principia mathematica - whitehead & russell. pag 141-160

8/2/2019 Principia Mathematica - Whitehead & Russell. Pag 141-160

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SECTION A] THE LOGICAL PRODUCT OF TWO PROPOSITIONS 119*3'45, 1 - : . p ::>q : :>: p . r . ::>. q . r

This principle show s that w e may multiply both sides of an im plicationby a common facto r; hence it is called by Peano the" princip le of the facto r,"W e shall refer to it as "Fact," It is the analogue, fo r multiplication , of theprim itive proposition *1'6,

Dem."'rI - Syll . ::> I - : . p : : > q : :>: q : :>'" r , : : > p ::> '" r :r

[Transp] ::> : '" (p : :>'" r) . ::> . '" (q ::> '" 1'):,[Id.(*101.*3Ol)]::> 1 - . Prop

*3'47, 1 - : . p ::>r . q ::>s . : : > : p . q . : > . r. SThis proposition, or rather its analogue for classes, w as proved by Leibniz ,

and ev iden tly pleased him , since he calls it "prreclarum theorerna *,"Dem.

I - *:3 '26. ::> I - : . p ::>r . 'I ::>s : :>: p ::>r :[Fact] ::> : p . q . ::>. r . q :[*3 '22] ::> : p q . ::>. q . rI - *3'27 ::> I - : . p::> r . q ::>s . : :>: q ::>s :[Fact] ::> : q . r . : :> s . r :[*3 '22] : : > : 'I . r . : :> . r . SI - (1) . (2) * :1 '03 . *2 'R3 . ::>

(1)

(2)

I - : . p ::>r . q ::>s . :> : p . q . : :> . r . S :. ::> I - Prop*3'48, 1-:. p ::>r q ::>s : :>: p V q . : :>. r V s

This theorem is the analogue of *3 '47,Dem.

I - *3'26 ::> I - : . p:> r , q ::>s , ::>: p : : > r:[Sum ] ::>: p v q . ::>. r v q :[Perm ] ::> : pvq .::> . qvrI - *3 '27 . ::> I - : . p ::>r . q ::>s . : :>: q ::>s :[Sum ] ::> : q v r . : :> . s V r :

(1 )

[Perm]I - (1 ) . (2 ). *2 '83 . ::>

I - : . p::> r . q ::>s : :>: p V q : > r V s :. ::> I - Prop: : > : q v r . : :>. r V S (2)

* Philosophical works, Gerhardt's edition, Vol. VII. p. 223.


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;M. EQUIVALENCE AND FORMAL RULES.Summary of *4 .In this number, we shall be concerned with rules analogous, more or less,

to those of ordinary algebra. It is from these rules that the usual" calculusof formal logic" starts. Treated as a "calculus," the rules of deduction arecapable of many other interpretations. But all other interpretations dependupon the one here considered, since in all of them we deduce consequencesfrom our rules, and thus presuppose the theory of deduction. One verysimple interpretation of the" calculus" is as follows: The entities consideredare to be numbers which are all either 0 or 1; "p:> q" is to have the value 0if p is 1 and q is 0; otherwise it is to have the value 1; ....... is to be 1 if Pis 0, and 0 if p is 1; P q is to be 1 if p and q are both 1, and is to be 0 inany other case; p v q is to be if P and q are both 0, and is to be 1 in anyother case; and the assertion-sign is to mean that what follows has thevalue 1. Symbolic logic considered as a calculus has undoubtedly muchinterest on its own account; but in our opinion this aspect has hitherto beentoo much emphasized, at the expense of the aspect in which symbolic logicis merely the most elementary part of mathematics, and the logical pre-requisite of all the rest. For this reason, we shall only deal briefly with whatis required for the algebra of symbolic logic.

When each of two propositions implies the other, we say that the two areequivalent, which we write" p = = q." We put*4'01. p = = q . =p :> q . q :> P Df

It is obvious that two propositions are equivalent when, and only when,both are true or both are false. Following Frege, we shall call the truth-value of a proposition truth if it is true, and falsehood if it is false. Thus twopropositions are equivalent when they have the same truth-value.It should be observed that, if p = = q, q may be substituted for p without

altering the truth-value of any function of p which involves no primitiveideas except those enumerated in *1. This can be proved in each separatecase, but not generally, because we have no means of specifying (with ourapparatus of primitive ideas) that a function is one which can be built up out


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SECTION A] EQUIVALENCE AND FORMAL RULES 121of these ideas alone. We shall give the name of a truth-fnrwtion to a functionf(p) whose argument is a proposition, and whose truth-value depends onlyupon the truth-value of.its argument. All the functions of propositions withwhich we shall be specially concerned will be truth-functions, i.e. we shallhave p = q . : : > f( p) = f(q)The reason of this is, that the functions of propositions with which we dealare all built up by means of the primitive ideas of *1. But it is not auniversal characteristic of functions of propositions to be truth-functions.For example, "A believes p" may be true for one true value of p and falsefor another.

