The subject to which I invite your attention this evening is a very recent development in mathematics which happens to be of great importance to philosophy.The question may perhaps be raised why a terrifying topic like mathematics as introduced into an Institute of Philosophy. There are two answers to this question. In the first place my talk this evening will not be a “mathematical lecture” in any ordinary sense of the term, and no mathematical knowledge whatsoever will be presupposed. In the second place, even if it were to be a mathematical lecture, such an audience as is here assembled tonight would find nothing terrifying in the prospect.

The four types of order whose inter-relations are considered in this paper may be called, for brevity, (1) serial order; (2) betweenness; (3) cyclic order; and (4) separation.We first recapitulate the known sets of postulates which define each of these types as an abstract system, and recall the usual geometric interpretation of each type; we then develop the way in which each of these four types may be defined in terms of each of the other three.(For convenience of reference, the numbering of the postulates in earlier publications has been retained.)1. Serial order.A system (K, R), where if is a class of elements A, B, C, • ■ ■ , and R(AB), or simply AB, is a dyadic relation, is called a "system of serial order" when and only when the following four postulates are satisfied.In each of these postulates it is understood that distinct letters represent distinct elements of K. [The notation " =0" means "is false"; the "horseshoe", d , means "If . . .then"; the "wedge," v, means "or" (in the sense of "at least one"); and the "dot," ., means "and."Dots, singly or in groups, serve also as punctuation marks.]

IV.—INDEPENDENT POSTULATES RELATED TO C. I. LEWIS'S THEORY OF STRICT IMPLICATION1 Get access EDWARD V. HUNTINGTON EDWARD V. HUNTINGTON Search for other works by this author on: Oxford Academic Google Scholar Mind, Volume XLIII, Issue 170, April 1934, Pages 181–198, https://doi.org/10.1093/mind/XLIII.170.181 Published: 01 April 1934

Ilia. a+b = b+a.Illb.ab = ba.IVa.a+bc = (a+b)(a+c).IVb.a(b+c) =ab+ac.V. For each element a there is an element a' such that a+a' = u and aa' = z.VI.There are at least two distinct elements in the class K. From these postulates the following theorems are deduced in the paper cited.Vila.The z in Ha is unique.Vllb.The u in lib is unique.Villa.a+a = a.VHIb. aa = a.IXa.a+u=u.IXb. az = z.Xa. a+ab = a.Xb. a(a+b) =a.XL The element a' in V is uniquely determined by a. Xlla.a+b = (a'b')'.Xllb.ab = (a'+b')'.* The name Boolean algebra (or Boolean "algebras") for the calculus originated by Boole, extended by Schröder, and perfected by Whitehead seems to have been first suggested by Sheffer, in 1913.*1.71.If /> and ç are elementary propositions, p v q is an elementary proposition.*1.7.If p is an elementary proposition, ~p is an elementary proposition.*4.31.}-:pvq-= -qvp.*4.33.h: (p v q) v r• = ■ p v (a vr).*4.25.\-: p ■= -pvp.*4.5.\-: p-q-= -~(~p v~q).*4.42.h: • p-= :pq-v -p-~q.If now we call the class of "elementary propositions" the class K, and write p+qîoi pvq, and p' for ~p, these propositions become the following: *1.71.If p and q are in the class K, then p+q is in the class K. *1.7.If p is in the class K, then p' is in the class K. *4.31.p+q^q+p.*4.33.(p+q)+r^p+(q+r).*4.25.p=p+p.H.5.pq = (p'+q')'.*4.42.p=pq+pq'.But these propositions are precisely the same as the postulates of our fourth set for Boolean algebra, except that the sign = occurs in place of the sign =.It remains, therefore, to examine the properties of the sign m as used in the Principia, in comparison with Postulates A, B, C, D governing the use of the sign =.Here it is necessary to distinguish between the formal statements and the informal statements in the Principia.Among the formal statements we find *4.2.[-.p^p, which is the same as Postulate A.Another formal statement (in view of *4.01 and *1.01) is *4.21.h: (p = q)'-v.(q=p).In accordance with the informal statement under (6) in *1, this formula means ap = q is false or q^p is true," and this in turn means "If p=q, fhenq=p" which is the same as Postulate B.

Journal Article A SIMPLIFICATION OF LEWIS AND LANGFORD'S POSTULATES FOR BOOLEAN ALGEBRA Get access EDWARD V. HUNTINGTON EDWARD V. HUNTINGTON Harvard University Search for other works by this author on: Oxford Academic Google Scholar Mind, Volume XLII, Issue 166, April 1933, Pages 203–207, https://doi.org/10.1093/mind/XLII.166.203 Published: 01 April 1933

The proof of 5 from 1, 2, 3, 4, 6, 7 is as follows.* 6a. If a+b is in T and a not in T, then b is in T. (From 6.) 6b. If a not in T and b not in T, then a+b not in T. (From 6.) 7a. If a is in T, then a' is not in T. (From 7.) 3a. If b is in T, then a+b is in T. For, by 7a, b' is not in T. But by 3, b'+(a+b) is in T. Hence by 6a, a+b is in T. 4a. If bisin T, then b+a is in T. For, by 3a, a+b is in T, whence by 7a, (a+b)' is not in T. But by 4, (a+b)'+(b+a) is in T. Hence by 6a, b+a is in T. 5a. If a is not in T, then a' is in T. For, suppose a' not in T. Then by 6b, a'+a not in T, whence by 6b, a' +(a'+a) not in T, contrary to 3. 5. If a, b, etc. are in K, then (b'+c)'+ [(a+b)'+(a+c)] is in T. Case 1; a in T. By 4a, a+c is in T. Hence the theorem, by 3a (twice). Case2:bin T. By7a, b'isnotin T. If c is in T, then by 3a, a+c is in T, whence the theorem, by 3a (twice). If c is not in T, then by 6b, b' +c is not in T, whence by 5a, (b'+c)' is in T, whence the theorem, by 4a. Case 3: a not in T and b not in T. By 6b, a+b not in T, whence by 5a, (a+b)' is in T. Hence the theorem, by 4a and 3a. The proof is thus complete. It can also be shown that 1, 2, 3a, 4a, 5a, 6a, 7 form a set of independent postulates equivalent to the set 1, 2, 3, 4, 6, 7.

in which the number of postulates is reduced to five. Moreover, the new postulate 9 itself is easier to remember and more convenient to handle than any of the other postulates 1-8. The addition of this new postulate makes desirable an extension of the discussion of the earlier paper so as to include all thirteen of the basic postulates; and this extension has been made in the present paper. Finally, the postulates of the new set (12) are shown to be completely independent in the sense of E. H. Moore. (In regard to the other sets, a

(1919). Mathematics and Statistics, with an Elementary Account of the Correlation Coefficient and the Correlation Ratio. The American Mathematical Monthly: Vol. 26, No. 10, pp. 421-435.

(1918). Bibliographical Note on the Use of the Word Mass in Current Textbooks. The American Mathematical Monthly: Vol. 25, No. 1, pp. 1-15.