principia mathematica - whitehead & russell. pag 161-180

8/2/2019 Principia Mathematica - Whitehead & Russell. Pag 161-180

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SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 139In the second line of the above proof, " f " . ) c p y v C P . 1 / "is taken as the value,

for the argument y , of the function" f " . ) X v c p y , " where x is the argument.A similar method of using *9'1 is employed in most of the followingproofs.

*1'1l is used, as in the third line of the above proof, in almost all stepsexcept such as are mere applications of definitions. Hence it will not befurther referred to, unless in cases where its employment is obscure orspecially important.*9'21. I - : . ( x ) c p x : > t x . : > : ( x ) c p x . : > . ( x ) . " , " x

I . e . if c p x always implies t x , then" c p x always" implies" tx always."The use of this proposition is constant throughout the remainder of thiswork.

Dem.I - *2'08.1 - . (1).*9'1.

I - (3) . *9'13.[(4).(*906)J[(5).(*1'01.*9'08)][(6).(*908)J[(7).(*101)J

: > I - : c p z : > ' o / z : > c p z : > ' o / z: > I - : ( : H : Y ) : c p z : > ' o / z : > c p y : > ' 0 / 2

: > 1-:: (2):: (:H:x): . ( 3 : Y ) : c p x :> 'o / x . : > . c p y : > " , " zI - : : ( z ) : : ( 3 : x ) : . c p x :> 'o / x . : > : ( 3 Y ) c p y : > " , " zI - : . ( : a x ) f " .)( c p x : > ' o / x ) : v : ( z ) : ( 3 Y ) f " . )C P Yv ' o / zf - : . ( :a x ) " - ' ( c p x : > ' o / a ; ) : v : ( 3 :Y ) " -' C P Y . v . ( z ) tf - : . ( x ) c p x:> 'o / x . : > : ( y ) . C P Y : > ( z ) 'o / z

(1)(2)(3)(4)(5)(6)(7)

This is the proposition to be proved, since" ( y ) C P Y " is the same propo-sition as " ( x ) . c p x , " and" ( z ) . 'o /z " is the same proposition as " ( x ) . 'o / x . "*9'22. f - : . ( x ) . c p x :> 'o / x : > : ( 3 : x ) c p x . : > ( 3 x ) ' o / x

I . e . if c p x always implies ' o / x , then if c p x is sometimes true, so ISThis proposition, like *9'21, is constantly used in the sequel.Dem.

I - , *2'08.I - (1) . *9'1 .I - (2) . *9'1 .I- (3) . *9'13 .[(4).(*9'06)]

: > I- : C P Y : > ' o /Y : > . C P Y : > ' f r y: > I - : ( 3 : z ) : C P Y : > ' o / y : > . C P Y : > ' o / z: > I - : . ( 3 x ) : . ( 3 : z ) : c p x :> 'o / x . : > . C P y : > 'o / z: > I - : : ( y ) : : ( :H : x ) :. ( 3 : z ) : c p x :> t x . : > C P Y : > ' o / z

I - : : ( y ) : : ( 3 x ) : . c p x :> 'o / x . : > : ( :H : z ) C P Y : > ' o / z

t.

(1)(2)(3)(4)(5)

[(5).(*1'01.*9'08)J I - : : ( 3 x ) . " ' ( c p x :> 'o / x ) : v : ( y ) : ( 3 : z ) . c p y :> 'o / z (6 )[(6).(*1'01.*9'07)] 1-:: ( :H : x) " - ' ( c p x : > ' o / x ) : v: ( y ) . " 'C P y . v , ( 3 : z ) . 'o / z (7)[(7).(*1'01.*9'01'02)] 1-:. ( x ) . c p x : > 'o / x . : > : ( :H :Y ) . C P y . : > . ( :H : z ) , 'o / z


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140 MATHEMATICAL LOGIC [PART IThis is the proposition to be proved, because ( ~ Y ) c p y is the same pro-

position as ( ~ x ) c p x , and ( ~ z ) v z is the same proposition as ( ~ x ) v x .*9'23. r: ( x ) c p x . ~ ( x ) c p x*9'24. r: ( ~ a : ) c p a : . ~ ( ~ a : ) c p a : [Id. *9'1:3'22]*9'25. r:. ( x ) . p v c p a : . ~ : p . v , ( x ) . c p x [*9'23. (*9'04)]

We are now in a position to prove the analogues of *1'2-'6, replacingone of the letters p , q , r in those propositions by ( a :) c p x or ( ~ x ) c p x . Theproofs are given below.*9'3. r :. ( x ) c p a : V ( x ) c p a : : ~ ( x ) c p x

Dem.r . *1'2. ~ 1 - . c p a :v c p a : . ~ . c p a : (1)r.(1).*91. ~ I - : ( ~ y ) : c p x v c p y . ~ . c p x (2)1 - . (2). *9'1:3. ~ 1 - : . ( x ) : . ( ~ Y ) : c p a :v c p y . ~ . c p a : (3)[(3).(*9'05'01'04)] 1 - : . ( x ) : . c p x . v. (y). c p y : ~ . c p a : (4)r . (4) . *9'21 . ~ I - : . ( x ) : c p x V (y) c py : ~ . ( x ) c p a : (5)[(5).(*9'03)] I - : . ( x ) c p x . V ( Y ) c p y : ~ ( x ) c p x : . ~ I - Prop

*9'31. r:. ( ~ x ) . c p x v ( : E { a : ) c p x : ~ ( : E { x ) c p xThis is the only proposition which employs *9'11.Dem.

r . *9-11'1:3. ~ I - : ( y ) : c p a :v c p y ~ . ( : E { z ) c p z (1)[(1).(*9'0:3'02)] 1 - : ( : E {Y ) . c p x v c p y . o , ( 3 : z ) . c p z (2)I - (2) . *9'13 . ~ I - : ( x ) : ( : E {y ) c p x v c p y ~ ( : E { z ) cp z ( : 3 )[(3).(*9'0:3'02)] 1 - : . ( : E { x ) : ( : E L Y ) c p x v c p y : ~ ( : E { z ) c p z (4)[( 4).(*9'05'06)] r:. ( : E { x ) c p x . V ( : . f l y ) C P , Y : ~ ( :. f l z ) c p z

