Abstract
In a series of papers (Fine et al., 1982; Fine, Noûs28(2), 137–158; 1994, Midwest Studies in Philosophy, 23, 61–74, 1999) Fine develops his hylomorphic theory of embodiments. In this article, we supply a formal semantics for this theory that is adequate to the principles laid down for it in (Midwest Studies in Philosophy, 23, 61–74, 1999). In Section 1, we lay out the theory of embodiments as Fine presents it. In Section 2, we argue on Cantorian grounds that the theory needs to be stabilized, and sketch some ways forward, discussing various choice points in modeling the view. In Section 3, we develop a formal semantics for the theory of embodiments by constructing embodiments in stages and restricting the domain of the second-order quantifiers. In Section 4 we give a few illustrative examples to show how the models deliver Finean hylomorphic consequences. In Section 5, we prove that Fine’s principles are sound with respect to this semantics. In Section 6 we present some inexpressibility results concerning Fine’s various notions of parthood and show that in our formal semantics these notions are all expressible using a single mereological primitive. In Section 7, we prove several mereological results stemming from the model theory, showing that the mereology is surprisingly robust. In Section 8, we draw some philosophical lessons from the formal semantics, and in particular respond to Koslicki’s (2008) main objection to Fine’s theory. In the appendix we present proofs of the inexpressibility results of Section 6.
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References
Casati, R., & Varzi, A.C. (1999). Parts and places: the structures of spatial representation. Cambridge: MIT Press.
Cotnoir, A.J. (2013). Beyond atomism. Thought, 2(1), 67–72.
Cotnoir, A.J. (2014). Universalism and junk. Australasian Journal of Philosophy, 92, 649–664. https://doi.org/10.1080/00048402.2014.924540.
Cotnoir, A.J., & Bacon, A. (2012). Non-wellfounded mereology. The Review of Symbolic Logic, 5(2), 187–204.
Evnine, S. (2016). Making objects and events: a hylomorphic theory of artifacts, actions, and organisms. Oxford: Oxford University Press.
Fairchild, M. (2017). A paradox of matter and form. Thought, 6, 33–42.
Fine, K. (1977). Properties, propositions and sets. Journal of Philosophical Logic, 6(1), 135–191.
Fine, K. (1982). Acts, events, and things. In Leinfellner, W., Kraemer, E., Schank, J. (Eds.) Language and ontology. Proceedings of the 6th international Wittgenstein symposium (pp. 97–105). Vienna: Hölder-Pichler-Tempsky.
Fine, K. (1994). Compounds and aggregates. Noûs, 28(2), 137–158.
Fine, K. (1999). Things and their parts. Midwest Studies in Philosophy, 23, 61–74.
Fine, K. (2005). Our knowledge of mathematical objects. In Gendler, T.Z., & Hawthorne, J. (Eds.) Oxford studies in epistemology (pp. 89–109). Oxford: Clarendon Press.
Fine, K. (2006). Relatively unrestricted quantification. In Rayo, A., & Uzquiano, G. (Eds.) Absolute Generality, chapter 2 (pp. 20–44). Oxford: Oxford University Press.
Fine, K. (2007). Response to Kathrin Koslicki. Dialectica, 61, 161–166.
Fine, K. (2010). Towards a theory of part. The Journal of Philosophy, 107, 559–589.
Johnston, M. (2006). Hylomorphism. The Journal of Philosophy, 103(12), 652–698.
Koslicki, K. (2007). Towards a neo-Aristotelian mereology. Dialectica, 61(1), 127–159.
