Now, given the stakes, Heller’s argument goes as follows; if one is to reject the truth of the undesirable alternatives Log in to see images! through (E) and still maintain the truth of three-dimensional theory, one is necessarily lead to a logical contradiction. The premises that the Heller’s argument is based upon follow, as presented and simplified by Michael Loux: Given the entity Descartes at time t1, which is a fully intact Descartes, and Descartes(-) at time t1, which is all of Descartes minus his left hand, and given that at a certain time t2, Descartes had his left hand amputated, the following must be true.

(I) Descartes at t1 is numerically identical with Descartes at t2, based on the belief that a person can survive an amputation as the same person.

(II) Descartes(-) at t1 is numerically identical with Descartes(-) at t2, based on the fact that nothing happened to Descartes(-).

(III) Descartes at t2 is numerically identical with Descartes(-) at t2, based on the fact that both are composed of exactly the same matter and occupy exactly the same space.

(IV) Descartes(-) at t1 is numerically identical with Descartes at t1, based on the transitivity of identity.

However:

(V) Descartes(-) at t1 is not numerically identical with Descartes at t1 by the

true Principle of the Indiscernibility of Identicals. (Loux, 238-240)

Thus, on Heller’s account, the endurantist must accept the truth of (I) through (III) as they are all based on simple principles of numerical identity, and given these premises, (IV) must be true by the transitivity of identity, which as mentioned above, is one of the three main principles of identity conditions. However, (V) is necessarily true also, because Descartes and Descartes(-) are different, distinct objects that have different shapes, mbumes, etc. It is thus that one finds that finishes Heller’s trap is complete, and apparently forces the endurantist into admission of believing contradictory truths, or abandonment of previously held views.

How then does Heller seek to avoid the contradiction? It is not immediately clear that perdurantism provides a suitable answer to the problem at hand, and it would certainly be problematic if Heller’s argument defeated both theories. According to the perdurantists, Neither (I) nor (II) must be held to be necessarily true, because Descartes and Descartes(-) are separate, distinct space-time worms. Descartes at t1 is not numerically identical to Descartes at t2 because both instances of Descartes are specific, different time slices from the aggregate whole of the Descartes space-time worm. As Loux explains it:

As perdurantists see things, Descartes is an aggregate if temporal parts; and his persistence over time is a matter of his having different temporal parts existing at different times. Descartes [at t1] and Descartes [at t2] are just such temporal parts. On the perdurantists’ view, then, Descartes’ making it through the amputation does not involve the numerical identity of Descartes [at t1] and Descartes [at t2]; it involves their standing in the weaker relation of being parts of a single space-time worm. (240)

The same holds true in the case of the temporally distinct Descartes(-) instances. That which guarantees sameness of personhood, the perdurantists claim, is simply that the two Descartes time slices are a part of the same space-time worm, different points on the 4-D axes of Descartes lifespan. To explain the overlap between Descartes and Descartes(-), the perdurantists claim that the two distinct space-time worms merge at t2 with no problems whatsoever. By thus avoiding premises (1) and (2), the perdurantist is capable of escaping the contradiction by simply not worrying about numerical identity.