*9'32. 1 - : . q ~ : ( x ) c p x v qDem. r.*l'3. ~ I - : . q . ~ : c p x . v . q

I - (1). *9'13 . ~ r :. ( x ) : . q ~ : c p x . v q[*9'25] ~ r:. q . ~ : ( x ) : c p a : v q[(2).(*9'03)] I - : . q ~ : (x ) cp x v q

*9-33. 1 - : . q . ~ : ( ~ ~ x ) . c p x . v , q [Proof as above]

(1)

(2)


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SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 141*9'34. 1-:. (X ). epX . : : > : p . v . (x ) epX

Dem.I - *1'3 . : : > I - : epx : : > P v epx (1)I - (1) . *9'13. : : > I - : (x ) : epx . : : > P v epx (2)I - (2). *9'21. : : > I- : (x ) e px .::> . (x ). p v epx (3)I - (3). (*9'04) . : : > I - Prop

*9'35. 1-:. (J Ix ). epx .::> : p . v. (J Ix ). epx [Proof as above]*9'36. 1-:. p v (x ) epx : : : > : (x ) epx v p

Dem.I - *1'4. : : > I-:p v e px .::> . e px v P (1 )I - (1). *9'13'21. : : > I - : (x ) p v epx . : : > . (x ) epxv P (2)I - (2) . (*9'03'04) . : : > I - Prop

*9'361. 1-:. (x ) epx v p : : : > : p . v . (x ) epx [Similar proof]*9'37. 1-:. P> v. (3 :x ). epx : : : > : (J Ix ) . ep .x . V P [Similar proof]*9-371. 1-:. (J Ix ) epx v . p : : : > : p v , (J Im ) e px [Similar proof]*9-4. I - : : P : v: q . v (x ). epx :. : : > : . q : v: p v . (x ). epx

Dem.I - *1'5 . *9'21. : : > I - : . (x ) : p v . q v ep.x: : : > : (x ) : q . v . p v epx (1 )1 - . (1) . (*9'04). : : > 1 - . Prop

*9'401. 1-:: p : v: q . v. (J Ix ). epx :.::> :. q : v : p v. (J Ix ) ep .x*9'41. 1 - : : p : v : (x ) epx v . r :. : : > : . (x ) e px : v : p v r*9'411. 1-:: p : v : (J I:v ) e px . v. r:.::> :. (3 :x ) epx : v: p v r*9'42. 1 - : : (x ) epx : v : q v r :. : : > : . q : v: (x ) epx . v. r*9'421. 1-:: (3 :x ) epx : v: q v r :. : : > : . q : v : (3x) . ep .x . v. r*9'5. I - ::p ::> q.::> :.p . v. (x ). cpx : : : > : q . v. (x ). epx

Dem.1 -.*1 '6 . ::> I-:.p ::>q .::> :pvcpy. ::> .qvepy (1)1 - . (1). *9'1. (*9'06). : : > I - :.p ::> q .::> : (J Ix ): pv s:, qv cpy (2)I - (2) . *9'13. (*9'04). : : > I - : : p : : > q . : : > : . (y) :. (J Ix ) : p v epx. : : > q v y (3 )[(3 ).(*9 -08)] 1 -::p ::>q .::> :.(J Ix ).",(pvepx).v .(y).qvepy (4)[(4).(*9-01)] I - : : p ::> q . : : > : . (x ) p v epx . : : > . (y) q v epy (5 )[(5).(*9'04)] I - : : p ::> q . : : > : . p . v. (x ). cPx : : : > : q v . (y). epy*9-501. 1-:: p ::> q . : : > : . p . v. (3 :x ). epx : : : > : q v . (3x) cpx [As above]

[As above][As above][As above][As above][As above]

*9'51. I - ::p .::> . (x ). epx :::> :. pv r.::> : (x ). epx . v. rDem.

I - *1'6 . : : > I - : . p : : > e p x : : > : p v r : : > epx v r (1)I - (1). *9'13'21. : : > I - : : (x ) p : : > epx . : : > : . (x ) : p v r. : : > epx v r (2)I- (2) . (*9'03'04) . : : > I - Prop


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142 MATHEMATICAL LOGIC [PART I*9'511. 1-::p.::>.('JIx).cpx:::>:.pvr.::>:('JIx).cpx.v.r [As above]*9'52. 1 - : : (x ). cpx. : : > . q : : : > : . (x). cpx. v. r: : : > . qv r

Dem.I- *1'6 . : : > I - : . cpx : : > q : : > : cpx v r : : > q v r (1)I- (1). *9'13'22. : : > I - : : ('J Ix). cpx::> q : : > : . ('J Ix) : cpx v r. ") . q V 11 (2)I - (2) . (*9 '05 '01) . : : > I- :: (x) cpx : : > q : ") :. (x ) cpx V r ") q v r (3)I - (3) . (*9'03) . ") I- Prop

*9'521, 1 - : : ('J Ix). cpx."). q : ") :. ('J Ix). cpx. v. r:"). qvr [As above]*9'6, (x). cpx, ",(x). cpx, ('J Ix) cpx and "'('J Ix). cpx are of the same type.