The principal propositions of this number are the following:*4'1. I - : p : : > q . = . ' " q : : > ' " p~11. 1 - : p = q : : : : : : : ' " p : : : : : : : " - ' q

These are both forms of the" principle of transposition."~13. I - - P = . . . . . . , ( " - ' p)

This is the principle of double negation, i.e. a proposition is equivalent tothe falsehood of its negation.*4'2. I- P : :: :: ::*4'21. 1 - : p :::::::. = 0 q = P*4'22. 1 - : p : :: :: :: q = r . : : > P = r

These propositions assert that equivalence lR reflexive, symmwtl'ical andtransitioe.*4'24. 1 - : p 0 = . pop*4'25. 1 - : p 0 = . p v p

I.e. p is equivalent to " and p" and to ,(p or p," which are two forms ofthe law of tautology, and are the source of the principal differences betweenthe algebra of symbolic logic and ordinary algebra.~3. I - : p . q . = 0 q . P

This is the commutative law for the product of propositions.*4'31. 1 - : P v q 0 = . q v PThis is the commutative law for the sum of propositions.The associative laws for multiplication and addition of propositions,

namely~32. 1 - : (p . q) r. :::::::p . (q . r)*4'33. I-:(pvq)vr.=.pv(qvr)

The distributive law in the two forms


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122 MATHEMATICAL LOGIC [PART I~4. I - : . p . q V r . = = : P q V P l'~41. 1-:. p V q . 1" : = = p V q . p V r

The second of these forms has no analogue in ordinary algebra.~71. 1-:. P : : > q = = : p = = p qI.e. p 'implies q when, and only when, p is equivalent to p. q. This

proposition is used constantly; it enables us to replace any implication byan equivalence.*4'73. 1-:. q . :J : p = = . p . q

Le. a true factor may be dropped from or added to a proposition withoutaltering the truth-value of the proposition.

~'Ol, P = = q =.p :J q q : : > P Df~02. P = = q = = r . =.p = = q q = = r Df

This definition serves merely to provide a convenient abbreviation.*4'1,*4'11.~12.*4'13,*4'14,*4'15,~'2,*4'21,

l-:p:J q. = = . " - ' q:J "-' PI- :p==q.::::."- 'p=="- 'qI - :p== "-' q . = = . q== "-' pI-.p=="'(" 'p)I - : . p . q . : : > r: = = : p "-' r . : : > " - ' qI - : . p . q . : : > " - ' l':= = : q . 1'. : : > " - ' pI - .p==pI- :p==q.==.q==p

*4'22, I - : p = = q . q = = r . : : > P = = rDem.

[*2'16'17][*2'16'17 . *3'47'22][*2'0:1-15][*2'12'14][*3':37 . *4'13][*3'22 . *4'13'14][Id. *3'2][*3'22]

I - *3'26 . :J I - : p = = q q = = r :J . p = = q [*3'26] : : > p::> q (1)I - *3'27 . :J I - : p = = q q = = r . :J . q = = 'I' [*3'26] :J . q::> r (2)I - (1) . (2) . *2'83 :J I - : p = = q q ::::r. :J . P : : > r (3)I - *3'27 . : : > I - : p = = q q = = r . : : > q = = r .[*3:27] :J.r::>q (4)I - *3'26 . : : > I- : p = = q . q = = r . : : > P = = q .[*:3 '27] : :>.q:Jp (5 )I - (4) . (5) . *2'83 . :J I - : p = = q . q = = r . :J . r :J p (6)I - (3) . (6) . Comp , :J I - Prop


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SECTION A] EQUIVALENCE AND FORMAL RULES 123Note. The above three propositions show that the relation of equivalence

is reflexive (*4'2), symmetrical (*4'21), and transitive (*4'22). Implicationis reflexive and transitive, but not symmetrical. The properties of beingsymmetrical, transitive, and (at least within a certain field) reflexive areessential to any relation which is to have the formal characters of equality.*4'24. 1 - : p . = = . p .p

Dem.I - *3'26 . ) I - : p p . ) . PI - *3'2. ) I - : . p . ) : p ) .p. p .:[*2'43] ) I - : p ) . P P1-.(1).(2).*3'2.) 1-. Prop

*4'25. 1-:p . = = . p V P [Tau t . Add ~ ]Note. *4'24'25 are two forms of the law of tautology, which is what chiefly

distinguishes the algebra of symbolic logic from ordinary algebra.

(1)

(2)

*4'3. I - : p . q . = = . q p [*3'22]Note. Whenever we have, whatever values p and q may have,

1>(p , q).). 1> (q , p ),1>(p , q). = = . 1> (q , p ).

For 1 c J > (p , q ). ) 1 > (q , p ) q , p ) : c J > (q , p ) ) 1 > (p , q ).p, q

we have also

*4'31. 1-: p v q = = . q v p [Perm]*4'32. 1-: (p q) T = = . p . (q . 1')

Dem.I - *4'15. ) I - : . p . q . ) '" T: = = : q r . ) '" p :[*4 '12 ] = = : p . ) . '" (q . r) (1 )1-. (1). *4'1l . ) I - : ' " (p. q. ) . ,....,) = = . , . . . . , [p. ) . '" (q. T ) } :[(*1'01.*3'01)]) 1-. Prop

Note. Here "(1)" stands for "I-:.p.q.)."'r:==:p.:> .......(q.r),"which is obtained from the above steps by *4'22. The use of *4'22 willoften be tacit, as above. The principle is the same as that explained inrespect of implication in *2'31.*-4'33. I-:(pvq)vr.==.pv(qvr) [*2'31'32]

The above are the associative laws for multiplication and addition. Toavoid brackets, we introduce the following definition:


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124 MATHEMATICAL LOGIC [PART I*4'34. P > q . r .=p . q ) r Df~36. 1-:. p = = q . J : p r . = = . q . r*4'37. 1-:. p = = q . J : p v T = = . q v r*4'38. 1-:. p = = r . q = = S J : p . q = = . r . S~39. I-:. p = = r . q = = s , J : p v q . = = . r V s*4'4. I-:. p . q v r . = = : p . q . v p . r

This is the first form of the distributive law.

