Sheffer Quantifies Over Possible Universes (1922)

With C, a “necessary condition” for (5), Giving, our task would be to see if there is at least one universe of discourse where “x . ~igx” fails to hold, that is, where there is a governor of Massachusetts elected 1900-1915 for less than one term. B is not a necessary condition for Giving, because it does not state that the connection holds for every possible “universe,” we might select another from the neutral realm of all possible ones. It does so happen that all governors of Massachusetts 1900-1915 were elected for more than one year, this being the nature of gubernatorial elections in the state. Thus not-B, that is, C, is a necessary condition for Giving, according to Sheffer. Intuitively, the very idea of this relation is that it is true for every possible universe.

Obviously, there are vaguenesses to be straightened out: this is Sheffer’s point. He goes on to elaborate a variety of relations. “Inductively Implies” is exemplified by the fact that “x is the distance between particles A, and A₂” “produces” “the attraction between A] and A₂ is proportional and inverse square of x.” He next suggests glossing Giving as “purely accidental” or “Conventional Implication,” in the sense that “the rules of the game might just as well have been otherwise.”

He also points out that the following are not necessary conditions for Comprising, Involving, Producing, and Giving:

D: (U)(x) ~x

E: (U)(x) igx

for if we substitute into the matrix in B (~[^x . ) the concepts x =

“x is a five-sided plane Euclidean triangle” and igx = “x is a horse,” their conjunction gives a uniformly false result, making B hold, whereas their combined substitution into D is true. If we substitute for “igx” in E the concept “x is either a triangle or not a triangle,” because it is true for every universe and every x, we can make E hold, whereas substituting in for “ipx” in E “x is a horse” yields an “absurd” result, namely that there are horses in every possible universe (of discourse). Sheffer concludes that B, not-D and not-E are “necessary and sufficient conditions for Involving” (18).

These difficulties lead Sheffer to emphasize that one must continually attach additional conditions to Comprising or else one “gets funny results.” Substituting into B a contradictory concept (“x is a triangle and not a triangle”) for one of the propositional functions may yield, without other conditions being met, a condition of “pseudo-involving,” and this is true also when A holds, but when one of not-D or not-E doesn’t hold.

In this way, Sheffer rearticulates Lewis’s worries about the “paradoxes” of implication, but he presents them in the context of a higher-order theory. Substituting into C “x is a governor of Massachusetts in 1900-1915 elected for a term of 5 years” for “fix” and for “if/x” “x is a horse” yields a “foolish” result, namely that there is a universe in which for all x, nothing is a governor elected in Massachusetts for 5 years (the term being four years by “convention”) and also not a horse. This is the relation of “pseudo-comprising” (later, in canvassing “strange logics,” Sheffer will discuss “pseudo-negation”). One gets to “pseudo-giving” in the same case by similar substitutions into two other conditions:

F: (U)(x) -fix

G: (U)(x) i//x^5s

Because, presumably, it is “foolish” - that is, wholly unuseful, one is tempted to say - to state that there is a universe in which nothing is a governor elected in Massachusetts between 1900 and 1915, and similarly to state that there is a universe in which there are horses.

We see here a kind of combined problem of types and of modality. But Sheffer’s point lies elsewhere. “We are after real implication,” he says, and he shows how difficult this quest will be. Clearly the conditions he has formulated, he says, are “not deductive implication” (21). Russell’s propositional functional “implication” is formal implication, and Sheffer takes Russell to have meant with his

(x): fix => tf/x

“for every x in at least one universe of discourse” (1922,21). Sheffer says that Russell has “such a wide definition” and “makes certain sweeping statements” claiming “to have no exceptions” that what he says is just false. Russell’s Principia notation “has symbolic generality,” but it is “not deductive or mathematical implication.” In sum, we should “object to this definition” of “implies,” but not from the point of view of getting

Sheffer, Lewis, and the “Logocentric Predicament” 85 neat theorems. Rather, one should avoid the “fallacy” of thinking that “deductive implication” has been analyzed by Russell.

Here, Sheffer says, Lewis has a point. Lewis would, Sheffer thinks, have gone to B if he had considered “implication between propositional functions,” but he did not, basing his criticisms of Russell (which are “just,” according to Sheffer) on propositional logic alone (1922,22). Lewis therefore also failed, however, to avoid “pseudo-involving.” Sheffer infers that Lewis’s “strict implication” is neither the deductive implication of mathematics, nor deductive implication for the logician in general (22). Sheffer says,

In Principia, [we have a] set of postulates for implication of propositional functions and propositions.

In Lewis, [we have a] set of postulates for his propositional implication. [A] set of postulates for Involving alone - [we face] certain difficulties. [We must] ask [a] prior question - Is it possible? Can you work out postulates for deductive implication?

It can’t be done (1922, 22).

The only path forward, Sheffer thinks - and he tries some faltering steps in this direction in the lectures - is to try to formulate wider ways of encompassing “implication.” He points out that Lewis “tried to amend the situation” in a certain direction with “strict implication,” which is an “improvement” over Russell. Sheffer formulates it this way:

B': (U)~(p.~q)

This is an “improvement” over Russell’s material implication, which might be rendered this way, as saying that in some universe or other the combination p and not-q is false:

C': (UJ~(p.~q)

Sheffer regards Russell’s C' as the “highest common factor” of all of the relations of “implication” he discusses (24). But he points out that there is “a very subtle difficulty” to be faced:

[T]he mathematician says, in order not to have to give exceptions, where [he] says, “has 2 roots” [that if there is one], take it twice. [The mathematician] has a right to do it, since he assumes logic.

But when [a] man writing on foundations of logic says so, [he] is in danger of saying something fallacious.

Russell’s desire for [a] general definition which won’t have exceptions: that has pitfalls. Russell doesn’t use what he says ... a false proposition implies every proposition - Paradoxical.

(1922,24)

Sheffer goes on to propose that only the algebraical approach can gain the necessary comprehensiveness, and he develops a combination of operational logic and algebraical structures, melding a “newfangled set of postulates for Boolean algebra, for one interpretation” (1922, 26). He points out that multiple interpretations will be possible: in terms of classes, integer structures, propositional logic, or even just the truth-values themselves. He writes out a truth table, crediting Frege with the idea (and probably having read Post’s work on truth tables very recently) (27).

He then revisits the issue of method, distinguishing between two necessary moments in the project of logic. There is the “descriptive” aspect of postulates - the statements of facts within logic - and the “prescriptive” aspects of postulates, which is “normative.” This anticipates his later framing of the “logocentric predicament” in his review of the second edition of Principia, the passage with which we began this essay (1926,228; quoted earlier). As Sheffer sees it, the issue of “notations” must be separated from the possible “loci” or interpretations of these “forms,” and “these two phases in turn must be kept separate” from a discussion of the “conditions” that make the first two phases “significant and valid.” In his 1926 review, we see several of the issues that appear in the 1922 lectures briefly alluded to. Sheffer says, for example, that so far as the primitive propositions of Principia are concerned, “H p" “may always be interpreted as “p is necessary,” since that is really what a “primitive” proposition means. “This,” he adds, “has an important bearing on the analysis of the “real” meaning of implication” (1926, 230n).