The case of infinitely many propositions following from a single one

Is it impossible that infinitely many propositions should follow from a single one-in the sense, that is, that we might go on ad infinitum constructing new propositions from a single one according to a rule?[1]

Modal interpretation:

Is the following a valid argument, or rather, is it a possible argument:

◊ (p→q /\ (q→r) /\ (r →s)…n)))

Another interpretation:

◊ (p→(q /\ r /\ s…n))

In English: “it is possible that p implies q and q implies r and r implies s ad infinitum”

Now, the justification for the ‘ad infinitum’ is the idea that each the first proposition, p, is the first in a series of logical implications.

I think an example of this idea might be the following:

1. In order to move to the door of this room I need to travel the distance between this chair and the exit for this room.

1. The distance between this chair and the exit is roughly 6 feet.

1. It is necessary that in order to travel from this chair to the room exit, I must also travel from this chair to half of the distance to the exit, which is roughly 3 feet.

1. It is necessary that in order to travel from this chair to ‘half of the distance to the exit’, I must also travel to ‘half of the half of the distance to the exit’, which is roughly 1.5 feet.

You get the picture: a single proposition functions as a rule for the infinite regress that follows.

Wittgenstein’s response to this language game:

Suppose that we wrote the first thousand propositions of the series in conjunction.  Wouldn’t the sense of this product necessarily approximate more closely to the sense of our first proposition than the product of the first hundred propositions? Wouldn’t we obtain an ever closer approximation to the first proposition the further we extended the product? And wouldn’t that show that it can’t be the case that from one proposition infinitely many others follow, since I can’t understand even the product with 10¹⁰ terms?

We imagine, perhaps, that the general proposition is an abbreviated expression of the product.  But what is there in the product to abbreviate? It doesn’t contain anything superfluous?[2]

In a nutshell, Wittgenstein opposes the idea that any proposition p can serve as a rule for an infinite regress (or a series of infinitely many propositions following from p) from that proposition.

The assumption in the line of thought Wittgenstein opposes here is that any proposition p is inherently indeterminate; more carefully constructed, the idea is that it’s always possible to infer q or r or s, for instance, from any p.

For convenience I’ll call this view propositional indeterminism.

The logic of propositional indeterminism

Our idea that a proposition is indeterminate and thus may always be inferred from springs from our idea of language as a calculus.

Consider

P: (10+4)=Q: 14

And

P: (10+4)=Q: 14 or R: (7*2)

And

P: 10+4= Q: 14 or R(7*2) or S(5*2+4)

Let’s translate this into a logical argument.  To make it easier, I treated both addition and multiplication as logical conjunctions.  I also added inferences to make the argument valid.

1. If p, then q
2. If q, then (r and s)
3. If q then (t and u)[3]
4. If q then (r and s) or (t and u)

Wittgenstein, in effect, is saying that the expression ‘10+4′ is not coextensive with respect to the expression ‘14′.  It is in some circumstances, but not in the context of using their coextensionality as a foundation for an infinite regress. By itself, the proposition which says that ‘if 10+4′ then ‘14′ is a valid construction via the rules of arithmetic.  However, the expression doesn’t also say that “14 and 7 multiplied by 2 are identical”.  That they are identical can be expressed in another proposition, but it is not guaranteed or logically necessitated by the distinct expression that 10+4 also means 14.

[2] Wittgenstein, PG 250

[3] I realize that treating both multiplication and addition as a logical conjunction is problematic but that it is problematic is not relevant to this discussion