[Fact. *3'47][Sum. *3'47][*3'47 . *4'32 . *:3'22][*3'48'47 . *4'32 . *3'22]

Dem.I- . *3'2 . J I-:: p J : q . J . P . q :. p . J : r J . P . r ::[Comp] J I-:: p . J :. q . J . P . q : r . J . P . r :.[*3 '48] J :. q v r . J : p . q . v . p . r (1)1-. (1). Imp. J I - : . P q v r , J: P > q. v. p. r (2 )I - *3'26 . J I - : . p . q . J . P : P . r . J . P :.[*3'44 J J I-:. p q v . p r :J . p (3)1-. *3'27. J 1-:. r - q. J. q: - r , J . r:.[*3'48] J I-:. p q v . p . r : J . q v r (4)I- . (3) . (4) . Comp . J I-:. p . q v . p . l':J .p . q v r (5)1-.(2).(5). JI-.Prop

*4'41. I-: .p.v.q.r:==.pvq.pvrThis is the second form of the distributive law-a form to which there

is nothing analogous in ordinary algebra. By the conventions as to dots," " "()". v . q . r means p v q. r .Dem.

1-.*326.Sum. JI-:.p.v.q.r:J.pvq (1)I - *3'27 . Sum. J I - : . p v . q . r: J . P v r (2)I- (1). (2) . Comp . J I - : . p . v . q. r: J . p v q p v r (3)I - *2' 53 . *:3'47. J I-:. p v q . p v r J : '" p J q , ....., J T :[Comp] J : '" p . J . q . r :[ *2 ' 54 ] J :p . v . q. r (4)I- . (3) . (4) . J I-. Prop

~42. 1-:. p . = = : p . q v . p '" qDem.

I-. *3 '21. J I-:. q v '" q J : p J P q v '" q :.[*2 '11] J I-:p . ,. qv '" q (1)I- . *3'26 . J I- : p q v '" q . J P (2 )I- (1) . (2) . J I-:. p . = = : p q v '" q :[*4'4] = = : p . q v . p . ' " q :. J I-. Prop


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SECTION A] EQUIVALENCE AND FORMAL RULES 125ri43. 1-:. p . = = : p v q p v ,....,q

Dem.I - *2'2 .[Comp]

:>1- :p . :> .pvq :p . :> .pV I - : . ' " p :> '1 :> : . . . . . . ,p : > " - ' q :> P :.p[Imp][*2',53.*3'47]r . (1) . (2) .

:> 1-:. '" p :> '1 ' '" p :> '" '1 ' :> p :.:> I - : . p v '1 P v "-' '1 :> P:> I - Prop

(2)

*4'44. 1-:. p . = = : P v p . qDem.

I - *2'2 . :> I - : . p . :> : p . v . p . '1I - Id . *3'26 . :> I - : . p :> p : p . ' 1 ' :> p :.[*:3'44] :> I - : . p . v . p . q : :> . PI - (1) . (2) . :> I - Prop

(1)

(2)

*4'45. 1 - : p . = = . p p v q [*3'26. *2'2]The following formulae are due to De Morgan, or rather, are the propo-

sitional analogues of formulae given by De Morgan for classes. The firstof them, it will be observed, merely embodies our definition of the logicalproduct.*4'5. I - : p '1 = = . " - ' ( ' " p v "-' ( 1 ) [*4'2. (*:3'01)J*4'51. I - : "- ' (p q ) = = . " - ' p v "-' '1 [*4.-512JM52. I - : p ......,1 = = ' " ( . . . . . . , p v q ) [ *4'5 "-',/'. *4'13 J*4'53. I - : " '(p ." ''1 ).== ......,p vq [*4'52'12]M54. I - : '" p . q . = = ' " (p v'" q ) [ * 4 " ) ' : : 'p l !. *4'13]*4'55. I - : '" (" - ' p . q ). = = p v '" '1 [*4'.54'12]*4'56. I - : ......,. "-' '1 = = . . . . . . , (p v ( 1 ) [*4';')4 ~ '1 *4'1:3J*'57. I - : , . . . . , ( " - ' p " " q ) . = = p v q [*4'56'12]

The following formulae are obtained immediately from the above. Theyare important as showing how to transform implications into sums or intodenials of products, and vice versa. It will be observed that the first of themmerely embodies the definition *1'01.


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126~6. 1 - :~61. 1 - :~62. 1 - :~63. 1 - :~64. 1 - :*4'65. 1 - :*4'66. 1 - :

MATHEMATICAL LOGIC [PART Ip')q.==.r-.>pvq

r-.> (p ') q) . = = . p "-' 'L

"-' (p ') r-.> q) = = . p . qr-.>p')q.==.pvq

"-' ( '" p ') q) . = = . ' " p . r-.> qr-.>p')r-.>q.==.pv"-'q

~67. 1 - : '" ( " - ' p ') r-.> q) . = = . " - ' p q*4'7. I - : . p ') q. = = : p. '). p. q

Dem.

[*4'2. (*1'01)][*4'6'11'52][ * 4 ' 6 r-.>qJ[*4'62'11'5][*2-5a'54][*464-11.56][*4'64 "'(/ ][*4'66-11'54]

1 - . *3'27 . Sy11. ') I - : . P > ' ) P q: ') . p ') q (1)I - Comp ') I - : . p ') p p ') q ') : p . ' ) . P q:.[Exp] ')I-::p')p.'):.p')q.'):p.').p.q::[Id] ') I - : . p ') q . ' ) : p . ') . P q (2)I - (1) . (2) . ') I - Prop

*4'71. 1 - : . p ') q . = = : p . = = . p. qDem.

I - *3'21 .[*3'26]I - *3'26.

