Variable theory

Question

I wanted to know if there's any alternative variable theory to Russell's used in his Principia. I mean, the modern definition of "variable" and "constant" still follows his works? I try to search on the web but I didn't found anything, so if you guys know some alternative variable theory also, I would be glad to know about. Thanks for your time, have a nice day.

Answer 1

An in depth analysis of the development of pre-Principia Russell's logic is in

Gregory Landini, Russell's hidden substitutional theory (1998); see all PART I : THE UNRESTRICTED VARIABLE, and in Ch.2 the subchapter : The Analysis of the Variable (page 63-on).

From a "technical" point of view, the main problem is that PM's logic is "sloppy" (compared to Frege's and modern "standard") regarding syntactical specifications.

In addition, there is no semantics into PM, simply because logic (according to Russell) is the "most general" theory: its "laws" (i.e.logical truth) are laws which "apply" to everything.

Thus, I think, the initial explanation of the role of variables is unsatisfcatory because it mix up "sloppy" syntactical considerations and (missing) semantical aspects.

In addition, there are no use of "schema": laws of propositional calculus are $p \supset q$, with $p,q$ propositional letters, instead the "modern one" : $\mathcal A \rightarrow \mathcal B$, with $\mathcal A,\mathcal B$ schematics; but an explicit rule of substitution is missing.

But if you read the "formal" part of this magnum opus, I think (but my "practice" with it is limited) that you will not find significant differences regarding the way the "machine works".

The formula are sometime hard to read, due to the old symbolism, and the proof are always too much "compressed", bur the derivations are of course "sound".

From a more "philosophical" point of view, we can compare a modern exposition; see Joseph Shoenfield, Mathematical Logic (1967), page 12 :

[in order to express some "interesting" mathematical facts] we introduce individual variables. These are like the variables in [mathematical] texts [like $x,y$ into an equation : $(x+y)^2=x^2+2xy+y^2$], except that they vary through the individuals [of the domain] instead of through the real numbers. Thus an individual variable can mean any individual, but its meaning remains fixed throughout anyone context.

A formula containing an individual variable has many meanings, one for each assignment of an individual as meaning to each individual variable in the formula. If we assert such a formula, we are asserting that all of its meanings are correct.

For Russell, there is only one domain : the "universe". Laws of logic are "universal" because they are not restricted to a certain domain of discourse.

Thus, in Principia's logic we have variables for propositions : the propositional letters $p, q, ...$ as in *2.11 : $\vdash . p \lor \lnot p$.

Vriables for functions : $f, g, ... \phi, \psi, ...$; variables for relations : $R, S, ...$; variables for classes : $\alpha, \beta, ...$.

But in Principia's universe there are also prpopositional functions : $\phi x$.

Classes and propositional functions are related : we have that :

$\vdash : . \alpha = \hat z (\phi z) . \equiv : x \in \alpha . \equiv_x . \phi x$,

i.e. $\alpha$ is identical with the class determined by $\phi \hat z$ when, and only when, "$x$ is an $\alpha$ " is formally equivalent to $\phi x$.

And this rises the "ontological" issue of propositional fucntions into Principia, which are "primitive" [they are "objects" of the universe], while class terms are only "abbreviations".