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Acknowledgements
This paper arose out of the weekly Metaphysics Research Group seminars at the Arché Philosophical Research Centre at the University of St Andrews. We are thankful to participants for their contributions. Thanks also to audience members at the the Metaphysics of Totality workshop and the University of Glasgow, especially Phillip Blum, Louis deRosset, Berta Grimau, Nathan Kirkwood, Anna-Sophia Maurin, Stephan Leuenberger, Alex Skiles, Naomi Thompson, Bruno Whittle, and Nathan Wildman. Thanks also to the Argument Clinic Group of the University of Lisbon, especially to Raimundo Henriques, José Mestre, Ricardo Miguel, and Diogo Santos. Thanks to two anonymous referees for this journal for their valuable comments. Special thanks to Gabriel Uzquiano and Maegan Fairchild for their comments on an earlier draft of this paper that led to many improvements. The research and writing of this paper was supported in part by a 2017–2018 Leverhulme Research Fellowship from the Leverhulme Trust.
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Appendix
Appendix
1.1 A.1 <-bisimulation
In this appendix we offer a proof of Theorem 2. The proof of this theorem will appeal to a notion of model indistinguishability, <-bisimulation. We begin by formulating this notion and proving that it indeed captures a sense in which models are indistinguishable vis-á-vis parthood.
Definition 43 (<-Bisimulation)
Where \(\mathscr{M}=\langle \mathscr T,\mathscr P,\mathscr d_{0},\kappa ,\leq ,\mathscr v\rangle \) and \(\mathscr{M}^{\bigstar }=\langle \mathscr T^{\bigstar },\mathscr P^{\bigstar },\mathscr d^{\bigstar }_{0},\kappa ^{\bigstar },\leq ^{\bigstar },\mathscr v^{\bigstar }\rangle \) are any LE models (based, respectively, on frames \(\mathscr F\) and \(\mathscr F^{\bigstar }\)), a function \(\pi :\mathscr B_{\kappa }\cup \bigcup _{n\in \mathbb {N}}\mathscr D^{n}_{\kappa }\cup \mathscr T\cup P\to \mathscr B^{\bigstar }_{\kappa }\cup \bigcup _{n\in \mathbb {N}}\mathscr D^{\bigstar ,n}_{\kappa }\cup \mathscr T^{\bigstar }\cup P^{\bigstar }\) is a <-bisimulation between \(\mathscr M\) and \(\mathscr M^{\bigstar }\), \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\), if and only if:
-
1.
For every \(\mathscr t\in \mathscr T\): \(\pi \restriction _{\mathscr T}\) is a bijection between \(\mathscr T\) and \(\mathscr T^{\bigstar }\);
-
2.
For every \(\mathscr t,\mathscr t^{\prime }\in \mathscr T\): \(\mathscr t \leq \mathscr t^{\prime }\) if and only if \(\pi (\mathscr t)\leq ^{\bigstar }\pi (\mathscr t^{\prime })\);
-
3.
For every \(\mathscr p\in \mathscr P\): \(\pi \restriction _{\mathscr P}\) is a bijection between \(\mathscr P\) and \(\mathscr P^{\bigstar }\);
-
4.
For every \(\mathscr t\in \mathscr T\): \(\pi \restriction _{\mathscr d_{\kappa ,\mathscr t}}\) is a bijection between \(\mathscr d_{\kappa ,\mathscr t}\) and \(\mathscr d^{\bigstar }_{\kappa ,\pi (\mathscr t)}\);
-
5.
For every \(\mathscr t\in \mathscr T\) and \(\mathscr p\in \mathscr P\): \(\pi \restriction _{\mathscr d_{\kappa ,\mathscr t,\mathscr p}}\) is a bijection between \(\mathscr d_{\kappa ,\mathscr t,\mathscr p}\) and \(\mathscr d^{\bigstar }_{\kappa ^{\bigstar },\pi (\mathscr t), \pi (\mathscr p)}\);
-
6.
For every \(n\in \mathbb {N}\):
-
(a)
\(\pi \restriction _{\mathscr D^{\bigstar ,n}_{\kappa }}\) is a bijection between \(\mathscr D^{n}_{\kappa }\) and \(\mathscr D^{\bigstar ,n}_{\kappa ^{\bigstar }}\);
-
(b)
For every \(\mathscr X\in \mathscr D^{n}_{\kappa }\): \(\pi (\mathscr X)(\pi (\mathscr t))=\{\langle \pi (\mathscr x_{1}),\ldots ,\pi (\mathscr x_{n})\rangle :\langle \mathscr x_{1},\ldots ,\mathscr x_{n}\rangle \in \mathscr X(\mathscr t)\}\)
-
(a)
-
7.