' ) I - : : p . q . ') p : ') :. p . ') . p q : ') : p = = . p q ::' ) I - : . p . ') .p q: ') : p = = . p q (1)' ) I - : . p. = = . p. q: ') : p. ') . p . q (2)(3)- (1) . (2) . ') I - : . p ') . p q : = = : p = = . p . q

I - (8). *4'7-22. ') I - PropThe above proposition is constantly used. It enables us to transform

every implication into an equivalence, which is an advantage if we wish toassimilate symbolic logic as far as possible to ordinary algebra. But whensymbolic logic is regarded as an instrument of proof, we need implications,and it is usually inconvenient to substitute equivalences. Similar remarksapply to the following proposition.~72. 1 - : . p ') q = = : q = = . p v q

Dem.I - *4'1 . ') I - : . p ') q. = = : ' " q ') "-' P :[*4'71 -~~' -~ P J = = : ' " q . = = . ' " q . ' " p :[*4'12] = = : q = = . ' " ( " - ' q . r-.> p) :[*4 ' 57 ] ==:q.==.qvp:[*4'31] = = : q. = = .pvq:. ') 1 - . Prop


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SECTION A] EQUIVALENCE AND FORMAL RULES 127* 4 '7 3 , 't-:.q .':):p .== .p .q [Simp.*4'71]

This proposition is very useful, since it shows that a true factor may beomitted from a product without altering its truth or falsehood, just as a truehypothesis may be omitted from an implication,~'74, ' t - : . " 'P ':) : q = = p v 'I [*2'21. *4'72]

[ p : = = : q r J . PDem.

I - *4'1'39 . ':) I - : . q ':)P . v . r J p : = = : " - ' p : : :> " - ' 'I . v "-'p : : : > r..J r :[*4 ' 78 ] ==:"""'p.:::>.r..Jqv,,-,r:[*2'15J =: : r..J ( r..J q V r..J 1') : : : > P :[*4'2.(*3'01)] =:: q. 1'.).p:.) 1 - . PropNote, The analogues, for classes, of *4'78'79 are false, Take, e,g, *4 ' 78 ,

and put p = English people, q = men, r = women, Then p is contained in qor r, but is not contained in q and is not contained in 1',*4'8, I - : p :::> '" P > =: "-' P [*2'01. SimpJ~'81. 1 - : " - ' p :: :> P> = = P [*2'18. SimpJ~'82, 1 - : p : : : > q . } J ) " - ' q. = = . " - ' p [*2'6,5 . Imp. *2'21 . CampJ*483,I-:p)q."-'p:::>q.=:.q [*2'61.Imp. Simp. CompJ

Note. *4'82'8:3 may also be obtained from *4'43, of which they arevirtually other forms,*4'84, 1 - : . p =: q . ':) : p : : : > r . = = . q : : : > r [*2'06. *3'47]*4'85, 1 - : . 1 ) = = q : : : > : r :: :> p . = = r ':) q [*2'05. *:3'47]~'86, 1 - : . p = = q ':) : p = = r . = = q = = r [*4'21'22]~'87, 1 - : . p q ':) r : = = : p : : : > q ) r : = = : q : : : > p:::> r : =: : q p : : : > r

[Exp . Comm . Imp J*4'87 embodies in one proposition the principles of exportation and

importation and the commutative principle,


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*5. ~IISCELLANEOUS PROPOSITIONS.Summary of *5.The present number consists chiefly of propositions of two sorts: (1) those

which will be required as lemmas in one or more subsequent proofs, (2) thosewhich are on their own account illustrative, or would be important in otherdevelopments than those that we wish to make. A few of the propositionsof this number, however, will be used very frequently. These are:*5'1. I - : p q : : > P = - q

I.e. two propositions are equivalent if they are both true. (The statementthat two propositions are equivalent if they are both false is *5'21.)*5'32. 1 - : . p. : : > q = = r: = - : P q . = - . p. r

Le. to say that, on the hypothesis p, q and r are equivalent, is equivalentto saying that the joint assertion of p and q is equivalent to the joint assertionof p and r. This is a very useful rule in inference.*5'6. I - : . p "" q . : : > T : = - : p . : : > q v TLe. "p and not-q imply r" is equivalent to "p implies q or r"Among propositions never subsequently referred to, but inserted for their

intrinsic interest, are the following: *5'11'12'13'14, which state that, givenany two propositions p, q, either p or ""P must imply q, and p must implyeither q or not-q, and either p implies q or q implies p ; and given any thirdproposition r, either p implies q or q implies r * .

Other propositions not subsequently referred to are *5'22'23'24; in theseit is shown that two propositions are not equivalent when, and only when,one is true and the other false, and that two propositions are equivalentwhen, and only when, both are true or both false, It follows (*.5'24) thatthe negation of "p. q. v . ......., '" q" is equivalent to "p . .......,l : v. q . ......,."*5'54'55 state that both the product and the sum of p and q are equivalent,respectively, either to p or to q.

The proofs of the following propositions are all easy, and we shall thereforeoften merely indicate the propositions used in the proofs.

* Cf. Schroder, Vorlesungen ilber Algebra der Logik, Zweiter Band (Leipzig, 1891), pp. 270-271, where the apparent oddity of the above proposition is explained.


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SECTION A] MISCELLANEOUS PROPOSITIONS 129*5'1. I - : p . q . ::> P = q*5'11. 1 - : p ::> q v . "' p ::>q*5'12. 1 - : p ::>q . v. p ::>"'q*5'13. 1 - : p : : > q v . q ::>P*5'14. 1 - : p ::>q v . q ::>r*5'15. 1 - : P = = q . v p = = "'q

Dem.

