The reason Tortoise couldn’t stop hypothesizing is that an implication never asserts its constituent propositions.
A. P is true.
B. if P is true then Q is true, (P implies Q).
B only asserts the implication. Even if both P and Q are false, B can still be true. One is tempted to say there is an implicit hypothesis C which asserts that A and B together imply Q:
C. If P is true and P implies Q, then Q is true.
But C, as an implication, can be true regardless whether A, B, or Q are true or not. One is tempted to say A, B, C together implies Q:
D. If A is true, B is true and C is true than Q is true.
Since D, again, can be true by itself and insufficient to reach Q, one would say A, B, C, D together implies Q. Thus the Tortoise’s regression.
The root of the problem is that, in all the newly accepted hypotheses, Q is never asserted. Each hypothesis only asserts the implication. An implication only asserts the relation of what proposition logically follows a given proposition. An implication does not assert the constituent propositions. That is why each hypothesis can be true by itself yet still insufficient to move any closer to Q.
In order to stop the regression, we need to distinguish inference from implication. An inference deduce one true proposition from another true proposition instead of simply imply one unasserted proposition from another unasserted proposition.
*1.1 and *1.11 in PM is the primitive idea that converts asserted implications into inferences:
*1.1 Anything implied by a true elementary proposition is true. Pp
Notice that *1.1 and *1.11 are not hypotheses. They are general statements and whatever inference can be made in virtue of *1.1 or *1.11 has already been contained by *1.1 or *1.11. In other words, the inference discovers nothing new and asserts nothing that has not been asserted by *1.1 and *1.11. *1.1 and *1.11 simply help the reader to “see” particular cases. Take the Tortoise’s case for example:
(A) Things that are equal to the same are equal to each other.
(B) The two sides of this triangle are things that are equal to the same.
(Z) The two sides of this triangle are equal to each other.
Both A and B are true means this is a particular case of *1.1. Since 1.1 is a general assertion that asserts all its individual cases, we know Z is asserted.
Another example illustrates that 1.1 is not a hypothesis:
A. Pigs Fly.
B. If Pigs Fly then I am pope.
C. If pigs fly and pigs fly implies I am pope then I am Pope.
Z. I am Pope.
Both B and C assert only implications that are true and can hold on its own but, nevertheless, assert none of its constituent propositions. Since A is false, A and B together does not make a case that belongs to the class of cases asserted by 1.1, thus Z cannot be accepted.
Russell, Bertrand. §38. The Principles of Mathematics. New York. London: W.w. Norton & Company, 1996
Alfred North Whitehead. Russell, Bertrand. *1.1. Principia Mathematica 1st ed. Merchant Books, 1910
Carroll, Lewis. What the Tortoise Said to Achilles. Mind, 1895.