3fr Van Horn, An Axiom in Sipnbolic Lo(jio 25 



*1.H When <^x can be asserted, where x is a real variable, 

 and ' (fjxD yfr x ' can be asserted, where x is a real variable/then yjrx 

 can be asserted, where x is a real variable. Pp. 



*1.2 h : pvp.D .p Pp. 



*1.3 \- : q.D .pv q Pp. 



* 1.4 [■ : pv q .0 .qv p Pp. 



* 1 .5 \- : py/ (qv r).D .qv {pv r) Pp. 



*1.6 h: .q'^r.D-.pvq.D.pyr Pp. 



*1.7 If j9 is an elementary proposition, ~ p is an elementary 

 proposition. Pp. 



*1.71 If p and (/ are elementary propositions 'pvq' is an 

 elementary proposition. Pp. 



*1.72 If ^p and i/rj? are elementary prepositional functions 

 which take elementary propositions as arguments, ' (f) pv -ylrp' is an 

 elementary prepositional function. Pp. 



These are all the primitive propositions that are needed for the 

 development of the theory of deducti(jn, as applied to elementary 

 propositions, according to Russell's method of treatment. 



It is my purpose to show that by means of my axiom 

 Russell's primitive propositions *1.2 to *1.7l can be demon- 

 strated. I do this by starting at the very beginning and 

 developing the immediate consequences of three of the axioms 

 which I lay down as the basis of the theory of deduction as applied 

 to elementary propositions. The resulting deductive development 

 at length reaches a point where it includes among its theorems 

 Mr Russell's seven pi'imitive propositions and two others that can 

 take the place of his definitions of Implication and Conjunction. 

 Altogether I prove seventeen theorems. Some of these theorems 

 occur as propositions in the first volume of the Principia. Al- 

 though many more theorems can be proved as simply as the ones 

 given, to economize space I shall stop at the point where my 

 development of Mathematical Logic includes the nine theorems 

 mentioned above. 



I will now state the three axioms used in this paper. The 

 first is * 1.1 given above, the last is my axiom as already enunciated. 



Axiom 1. Anything implied by a true elementary proposition 

 is true. 



Axiom 2. Ifp and q are elernentary propositions, then " p Aq' 

 is an elementary proposition. 



Axiom 3. If p cund q are of the same truth-value, then ' p Aq' 

 is of the opposite truth -value ; hut if p and, q are of opposite truth- 

 values, then ' p Aq' is true. 