[*3 '4 '22J[* 2'5 1'5 4 ][*2 '52 '54J[*2 '521][S im p. T ransp . *2 '21]

I - *4 '61 . ::>I - : '" (p : : > q ) ::>. P '" q [*5 '1 ] ::>. p = - "-J q :[*2 '54J : :>1-:p::>q.v.P="'-JqI - *4 '61 . ::> I - : "-J (q ::>p) ::>. q "'-Jp .[*5 '1][*4 '12]

(1)

::>. q = = "'-Jp.::>.p=-"'-Jq:

[*2 '54J : :>1-:q::>p.v.P=="'-JqI - (1 ) . (2 ) . *4 '41 . ::>I - Prop

*5'16, I - ."'-J(p = = q. P ="'-Jq )Dem,

(2)

1 - . *3'26. : :> I-:p = - q .p ::> rvq .::> .p ::> q .p ::> "'-Jq .[*4 '82 ] ::>- r - (1 )I - *3 '27 . ::>I - : p = = q p ::> "'q : :>. q ::>p . p ::> " 'q .[Syll] : :>.q::>",q.[AbsJ ::>.r'Vq (2 )1 - . (1). (2 ). Com p v D I - :p== q.p ::> "'q .::> ."'-JP ."'q .[*4 '65 g,p ] ::> ."'-J ("'q ::>p) (3)p, q1 - . (3 ). Exp . ::>I - :. p = = q . ::> :p ::> "'-Jq . ::>."'( rv q::> p ):[Id .(* l'01)] ::>: rv(p ::> "'q ) v ." '- J( " '- Jq ::>p ) :[*4 '51. (*401)J

*5 '17, I - :p vq .r'V(p . q ). = .p==r'VqDem.

1-.*4 '64 '2 1. ::> I-:pvq .==.r-.Jq ::>p (1)1 - . *4 '63 . T ransp , ::>I - :" '-J (p . q ). = .p ::> "'-Jq (2 )I - (1 ). (2 ). *4 '38 '21 . : : > I - Prop

&&~ 9


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130 MATHEMATICAL LOGIC

*5 ' 19 , r .f')(p == f')p )*5'21. r : f')P ."'q . : : > P = = q

[PART I

[*,5'15'16. *5'17 P = s. ]J _= =ro .;q ]p, q

[*5'18 ~. *4'2][*5'1 . *4'11]*5'22. r : . "'(p - = q) . = = : p . "'q . v q ."'P [*4'61'51'39]*5'23. r : . p = = q. = = : p . q v . "'p. "'q [*5'18. *5'22 ~q . *4'13'36 ]*5 ' 2 4 , r :."'(p. q. v ."'p ."'q). = = : p ."'q. v , q ''''P [ * 5 ' 2 2 ' 2 3 ]*5'25, r:. p v q = = : p : : > q . : : > q [*2 '62 '68]

From *5'25 it appears that we might have taken implication, instead ofdisjunction, as a primitive idea, and have defined "p v q" as meaning"p : : > q : : > q," This conrse, however, requires more primitive propositionsthan are required by the method we have adopted,*5 ' 3 , r : . p q : : > r : = = : p . q : : > P . r [Simp. Comp . Syll]*5'31, r: . r. p ::> q : : : > :p . : : > . q . r [Simp. Comp]*5'32, r : . p : : > q = = r : = = : p . q . = = . p . r [*4'76. *3'3'31 . *5'3]

This proposition is constantly required in subsequent proofs,*5'33, r : . p . q : : > r . = = : p : p . q . : : > r*5'35, 1-:. p : : > q . p : : > r . : : > : P : : > q = = r*5'36, r: p p = = q = = . q . p = = q*5 ' 4 , r : . .p . : :> .p : :>q :== .p ::>q*5 ' 41 , r : . p : : > q : : > P : : > r : = = : p . : : > q : : > r*5'42, r:: p : : > q : : > r : = = ' : . p . : : > : q . : : > P . r

[*4'73'84. *5'32][Comp . *5'1][Ass. *4'38][Simp. *2'48][*2'77'86][*5'3 . *4'87]

*5'44, 1-:: p ::> q . : : > : . p::> r , = = : p . : : > . q . r [*4'76. *5'3'32]*5 ' 5 , r : .p.: :>:p::>q.==.q [Ass. Exp.Simp]*5'501, r : . p . : : > : q . = = . p = = q [*5'1 . Exp . Ass]*5'53, r:. p v q v r : : > s : = = : p::> s. q::> s. r : : > s [*4'77]*5'54. r:. p q = = . p : v : p q = = . q [*4'73 . *4-44 . Transp . *5'1]*5'55, r:. p v q = = . p : v : p v q = = . q [*1'3 . *5'1 . *4'74]*5 ' 6 , r : . p - r - .o , r: = = : p .o . q V?' [*4'87 ~q . *4'64'85]*5'61. r : p v q '" q . = = . p . '" q [*4'74 . *5'32]*5'62, r : . p q V .,.....,q : = = p v,.....,

lJ


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SECTION A] MISCELLANEOUS PROPOSITIONS 131*5'63, f - : . p v q = = : p v ,"""p q*5'7, f - : . p v r . = = . q v r : = = : r , v . p = . q [*4 '74 . *1 '3 . *5 '1 *4.37]*5.71. 1 - : . q ::>.....,r.::>: p v q . r = = . p r

In the fo llow ing proof, as always henceforth, "H p" m eans the hypothesiso f the proposition to be proved.

Dem,I - *4'4 ::> I - : . p v q . r = = : p r . v . '1 r (1 )1 - . *4'62'51.::> f - : : Hp , ::>:'''''''('1' r) :.[*4 '74] ::>:. p r v . '1 r : = . : p . r (2 )1 - . (1) . (2). *4 '22. ::> I - Prop

*5'74, 1 - : . p ::> '1 = = r : = = : p ::>'1 = = . p ::>rDem,

f- *,) '41 . ::> f - : : p ::>'1 ::> p ::>r : = . : p . ::> '1 ::>r :.p ::>r ::> p ::>'1 : = = : p ::>. r ::>'1 (1 )

I - (1) . *4'38 ::> f - : : p ::>'1 = = . p ::>r . = = : . p : :>. '1 ::>r : p ::> r ::> ' 1 : .[*4 '76] = = :.p.::> . '1= r:: ::> f -. Prop

*5'75, f - : . r : :>1 - : . Hp.: :>: P


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SEOTION B.THEORY OF APPARENT VARIABLES.

