Abstract
The fifth chapter sets up the challenge of insufficient binding as a relatively new problem for 4-dimensionalism, and in particular for the stage theory. I argue that the ontology of temporal counterpart relations is underspecified. I explicate this claim discussing various theories of persistence, which can be explicitly or implicitly understood as versions of the stage theory. The chapter ends my discussion of 4-dimensionalism and its challenges.
The sensible fourdimensionalist will claim that current temporal parts are caused to exist by previous temporal parts.
(Sider 2001, 217)
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Notes
- 1.
For a similar analysis, see Heller (1984, section 5).
- 2.
For a more detailed analysis I refer to Sect. 2.3.2 of the first part.
- 3.
Additions in […] by the author.
- 4.
Additions in […] by the author.
- 5.
The following figures are inspired by Jonathan Schaffer’s article on The Metaphysics of Causation (2016, n.p.).
- 6.
I would like to thank Roland Poellinger from the Munich Center for Mathematical Philosophy for discussing these concepts with me.
- 7.
Additions in italics and […] by the author.
- 8.
Additions in […] by the author.
- 9.
There are limits to this presentation and dismissal of EOSR. Christina Schneider has pointed out that there might be special structural relations that in some sense they act as closing function to structural systems. If I understand this idea correctly, then those systems would be self-implementing. French has developed a similar idea:
[French] emphasized how invariance and constancy of relations literally create objects: they play the role of conditions of accessibility which immediately become conditions of objects of experience. (Morganti 2004, 88)
If this idea could be further developed, my dismissal EOSR would not be sufficiently warranted. However, the proponents of EOSR admit that this would include an extreme revision of everyday ontology:
Let us consider the idea that relations need relata. On this view, a purely structural description of the world is impossible in principle because structures are defined by relations, but the latter presuppose related individuals. This objection is flawed because relations obviously need relata only in the traditional view of reality, where by assumption we take individual things as fundamental. French and Ladyman have no difficulty in replying that their proposal requires a deep revision of our conceptual apparatus. They also show that such a revision is possible by referring to already existing work. (Morganti 2004, 98)
Nevertheless, such a revision would not undermine my project substantially. The revised concept of relations would have to be of a completely different kind, than the relations we normally work with in scientific theories. I therefore assume that those reconceived relations would turn out to behave similarly to the carriers, for which I am going to argue at the end of this section. Ultimately, I suppress this possibility for the sake of a concise argument – but it must be mentioned, at least in an (admittedly lengthy) footnote.
- 10.
Where C(…) denotes the predicate ‘…is a cat’ and M(…) denotes the predicate ‘…sits on a mat’.
- 11.
- 12.
Here you will find a loosely formal substantiation of the skolem theorem for a model with a denumerable universe:
Consider a first-order-logic model \(\mathcal{L}\) with a denumerable universe, such that \(\mathcal{L}\) can be written as quadruple 〈 \(\mathbb{K}\), \(\mathbb{F}\), \(\mathbb{P}\), ar 〉. \(\mathbb{K}\) denotes the finite set of constants. \(\mathbb{F}\) denotes the set of all the functional terms, such like the normal operation ‘+’. \(\mathbb{P}\) denotes the set of the predicates and ar the arity of \(\mathcal{L}\).
Consider some mapping from the variables of \(\mathcal{L}\) to the predicates. For example Φ:x → Px, where P\(\in \mathbb{P}\). Surely, a sentence of the form ‘Px’ is an item of \(\mathcal{L}\). Let us, therefore, call \(\mathcal{L}\) the respective maximal model.
Nevertheless, when considering models we are interested in the preservation of truth-values of propositions. Thus, we can define two models to be equivalent (Newman 1928, 139), if they range over the same set of constants and if they give rise to the same distribution of truth-values.
Further, consider a model \(\mathcal{L}\)* such that \(\mathcal{L}\)* is identical to \(\mathcal{L}\) only \(\mathcal{L}\)* has no free variables. This implies that this model reaches only over the set of constants. We can call \(\mathcal{L}\)* the \(\mathbb{K}\)-model. Trivially, \(\mathcal{L}\)* is finite, because the set \(\mathbb{K}\) was defined to be finite.
But: \(\mathcal{L}\) and \(\mathcal{L}\)* give rise to the same distribution of truth values. Why? Well, just because for any formula Px it holds true, that only the respective constants bring about truth-values – for example Pa. And since \(\mathcal{L}\) and \(\mathcal{L}\)* differ only in their free variables, this means that for any formula it holds true that if the formula is true in \(\mathcal{L}\), then it is true in \(\mathcal{L}\)* (and vice versa). This means that \(\mathcal{L}\) and \(\mathcal{L}\)* are equivalent in the sense discussed above.
Q.E.D. This proofs Skolem’s theorem.
- 13.
Additions in […] by the author.
- 14.
The following argument has been developed by David Oderberg (2012, 6–9). I will merely give a re-formulation of it here.
- 15.
Additions in […] by the author.
- 16.
Personally, I do not think that Balashov’s definition of exdurance is consistent. To me it seems utterly unclear, how some thing can be said to be exactly located and LOCATED multiply at the same time. Because of this problem, I will not follow Balashov’s line of reasoning within the context of the following considerations – in particular Chap. 3 of this essay. Marc Andree Weber has pointed me to another serious problem: It seems like certain seemingly equivalent definitions of Balashov’s are not logically equivalent: for example D12 and D13.
- 17.
The paper can be made available by the author.
Another important remark: In the following section, I am going to use a fairly technical, mathematical terminology. Thus, terms like ‘field’ (in German: Körper) are defined by their mathematical meaning.
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Jaskolla, L. (2017). The Challenge of Insufficient Binding. In: Real Fourdimensionalism. Philosophical Studies Series, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-319-65927-5_5
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