For every individual constant σ: \(\pi (\mathscr{v}(\sigma ))=\mathscr{v}^{\bigstar }(\sigma )\);
-
8.
For every n-ary predicate ζ, for every \(n\in \mathbb {N}\): \(\pi (\mathscr v(\zeta ))=\mathscr v^{\bigstar }(\zeta )\);
-
9.
For every \(\mathscr t\in \mathscr T\):
\(\pi (<^{\mathscr t}_{\mathscr F})=\{\langle \pi (\mathscr x),\pi (\mathscr y)\rangle :\langle \mathscr x,\mathscr y\rangle \in <^{\mathscr t}_{\mathscr F}\}=<^{\pi (\mathscr t)}_{\mathscr F^{\bigstar }}\).
A first important result of the appendix is a theorem to the effect that <-bisimilar models are indiscernible vis-à-vis parthood, in the minimal sense that the same formulae of \(\mathrm {L^{<}_{E}}\) are true in <-bisimilar models:
Theorem 1
For all models \(\mathscr M\) and \(\mathscr M^{\bigstar }\) (based, respectively, on frames \(\mathscr F\) and \(\mathscr F^{\bigstar }\) ) and function π such that \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\) , and for every formula φ of \(\mathrm {L^{<}_{E}}\) :
\(\mathscr M,\mathscr t,\mathscr g\vDash \varphi \) if and only if \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \varphi \) ,
The proof of this theorem will rely on some auxiliary definitions and lemmas. We start by defining the image of a variable-assignment under a bisimulation.
Definition 44
For all models \(\mathscr M\) and \(\mathscr M^{\bigstar }\) (based, respectively, on frames \(\mathscr F\) and \(\mathscr F^{\bigstar }\)), if \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\), then, for every variable-assignment \(\mathscr g\) based on \(\mathscr F\), \(\pi (\mathscr g)\) is that function such that, for every individual variable, second-order variable or spatial variable ρ,
\(\pi (\mathscr g)(\rho )=\pi (\mathscr g(\rho ))\).
The following lemma shows that the image of a variable-assignment under a <-bisimulation is itself a variable-assignment:
Lemma 13
For all models \(\mathscr M\) and \(\mathscr M^{\bigstar }\) (based, respectively, on frames \(\mathscr F\) and \(\mathscr F^{\bigstar }\) ), if \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\) , then \(\pi (\mathscr g)\) is a variable-assignment based on \(\mathscr F^{\bigstar }\) , for every variable-assignment \(\mathscr g\) of \(\mathscr F\) .
Proof of Lemma 13
Suppose that \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\) where, \(\mathscr M\) is based on \(\mathscr F\) and \(\mathscr M^{\bigstar }\) is based on \(\mathscr F^{\bigstar }\). Let \(\mathscr g\) be an arbitrary variable-assignment based on \(\mathscr F\).
Take an arbitrary individual variable v. We have that \(\pi (\mathscr g)(v)=\pi (\mathscr g(v))\). Since \(\mathscr g(v)\in \mathscr B_{\kappa }\), by Definition 26, it follows that \(\mathscr g(v)\in \mathscr d_{\kappa ,\mathscr t}\), for some \(\mathscr t\in \mathscr T\), by Definition 7. So, \(\pi (\mathscr g)(v)=\pi (\mathscr g(v))\in \mathscr d^{\bigstar }_{\kappa ^{\bigstar },\pi (\mathscr t)}\), by Definition 43. Therefore, \(\pi (\mathscr g)(v)\in \mathscr B^{\bigstar }_{\kappa ^{\bigstar }}\), by Definition 7.
