Mr Wiener, Studies in Synthetic Logie. 15 
m-adic and n-adic relations exist, we shall have to permit a certain 
logical laxity in our symbolism. Though our theorems really 
demand a separate, though precisely parallel, proof when the 
relations dealt with are m-adic and when they are n-adic, we shall 
have to treat these proofs as one. Though every relation holds 
among a definite set of terms, we shall permit dots to fill the 
places of an indefinite number of these. Though the analogues of 
A, Y, §, etc. are different with each different sort of relation with 
which they have to do, we shall represent them all by the symbols 
we use in the case of binary relations. To the reader acquainted 
with symbolic logic, there will be no difficulty in reducing any 
particular case of the theorems I prove to a strictly rigorous 
form. 
§2. Let us write the proposition, ‘a,, d:,..., @, are in the 
n-adic relation R, as R {a,, a, ..., Gn}. I shall call a property of 
an n-adic relation, R, an n-transitinty of R when it can be written 
in the form 
(1) (qb, bs, D005 by) 0 Tr {ai, Az, Ug, +++, An, b;, bs, g00%) by = 
De iy ce Ge, 0 LO Gio Gay cee Gal 
where 7’, is the logical disjunction of a number of expressions in 
the form 
Bleep ---s Gn} - F ei,04', 1,001) « BG", 044, .-.30n"} 
R {c,, Gy e@eey Gale 
where / is not necessarily the same in each of these expressions, 
NEC IOC) 6 Co 9 oo 5. Cn p coe Ci Cay one Cn, which are not all 
distinct from one another, are to be found among ay, ds, ..., dn, 
b,,b,,...,b,;. Ordinary binary transitivity is an example of a 
2-transitivity ; the property of ‘betweenness, which may be 
written 
(qd): abd.bde.v.abd.bed.v.adc.dbec. 
v.abd.acd.bac.v.dab.dac.bac.v. bea: Dap,» abe, 
is a 3-transitivity; the property of Vailati’s separation-relation, 
which may be written 
(qe): ab||dc.v.cd||ab.v.ab|lec.ae||cd : Da,s,c,a+ ab|| cd, 
is a 4-transitivity. From these examples it is obvious that the 
transitivity-properties of relations are of very great logical 
interest, and that a method which shall point out significant 
analogies between the various sorts of transitivity is not without 
importance. 
One property which all sorts of mn-transitivity have in 
common is this: if R is any n-adic relation whatever, then it is 
always possible, given any particular form of n-transitivity, to 
construct ina perfectly determinate manner a relation, R’, 
