Mr Van Horn, An Axiom in Symholio Loyic 27 



Theorem 5 



If 2) is false, ' p A q' is always true. 



Bern. 



Two possibilities can occur : either q true, or q false. In either 

 case ' p A q' IB true by Ax. 3. 



Theorem 6 



If q Is false, ' 2J A q' is alivays true. 

 Proof similar to that of preceding theorem. 



Theorem 7 

 llie jyropositions ' p A q' and ' q A p' Jmve the same triUh-ualue. 



Deni. 



li' p and q are of the same truth- value then, by Ax. o, ' p A q' 

 and ' q A p' are both of the opposite truth-value. If p and q are 

 of opposite truth-values then, by Ax. 3, ' p A q' and ' q A p' are 

 both true. Hence the theorem. 



Theorem 8 

 The proposition 



f"^ p A 1^ {(^ q A f^ r) 



is true if any one or more of the propositions p, q, r are true; but 

 if all of these propositions are false then the proposition 



~ jj A '^ {^ q A ~ r) 

 is false. 



Dem. 



Eight possibilities can occur : 



1° : p, q, r all true. Then (Th. 3) ~ ^j, ~ q, ~ r are all false. 

 Hence (Ax. 3) ' ~ 9 A ~ r ' is true. Hence (Th. 3) ~ (~ g- A ~ r) 

 is false. Hence (Ax. 3) the proposition '~ jo A ~ (~ </ A ~ /•)' 

 is true in this case. 



2^ : jj and q true, but r false. By Th. S, r^ p and ~ q are 

 false, while ~ r is true. Hence (Ax. 3) ' ~ r/ A ~ ?■ ' is true. 

 Hence (Th. 3) ~ (~ (/ A ~ r) is false. Hence (Ax. 3) the 

 proposition is true in this case. In a similar manner in the 

 following cases : 



3° : j) true, q false, r true ; 



4° : ]) false, q, r true ; 



o" : p true, q, r false ; 



6° : J) false, q true, r false ; 



7° : p, q false, r true ; 

 we have ' ^' j) A ^ (■-- </ A ^^ /•) ' true. 



But in 8" : p, q, r false, we have ~ jj, r^ q, ^ r all true, by 