[*9'131, (7) and (8)]*9'61, If cp~ and v~are elementary functions of the same type, there is afunction cp~ v v~ . -

By *9'14'15, there is an a for which " ' i r a , " and therefore " c p a , " aresignificant, and therefore so is " cpa v ta," by the primitive idea of disjunc-tion. Hence the result by *9'1.5,

The same proof holds for functions of any numher of variables.*9'62, If c p ( ~ , y ) and Vz are elementary functions, and the x-argument tocp is of the same type as the argument to V' there are functions

(y) . cp(~ , y) v ' i r ~ , ('J Iy). cp (~, y) v . 'o /~.Dem.By *9'15, there are propositions c p ( , T , b ) and v a , where by hypothesis a :and a are of the same type. Hence by *9'14 there is a proposition cp (a, b),

and therefore, by the primitive idea of disjunction, there is a propositionc p (a, b) v 'o/a, and therefore, by *9'1.5 and *9'03, there is a proposition(y). cp (a, y) v . t a o Similarly there is a proposition ('J Iy). cp (a, y) v . ,y.a,Hence the result, by *'9'15.*9'63. If cp ( . v , y), ' 0 / ( ~ , y) are elementary functions of the same type, thereare functions (y). c p ( ~ , y). v , ( z ) 0 / ( ~ , z ) , etc, [Proof as above]

We have now completed the proof that, in the primitive propositions of*1, anyone of the propositions that occur may be replaced by (x ). cpx or('J Ix) cp x. Itfollows that, by merely repeating the proofs, we can show thatany other of the propositions that occur in these propositions can be simul-taneously replaced by (x).,y.x or ('JIx).,y.x, Thus all the primitive propositionsof *1, and therefore all the propositions of *2-*5, hold equally when someor all of the propositions concerned are of one of the forms (x ) cpx, ('J Ix) cpx,which was to be proved.

It follows, by mere repetition of the proofs, that the propositions of*1----*5 hold when p, q, r are replaced by propositions containing any numberof apparent variables,


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*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENTVARIABLE.

Summary of *10.The chief purpose of the proposrtions of this number is to extend to

formal implications (i.e. to propositions of the form ( . 1 ' : ) . x : : > 'o/x) as many aspossible of the propositions proved previously for material implications, i.e.for propositions of the form p : : > q. Thus e.g. we have proved in *3'33 thatP : : > q q : : > r . : : > P : : > r.

Put p = Socrates is a Greek,q =Socrates is a man,r = Socrates is a mortal.

Then we have "if' Socrates is a Greek' implies 'Socrates is a man,' and, Socrates is a man' implies' Socrates is a mortal,' it follows that' Socrates isa Greek' implies' Socrates is a morta1.'" But this does not of itself provethat if all Greeks are men, and all men are mortals, then all Greeks aremortals.

Putting a ; . =.x is a Greek,'o/a: . =.x is a man,Xx . =x is a mortal,

we have to prove(x ) x ::> 'o /x : (x ) 'o /x ::> xc : : : > : (x ) x ::> X x.

It is such propositions that have to be proved in the present number. Itwill be seen that formal implication (x).x: :> 'o/x) is a relation of two functions~and ' o / ~ . Many of the formal properties of this relation are analogous toproperties of the relation " p : : > q" which ex~resses material implication; it issuch analogues that are to be proved in this number.

We shall assume in this number, what has been proved in *9, that thepropositions of *1-*5 can be applied to such propositions as (x ). x and(s :x ) . x. Instead of the method adopted in *9, it is possible to takenegation and disjunction as new primitive ideas, as applied to propositionscontaining apparent variables, and to assume that, with the new meanings ofnegation and disjunction, the primitive propositions of *1 still hold. If this


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144 MATHEMATICAL LOGIC [PART Imethod is adopted, we need not take ( 3 : x ) . < l> x as a primitive idea, butmay put*10'01. ( 3 : x ) . < l> x . =." , ( x ) . " '< I> x Df

In order to make it clear how this alternative method can be developed,we shall, in the present number, assume nothing of what has been proved in*9 except certain propositions which, in the alternative method, will beprimitive propositions, and (what in part characterizes the alternative method)the applicability to propositions containing apparent variables of analoguesof the primitive ideas and propositions of *1, and therefore of their conse-quences as set forth in *2-*5,

The two following definitions merely serve to introduce a notation whichis often more convenient than the notation ( x ) . < l> x : : > tx or ( x ) < l> x = = t x .*10'02. c p x : : > x t x . =. ( x ) . < l> x : : > tx Df*10'03. < l > x = = x t x =. ( x ) . < l> x = = tx Df

The first of these notations is due to Peano, who, however, has no notationfor ( x ) < l> x except in the special case of a formal implication.

The following propositions (*10'1'11'12'121'122) have already been givenin *9, *10'1 is *9'2, *10'11 is *9'13, *10'12 is *9'25, *10-121 is *9'14, and*10'122 is *9'15. These five propositions must all be taken as primitivepropositions in the alternative method; on the other hand, *9'1 and *9'11 arenot required as primitive propositions in the alternative method.

The propositions of the present number are very much used throughoutthe rest of the work. The propositions most used are the following:*10'1. I - : ( x ) < l> x : : > < l > y

Le. what is true in all cases is true in anyone case.*10'11. If < l > y is true whatever possible argument y may be, then ( x ) < l> x istrue. In other words, whenever the propositional function < l > y can be asserted,so can the proposition ( x ) . < l> x .*10'21. 1 - : . ( x ) . p : : > c p x = = : p . : : > ( x ) < l> 1 C*10'22. 1 - : . ( x ) . < l> x : ( x ) tx : = = . ( x ) < l> x tx

The conditions of significance in this proposition demand that c p and tshould take arguments of the same type.*10-23. 1 - : ( x ) < l> x : : > p . = = . ( 3 : x ) c p x : : > P

I.e. if < l > x always implies p , then if < l > x is ever true, p is true.*10'24. 1 - : < l > y . : : > ( ~ x ) < l> x

I . e . if < l > y is true, then there is an x for which < l > x is true. This is thesole method of proving existence-theorems.


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SECTION B] THEORY 01


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146 MATHEMATICAL LOGIC [PART I*10'12. f - : . (x ) .pv cpx .:> :p.v.(x ) . c px [*9'2,5]

According to the definitions in *9, this proposition is a mere exampleof "q : > q, " since by definition the two sides of the implication are differentsymbols for the same proposition. According to the alternative method, onthe contrary, *10'12 is a substantial proposition.*10'121. If" cpx II is significant, then if a is of the same type as x , " cpa" issignificant, and vice versa. [*9'14]

It follows from this proposition that two arguments to the same functionmust be of the same type; for if xand a are arguments to c p~, " c px " and "cpa"are significant, and therefore x and a are of the same type. Thus the aboveprimitive proposition embodies the outcome of our discussion of the vicious-circle paradoxes in Chapter IIof the Introduction.*10'122. If,for some a, there is a proposition cpa, then there is i,' functioncp~, and vice versa. [*9'15 ]*10'13. If c p ~ and 't~take arguments of the same type, and we have "f- .cpx"and" f - tx," we shall have" f - cpa; 't x ."

