Principia Mathematica - Whitehead & Russell. Pag 121-140

8/2/2019 Principia Mathematica - Whitehead & Russell. Pag 121-140

1/20

SECTION A] PRIMITIVE IDEAS AND PROPOSITIONS 99true." This is a true proposition, but it holds equally when p is not trueand when P does not imply q. Itdoes not, like the principle we are concernedwith, enable us to assert q simply, without any hypothesis. We cannot \express the principle symbolically, partly because any symbolism in which pis variable only gives the hypothesis that p is true, not the fac t that itIS true*.

The above principle is used whenever we have to deduce a proposit ionfrom a proposit ion. But the immense majority of the assertions in thepresent work are assertions of propositional functions, i.e. they contain anundetermined variable. Since the assertion of a propositional function is adifferent primitive idea from the assertion of a proposition, we require aprimitive proposition different from *1'1, though allied to it, to enable us todeduce the assertion of a propositional function (( ' lrx" from the assertions ofthe two propositional functions "cpx" and "cpx: :> ' lr x ." This primitiveproposition is as follows:*1'11. When cpx can be asserted, where to is a real variable, and cpx::>'lrx canbe asserted, where ai is a real variable, then 'lrx can be asserted, where x isa real variable. Pp.

This principle is also to be assumed for func~.ioRs'-of;everal variables.Part of the importance of the above primitive proposition is due to the

fact that it expresses in the symbolism a result following from the theory oftypes, which requires symbolic recognition. Suppose we have the two asser-tions of p ro po sitio na l fu nc tio ns " f - . cpx" and" f - cpx::>lrx"; then the" a:" in cpx isnot absolutely anything, but anything for which as argument the function "x"is significant; similarly in "cpx: :> yx " the ai is anything for which" cpx::>yx"is significant. Apart from some axiom, we do not know that the x's for which" cpa; : > 'Ira;" is significant are the same as those for which" cpx" is significant.The primitive proposition *1'11, by securing that, as the result of the asser-tions of the propositiona l functions "cpa:" and "cpx ~ 'lrx ," the propositionalfunction tx can also be asserted, secures partial symbolic recognition, in theform most useful in actual deductions, of an important principle whichfollows from the theory of types, namely that, if there is anyone argument afor which both" cpa" and" ya" are significant, then the range of argumentsfor which "cpx" is significant is the same as the range of arguments forwhich" ' lrx" is significant. It is obvious that, if the propositional function"cpx : : > ya:" can be asserted, there must be arguments a for which" cpa::>ya"is significant, and for which, therefore, " cpa" and "ya" must be significant.Hence, by our principle, the values of a: for which" cpx" is significant are thesame as those for which "ta:" is significant, i.e. the type of possible argu-ments for 4 > ~ (cf. p. 15) is the same as that of possible argu ments for y~. The

* For further rema.rks on this principle, cf. Principles of Mathematics, 38.7-2

jr\A /


2/20

100 MATHEMATICAL LOGIC [PART Iprimitive proposition *1'11, since it states a practically important con-sequence of this fact, is called the" axiom of identification of type."

Another consequence of the principle that, if there is an argument a forwhich both c p a and ' o / a are significant, then c p a ; is significant whenever ' o / a ; issignificant, and vice versa, will be given in the" axiom of identification ofreal variables," introduced in *1'72. These two propositions.wl tL f and ,d'72,give what is symbolically essential to the conduct of demonstrations inaccordance with the theory of types.

The above proposition *1'] 1 is used in every inference from one assertedpropositional function to another. We will illustrate the use of this propositionby setting forth at length the way in which it is first used, in the proof of*2'06. That proposition is

" I - : 0 p : : : > q 0 : : : > : q : : : > r 0 : : : > 0 p : : : > r."We have already proved, in *2'05, the propositionI- : 0 q : : : > r 0 : : : > : p : : : > q 0 : : : > P : : : > r.

It is obvious that *2'06 results from *2'05 by means of *2'04, which isI - : 0 p o : : : > 0 q : : : > r : : > : q 0 : : : > 0 p : : : > r.

For if, in this proposition, we replace p by q:> r, q by p : : : > q, and r by p : : : > r,we obtain, as an instance of *2'04, the proposition

I - : : q : : : > r 0 : : : > : p : > q 0 : > 0 P : : : > r : : : : > : 0 p : : : > q 0 : : : > : q : : : > r 0 : : : > 0 P : : : > r (1),and here the hypothesis is asserted by *2'05. Thus our primitive propo-sition *1'1l enables us to assert the conclusion.*1'2. I- :pvp 0 : : : > o p Pp.

This proposition states: "If either p is true or p is true, then p is true."It is called the" principle of tautology," and will be quoted by the abbreviatedtitle of " T h - ~ . " It is convenient, for purposes of reference, to give names toa few of the more important propositions; in general, propositions will bereferred to by their numbers.*1'3. I - : q 0 : : : > 0 p v q Pp.

This principle states: "If q is true, then 'p or q ' is true." Thus e.g. if q is"to-day is Wednesday" and p is "to-day is Tuesday," the principle states:"If to-day is Wednesday, then to-day is either Tuesday or Wednesday." Itis called the" principle of addition," because it states that if a proposition istrue, any alternative may be added without making it false. The principlewill be referred to as " Add."--*1'4. 1 - : p v q 0 : : : > 0 q v p Pp.