Take an arbitrary n-ary second-order variable V. We have that \(\pi (\mathscr g)(V)=\pi (\mathscr g(V))\). Since \(\mathscr g(V)\in \mathscr D^{n}_{\kappa }\), by Definition 26, it follows that \(\pi (\mathscr g)(V)=\pi (\mathscr g(V))\in \mathscr D^{\bigstar ,n}_{\kappa ^{\bigstar }}\), by Defn. 7.
Take an arbitrary spatial variable s. We have that \(\pi (\mathscr g)(s)=\pi (\mathscr g(s))\). Since \(\mathscr g(s)\in \mathscr P\), by Definition 26, it follows that \(\pi (\mathscr g)(s)=\pi (\mathscr g(s))\in \mathscr P^{\bigstar }\), by Definition 7.
Therefore, \(\pi (\mathscr g)\) is a variable-assignment based on \(\mathscr F^{\bigstar }\). □
The next lemma shows that values under variable-assignments are preserved under <-bisimulations:
Lemma 14
For all models \(\mathscr M\) and \(\mathscr M^{\bigstar }\) (based, respectively, on frames \(\mathscr F\) and \(\mathscr F^{\bigstar }\) ), if \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\) , then, for every individual constant, individual variable, n-ary predicate or n-ary second-order variable σ of \(\mathrm {L^{<}_{E}}\) :
\(\pi (\mathscr v^{\mathscr g}(\sigma ))=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\) .
Proof of Lemma 14
Suppose σ is an arbitrary individual constant or n-ary predicate. Then, \(\mathscr v(\sigma )=\mathscr v^{\mathscr g}(\sigma )\) and \(\mathscr v^{\bigstar }(\sigma )=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\), by Definition 27. By Definition 43, \(\pi (\mathscr v(\sigma ))=\mathscr v^{\bigstar }(\sigma )\). So, \(\pi (\mathscr v^{\mathscr g}(\sigma ))=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\).
Suppose σ is an individual variable or an n-ary second-order variable. Then, \(\mathscr g(\sigma )=\mathscr v^{\mathscr g}(\sigma )\), by Definition 27. By Lemma 13, \(\pi (\mathscr g)\) is a variable-assignment based on \(\mathscr F^{\bigstar }\). Furthermore, by Definition 27, \(\pi (\mathscr g)(\sigma )=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\). Hence, \(\pi (\mathscr v^{\mathscr g}(\sigma ))=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\).
So, either way, \(\pi (\mathscr v^{\mathscr g}(\sigma ))=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\). This concludes the proof of Lemma 14. □
We now turn to the proof of Theorem 7:
Proof of Theorem 7
The proof is by induction on the complexity of a formula of \(\mathrm {L^{<}_{E}}\).