Dem.By repeated use of 9-61'62-6:3-1:31 (3), there IS a function " 'cp~ v . . . . . . , t ~ .

Hence by *2'11 and *3-01,f - : " 'cpa; v "'ta;. v c px . tal (1)f - (1). *2'32. (*1'01). : > f - : . cpx . : > : tx .:> . cpx . tal (2)f - (2) . *9-12. : > f - Prop

*10'14. f - : . (a ;) . c px : (a: ) tx : : > 1 >y . tyThis proposition is true whenever it is significant, but it is not always

significant when its hypothesis is significant. For the thesis demands that1 > and tshould take arguments of the same type, while the hypothesis doesnot demand this. Hence, if it is to be applied when 1 > and tare given, orwhen 't is given as a function of 1 > or vice versa, we must not argue from thehypothesis to the thesis unless, in the supposed case, c p and ttake argumentsof the same type.

Dern.f - *10-1 .f - *10'1.

: > f - : (a;) .cpa;.:>.cpy: > f - : (a;). t.:> . t y

(1)(2)

f - . (1). (2). *10'13. : > f - : ( a; ). c pa; . : > cpy: (x ) . tx . : > ty :[*3'47J : : > f - : . (a;). c pa;: ( x). tx : : > . cpy . ' o / y : . : > f - Prop (I


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SECTION BJ THEORY OF ONE APPARENT VARIABLE 147

Dem.*10'2. I - : . (x) . p v x. = = : p. v , (ox). x

I - *10'1 . *1'6 . ) I - : . p v (ox). ,x : ) . P v y :.[*10'l1J ) 1-:. (y) :.p. v. (x). x:).p v y:.[*10'12J ) I - : . p . v . (x) . x : ) . (y) .p v y1-.*10'12.I - (1) . (2) .

) I - : . (y) . p v y . ) :p . v (x) . x) 1-.Prop.

(1)(2)

*10'21. 1-:. (x) .p) x. = = : p . ) . (x). x [*10'2 7J]This proposition is much more used than *10'2.

*10'22. 1-:. (x) . x. -rx. = = : (x). x : (x) . -rxDern. I - *10'1 .

[*3'26J[*10'11J[*10'21JI - (1) . *:3-27.[*lO'11J

) I - : (x). x. -r;c. ) . y. + s -) .y:

) I - : . (y) : ( ; 1 ; ) x . -rx . ) !J :.) I - : . (a:) . a; . tx. ) . (y). y) I - : . (x) . x . -raJ. ) . -rz :.) I - : . (z) : (x). (J;. -rx. ) . -rz :.

(1)

(2)

[*10'21J ) I - :.(x).x.ta;.).(z).tz (3)1-. (2). (3). Comp v D I - : . (x). x. tx. ): (y). y: (z). tz (4)I - *10'14'11 .[*10'21J1-.(4).(5).

) I - : . (y) :. (x). ,),: (o j ) t' : ) !J . ty:.) I - : . (x). J~: (;)') . tx :) . (y). y . -rY (.5)) I - Prop

The above proposition is true whenever it is significant; but, as waspointed out in connexion with *10'14, it is not always significant when"(x) x : (x) . tx" is significant.*10'221. If x contains a constituent X (x, y, z, ... ) and -rx contains a con-stituent X (x, u, v, ... ), where X is an elementary function and y, z, ... u, v , ...are either constants or apparent variables, then 5; and t~take argumentsof the same type. This can be proved in each particular case, though notgenerally, provided that, in obtaining and - r from X, X is only submittedto negations, disjunctions and generalizations. The process may be illustratedby an example. Suppose x is (y).X(x, y).). ex, and tx is f o : , ) . (y). X (x, y).By the definitions of *9, x is (3:Y). "'X (x, y) vex, and 'lrx is (y). '" fxvx (x, y).Hence since the primitive ideas (x). Fo: and (3:x) .Ftc only apply to functions,there are functions "'X (5; , y ) v e~ , ",;5; v X (5; , y ) . Hence there is a proposi-tion " 'X (a, b ) v ea. Hence, since "p v q" and" r " o . J p " are only significant

10-2


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148 MATHEMATICAL LOGIC [PART Iwhen p and q are propositions, there is a proposition X (a, b ) . Similarly, forsome u and v, there are propositions f' ) - f u v X (u, v) and X ('It, v). Hence by*9'14, u and a, v and b are respectively of the same type, and (again by *9'14)there is a proposition f' ) fa v X (a, b ) . Hence (*9'15) there are functionsf ' ) X (a, fj ) v B a , f ' ) fa v X (a, fj), and therefore there are propositions

(3 :Y ) f ' )X (a, y) v ea, (y) . P

Dem.1 - . *4'2. (*9'03). J 1 - : . ( a : ) .,,-,cpx V p. = = : ( x ) .r ..J c pa ;. v .p :[(*9'02)J = = (3 :x ) . c px . :> . P (1)I - (1) . (*1'01). J I - Prop

In the above proof, we employ the definitions of *9. In the alternativemethod, in which (3 :x ) cpx is defined in accordance with *10'01, the proofproceeds as follows.*10'23. I - : . (.: t) cpx :> p = = : (3 :x ) . c px :> . Pu-

1 - . Transp. (*10'01).:> I - : . ( 3 " a : ) . cpx . :> .p: = = :" 'p.:> . (x ) .,,-,cpx :[*10'21J = = : (x ) :"-'p.: > .,,-,c px : (1)[*10'lJ[TranspJ[*10'l1J[*10'21JI - *10'1 .[TranspJ[*1O'11'21J[(l)JI- (2) . (3) .

