98 MR. JOSEPH JOHN MURPHY ON AN 



are not generally true of numbers, the conclusions of 

 these syllogisms are not all true in arithmetic, except 

 when L has the value of unity, of which the interpretation 

 is that the enclosure is co-extensive with its includent. 

 Those conclusive forms are doubly underlined which are 

 true in arithmetic for all values of L ; those are singly 

 underlined which are true in arithmetic only when L has 

 the value of unity ; and the inconclusive ones are not 

 underlined. It will be seen that eight out of the sixteen 

 are doubly underlined. 



I. L.L=L 9. L°.L=L 



10. L° .L-' = U 



II. L°.L°=L° 



2. 



L.L-'=L° 



3- 



4. 



L.L°=L'' 



L. {L-'Y=L 



5. 



L-'.L={L-'Y 



6. 



L-'.L-'=L-' 



7- 



L-',L°=L-' 



8. 



L-'. {L-'Y={L-'Y 



12. L°.{L-y = L° .{L-')° 



13. { L-r-L={L-r 



14. [L-'Y .L-'=L-' 



15. {L-'Y ■^°={^~T-L° 



The truth and the applicability of the fundamental 

 equation of this system, namely 



do not depend on the interpretation of L as inclusion ; 

 this equation is merely the expression of the transitiveness 