-
φ is ζσ1…σn:
\(\mathscr M,\mathscr t,\mathscr g\vDash \zeta \sigma _{1},\ldots ,\sigma _{n}\) iff \(\langle \mathscr v^{\mathscr g}(\sigma _{1}),\ldots ,\mathscr v^{\mathscr g}(\sigma _{n})\rangle \in \mathscr v^{\mathscr g}(\zeta )(\mathscr t)\), by Definition 28, iff \(\langle \pi (\mathscr v^{\mathscr g}(\sigma _{1})),\ldots ,\pi (\mathscr v^{\mathscr g}(\sigma _{n}))\rangle \in \pi (\mathscr v^{\mathscr g}(\zeta ))(\pi (\mathscr t))\), by Definition 43, iff \(\langle \mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma _{1})),\ldots ,\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma _{n}))\rangle \in \mathscr v^{\bigstar ,\pi (\mathscr g)}(\zeta ))(\pi (\mathscr t))\), by Lemma 14, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \zeta \sigma _{1} \ldots \sigma _{n}\), by Definition 28;
-
φ is Loc(s)σ:
\(\mathscr M,\mathscr t,\mathscr g\vDash Loc(s)\sigma \) iff \(\mathscr v^{\mathscr g}(\sigma )\in \mathscr d_{\kappa ,\mathscr t,\mathscr g(s)}\), by Definition 28, iff \(\pi (\mathscr v^{\mathscr g}(\sigma ))\in \{\pi (\mathscr x):\mathscr x\in \mathscr d_{\kappa ,\mathscr t,\mathscr g(s)}\}\), by Definition 43, iff \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\in \mathscr d^{\bigstar }_{\kappa ^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)(s)}\), by Lemma 14, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash Loc(s)\sigma \), by Definition 28;
-
φ is σ1 = σ2:
\(\mathscr M,\mathscr t,\mathscr g\vDash \sigma _{1}=\sigma _{2}\) iff \(\mathscr v^{\mathscr g}(\sigma _{1})\in \mathscr d_{\kappa ,\mathscr t}\) and \(\mathscr v^{\mathscr g}(\sigma _{1})=\mathscr v^{\mathscr g}(\sigma _{2})\), by Definition 28, iff \(\pi (\mathscr v^{\mathscr g})(\sigma _{1})\in \{\pi (\mathscr x):\mathscr x\in \mathscr d_{\kappa ,\mathscr t}\}\) and \(\pi (\mathscr v^{\mathscr g}(\sigma _{1}))=\pi (\mathscr v^{\mathscr g}(\sigma _{2}))\), by Definition 43, iff \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma )\in \mathscr d^{\bigstar }_{\kappa ^{\bigstar },\pi (\mathscr t)}\) and \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma _{1})=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma _{2})\), by Lemma 14, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \sigma _{1}=\sigma _{2}\), by Definition 28;
-
φ is ζ1 = ζ2:
\(\mathscr M,\mathscr t,\mathscr g\vDash \zeta _{1}=\zeta _{2}\) iff \(\mathscr v^{\mathscr g}(\zeta _{1})=\mathscr v^{\mathscr g}(\zeta _{2})\), by Definition 28, iff \(\pi (\mathscr v^{\mathscr g}(\zeta _{1}))=\pi (\mathscr v^{\mathscr g}(\zeta _{2}))\), by Definition 43, iff \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\zeta _{1})=\mathscr v^{\bigstar ,\pi (\mathscr g)}(\zeta _{2})\), by Lemma 14, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \zeta _{1}=\zeta _{2}\), by Definition 28;
-
φ is ρ1 < ρ2:
\(\mathscr M,\mathscr t,\mathscr g\vDash \rho _{1}<\rho _{2}\) iff \(\mathscr v^{\mathscr g}(\rho _{1})\) and \(\mathscr v^{\mathscr g}(\rho _{2})\) are defined and \(\mathscr v^{\mathscr g}(\rho _{1})<^{\mathscr t}_{\mathscr F}\mathscr v^{\mathscr g}(\sigma _{2})\), by Definition 28, iff \(\pi (\mathscr v^{\mathscr g}(\rho _{1}))\) and \(\pi (\mathscr v^{\mathscr g}(\rho _{2}))\) are defined and \(\pi (\mathscr v^{\mathscr g}(\rho _{1}))<^{\pi (\mathscr t)}_{\mathscr F^{\bigstar }}\pi (\mathscr v^{\mathscr g}(\rho _{2}))\), by Definition 43, iff \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\rho _{1})\) and \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\rho _{2})\) are defined and \(\mathscr v^{\bigstar ,\pi (\mathscr g)}(\sigma _{1})^{\pi (\mathscr t)}_{\mathscr F^{\bigstar }}\mathscr v^{\bigstar ,\pi (\mathscr g)}(\rho _{2})\), by Lemma 14, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \rho _{1}<\rho _{2}\), by Definition 28;