This principle s~ates that "p or q" implies "q or p. It states thepermutative law for logical addition of propositions, and will be called the"principle of permutation." It will be referred to as "J~_~IE. '"


3/20

SECTION A] PRIMITIVE IDEAS AND PROPOSITIONS 101*1'5. 1 - : p v (q v r). J.q v (p V 1 ') Pp.

This principle states: "If either p is true, or 'q or r' is true, then eitherq is true, or 'p or r' is true." It is a form of the associative law for logicaladdition, and will be called the" associative principle." Itwill be referred toas "~" The proposition

p v (q V?') J .(p v q) v r,which would be the natural form for the associative law, has less deductivepower, and is therefore not taken as a primitive proposition.*1'6. f -: . q Jr. J :p vq . J .P v r Pp.

This principle states: "If q implies r , then' p or q' implies' p or r.'" Inother words, in an implication, an alternative may be added to both premissand conclusion without impairing the truth of the implication. Theprinciple will be called the" principle of summation," and will be referred toas "Sum."--_.*1'7. If p is an elementary proposition, < " o J p is an elementary proposition. Pp.*1'71. If p and q are elementary propositions, p v q is an elementary pro-position. Pp .

. *1'72. If c p p and - t p are elementary propositional functions which takeelementary propositions as arguments, cppvtp is an elementary propositionalfunction. Pp.

This axiom is to apply also to functions of two or more variables. It iscalled the" axiom of identification of real variables." It will be observedthat if c p and tare functions which take arguments of different types, thereis no such function as " c p x v tx," because c p and tcannot significantly havethe same argument. A more general form of the above axiom will be givenin *9.

The use of the above axioms will generally be tacit. It is only throughthem and the axioms of *9 that the theory of types explained in the Intro-duction becomes relevant, and any view of logic which justifies these axiomsjustifies such subsequent reasoning as employs the theory of types.

This completes the list of primitive propositions required for the theoryof deduction as applied to elementary propositions.

,l


4/20

*2. IMMEDIATE CONSEQUENCES OF THE PRIMITIVE PROPOSITIONS.S-ummary of *2 .The proofs of the earlier of the propositions of this number consist simply

in noticing that they are instances of the general rules given in *1. In suchcases, these rules are not premisses, since they assert any instance of them-selves, not something other than their instances. Hence when a general ruleis adduced in early proofs, it will be adduced in brackets*, with indications,when required, as to the changes of letters from those given in the rule tothose in the case considered. Thus "Taut "'l!_" will mean what "Taut" becomespwhen "'P is written in place of p. If" Taut "'P" is enclosed in square, , pbrackets before an asserted proposition, that means that, in accordance with"Taut," we are asserting what" Taut" becomes when "'P is written in placeof p. The recognition that a certain proposition is an instance of somegeneral proposition previously proved or assumed is essential to the processof deduction from general rules, but cannot itself be erected into a generalrule, since the application required is particular, and no general rule canexplicitly include a particular application.

Again, when two different sets of symbols express the same proposition invirtue of a definition, say *1'01, and one of these, which we will call (1), hasbeen asserted, the assertion of the other is made by writing" [(1).(*1'01)]"before it, meaning that, in virtue of *1'01, the new set of symbols asserts thesame proposition as was asserted in (1). A reference to a definition isdistinguished from a reference to a previous proposition by being enclosedin round brackets.

The propositions in this number are all, or nearly all, actually needed indeducing mathematics from our primitive propositions. Although certainabbreviating processes will be gradually introduced, proofs will be given veryfully, because the importance of the present subject lies, not in the propo-sitions themselves, but (1) in the fact that they follow from the primitivepropositions, (2) in the fact that the subject is the easiest, simplest, and mostelementary example of the sym bolic method of dealing with the principles ofmathematics generally. Later portions-the theories of classes, relations,cardinal numbers, series, ordinal numbers, geometry, etc.-all employ thesame method, but with an increasing complexity in the entities and functionsconsidered.

* Later on we shall cease to mark the distinction between a premiss and a rule according towhich an inference is conducted. Itis only in early proofs that this distinction is important.


5/20

SECTION A] IMMEDIATE CONSEQUENCES 103The most important propositions proved in the present number are the

following:*2'02. 1 - : q . : : > P : : > q

I.e. q implies that p implies q , i.e. a true proposition is implied by anyproposition. This proposition is called the "principle of simplification"(referred to as "Simp "), because, as will appear later, it enables us topass from the joint ass~tion of q and p to the assertion of q simply.When the special meaning which we have given to implication is remem-bered, it will be seen that this proposition is obvious.*2'03.*2'15.*2'16.*2'17.