*9. EXTENSION OF THE THEORY OF DEDUCTION FROM LOWER TOHIGHER TYPES OF PROPOSITIONS.

Summary o f *9.In the present number, we introduce two new primitive ideas, which

may be expressed as "cpx is always* true" and" cpa; is sometimes * true,"or, more correctly, as "cpx always" and "cpx sometimes." When weassert "cpx always," we are asserting all values of cp1:, where "cpJj I, meansthe function itself, as opposed to an ambiguous value of the function (cf.pp. 15, 42); we are not asserting that c f > x is true for all values of ai, because,in accordance with the theory of types, there are values of a : for which" cpx"is meaningless; for example, the function cp1: itself must be such a value.We shall denote" cpx always" by the notation(x ) cpx ,where the" (x)" will be followed by a sufficiently large number of dots tocover the function of which" all values" are concerned. The form in whichsuch propositions most frequently occur is the" formal implication," i.e. sucha proposition asi.e. "c px always implies ,y.x."universal affirmative "allproperty ,y."

We shall denote " cpx sometimes" by the notation(~x ) cpx .

(x ) : cpx :l.yx ,This is the form III which we express the

objects having the property cp have the

Here "a" stands for" there exists," and the whole symbol may be read"there exists an x such that cpx."

In a proposition of either of the two forms (x ) cpx , (~x ) cpx , the to iscalled an a ppa ren t variab le. A proposition which contains no apparentvariables is called "elementary," and a function, all whose values are

* We use "always" as meaning "in all cases," not "at all times." A similar remark appliesto "sometimes."


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SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 133elementary propositions, is called an elementary function, For reasonsexplained in Chapter II of the Introduction, it would seem that negationand disjunction and their derivatives must have a different meaning whenapplied to elementary propositions from that which they have when appliedto such propositions as (x ) cpx or (~x) . cpx , Ifcp~ is an elementary function,we will in this number call (x ). cpx and (a x ). cpx "first-order propositions."Then in virtue of the fact that disjunction and negation do not have thesame meanings as applied to elementary or to first-order propositions, itfollows that, in asserting the primitive propositions of *1, we must eitherconfine them, in their application, to propositions of a single type, or wemust regard them as the simultaneous assertion of a number of differentprimitive propositions, corresponding to the different meanings of "dis-junction" and "negation," Likewise in regard to the primitive ideas ofdisjunction and negation, we must either, in the primitive propositions of *1,confine them to disjunctions and negations of elementary propositions, or wemust regard them as really each multiple, so that in regard to each type ofpropositions we shall need a new primitive idea of negation and a newprimitive idea of disjunction. In the present number, we shall show how,when the primitive ideas of negation and disjunction are restricted toelementary propositions, and the p, q, r of *1----*5 are therefore necessarilyelementary propositions, it is possible to obtain definitions of the negationand disjunction of first-order propositions, and proofs of the analogues, forfirst-order propositions, of the primitive propositions *1'2-'6. (*1'1 and*1'11 have to be assumed afresh for first-order propositions, and the analoguesof *1'7'71'72require a fresh treatment.) It follows that the analoguesof the propositions of *2-*5 follow by merely repeating previous proofs. Itfollows also that the theory of deduction can be extended from first-orderpropositions to such as contain two apparent variables, by merely repeatingthe process which extends the theory of deduction from elementary to first-order propositions, Thus by merely repeating the process set forth in thepresent number, propositions of any order can be reached, Hence negationand disjunction may be treated in practice as if there were no difference inthese ideas as applied to different types; tha.t is to say, when " "'p" or"p v s " occurs, it is unnecessary in practice to know what is the type ofp or q, since the properties of negation and disjunction assumed in *1 (whichare alone used in proving other properties) can be asserted, without formalchange, of propositions of any order or, in the case of p v q, of any two orders.The limitation, in practice, to the treatment of negation or disjunction assingle ideas, the same in all types, would only arise if we ever wished toassume that there is some one function of p whose value is always r - . J p ,whatever may be the order of p, or that there is some one function of p andq whose value is always p v q, whatever may be the orders of p and q. Suchan assumption is not involved so long as p (and q) remain real variables,


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134 MATHEMATICAL LOGIC [PART Isince, in that case, there is no need to give the same meaning to negationand disjunction for different values of p (and q), when these different valuesare of different types. But if p (or q) is going to be turned into an apparentvariable, then, since our two primitive ideas (x). cpx and (:3 :x ). c px both de-mand some definite function cp, and restrict the apparent variable to possiblearguments for cp, it follows that negation and disjunction must, whereverthey occur in the expression in which p (or q) is an apparent variable, berestricted to the kind of negation or disjunction appropriate to a given typeor pair of types. Thus, to take an instance, if we assert the law of excludedmiddle in the form "I-.pv"'p"there is no need to place any restriction upon p: we may give to p a valueof any order, and then give to the negation and disjunction involved thosemeanings which are appropriate to that order. But if we assert" I - (p). p v",p"it is necessary, if our symbol is to be significant, that" p v"'p" should bethe value, for the argument p, of a function cpp; and this is only possible ifthe negation and disjunction involved have meanings fixed in advance, andif, therefore, p is limited to one type. Thus the assertion of the law ofexcluded middle in the form involving a real variable is more general thanin the form involving an apparent variable. Similar remarks apply generallywhere the variable is the argument to a typically ambiguous function.