:> : . < '..!cpx:.:> I - : . (x ): c px .:> p: J :(x ) :" 'p.:> .",cpx :

:>:(3:x).cpx. :>.p (3)

(2)

:> I - PropWhenever we have an asserted proposition of the form p :> cpx , we can

pass by *10'11'21 to an asserted proposition p : : > (x ). cpx . This passage isconstantly required, as in the last line but one of the above proof. It willbe indicated merely by the reference" *10'11'21," and the two steps which itrequires will not be separately put down.


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SECTION B] THEORY OF ONE APPARENT VARIABLE 149*10'24. r: c p y . : > ( ~ x ) c p x

This is *9'1. In the alternative method, the proof is as follows.Dem . r. *10'1. : > r: ( x ) . , , - , c p x .: > . , , - , c p y :

[Transp] : > r : c p y . :> . " - ' ( x ) " - ' c p x :[(*10'01)] : > r .Prop

*10'25. r: ( x ) . c p x . o , ( ~ x ) . c p , r : [*10'1'24]*10'251. r: ( x ) " - ' c p x : > ' " (C x ) c p , T } [*10'2;). Transp]*10'252. r : " - ' ( ~ x ) . c p x } . = = . ( x) . , , - , c p x [*4'2. (*9'02)]*10'253. r : , , - , ( x ) c p x } . = = . ( ~ x) . , , - , c p x [*4'2. (*9'01)]

In the alternative method, in which ( ~ x ) . c p x is defined as in *10'01,the proofsof*10'252'253 are as follows.*10'252. r : . . . . . , {(~x ) . c p x ) = = (x). , , - , c p a : [*4'13. (*10'01)]*10'253. r :",{(x). c p a : } = = . (~x~) . " - ' c p a :

Dem , r .*10'1. :> r : (x) . c p . T . : > c p y [*2'12] : > . " ' ( " 'c p y ) :[*10'11'21]:> r: ( ,T ) . c p a : . :> . ( y ) ." ' ( " 'c p y ) :L Transp] :> r : '" [ ( y ) ' " ( ' " c p y ) } :>.'" [ (a :) c pa : :[(*10'01)] :>r: (~y) . r - . . > c p y . : > ",{(x) . c p a : } (1)r . *10'1. :>r : ( y ) . " - ' ( " 'c p y ) :>."'( " - ' C P a : ) .[*2'14] :>. c p a : :[*10'11'21] :> r : ( y ) " - ' ( ' " c p y ) :>. ( x ) c p a : :[Transp] :>r : '" [ ( a :) c p t r } : > ' " [ ( y ) " - ' ( '" c p y ) } [(*10'01)] :> . ( ~ y ) . " , c p y (2)r. (1). (2) : > r. Prop

*10'26. r:. ( z ) c p z :> tz :c p a : : : > ; ' o / x [*10'1. Imp]This is one formof the syllogism in Barbara. E.g. put c p z . =. z is a man,tz .=.z is mortal, x =Socrates. Then the proposition becomes:"If all men are mortal, and Socrates is a man, then Socrates is mortal."Another form of the syllogism in Barbara is given in *10'3. The two

forms, formerly wrongly identified, were first distinguished by Peano andFrege.*10'27. r:. ( z ) c p z : > O /Z :> : ( z ) c p z . :> ( z ) 'o / z

This is *9'21. In the alternative method, the proof is as follows.


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150 MATHEMATICAL LOGIC [PART IDem.

1-. *10'14.[AssJ[*10'1 J[*10'21J

:> I - : . (z). cpz:> ' frz: (z) cpz : :> . c py:> 'fry . c py :> . 'fry : .

:> I - : . (y) :. (z). cpz:> ' frz : (z) cpz : :> . 'fry:.:> I - : . (z) cpz:> 'frz : (z) cpz : :> . (y) 'fry (1)

1-. (1). Exp.:> 1-. Prop*10271. 1-:. (z) cpz = = 'frz. :> : (z). cpz. = = (z) 'frz

Dem. 1-.*10'22. :> 1-:. Hp.:>:(z).cpz:>'irz:[*10'27J :> : (z) . cpz :> . (z). tz (1)I - *10'22. :> I - : . Hp . :> : (z). tz:> z:[*10'27J :> : (z) . 'frz. :> . (z). cpz (2 )I - (1). (2). Cornp , :> I - Prop

*10'28. 1-:. (x ) c px :> 'frx . :> : (~x ) cpx . :> . (~x ) 'frxThis is *9'22, In the alternative method, the proof is as follows.Dem.I- *10'1. :> I - : . (x ) . c px :> 'frx . :> . cpy :> ' iry .[Transp] :> . '" 'fry :>'" c py : .[*10'l1'21J:> f-:. (x ). c px :> 'frrc .:> : (y) "''fry :> "'c py:[*10'27J :>: (y ) . " "'fry .:>. (y ) .f"o.Jcpy:[Transp] :> : (~[ y). c py. :>. (~y). 'fry:. :> f-. Prop

*10'281. f-:. (x ). cpx = = tx .:> : (~x ). cpx . = = . (~'1:). 'frx [*10'22'28. CompJ*10'29. 1-:. (x ). cpx :> 'frx : (x ). cpx :> X x : = = : (x ): cpx :> tx . X x .

Dem.I - *10'22. :> f- :. (x ) cpx :> t;1: : (x ) cpx :>X x :

= = : (x ) : x :> tx. cpx :>X x (1)I - *4'76. :> I - : . cpx :> tx . cpx :> X x . = = : cpx . :> . tx . Xx : .[*10'l1J :> 1-:. (x ): . cpx :> tx . x :> X x . = = : cpx .:> . tx X x :.[*10'271J :>1-:. (x ): cpx :> tx . cpx :> X .x : = = : (x ): c px .:> . tx. X x (2 )I - (1) . (2) :> f - PropThis is an extension of the principle of composition.

*10'3. I - : . (x ). cpx :> 'frx : (x ). tx:> X x : :> . (x ) cpx :> X xThis is the second form of the syllogism in Barbara.Dem.