I - : p J "'q. ) . q ) "'p1 - : "'p) q. J. c--vq) PI-:p)q.). "'q)"'-lP1-:"'q)"'p,),pJq

These four analogous propositions constitute the" principle of transpo-sition," referred to as "Transp." They lead to the rule that in an implicationthe two sides may be interchanged by turning negative into positive andpositive into negative. They are thus analogous to the algebraical rulethat the two sides of an equation may be interchanged by changing thesIgns.*2'04. 1 - : p . ) . q J r : ) : q . ) . P J r

This is called the" commutative principle" and referred to as "Qomm."It states that, if T follows from q provided p is true, then r follows from pprovided q is true.*2'05. 1 - : . q ) r ) : p ) q . ) . P ) r*2'06. 1 - : . p J q . ) : qJr.). p)~

These two propositions are the source of the syllogism in Barbara (as willbe shown later) and are therefore called the "principle of the syllogism"(referred to as "Syll "). The first states that, if r follows from q, then if qfollows from p, r follows from p. The second states the same thing with thepremisses interchanged.*2'08. I - P : : > P

I.e. any proposition implies itself. This is called the" principle of identity"and referred to as "Td." It is not the same as the" law of identity" (" to isidentical with a;"), but the law of identity is inferred from it (cf. *13'15).*2'21. 1 - : '" p : : > P : : > q

I.e. a false proposition implies any proposition.The later propositions of the present number are mostly subsumed under

propositions in *3 or *4, which give the same results in more compendiousforms. We now proceed to formal deductions.

)


6/20

104 MATHEMATICAL LOGIC [PART I*2'01. 1-: p : : > " " p : : > " " P

This proposition states that, if p implies its own falsehood, then p is false.It is called the" principle of the reductio ad absurdum," and will be referredto as "Abs."* The proof is as follows (where" Dem." is short for demon-stration ") :

Dem.[ Taut ; p ] I - : f ' o . J p v f ' o . J p . : : > f ' o . J p[(1).(*1'01)] 1-: p -:, : : > .""p

*2'02. 1- : q . : : > P : : > q

(1)

Dem.[Add ;p J I - : q . : : > ""P v q[(1).(*1'01)] I-:q.::>.p::>q

*2'03. I - : p : :> " " q . J . q J ""PDem.

(1)

[Perm "'}J, ""q] I - : ""P v ""q : : > "' q v ""Pp, q[(1).(*1'01)] I-:p -:, o , q J""p

*2'04. I - : . p J . q::> r : : : > : q J.P : : > r

(1)

Dem .[Assoc('o.,)P' ( ' o . . J q ] I - : . = v ( " " q v r). J. ( ' o . , ) q v ( ' o . , ) p v r) (1)p , q[(1).(*101)J I - - - J. q::> r: J: q.::>. P Jr

*2'05. 1-:. q : : > r . : : > : p : : > q . J.P : : > rDem .

[ Sum ; p J 1-:. q : : > r . J :'"p v q : : > " " p v r (1)[(1).(*1'01)] 1-:. q::> r , J :p::> q . : : > . p::> r

*2'06. 1-:. p J q . : : > : q Jr. : : > P : : > rDem.

[Com m q Jr, p : : > q, p : : > r J I - : : q : : > r . J:p : : > q . : : > p : : > r :.p, q, r : : > : . p : : > q J:q : : > r : : > P : : > r (1). "I - : . q jr .J :p : : > q . : : > P J r*2'05][(1).(2).*1'11]

(2)I - : . p J q . J : q::> r . J .P J r

* There is an interesting historical article on this principle by Vailati, "A proposito d' unpa~so del Teeteto e di una dimostrazione di Euclide," Rivista di Eiloeofia e I Ic ie nze a ffin e, 1904.


7/20

SECTION A] IMMEDIATE CONSEQUENCES 105In the last line of this proof, "(1). (2) . *1'11" means that we are

inferring in accordance with *1'11, having before us a proposition, namelyp J q . J : q J r . J . P J r , which, by (1), is implied by q Jr . J : p J q . J . P J r,which, by (2), is true. In general, in such cases, we shall omit the referenceto *11l.

The above two propositions will both be referred to as the" principle ofthe syllogism" (shortened to "Syll "), because, as will appear later, thesyllogism in Barbara is derived from them.

Here we put nothing beyond "*1'3l!_ ," because the proposition to beqproved is what *1'3 becomes when p is written in place of q.*2'08, r. P J P

Dem.[*2'05 p _ _ v p , p J r::P v r - J P : J :. p J P v P : J P J P (1)q, r[Taut] r :pvp . J .p[(1).(2).*1'11] r : .p .J .pvp:J .pJp[2'07] r :p . J . p v P[(8).(4).*1'11] r.p JP

*2"1. r. r- -Jpvp [Id. (*1'01)]*2'11. r. p v"'P

(2)(3)(4)

Dem.[Perm "'P, p J r :"'P vP J .p v"'Pp, q[(1).*2'1.*1'11] r .p vr--JP

(1)

This is the law of excluded middle.

Dem.[*2'U;P] r . r "oJpvr "oJ ( r "oJp )[(1).(*1'01)] r .pJ",(" ,p)

(1)


8/20

106*2'13. I - P v . . . .. . , { . . . . . . , ( . . . . . . , p ) }

This proposition is a lemma for *2'14, which, with *2'12, constitutes theprinciple of double negation,

MATHEMATICAL LOGIC [PART I

Dem.[Sum " " " ' p , " " ' J ~ ( ~ - 1 ? ) J I - : . . . . . . , p . ) . . . . . . , ! . . . . . . , ( . . . . . . , p ) } ) :q, r

p v,,-,p .) .p v" '{ ......, (......,p )} (1)[*2'12 ~ p J I - : . . . . . . , p . ) . " '{" '(""" 'p)} (2)[(1).(2).*1'11] I - :p v" -'p .) .p v......,{ ......,(" 'p )} (3)[(3).*2'11.*1'11] I - :pv"-,{",(,,,p)}

'. *2'14. 1 - . " '( "'p) ) pDem.