In what follows the single letters p and q will represent elementa1 'Y pro-positions, and so will "cpx," "y x," etc. We shall show how, assuming theprimitive ideas and propositions of *1 as applied to elementary propositions,we can define and prove analogous ideas and propositions as applied topropositions of the forms (x) cpx and (:3 :x ). c px . By mere repetition of theanalogous process, it will then follow that analogous ideas and propositionscan be defined and proved for propositions of any order; whence, further, itfollows that, in all that concerns disjunction and negation, so long as propo-sitions do not appear as apparent variables, we may wholly ignore thedistinction between different types of propositions and between differentmeanings of negation and disjunction. Since we never have occasion, inpractice, to consider propositions as apparent variables, it follows that thehierarchy of propositions (as opposed to the hierarchy of functions) will neverbe relevant in practice after the present number.

The purpose and interest of the present number are purely philosophical,namely to show how, by means of certain primitive propositions, we candeduce the theory of deduction for propositions containing apparent variablesfrom the theory of deduction for elementary propositions. From the purelytechnical point of view, the distinction between elementary and other propo-sitions may be ignored, so long as propositions do not appear as apparentvariables; we may then regard the primitive propositions of *1 as applying


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SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 135to propositions of any type, and proceed as in *10, where the purely technicaldevelopment is resumed.

It should be observed that although, in the present number, we provethat the analogues of the primitive propositions of *1, if they hold for propo-sitions containing 11 , apparent variables, also hold for such as contain n+ 1,yet we must not suppose that mathematical induction may be used to inferthat the analogues of the primitive propositions of *1 hold for propositionscontaining any number of apparent variables. Mathematical induction is amethod of proof which is not yet applicable, and is (as will appear) incapableof being used freely until the theory of propositions containing apparentvariables has been established. What we are enabled to do, by means of thepropositions in the present number, is to prove our desired result for anyassigned number of apparent variables-say ten-by ten applications of thesame proof. Thus we can prove, concerning any assigned proposition, that itobeys the analogues of the primitive propositions of *1, but we can only dothis by proceeding step by step, not by any such compendious method asmathematical induction would afford. The fact that higher types can only bereached step by step is essential, since to proceed otherwise we should needan apparent variable which would wander from type to type, which wouldcontradict the principle upon which types are built up.

Definition of Negation. We have first to define the negations of (x ). cpxand (J Ix ). cpx . We define the negation of (x ) . cpx as (J Ix ). " 'c px , i.e. "it isnot the case that cpx is al ways true" is to mean "it is the case that not-cpxis sometimes true." Similarly the negation of (J Ix ). cpa :is to be defined as(x ) " ,c px . Thus we put*9'01. ",{(x ). cpx }. =. (J Ix ) .",c px Df*9'02. "'{(J Ix). cpx }. =. (x ) .",c px Df

1'0 avoid brackets, we shall write ",(x ). cpx in place of '" {(x ) . cp x}, and" '(J Ix ). cpx in place of '" {(J Ix ) . cp x}. Thus :*9'011. ",(x ). cpx. = . ......,(x ). cpx } Df*9'021. "-'(J Ix). cpx . = ."'{(:fIx) . cpx } Df

Definition of Disjunction. 1'0 define disjunction when one or both of thepropositions concerned is of the first order, we have to distinguish six cases, asfollows:*9'03. (x ). cpx v . p : =. (x ) cpx v P Df*9'04. p . v . (x) cpx : =. (x ). p v cpx Df* 9'0 5 . (J Ix ).c p x.v .p := .('J Ix ).c p xv p Df*9 '06 . p .v.('J Ix ).cpx : = .(J Ix ).pvcpx Dr*9 '07 . (x ) c px . V ('J Iy ) ty :=: (x ) : ('J Iy) . cpx v ty Df*9'08. ( 'JIy). ty. v . (x ) . cpx : =: (x ) : ('J IY ) . ty v cpx Df


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136 MATHEMATICAL LOGIC [PART I(The definitions *9'07'08 are to apply also when c p and" are not both

elementary functions.) .In virtue of these definitions, the true scope of an apparent variable is

always the whole of the asserted proposition in which it occurs, even when,typographically, its scope appears to be only part of the asserted proposition.Thus when (~r :v ) cpx or (x ) c px a ppea rs as part of an asserted proposition, itdoes not really occur, since the scope of the apparent variable really extendsto the whole asserted proposition. It will be shown, however, that, so faras the theory of deduction is concerned, (~x ). cpx and (x ). c px behave likepropositions not containing apparent variables.

The definitions of implication, the logical product, and equivalence are tobe transferred unchanged to (x ) cpx and (~x ) c px .

The above definitions can be repeated for successive types, and thus reachpropositions of any type. ..

P rim itive P ro po sitio ns. The primitive propositions required are six innumber, and may be divided into three sets of two. We have first twopropositions which effect the passage from elementary to first-order propo-sitions, namely

Pp*9'11, ~: cpx v cpy ) (~z). c pz Pp

Of these, the first states that, if cpx is true, then there is a value of cp-Zwhich is true; i.e. if we can find an instance of a function which is true, thenthe function is "sometimes true." (When we speak of a function as " some-times" true, we do not mean to assert that there is more than one argumentfor which it is true, but only that there is a t lea st one.) Practically, theabove primitive proposition gives the only method of proving "existence-theorems": in order to prove such theorems, it is necessary (and sufficient) tofind some instance in which an object possesses the property in question. Ifwe were to assume what may be called "existence-axioms," i.e. axioms stating(~z) cpz for some particular cp, these axioms would give other methods ofproving existence. Instances of such axioms are the multiplicative axiom(*88) and the axiom of infinity (defined in *120'03). But we have notassumed any such axioms in the present work.