-
φ is ¬ψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash \neg \psi \) iff \(\mathscr M,\mathscr t,\mathscr g\not \vDash \psi \), by Definition 28, \(\mathscr M,\mathscr t,\mathscr g\not \vDash \psi \) iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\not \vDash \psi \), by I.H., \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\not \vDash \psi \) iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \neg \psi \), by Definition 28;
-
φ is ψ ∧ χ:
\(\mathscr M,\mathscr t,\mathscr g\vDash \psi \wedge \chi \) iff \(\mathscr M,\mathscr t,\mathscr g\vDash \psi \) and \(\mathscr M,\mathscr t,\mathscr g\vDash \chi \), by Definition 28, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \psi \) and \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \chi \), by I.H., iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \psi \wedge \chi \), by Definition 28;
-
φ is ∀vψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash \forall v\psi \) iff for all \(\mathscr x\in \mathscr d_{\kappa ,\mathscr t}\): \(\mathscr M,\mathscr t,\mathscr g[v/\mathscr x]\vDash \psi \), by Definition 28, iff for all \(\mathscr x\in \mathscr d_{\kappa ,\mathscr t}\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g[v/\mathscr x])\vDash \psi \), by I.H., iff for all \(\mathscr x\in \mathscr d_{\kappa ,\mathscr t}\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)[v/\pi (\mathscr x)]\vDash \psi \), by Definition 44, iff for all \(\mathscr x\in \mathscr d^{\bigstar }_{\kappa ^{\bigstar },\pi (\mathscr t)}\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)[v/\mathscr x]\vDash \psi \), by Definition 43, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \forall v\psi \), by Definition 28;
-
φ is ∀Vψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash \forall V\psi \) iff for all \(\mathscr X\in \mathscr D^{n}_{\kappa }\): \(\mathscr M,\mathscr t,\mathscr g[V/\mathscr X]\vDash \psi \), by Definition 28, iff for all \(\mathscr X\in \mathscr D^{n}_{\kappa }\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g[V/\mathscr X])\vDash \psi \), by I.H., iff for all \(\mathscr X\in \mathscr D^{n}_{\kappa }\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)[V/\pi (\mathscr X)]\vDash \psi \), by Definition 44, iff for all \(\mathscr X\in \mathscr D^{\bigstar ,n}_{\kappa ^{\bigstar }}\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \psi [V/\mathscr X]\), by Definition 43, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \forall V\psi \), by Definition 28;
-
φ is ∀sψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash \forall s\psi \) iff for all \(\mathscr p\in \mathscr P\): \(\mathscr M,\mathscr t,\mathscr g[s/\mathscr p]\vDash \psi \), by Definition 28, iff for all \(\mathscr p\in \mathscr P\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g[s/\mathscr p])\vDash \psi \), by I.H., iff for all \(\mathscr p\in \mathscr P\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)[s/\pi (\mathscr p)])\vDash \psi \), by Definition 44, iff for all \(\mathscr p\in \mathscr P^{\bigstar }\): \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)[s/\mathscr p]\vDash \psi \), by Definition 43, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash \forall s\psi \), by Definition 28;
-