[(1).*2'13.*1'11][(2).(*1'01 )]

I - ."'{"'("'p)}vpI - " - '( "-'p) ) p

(1)(2)

*2'15, 1 - : ' " p ) q ) . ......, ) pDem.

[*2'05'::'P' "-'( ~ - q ) J 1 - : . q ).. .. .. , ( . .. .. .,q) .): "'P ) q ) ......,p ) " '-'( "'q) (1)p, r[*2'12 ~ J[(1 ).(2).*1'11]

I - q ) "'( "'q)1 - : "-'p)q. ),~p)"""'("""'q)

(2)(3)

[*2'05 f " ' , . J P ) q, " 'p ) rv( """ 'q),rvq ) rv( " ' P ) ] I - : :p, q, rrvp ) rv( " 'q). ) rvq) "-'( r-.Jp ) : ) : .r-.Jp ) q.) . .....,p ) < '.J ( 'q): ): "-'p ) q ) r -.Jq) "-'( " -'p ) (7 )

[(4).(7).*1'11] 1 - : . "-'p ) q .) ." 'p ) " '(< '.Jq):):"rP ) q . ) . "'q) "-'("'p) (8)


9/20

SECTION A] IMMEDIATE CONSEQUENCES 107[(3).(8).*1'11] I - : < '.Jp ) q . ) " "q ) < ' . J ( : ('"P ), _~9') P 1 1 - : : ' " q ) '" ( < ' . J p) ) ....., ) p :P , q , r J): .....,p) q . ) < '.Jq ) "-'("""p) : ) : ""p ') q .) ""q ') P (10)

(9)

[(6).(10).*1'11] I - : . "'P ') q . o , " -Oq ' ) < ' . J ( "-O p) : ') :"-O p') q ') "'q ') P (11)

Note on the proof of *2'15, In the above proof, it will be seen that (3),(4), (6) are respectively of the forms PI') P 2 ' P 2 ' ) P 3 , P 3 ' ) P 4 , where PI') P 4 isthe proposition to be proved, From PI) P 2 , P2) P3 ' P3 ') P4 the propositionP I ') P 4 results by repeated applications of lld'05 or *2'06 (both ~of which arecalled "Syll "). It is tedious and unnecessary to repeat this process everytime it is used; it will therefore be abbreviated into

"[Syll] I - (a) . (b) . (c). ') I - (d),"where (a ) is of the form PI')P2 , (b ) of the form P2) P3, (c ) of the form P3 ') P4 ,and (d) of the form PI) P 4 ' The same abbreviation will be applied to asorites of any length.

Also where we have" I - PI" and" f-PI ') P 2 , " and P 2 is the proposition tobe proved, it is convenient to write simply

"f-']h'')[etc.]where "etc." will be a reference to the previous propositions in virtue ofwhich the implication "PI') P2 " holds. This form embodies the use of*1'11 or ,d'l, and makes many proofs at once shorter and easier to follow.It is used in the first two lines of the following proof.

*2'16. 1 - : P ) q . ') < '.Jq ) r-JpDem.

[*2 '05) l-:p)q.).p).....,(


10/20

108 MATHEMATICAL LOGIC [PART INote. The proposition to be proved will be called "Prop," and when

a proof ends, like that of *2'16, by an implication between asserted propo-sitions, of which the consequent is the proposition to be proved, we shallwrite " f - . etc. J f - Prop". Thus" J f - Prop" ends a proof, and more orless corresponds to "Q. E.D."*2'17. f - : . . . . . . , q J ......, J . P J q

Dem.[*2'03 rvq , p ]1 - : "'q J " 'p . J P J " '( " 'q) (1)p , q _[*2'14] f - : " ' ( "'q) J q: J[*2'05][Syll]

f - : p J rv( " 'q). J . p J qI- (1) . (2) . J f - Prop

(2)

*2'15, *2'16 and *2'17 are forms of the principle of transposition, andwill be all referred to as "Transp."*2'18. f-:... . . . ,pJp.J.p

Dem. [*2'12] I - .p J " ,(" ,p ). J[*2'05] f - "'P J P> J . . .. . .,pJ. . . .. . ,(""p ) (1)l*2'01 7 J f - : " "" 'p J rv( " 'p ). J . " '( " 'p ) (2)[Syll] 1 - . (1). (2). J f - : " 'P J P J . " '( " 'p ) (3)[*2'14] f - . " ' ( " 'p ) J p (4)[Syll] f - (3) . (4) . J I- Prop

This is the complement of the principle of the reductio ad absurdum. Itstates that a proposition which follows from the hypothesis of its own false-hood is true.*2'2. f - : p . J .p v q

Dem. f - Add. J I- : p . J q v p[Perm] I- : q v p . J .P v q[Syll] 1 - . (1) . (2) . J f - Prop

*2'21. I- : "'P . J p J q [*2'2 ~p ]The above two propositions are very frequently used.

*2'24. f - : p J . r - . > p J q [*2'21. Comm]

(1)(2)


11/20

SECTION A] IMMEDIATE CONSEQUENCES 109

*2-25. 1 -: p : v : p v q . : : > qDem.