The second of the above primitive propositions is only used once, inproving (~z) cpz v . (~z) cpz: ) (~z) cpz, which is the analogue of *1'2(namely p v p. ). p) when p is replaced by (~z). c pz . The effect of thisprimitive proposition is to emphasize the ambiguity of the s required inorder to secure (~z). cpz. We have, of course, in virtue of *9'1,

cpx . ) (~z) cpz and cpy . ) (~z) c pz.But if we try to infer from these that cpa;v cpy ) (~z). cpz, we must use the


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SECTION B] EXTENSION OF THE THEORY OF DEDUCTION 137proposition q::> P r::> p : : > q v r : : > p , where p is (az). z. Now it will befound, on referring to *4:77 and the propositions used in its proof, that thisproposition depends upon *1'2, i.e. p v p : : > p. Hence it cannot be used byus to prove (ax) . x. v (ax). x : : : > (ax) . x, and thus we are compelledto assume the primitive proposition *9'11.

We have next two propositions concerned with inference to or frompropositions containing apparent variables, as opposed to implication. First,we have, for the new meaning of implication resulting from the abovedefinitions of negation and disjunction, the analogue of *1'1, namely*9'12. What is implied by a true premiss is true. Pp.

That is to say, given" r . p" and" r . p : : > q," we may proceed to " r . q,"even when the propositions p and q are not elementary. Also, as in *1'11,we may proceed from "r. x" and "r. x : : > 'o/x" to "r. ,yx," where o : is a

real variable, and and '0 / are not necessarily elementary functions. Itis inthis latter form that the axiom is usually needed. It is to be assumed forfunctions of several variables as well as for functions of one variable.

We have next the primitive proposition which permits the passage froma real to an apparent variable, namely" when y may be asserted, where ymay be any possible argument, then (x). x may be asserted." In otherwords, when y is true however y may be chosen among possible arguments,then (x) a ; is true, i.e. all values of are true. That is to say, if we canassert a wholly ambiguous value y, that must be because all values are true.We may express this primitive proposition by the words: "What is true inany case, however the case may be selected, is true in all cases." We cannotsymbolise this proposition, because if we put" r : y . : : > (x) . x"that means: "However y may be chosen, y implies (x) . x," which IS IIIgeneral false. What we mean is: "If y is true however y may be chosen,then (x). x is true." But we have not supplied a symbol for the merehypothesis of what is asserted in " r . y," where y is a real variable, and it isnot worth while to supply such a symbol, because it would be very rarelyrequired. If, for the moment, we use the symbol [y] to express thishypothesis, then our primitive proposition is

r : [y] . : : > (x) . x Pp.In practice, this primitive proposition is only used for inference, not forimplication; that is to say, when we actually have an assertion containinga real variable, it enables us to turn this real variable into an apparentvariable by placing it in brackets immediately after the assertion-sign,followed by enough dots to reach to the end of the assertion. This processwill be called "turning a real variable into an apparent variable." Thus wemay assert our primitive proposition, for technical use, in the form:


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138 MATHEMATICAL LOGIC [PART I*9'13. In any assertion containing a real variable, this real variable may beturned into an apparent variable of which all possible values are asserted tosatisfy the function in question. Pp.

We have next two primitive propositions concerned with types. Theserequire some preliminary explanations.

Primitive Idea : Individua l. We say that x is "individual" if a : is neithera proposition nor a function (cf. pp. 53, 54).*9'131. Defin itio n o j " being o f the same type." The following is a step-by-step definition, the definition for higher types presupposing that for lowertypes. We say that u and v "are of the same type" if (1) both areindividuals, (2) both are elementary functions taking arguments of the sametype, (3) u is a function and v is its negation, (4) 'M is cp5 :or '0/5:, and v iscp5 :v '0/5:, where cpS; and '0/5: are elementary functions, (5) u is (y). cp (5:, y )and v is (z) ,,(5:, z), where cp (5:, y), ,,(5:, y) are of the same type, (6) bothare elementary propositions, (7) u is a proposition and v is "'u, or (8) u is(x) . cpx and 1 1 is (y ) . 'o /y , where cp5 :and ,,5: are of the same type.

Our primitive propositions are:*9'14. If" cpx" is significant, then if x is of the same type as a , " cpa " ISsignificant, and vice versa. Pp. (Cf. note on *10'121, p. 146.)*9'15. If, for some a , there is a proposition cpa , then there is a function cp5: ,and vice versa. Pp.Itwill be seen that, in virtue of the definitions,

(x ) cpx . :> . p means ",(x ). cpx . v . p, i.e. (3 :x ). '" cpx . v . p,i.e. (~x ) . '" cpx v p, i.e. (~x ). cpx :> p(~x). cpa ; . :> . P means "'(~x). cpa ;. v . p, i.e. (x) .",c px . v . p,

i.e. (x ) "'c pa ; v p, i.e. (x ). cpx :> pIn order to prove that (x).cpx and (~x ) .cpx obey the same rules of deductionas cpx , we have to prove that propositions of the forms (x ) cpx and (~x ) . cpxmay replace one or more of the propositions p, q, r in *1'2-'6. When thishas been proved, the previous proofs of subsequent propositions in *2-*5become applicable. These proofs are given below. Certain other proposi-tions, required in the proofs, are also proved.*9'2. r : (x ) cpx . :> . cpy

The above proposition states the principle of deduction from the generalto the particular, i.e. "what holds in all cases, holds in anyone case."

Dem.I - *2'1 . :> r . " 'cpy v cpyI - *9'1 . :> r : " 'cpy v cpy . :> (~x ). "'cpa ; v cpyr. (1). (2). *1'11 .: > r. (~x ) .",cpx v cpy[(3).(*905)] I - : (~x ) '" cpx v . C P Y[(4).(*9'01.*1'01)] r : (x ). c px .:> . C P Y

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

principia mathematica - whitehead & russell. pag 141-160

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