.973

** E:F(x)

.

N:F(x)=N:(F(x) &:G(x))+N~(F(x) &: ,.....,G(x))

Let 8(n) denote the positive difference between the two sides of the equation. The proved equation l-.:-p={I-.:-(p+q)}+[I-,:-{p-l-(I-.:-q)}] suffices both to provc 8(0)=0 and 8(n)=8(n-+-I). 3.974

N:{F(x) v G(x)}=N:F(x) +N:(,...., F(x) &: G(x)).

Proof as for above, using the equation, l-.:-pq=(I-.:-p)+[I-.:-{q+(l-'-p)}] N:(F(x) &: G(x)) <: N:F(x)

3.975

In the equation {1-.:-(a-.:-b)}([a+(I-'-(p+q))]-'-[b+(l-'-p)]}=O which is proved by applying schema 2.7 to the variable q, we take N:(.P(x) &: G(x)) for a, N:F(x) for b, F(n+ I) for p and G(n+ I) for q. z » n--3>- {N:(y=x)=O}

3.976

Let P(n) denote the implication; then P(O) is equivalent to {l-'-(I-'-x)} (l-'-x)=O.

Since x> n-tl--3>-x > n and x> n+I-*(I-.:-!x,n+l!)=O therefore and so P(n)

P(n)

--3>-

&: x>n+

{x > n-t-l

-+ [N~(y=x)=O]}

1 --3>- {N~(y=x)+(I-'-lx, n+ I\)}=O

76

THE LOGICAL CONSTANTS

P(n) -+ {x>n+ 1 -+ N;+l(y=x)= O}

whence

P(n) -+ P(n+ I) .

i.e.

N;(y=n) = 1

3.977

For n=O this is simply 1-'-(0-'-0)=1 ; and N~+l(y=n+ I) =N;(y=n+ I) + (I-'-In+ I, n + 11)= I, by 3.976.

N;+'(y=r) = 1

3.978

For n=O this reduces to the previous theorem; furthermore N~+'+l(y=r)=N;+'(y=r)

+ {I-'-In+r+ I, r\}= N;+'(y=r). N~(y=x) =

It follows that

n;>x -+

for

n;>x -+ n=x+(n-'-x)

1,

-+ N~(y=x)=N~+(n-'-")(y=x)=1 , by 3.978.

From

x>n -+ {N;(y=x)=O}

follows

x>n -+ {N;(y=x) < I}

and from

x
follows

xn)

which proves and since (x>n) conclude

v

v

(x< n) -+ N;(y=x),;;;;, 1

(x,;;;;,n) is provable by double recursion, we

3.979

3.980 Since and

N:(x
~

~

(x
v

(x=n+ I)

(x=n+ 1) & ,...".,

the result follows by 3.971 and 3.974.

3.981

N:(x
For n = 0 this is evident; and

(x
77

THE LOGICAL CONSTANTS

N~+l(x,,;;;n+1) =N:+

1(x,,;;;n)+N:+l(x=n+

1) , by 3.980,

= N:(x,,;;;n) + 1, whence the result follows.

3.982

by 3.977,

x<.nBrf(x)=O-'?- [NZ{y=x Brf(y)=O}=I]

We consider first the case n=O. Since

x,,;;; 0

{f(x)=O

therefore

x<.O Br f(x)=O

-'?-

/(0)=0

and so

x,,;;; 0 Br /(x)=O

-'?-

x+f(O)=O

whence

x
-'?-

-'?-

f(O)=O}

which proves the case n= O.

By 3.972 [N;{y=x

Br

and so E;{y=x Br f(y)

f(y)=O}=O] =

whence x,,;;;n Br /(x)=O But

N~{y=x Br

O} -'?-

-'?-

-'?-

A~

N~{y=x Br

N~{y=x Br

f'J

{y=x Br /(y)=O}

f(y)

=

O}>O, by 3.965,

f(y)=O}>O, by 3.966.

!(y)=O}";;; 1 , by 3.979 and 3.973

and therefore x,,;;;n Br f(x)=O

-'?-

N:{y=x Br f(y)=O}= 1.

3.983

{NZ(f(y) From

=

0) = k+ I}

-<.+

E:U(x) = 0 Br N:(f(y) = 0 Br yi'x) = k}.

N~(f(y)=O)=N:(f(y)=OBr

follows x
=

Yi'x)+N:(/(y)=O Br y=x)

0 -+ N:(f(y) =0) = N:(f(y) = 0 .. y~x) + 1

whence x,,;;;n" f(x)=O -+ {N:(f(y)=O Br yi'x)=k -'?- N:(f(y) = 0) = k + I} i.e, x,,;;;n Br f(x)=O Br {N:(f(y) = 0 Br y~x)=k} -+

whence

{N:(f(y)=O)=k+ I}

78

THE LOGICAL CONSTANTS

Oonversely from

follows x.c;;;n & !(x)=O ~ {N:(f(y)=O)=k+ 1 ~ N:(f(y) = 0 & y#x)=k}

whence {N:(f(y)=O)=k+ I}

~

{(x.c;;;n & !(x)=O) ~ N:(f(y)=O & y#x)=k}

and so

{N~(f{y)=O)=k+

I} ->- [x.c;;;n

&

!(x)=O ~ {x.c;;;n & !{x)=O

&

N:(f(y) = 0

Y?'ox)=k}]

&

whence, by 3.951, {N:(f{y)=O)=k+ I} ~ {E:{x .c;;;n & !(x)=O) ->- E~(x.c;;;n & !(x)=O & N:(f(y) = 0 & y#x)=k)}

and therefore, by 3.957, {N:(f(y) = 0) = k + I} ~ {E~(f(x) = 0) -;. E~(f(x) = 0 &

N:(f(y) = 0 & y#x)=k)}.

But N=(f(y)=O)=k+I->-E:(f(x)=O), by 3.972, and so N:(f(y)=O)=k+ 1 ~ E:{!(x)=O

lit

N:(f(y)=O & yolx)=k}

which completes the proof. It follows from 3.972 and 3.983 that {N~(f(t)=O)=I} +'7 E:{!(x)=O lit NW(t)=O & t#x)=O}
{N~(f(t) = 0) = 2} - E:{f(x)=O & N~(f(t)=O & t#x) +'7

E:{!{x) = 0

&

& N~(f(t)=O

=

I}

E:[f{y) =0 &yclx &

t#x & t#y)=O]}

+->-E:E:U(x)=O &/(y)=o &x#y 8< A7(f{t)=O ~ (t=x) v (t=y))};

{N~(f{t) = 0) = 3}

+'7

E~{!{x) = 0 & N~(f{t) = 0 & t?'ox) = 2}

-o(~ E~[f(x)

=0 & E:E=U(y) =0 &x¥y&x#z&y#z&

&. !(z) =

0

79

THE LOGICAL CONSTANTS

& AW(t)=O & t#x ~ (t=y) +-> E:E:E:{f(x)

=0

&

& A~(f(t) = 0

~

(t=x)

f(Y)=o &x#y&x#z&y#z y

&

y

(t=z»)}]

f(z)=o

(t=y)

y

(t=z»}

and so on. These equivalences show that the operator N~{f(x)=O} has the desired properties. Thus N:(f(x)=O) has the value unity if and only if there is a unique x from 0 to n for which f(x)=O. And N:U(x) =0) has the value 2 if and only if there is an x and y from o to n, x different from y, such that f (x) = 0 and f (y) = 0 and f (t) docs not vanish for any other value of t from 0 to n, and so on.