I- *2-1 : : > I - : f'.J(p V q) v (p v q) :[Assoc ] : : > I - : p v . {'" (p v q ) v q } : : : > I- Pro p

*2-26. I- :f'.JP : v : p::> q . o , q [*2-25 7*2-27. 1 - : . p . : : > : p : : > q . : : > q [*2-26]*2 -3 _ I-:p v(q vr).::> .p v(rvq )

Dem .[Perm ~ J, q I- :qvr.::>.rvq:[ qvr,rvq]Sum : : > 1 - : p v (q v r) . : : > .pv (rvq )q, r

*2'31_ 1 - : p v (q v r) . : : > (p v q) v rThis proposition and *2'32 together constitute the associative law for

logical addition of propositions. In the proof, the following abbreviation(constantly used hereafter) will be employed *; When we have a series ofpropositions of the form a::> b, b::> c, c ::> d, all asserted, and "a ::> d" is theproposition to be proved, the proof in full is as follows;

[Syll] 1 - : . a::>b.::>: i e .o . a ::> cI-:a.::>.bI-:b::>c.::>.a::>cI-:b.::>.cI-:a.::>.cI - : . ( ~ : : > c : : > : c : : > d : : > u. : : > dI- :c::>d.::>.a,::>dI-:c.::>.d

[(7).(8).*1'11] 1 - : a . : : > . d

(1)(2)(3)(4)(5)(6)(7)(8)

[(3).(4).*1'11][Syll][(.5 ).(6).*1-11]

It is tedious to write out this process in full; we therefore write simplyI-:a.::>.b.[etc.] o , c.[etc_] : : > d : : : > I- Prop,

where "a ::> d" is the proposition to be proved. We indicate on the leftby references in square brackets the propositions in virtue of which thesuccessive implications hold. We put one dot (not two) after Hb , " to show

* This abbreviation applies to the same type of cases as those concerned in the note to *2-15,but is often more convenient than the abbreviation explained in that note,


12/20

110 MATHEMATICAL LOGIC [PART Ithat it is b, not "a J b," that implies c. But we put two dots after d, toshow that now the whole proposition "a J d" is concerned. If" a J d" isnot the proposition to be proved, but is to be used subsequently in the proof,we put l-:a.J.b.

[etc.] J . c.[etc.] J. d

and then" (1)" means" a J d." The proof of *2'31 is as follows:Dem.

(1),

[*2'3J I - : p v (q v r) . J .P v (r v q) [ASSOC r, q ]q, r[Perm r, p _ v q ]p , q

*2'32. 1 - : (p v q) V T J . p v (q V r - )Dem.

J. r v (pv q).J . (p v q) v T: J I- Prop

[Perm p v q,_,!,] I - : (p v q ) V T. J . T V (p V q )p, q[ASSOC T, p ~ - Jp, q, r[*2'3]

*2'33. pvqvT.=.(pvq)vr DfThis definition serves only for the avoidance of brackets.

J.p v (rv q)J .p v (q v r) : J I- Prop

*2'36. 1-:. q J T J :p v q . J . T V PDem.

[Perm] l-:pvr.J.rvp:[syll p v q, p v T, r v PJ J I - : . p v q . J P v T : J :p v q J T V P (1)p , q , r[Sum] I - : . q Jr. J : p v q J . p v r (2)I - (1) (2) . Syll . J I - Prop

*2'37, f-:. q J r, J : q v p . J. p v r[Syll Perm. Sum]

*2'38. 1-:. q Jr. J : q v p J r v p[Syll. Perm. Sum]

The proofs of *2'37'38 are exactly analogous to that of *2'36. (We use" *2'37'38" as an abbreviation for "*2'37 and *2'38." Such abbreviationswill be used throughout.)


13/20

SECTION A] IMMEDIATE CONSEQUENCES 111The use of a general principle of deduction, such as either form of" Syll,"

in a proof, is different from the .use of the particular premisses to which theprinciple of deduction is applied. The principle of deduction gives thegeneral rule according to which the inference is made, but is not itself apremiss in the inference. If we treated it as a premiss, we should needeither it or some other general rule to enable us to infer the desired con-clusion, and thus we should gradually acquire an increasing accumulation ofpremisses without ever being able to make any inference. Thus when ageneral rule is adduced in drawing an inference, as when we write"[Syll] f - (1). (2) . J f - Prop," the mention of "Syll" IS only required inorder to remind the reader how the inference is drawn.

The rule of inference may, however, also occur as one of the ordinarypremisses, that is to say, in the case of "Syll" for example, the proposition"p J q . J : q Jr. J . p J r " may be one of those to which our rules ofdeduction are applied, and it is then an ordinary premiss. The distinotionbetween the two uses of principles of deduction is of some philosophicalimportance, and in the above proofs we have indicated it by putting the ruleof inference in square brackets. It is, however, practically inconvenient tocontinue to distinguish in the manner of the reference. We shall thereforehenceforth both adduce ordinary premisses in square brackets where con-venient, and adduce rules of inference, along with other propositions, inasserted premisses, i .e. we shall write e.g.