I - *10'22'221 . :> I- : Hp. :> . (x ) cpx :> 'irx tx :> X x .[Syll.*10'27J :> (x ) cpx :> X x: :>1-.Prop


13/20

SECTION B] THEORY OF ONE APPARENT VARIABLE 151*10'301. 1 - : . ( x ) c p x = = ' l r x : ( x ) " , " x = X x : : : > ( J : ) C P X= X x

Dem.I- *10'22 '221 : : > I - : . H p . : : > : ( ,x ; ) c p x = " , " ( J : " , , " x = X , v :[*422.*1027J : : > : ( x ) . c p , , : = X x : . : : > 1 - . Prop

In the second line of the proofs of *10'3 and *10'301, we abbreviate theprocess of proof in a way which is often convenient. In *10'3, the full processwould be as follows:

I- Syll :: >I- : c p a : : : > ' l r x . ' l r x : : > X ; J ~ ' : : > c p . x : : > X .v :[*10'l1J::> I- : ( : c ) : c p a : ) ' I r a :. ' I r . r : > x.r. ::>. c p a : : : > X i C :[*10'27J::> I- : ( : c ) x : : > " , , " x ' l r x ) X .~ : : : > ( x ) x ) X .x

The above two propositions show that formal implication and formalequivalence are transitive relations between functions.*10'31, 1 - : . ( x ) . x : : > ' l r x . ) : ( x ) : c p x X x . : : > . ' l r x X x

Dem,I- Fact. *10'll . ) 1 - : . ( ;1) :. c p . T ) " , " '1 ' ::>: c p . x . X i . ::>. " ," ,1 : X x (1)I- (1 ) . * 10 '27. ::>I- Prop

*10'311. 1 - : . ( x ) a ; = ' I r . x . ::>: ( x ) : c p x . X x = . ' l r x X xDem.

1 - . *4'36. *10 'll . ::>1-:. ( x ) : . c p x = ' I r . ' E . ::>: c p {r . X . v . = . ' I r : I' . X i V (1 )I- (1 ) . *10 '27. ::>I- Prop

The above two propositions are extensions of the principle of the factor.*10'32, 1 - : c p x = x ' l r x = = . ' I r .'E = x c p x

Del1'l .I- *10'22 . ::>I- : c p x = x " , " x . = = . c p . 1 : : : > x 0 / / " " , " x ) x c p .1 : '[*4'3J = . - o /a : : :> xc p . 1 ; c p x : :> x - 0 / ( 1 ;

This proposition shows that formal eq uivalence is symmetrical.*10'321. 1 - : c p x = x V x c p x = x X X ::> " , " x = x X ,V

Dem.I - * 10 ':3 2 Fact. ::>I- : Hp ) . - 0 / / / ; = x C P , 1 ) . c p x = x X x.[*10 ':301 J ) . ' I r .~ ; = x X 'C : :: >I- Prop

*10'322. 1 - : ' I r . x = x c p x X x = x r p .' C ::>. " ," x = x X xDem.

I- * 10':3 2 ::>I- : Hp ::>. " , x = x c p x . c p . x = x X X .[*10':301J ) . V x = x X x : : : > 1 - . Prop


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15 2 MATHEMATICAL LOGIC [PART I*10'33. 1-:. (x): cpx. p: = = : (x ). cpx :p

Dem.:::>~ :. (x ) : c px p : :::> cpy. p. (1)

[*3'27] :::>p (2)~ . (1) . *3'26.:::> 1-:. (x ): cpx. p::::>. cpy:[*10'11'21] :::>~ :. (x ): cpx . p: :::> (y). cpy (3 )~ (2) ( 3) . :::>~ :. (x ) : c px . p : :::> (y) cpy : p ( 4 )~ *10'1 [Fact][*10'11'21 ]~.(4) . (5) .

:::>~ :. (y) c py . :::> c px :.:::>~ :. (y) cpy : p : :::> cpx p :.: ::>~ : . (y) c py : p : :::> (x ) : cpx p:::>~ . P rop

(5)

*10'34. ~ :. (ax) cpx:::>P = = : (x ) cpx :::> pThis follows imm ediate ly from *9 '05'01 and *1'01. In the alte rnative

m e thod , the proof is as follows.Dem. ~ *4'2 (*10'01) . :::>

~:. (ax ) cpx :::>P = = : " " { ( x) "" (cpx :::>p) :[*4'61.*10'271] = = : " " { ( a ) : cpx . ""p) :[*10'33] = = : " " { (x) cpx : ""p} :[*4'53] = = : " " { (x ) cpx} V P :[*4'6] = = : (x ) cpx . :::> P

*10'35. ~ :. (ax ).p. cpx . = = :p: (ax ). cpxDem .

I - *3'26. :::>~ : p c px . :::> P :[*10'11 J :::>~ : (x) : p c px : : > P :[*10'23J :::> - : (ax) . p cpx :::> P (1)~ *3'27 [*10'l1J[*10'28J

:::>~:p.cpx.:::>.cpx:: ::>~ : (x ) : p cpx :::> cpx ::::> - : (ax) p . cpx . :::> (ax ) cpx (2)

~ *3'2 :::> - : . p . :::> cpx :::> p. cpx .[*1 0'1 1'2 1J:::> ~ :. p. :::>: (x ): cpx .:::>. p. cpx :[*10'28] :::> (ax ) . cpx . :::> (ax) . p c px (3 )~ ( 1) (2) (3) Im p. :::>~ P rop


15/20


16/20

154 MATHEMATICAL LOGIC [PART I*10'41, 1 - : . (x) cpa : v (x) ,yx: : : : > (x) cpx v ,y x

Dem,I- *10'1 . : : : > I - : (x ) cpx : : : > cpy [*2'2] : : : > cpyv,yy (1)I- *10'1 . : : : > I - : (x) V x. : : : > ,yy [*1'3] o , cpy v ,yy (2)1- . (1) . (2) . *10'13 . : : : > I - : . (x ) cpx . : : : > cpy v,yy : (x ) vx : : : > cpy V Vy:,[*3'44] : : : > I - : . (x) cpx V (x) vx: : : : > cpy V Vy[*10'l1'21] : : : > 1 - : . (x). cpx. v , (x). V x::::> . (y). cpyvVy:.:::> 1 - . Prop

