24 Mr Van Horn, An Axiom in Si/ntholic Logic 



properties and relations of the four fundamental operations. The 

 chief excellence of my method seems to reside in the fact that 

 proceeding as indicated above I have been able to prove as pro- 

 positions of mathematical logic some of the axioms hitherto laid 

 down at the basis of this logic. 



In its most satisfactory form the axiomatic basis of mathe- 

 matical logic has been stated by Bertrand Russell in the first 

 volume of the Principia Mathematical. In *1 of Vol. i., pp. 98-101 , 

 of the Principia will be found the primitive propositions required 

 for the theory of deduction as applied to elementary propositions. 

 I confine myself to these purposely, for it is here that I have 

 succeeded, I believe, in simplifying the axiomatic basis of 

 mathematical logic. 



Let p and q be any two elementary propositions. The four 

 fundamental operations give us (1) ~ p {not-p), (2) pv q (either p 

 or q), (3) j9 D q (p implies q), and {4<) p • q (both p and q). After 

 Sheffer, I define these four results in terms of a single undefinable 

 operation. I will call this undefinable operation Deltation. The 

 result of performing this operation upon two elementary propositions 

 p and q is symbolized, after Sheffer, 'pAq' (read " j) deltas q'). 

 The four fundamental operations of logic can be expressed as 

 logical functions of this parameter thus : 



Negation: ~p. = .jjAj9 D£ 



Disjunction : /;vg. = .~^jA~g Df 



Implication : pD q . = .p A <^ q Di 



Conjunction : p • q . = . '^ (p A q) Df. 



These definitions of the four fundamental operations of logic 

 as functions of the one undefined parameter, Deltation, are made 

 relevant to our discussion by means of the following axiom. 



Axiom. If p and q are of the same truth-value, then ' p A q ' 

 is of the opposite truth-value ; but if j) and q are of ojjposite truth - 

 values, then ' p A q' is true. 



For convenience of reference it might be well for me to state at 

 this point Russell's primitive propositions concerning elementary 

 propositions as he enunciates them in *1 of the first volume of 

 the Principia. 



*1.1 Anything implied by a true elementary proposition 

 is true. Pp|. 



t Whitehead and Russell, Princiina Mathematica, Vol. i. 1910, Vol. ii. 191'2, 

 Vol. III. 1913 (Cambridge University Press). 



X Eussell uses the letters "Pp" to stand for " primitive proposition, " as does 

 Peano. 