" f - (1) . (2) . Syll . J f - Prop""[Syll] f - (1). (2) . J f - Prop"ather than

*2'4. f - : . p v . p v q : J . p v qDem.

f - *2'31 J f - : . p . v . p v q : J : p v P v q :['l'aut.*238] J : p V q :. J f - Prop

*2'41. f - : . q v p v q : J .P v qDem.

[ARSOC 1 ! _ P , _ . ! . l . ] f - : . q . v . p v q : J :p . v q v q :p, q, r[Taut.Sum] J : P v q :. J f - . Prop

*2'42. f-:.",p.v'PJq:J'PJQ[*2'4 ;p]*2'43.*2'45.*2'46.

f-:.p.J.pJq:J.pJqf - : '" (p v q). J . "'pf - :"'(p v Q ) J . "'q

[*2'2. Transp][*1'3. Transp]


14/20

112 MATHEMATICAL LOGIC [PART I

*2'48, I- :"""'(p v q ). :> .p v"'q

. . . .*2'52, 1 - : f'oJ (p :> q) . :> "P :> f'oJ q*2'521. 1 - : f 'o J (p :> q ). :> . q :> P*2'53, 1 - : p V q :> ......,p:>q

Dem. 1 - . *2'12'3~. :> I- :p v 'l : :> . " - ' ( r. .op) v q : : > I - Prl'r*2'54. 1 - : f 'o Jp:> q .:> .p v q L*2'14'3~]*2'55. 1 - : . r..o p : > : p v q :> . q*2'56. 1 - : . ' " q : > : p v q :> p*2'6. I - : . ' " p : > q . : > : p :> q . : > q

Dem.I- :." 'p :> q .:> :f'.Jp V q .:> . q v q (1)

(2)Tall t 8 y 1 1 ] 1 - : . ' " p v q : > q v q : :> : ' " p V q :> q1 - . (1). (2) . Syll . :> I- :." 'p :> q .:> :r..op V q . : > . q : . : > f - Prop

*2'61. 1 - : . p:> q : > : "'p : > q . : > . q [*2'6. Comm]*2'62. f-:. p vq .:> : p:> q .:> . q [*2'53'6. 8yll]*2'621. f-:. p : > q : > : p v q :> . 'l*2'63, f-:. p v q . : > : "'p v q. : > q*2'64. f -: p v q . : > : p v......., : > P*2'65. f-:. p : > q :> : p : > f '> o )q : > - r - *2'67. f-:. p v q . :> q : :> P :> q

Dem.

[*2'62. Comm][*2'62][*2'63 'h_l! P e n l l ]p, q[*2'64 ; p ]

[*254.8yl1] 1 - : . p V q .:> . q :::> : "'p::> q .::> . q (1 )[*224.8yll] 1 - : . f'o Jp:> q ::>. q : ::> P ::>q (2)1 - . (1 ). (2). 8y11 .::> 1 - . Prop


15/20

SECTION AJ IMMEDIATE CONSEQUENCES 113

[*2'67 ;~J-:.p~q.~.q:~."'p~qI- (1) . *2'54 . ~ I - Prop -

I - : . p ~ 'I . ~ q : ~ : q ~ p . ~ . p

*2'68. 1 - : . p ~ q . ~ q : ~ . p v qDem.

*2'73. 1 - : . p ~ q . ~ : P V 'Iv r . ~ . q v r*2'74. 1 - : . q ~ p . ~ : P V q V r . ~ .p v r*2'75. 1 - : : p V q ~ :. p . v . q ~ r : ~ . p v r*2'76. 1 - : . p v . q ~ r : ~ : p v q ~ . p V r*2'77. 1 - : . p . ~ . q ~ r : ~ :p ~ q ~ . p ~ r*2'8. I - : . q V r . ~ : '" r V S ~ q v s

Dem.

(1)

l2'73 f j , - _ 'E Assoc. s y n ]1), q[ * 2 : t 4 ~q. *2'53'31 ][*2'7.5. Comm][ *2'76~~]

I - *2'53. Perm . ~ I - : . q v r. ~ :"'r ~q:[*2'38] ~: "'r v s , ~. 'I v ss, ~ 1 - . Prop

*2'81. 1 - : : q . ~ . r ~ s : ~ :. p v q . ~ : p V r . ~ p v sDem.

I - Sum . ~ I - : : q . ~ . r ~ s: ~ :. p V q . ~ : p . V .1' ~ s (1)1 - . *2'76. Syll . ~ I - : : p V q. ~: P v . r ~ S : . ~ : .

p v q . ~ : p V r ~ . p v s (2)I - (1) . (2) ~ I - Prop

*2'82. 1 - : . p v q V r ~ : p v......,r v s , ~ p v q v s[*2'8 . *2'81 q'!.~, ~r_v s, (j ' ! . _ _ 8 ]q, r, s

*2'83, 1 - : : p ~ q ~ r : ~ :. p . ~ . r ~ s : : > : p : > . q : > s[*2'82 """'p, " , q lp, q

*2'85, 1 - : . p v q ~ . p v r : : > : p V q ~ rDem. [Add.Syll] I-: .pvq.~.r::>.q~r (1)

I - * 2' 6 5 : > I - : :. .. .. .,p : > : . p v r , : > . r :.[Syll] : > : . p v q . : > P V r : : > : p v q : > r :.[(1).*2'83] :>: .pvq.~ .pvr: :>:q:>r (2)I - (2) . Comm . : > I - : . p v q ~ p v r : : > : .. .. .. ,p . : > q : > r :[*2'54] :>:p . v. q:> r:.:> 1 - . Prop