Observe that in the above proof the uses of *2'2 and *1'3 are onlylegitimate if cpy and Vy have overlapping ranges of significance, for other-wise, if y is such that there is a proposition cpy, it is such that there isno proposition Vy, and conversely.*10'411, 1 - : . cpx =x Xx. Vx =x ex : : : > : cpx V vx = x . X x vex

Dem,I - *10'14 . : : : > I - : . Hp . : : : > : cpx = Xx v x = ex:[*4'39J : : : > : cpxv V x. =. X /E vexI - (1) . *10'11'21 . : : : > I - Prop

*10'412. 1 - : cp x = x ,y x. = ." 'cpx = x""vx [*4'l1. *10'll'271]*10'413, 1 - : . cpx =x Xx. vx =x ex . : : : > : cpx : : : > vx =x Xx:::> exu-

1 - . *10'411'412 . : : : > I - : . Hp . : : : > : ""cpx V Vx. =x "'Xx V ex[(*1'01)] : : : > : cpx :::> Vx. =x ' X x:::> ex :. : : : > 1 - . Prop*10'414. 1 - : . cpx =x Xx. vx =x ex : : : > : cpx = vx =x Xx = ex

Dem.I - *10'413 Y ' . t , e 'X e *10'32 . : : : > I - : . Hp . : : : > : tx : : : > cpx =x ex:::> Xx (1)', 't', X '1 - . *10'413 . (1). *10'4. : : : > 1 - . Prop

(1)

The propositions *10'413'414 are chiefly used in cases where either X isreplaced by cp or e is replaced by ,y , in which case half the hypothesis becomessuperfluous, being true by *4'2,*10'42, 1-:. (a x) cpx v . (a x) . tx: = . (ax) cpx v tx

Dem,I - *10'22.[~'ll][*4'51'56.*10'271 ]

: : : > I - : . ( ~ ) ' " cpx : (x ) '" tx : = . (x ) "" cpx '" tx :.: : : > f - :.",{(x) . ""cp .:c : (x ) "'tx} . = .",{(x ) .",cpx "'tx} :.: : : > f - : . '" t (x ) .,...,;c px } v . '" {(x) '" tx} :=.'" f ( . : c ) .,...,;(cpx v tx)} :.: : : > I - : . (a x) cpx v . (ax) tx: = . (ax) cpx v vx :.

: : : > I - . Prop


17/20

SECTION B] THEORY OF ONE APPARENT VARIABLE 155This proposition is very frequently used. It should be contrasted with

*10'5, in which we have only an implication, not an equivalence.*10'43. 1 - : c p z = = z 'o /Z . c p x = = c p z = = z t ' o / X

Dem.I - *10'1 . J I - : c p z = = z 'o / z . J . c p x = = ' o / XI- (1) . *5';32. J 1 - . Prop

*10'5. I - : . ( ; H : x ) c p x . 'o /x , J : ( ; H : x ) c p x : ( ;H : a : ) ' o / : vDem.

(1)

I - *3'26 . *10'11 . J I- : ( x ) : c p x . ' o / x J . c p x :[*10'28J J I - : ( ; H : : ) c p x . 'o / x J . ( ;H : t c ) . c p .x (1)I - *3'27 . *10'11 . J I - : . (x ) : c p x . 'o /a : J . ' o / x :[*10 '28] J I - : ( ; H : x ) c p x . ' 0 / : 1 ' . J . ( ; H : x ) ' o / x (2)1 - . (1).(2). Cornp , J 1 - : . Prop

The converse of the above proposition is false. The fact that thisproposition states an implication, while *10'42 states an equivalence, is thesource of many subseq uen t differences between formulae concerning logicaladdition and formulae concerning logical multiplication.*10'51. I - : . " ' { ( ; H : x ) . c p x . 'o / x } = = : c p x . Jx "'ttV

Dem.I - *10'252 . J I - : . ' " t ( ; H : x ) c p x . 'o / x } = = : ( x ) " ' ( c p x 'o / x ) :[*4'51'62.*10'271 J = = : ( x ) : c p x J . " ' 'o / x : . J I - Prop

*10'52. 1 - : . ( ~ x ) c p x J : ( . 1 :) c P ; vJ P' = = pDern.

I - *5'5 . J I - : : Hp . J :. p . = = : ( 3 : x ) . c p x . J . P :[*10'23J = = : ( x ) . c p a : J p : : : : > 1 - . Prop

*10'53. 1 - : . " ' ( ; H : x ) c p x . J : c p x Jx ' o / xDem.

1 - . *2'21 . *10'11 . : : >I - : . ( x ) : . " - ' c p a : J : c p x . J . 'o / ,x : .[*10'27J J I - : . ( . x ) " ,c p x . J : ( x ) : c p x . J . ' o / x : .[*10'252J J I- :."-'(3:.1:). c p a : . J: ( x ) : c p x . J. ' o / x : . : : > 1 - . Prop

*10'541. 1 - : : c p y : : > y p v 'o / y : = = : p . v . c p y Jy ' o / yDem.

I - ~'2 . (*1'01) . : : > I - : . c p y . : : > y P v 'o / y : = = : ( y ) '" c p y v p v 'o /Y :[Assoc.*10271] = = : ( y ) . p v " ' c p y v ' o / y :[*10'2] = = : p. v. (y ) . " , c p y v ' o / Y :[(*I'OI)J = = : p . v c p y : : > y 'o / y : . : : > I - Prop


18/20

156 MATHEMATICAL LOGIC [PART IThe above proposition is only needed in order to lead to the following:

10'542. 1-:. q,y. :>y p:> ty : = = : p . : > . epy : >y t y [*10'541 ~ J ! - J_This proposition is a lemma for *84'43.

*10'55. 1-:. (R x) . epa: t x : ep:c :>x tx : = = : (R x) epx: cfxc :>x 'o/xDem.

