Mr Vail Horn, An Axiom in Si/nihohc Loffic ol 



conclusion in any implication without impairing the truth of the 

 implication. 



This completes the list of Mr Russell's primitive propositions 

 that I proposed for proof by means of my axiom, on the basis of 

 the definitions given in this paper of the four fundamental 

 operations of logic. 



I now propose to prove two propositions which can take the 

 place of his definitions of Implication f and Conjunction j, or Joint 

 Assertion. 



Theorem 16 



Dem. 



[Th. 4^^'"'^] h: p A -</.D.;9 A ~g (1) 



[Th. 10] D . ~ (~ p) A ~ ry (2) 



[(2). Def. Implica. Disjunc] h: pD q .D . ^^ pw q (3) 



[(1). Th. 10] h: ~(~j9)A ~(y.D.j)A ~ 7 (4) 



[(4). Def. Implica, Disjunc] h : '^py/q.D.pDq (5) 



[(3). (5). Def. Equiv.] h : theorem. 



Theorem 17 



h: p . q . = . ^ { '^ p W ^ q). 

 Dem. 



[Th. 4 *" ^^ ^ '^^ ] h: ~(jo Ary).D.~(p A (/) (1) 



[Th. 10] D.~[~(~y/) A ~(~^)](2) 



[(2). Def. Conjunc. Disjunc] I- : p . q . D . ^^ (^ p v r^ q) {S) 

 [(1). Th. 10] h: ~[~(~p)A ~(~5)].D.~(p Afy)(4) 



[(4). Def. Conjunc. Disjunc] h : '^ (^ p v ^ q) . "D . p .q (5) 

 [(3). (5). Def. Equiv.] h : theorem. 



With these theorems established the development of the 

 Principia Mathematica can proceed as given by its authors. 

 All that I have done is to reduce the number of axioms needed 

 for that development. 



Baptist College, 

 Rangoon, Burma. 



t Op. cit. Vol. I. p. 98, *1.01. + Ibid. p. 116, *3.01. 