*2'86. 1 - : . p ~ q . ~ . p ~ r : : > : - : > q:> r [*2'85 ;pJR. & W. 8


16/20

*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS.Swnrnary of *3.The logical product of two propositions p and q is practically the pro-

position" p and q are both true." But this as it stands would have to be anew primitive idea. We therefore take as the logical product the proposition'" ( ,...., v '" q ), i.e . "it is false that either p is false or q is false," which ISobviously true when and only when p and q are both true. Thus we put*3'01. P > q . =. ("00, ) ( '" p v '" q) Df

where "p q" is the logical product of P and q.*3'02. P :> q :> r . = . p :> q . q :> r Df

This definition serves merely to abbreviate proofs.When we are given two asserted propositional functions" f - c p x " and

" I- y . x , " we shall have" I- c p . ' C . y . x " whenever c p and V take arguments ofthe same type. This will be proved for any functions in *9; for the present,we are confined to elementary propositional functions of elementary pro-positions. In this case, the result is proved as follows:

By *1'7, " " " c p p and " ' V P are elementary propositional functions, and there-fore, by *1'72, '" c p p v '" V p is an elementary propositional function. Henceby *2'11, f - : ' " c p p v "" V p v . "" ( '" c p p v , .. .. ,+ ) .

Hence by * 2'32 and *1'01,f - : . c p p . :> : V p . : > '" ( " - ' c p p v "" ' o / p ) ,

i.e. by *3'01,I- :. c p p . :> : V p . : > c p p " , " p .

Hence by *1'11, when we have " I - . c p p " and "I-. V p " we have" f - . c p p . ' o /p . "This proposition is *3'03. It is to be understood, like *1'72, as applyingalso to functions of two or more variables.

The above is the practically most useful form of the axiom of identi-fication of real variables (cf. *1'72). In practice, when the restriction toelementary propositions and propositional functions has been removed, a con-venient means by which two functions can often be recognized as takingarguments of the same type is the following:Ifc p x contains, in any way, a constituent X ( x , y, z , . . . ) and ' o / x contains,

in any way, a constituent X ( x , u , v , . . . ) , then both c p x and ' o / x take arguments


17/20

SECTION A] THE LOGICAL PRODUCT OF TWO PROPOSITIONS 115of the type of the argument a ; in X (x, y, z, , .. ), and therefore both c p a : and' o / a ; take arguments of the same type. Hence, in such a case, if both c p a ; and' o / a ; can be asserted, so can c p a ; ' o / a ; ,

As an example of the use of this proposition, take the proof of *3'47. Wethere prove I - : . p : : > r , q : : > s : : > : P q.::> q r

I - : . p : : > r . 'I : : > s . : : > : q r. : : > r . s(1)(2)nd

and what we wish to prove isp : : > r . q : : > s . : : > : p g : : > r . s,

which is *3'47. Now in (1) and (2), p, g, r, s are elementary propositions(as everywhere in Section A); hence by *1'7'71, applied repeatedly,"p : : > r . q : : > r , : : > : p . 'l . : : > g 1" and "p::> r . q : : > s . : > : q . r . : > . r . s " areelementary propositional functions. Hence by *3'03, we have

I - : : p : > r . q : : > s . : > : p . q : > q . r :. p : > r . q : > 8 : > : q r . : > r s,whence the result follows by *3'43 and *3'33.

The principal propositions of the present number are the following:*3'2. I - : . p . : > : q. : > . P > q

I.e. "p implies that q implies p q ," i.e. if each of two propositions is true,so is their logical product.*3'26. 1 - : p . q . : > P*3'27, I - : p . q : > q

I.e. if the logical product of two propositions is true, then each of the twopropositions severally is true.*3'3. I - : . p . q . : > r : : > : p . : > q : > r

I,e. if p and q jointly imply r, then p implies that q implies r. Thisprinciple (following Peano) will be called" exportation," because q is" exp?rted ,.from the hypothesis. Itwill be referred to as " Exp."*3'31. 1 - : . p . : > q : > r : : > : p . q . : > r

This is the correlative of the above, and will be called (following Peano)"importation" (referred to as "Imp ").*3'35. 1 - : p . p : > q . : > qI.e. "if p is true, and q follows from it, then q is true," This will be calledthe" principle of assertion" (referred to as " Ass "). It differs from *1'1 bythe fact that it does not apply only when p really is true, but requires merelythe hypothesis that p is true,*3'43. 1 - : . p : > q . p : > r . : > : p . : > q . r

Le. if a proposition implies each of two propositions, then it implies theirlogical product, This is called by Peano the" principle of composition," Itwill be referred to as "Comp."

8-2


18/20

116 MATHEMATICAL LOGIC [PART I*3'45. 1 - : . p j q . j : p . r j . q . r

I.e. both sides of an implication may be multiplied by a common factor.This is called by Peano the" principle of the factor." It will be referred toas "Fact."*3'47, 1 - : . p j r . q j s . j : p . q. j . r . s

I.e. if p implies q and r implies s, then p and q jointly imply rand sjointly. The law of contradiction; "I-. '" (,p . '" p)," is proved in thisnumber (*3'24); but in spite of its fame we have found few occasions forits use.