I- *4'71 . : > I - : . cpx:> tx : > : epx ttr: = = q,x (1)1-. (1) . *10'll'27 . : >I - : . epx :>x tx : > : (x ) : q,x . ta: = = cpx :[*10'281] : > : (R x). epx . 'o/x . = = . (: : f[x) . cpx (2)I - (2) . *5'32 . : > I- Prop

This proposition is a lemma for *117'12'121.*10'56. 1-:. epx :>x tx : ( ::f[x ) . cpx . X x: : > ( : : f [x ) . 'o/x X x

Dem.I - *10'31 [*10'28]

: > I - : . q,x :>x tx : : > : cpx X x :>x tx X ,x :: > : (R x). cpx . X x . : > . (: : f[x) . 'o/x . X x (1)

I- (1) . Imp. : > I- PropThis proposition and *10'37 are used in the theory of series (Part V).

*10'57, 1-:. cpx :>x . 'o/x v Xx: : > : epx :>x 'o/x v. ( ::f[x ) . cpx . X xDem.J - *10'51 . Fact. : >I - : . epx :>x 'o/x v X x: " "( ::f [x ) . cpx X x: : > : cpx :>x . yx v X x: cpx :>x ""X x:[*10'29] : > : cpx :>x . 'o/x v X x . "" X x:[*5 '61] :>:cpx ,:>x''o/x (1)I - (1) . *5'6 . : > I- Prop


19/20

*11. THEORY OF TWO APPARENT VARIABLES.Summary of *11.In this number, the propositions proved for one variable in *10 are to be

extended to two variables, with the addition of a few propositions havingno analogues for one variable, such as *11'2'21'23'24 and *1l535567." < P (x, y)" stands for a proposition containing x and containing y; whenx and yare unassigned, < p (x, y) is a propositional function of a: and y. Thedefinition *11'01 shows that "the truth of all values of c p (x, y)" does notneed to be taken as a new primitive idea, but is definable in terms of "thetruth of all values of ,yx." The reason is that, when tc is assigned, c p (x, y)becomes a function of one variable, namely y, whence it follows that, forevery possible value of x, "(y) . c p (x, y)" embodies merely the primitive ideaintroduced in *10. But "(y ) c p (x, y)" is again only a function of one variable,namely oi, since y has here become an apparent variable. Hence the definition*1l'01 below is legitimate. We put:*11 '01. (x , y) c p ( { [ ' , y) = : (x ) : (y) c p V, y) Df*11 '02 . (x , y , z ) . < p (x , y , z ) = : (x ) : ( y , z ) . c p ( . ' L ' , y, a) Df*11 '03 . (ax, y) c p (x, y). = : (3:x) : (=fY) c p (x, y) Df* 11 '04. (3 : x , y, z) c p (x , y, z) = : (3 :a) : (3 :Y , z) . c p (:r, y, z) Df*11 '05 . c p (x , y) . Jx ,y ,y (x , y) : = : (a', y) : < p (x , y) . J ,y ( : L ' , y) Df*11 '06 . c p (a', y) = = z,y ,y (x , y) : = : (m , y) : c p (x , y) . = = . ,y ( t C , y) DfAll the above definitions are supposed extended to any number of variablesthat may occur.

The propositions of this section can all be extended to any finite numberof variables; as the analogy is exact, it is not necessary to carry the processbeyond two variables in our proofs.

In addition to the definition *1l'01, we need the primitive propositionthat "whatever possible argument x may be, c p (x , y) is true whateverpossible argument y may be" implies the corresponding statement with xand y interchanged. Either may be taken as the meaning of " < p (x, y) istrue whatever possible arguments o : and y may be."

The propositions of the present number are somewhat less used thanthose of *10, but some of them are used frequently. Such are the following:


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158 MATHEMATICAL LOGIC [PART I*11'1. I - : (x , y) cp (a:, y) . ) cp (z , w)*11'11. If cp (z , w) is true whatever possible arguments z and w may be,then (x , y) . cp (x , y) is true

These two propositions are the analogues of *10'1'11.*11'2. I- : (x , y) cp (x , y) = = . (y, x) cp (x , y)

I.e. to say that" for all possible values of x, cp (x, y) is true for all possiblevalues of y" is equivalent to saying" for all possible values of y, cp (x , y) istrue for all possible values of x."*11'3. I - :.p .). (x , y). cp (x , y): = = : (a:, y):p .). cp (x , y)

This is the analogue of *10'21.*11'32. 1-:. (x , y) : cp (x , y) ) t' (x , y) : ) : (x , y) . cp(x , y) . ) . (x , y). t' (x , y)

I.e. "if cp(x , y) always implies t' (x , y), then 'cp (x , y) always' implies't' (x, y) always.''' This is the analogue of *10'27. *1l'33'34'341 arerespectively the analogues of *10'271'28'281, and are also much used.*11'35. 1-:. (x , y) : cp (x , y) . ) P : = = : (3:x, y) . cp (x , y). ) P

I.e. if cp (x , y) always implies p , then if cp(x , y ) is ever true, p is true.This is the analogue of *10'23.*11'45. 1-:. (3:x, y) : p c p (x , y) : = = : p : (3:x , y) c p (x , y)

This is the analogue of *10'35.*11'54. 1-:. (3 :x , y) cpx."o /y. = = : (3 :x) . cpx : (3 :Y ) "O /Y

This proposition is useful because it analyses a proposition containingtwo apparent variables into two propositions which each contain only one." cpx " o /y " is a function of two variables, but is compounded of two functionsof one variable each. Such a function is like a conic which is two straizhtolines: it may be called an "analysable" function.*11'55. 1-:. (3 :x , y). cpx."o / (x , y) = = : (3:x) : cpx: (3 :Y ). t' ( . 1 7 , y)

I.e. to say" there are values of a : and y for which cpx. tx, y) is true" isequivalent to saying" there is a value of a : for which cpx is true and for whichthere is a value of y such that " 0 / (x, y) is true."*11'6. I - : : (3 :x):. (3 :Y ) cp (x , y) V y : Xx:. = = : . (3 :Y ):' (3:x).cp (x ,y). xx:"o /y

This gives a transformation which is useful in many proofs.*11'62. 1-:: cpx " 0 / (x , y). )x,y' X (x , y) : = = : . cpx. )x: t' (x , y). )11 X (x, y)

This transformation also is often useful.

principia mathematica - whitehead & russell. pag 161-180

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