*3'01. p. q . = . '" ('" p v '" q) Df*3'02, P j q j r . = . p j q q j r Df*3'03. Given two asserted elementary propositional functions" I- cpp " and" I- 'o/p" whose arguments are elementary propositions, we have I - c pp . 'o /P '

Dem.1 - . *1'7'72. *2'l1 . j I- : '" c f>pv"" 'o/P' v. '" ('" c pp v '" " 'o /p ) (1)1 - . (1) . *2'32. (*1'01) . j I- :. cpp . j : 'o /p . j. " '('" " c f> pv'"" 'o/p) (2)I - (2). (*3'03) . j I - : . p . j : 'o /P . j .p . tp (3)I - (3) . *1'1l . j I - Prop

*3'1. I- : p . q. j . rv (rv p V rv q) [Id . (*3'01)]*3'11. I - : '" ( rv p V rv q) j . p . q [Id . (*3'01)]*3'12, I -: ....., p.v . ....., q.v .p .q [ ,""p v.....,q]*211-p~-*3'13. I - : '" (p . q ) j . '" p V rv q [*3'1l .1'ransp]*3'14. I - : '" " p V rv q j . '" (p . q) [*3'1 . Transp]*3'2. I - : . p . j : q j p . q [*:3-12]*3'21. I - : . q j : p j . p . q [*3'2 . Comm J*3'22. I-:p.q.j.q.p

This is one form of the commutative law for logical multiplication. Amore complete form is given in *4'3.

Dem.[*3'13 q, p Jp ,q[Perm][*3'14JI - (1). Transp . j I - Prop

I- : '" (q . p) . j '"qv'" p.j ." 'pv"" 'q.j. '"(p . q) (1)


19/20

SECTION A] THE LOGICAL PRODUCT OF TWO PROPOSITIONS 117Note that, in the above proof, "(I)" stands for the proposition

"", (q. p). : > . ' " (p. q),"as was explained in the proof of *2'31.*3'24. I-.",(p.",p)

Dem,[*2'11 it] I - r-.J P V roo.; ( < ' o J p) . : >[ *3'14 ~ P ] 1 - . ~ (p < ' o J p)

The above is the law of contradiction.*3'26, 1 - : p . q . : : > P

Dem.[*2'02 q , p Jp, q[(I).(*IOI)J[*2'31]

I-:p . ::> .q ::>p (1)1 - : 1 - : . . . . . . , q :[Transp.Syll] : : > : p. : : > . q::> 1 ' : . : : > 1- . Prop

*3'31. 1-:. p . : : > q : : > r: : : > : p . q . : : > rDem.

[Id.(*IOI)] 1-:. p.::> . q ::> r:::> : '" p . V. < ' o J qvr:[*2'31] : : > : ' " p v '" q V r :

: : > : ' " ( ' " p v '" q) : : > r :: : > : p . q . : : > r :. : : > I - Prop


20/20

118 MATHEMATICAL LOGIC [PART I*3'33. 1 - : p:> q q : > r : > p:> r [Syll . Imp]*3'34. l-:q:>1.p")q.:>.p")r [Syll.ImpJ

These two propositions will hereafter be referred to as " Syll "; they areusually more convenient than either *2'05 or *2'06.*3'36. 1 - : p . p : > q " ) q [*2'27 . Imp]*3'37. 1 - : . p q : > r: ") :p roo.; r . : > roo.; q

Dem.I- Transp . : > I- : q ") r . : > . ' " r : > " " q :[Syll] J I - : . p . J . q J 1': J :p. : > ' " r J '" qI- Exp . ") I - : . p . q ") . r : "):p . " ) . q ") r

(1)(2)

1 - . Imp. J I - : . p . ") . ""' 1':> ""' q: "): p. ""'r. "). '" q (3 )I - (2). (1). (3). Syll . J I- Prop

This is another form of transposition.*3'4. I - : P q. J . P J q [*2'51 . Transp . (*1'01 . *3'01)]*3'41. 1 - : . p") 1'. "): p . q. J. r [*3'26. Syll]*3'42. 1 - : . q ") r . "):p . q. J . r [*3'27. Syll]*3'43. I - : . p :> q .p J r . J : P J. q r

Dem.1 - . *3'2. J 1 - : . q. J: r , J. q. r (1)1 - . (1). Syll. J 1 - : : p J q J:. p.:>: 1'. J. q 1':.[*2 '77] J:,pJr.J:p.:>.q.r (2)I - (2) . Imp. J I - Prop

*3'44. 1 - : . q J P r J p. J : q v r ") . PThis principle is analogous to *3'43. The analogy between *3'43 and

*3'44 is of a sort which generally subsists between formulae concerningproducts and formulae concerning sums.

Dem.I - Syll . ") I - : . " " ' q : > r . r : > p . J: ""' q J P :[*2'G] ") : q ") p J P1 - . (1). Exp . J 1 - : : ' " q") r . "):. r J P' "): q") p . ") . p:.[Comm.lmp] :>:.qJp.rJp.").p (2)I - (2) Comm . J I - : . q J P r J p. J: ""' q Jr. J . P :.[*253.Syll] ") I- Prop

(1)

Documents

Principia Mathematica - Whitehead & Russell. Pag 121-140