φ is Hψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash H\psi \) iff for all \(\mathscr t^{\prime }\in \mathscr T\) s. t. \(\mathscr t^{\prime }\leq \mathscr t\): \(\mathscr M,\mathscr t^{\prime },\mathscr g\vDash \psi \), by Definition 28, iff for all \(t^{\prime }\in \mathscr T\) s. t. \(\mathscr t^{\prime }\leq \mathscr t\): HCode \(\mathscr M^{\bigstar },\pi (\mathscr t^{\prime }),\pi (\mathscr g)\vDash \psi \), by I.H., iff for all \(t^{\prime }\in \mathscr T\) s. t. \(\pi (\mathscr t^{\prime })\leq \pi (\mathscr t)\): HCode \(\mathscr M^{\bigstar },\pi (\mathscr t^{\prime }),\pi (\mathscr g)\vDash \psi \), by Definition 43, iff for all \(t^{\prime }\in \mathscr T^{\bigstar }\) s. t. \(\mathscr t^{\prime }\leq \pi (\mathscr t)\): HCode \(\mathscr M^{\bigstar },\mathscr t^{\prime },\pi (\mathscr g)\vDash \psi \), by Definition 43, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash H\psi \), by Definition 28;
-
φ is Gψ:
\(\mathscr M,\mathscr t,\mathscr g\vDash G\psi \) iff for all \(\mathscr t^{\prime }\in \mathscr T\) s. t. \(\mathscr t\leq \mathscr t^{\prime }\): \(\mathscr M,\mathscr t^{\prime },\mathscr g\vDash \psi \), by Definition 28, iff for all \(t^{\prime }\in \mathscr T\) s. t. \(\mathscr t\leq \mathscr t^{\prime }\): HCode \(\mathscr M^{\bigstar },\pi (\mathscr t^{\prime }),\pi (\mathscr g)\vDash \psi \), by I.H., iff for all \(t^{\prime }\in \mathscr T\) s. t. \(\mathscr \pi (\mathscr t)\leq \pi (t^{\prime })\): HCode \(\mathscr M^{\bigstar },\pi (\mathscr t^{\prime }),\pi (\mathscr g)\vDash \psi \), by Definition 43, iff for all \(t^{\prime }\in \mathscr T^{\bigstar }\) s. t. \(\mathscr \pi (\mathscr t)\leq t^{\prime }\): HCode \(\mathscr M^{\bigstar },\mathscr t^{\prime },\pi (\mathscr g)\vDash \psi \), by Definition 43, iff \(\mathscr M^{\bigstar },\pi (\mathscr t),\pi (\mathscr g)\vDash G\psi \), by Definition 28.
□
The <-bisimulation relation will be used to prove that timeless parthood and immediate parthood are not definable in the E-theory solely in terms of parthood. Our proof will rely on the following theorem:
Theorem 8
Suppose that there are models\(\mathscr M\)and\(\mathscr M^{\bigstar }\)(based,respectively onframes\(\mathscr F\)and\(\mathscr F^{\bigstar }\)),\(\mathscr t\in \mathscr T\),\(\mathscr x_1,\ldots ,\mathscr x_n\in \mathscr d_{\kappa ,\mathscr t}\cup \bigcup _{n\in \mathbb {N}}\mathscr D^{n}_{\kappa }\)andπsuchthat:
-
1.
\(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\) ;
-
2.
\(\mathscr M,\mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \zeta v_1{\ldots } v_n\) ; and
-
3.
\(\mathscr M,\pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\not \vDash \zeta v_1{\ldots } v_n\) .
Then,ζis not explicitly definable in the E-theory solely in terms of<.
Proof of Theorem 8
Suppose that
-
(A)
There are models \(\mathscr M\), \(\mathscr M^{\bigstar }\), time \(\mathscr t\in \mathscr T_{\mathscr M}\), elements \(\mathscr x_1\), …, \(\mathscr x_n\in \mathscr d_{\mathscr M,\kappa ,\mathscr t}\cup \bigcup _{n\in \mathbb {N}}\mathscr D^{n}_{\kappa }\), and a function π such that:
-
(i)
\(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M^{\bigstar }\);
-
(ii)
\(\mathscr M, \mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \zeta v_1{\ldots } v_n\); and
-
(iii)
\(\mathscr M^{\bigstar }, \pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\not \vDash \zeta v_1{\ldots } v_n\).
-
(i)
Suppose also, for reductio, that ζ is explicitly definable solely in terms of <. Then:
-
(B)
There is some formula φ of \(\mathrm {L^{<}_E}\) (whose only free variables are v1, …, vn) such that, for every model \(\mathscr M\): \(\vDash _{\mathscr M} A\forall v_1\ldots \forall v_n(\varphi \leftrightarrow \zeta v_1{\ldots } v_n)\).
From (B) it follows that:
-
(1)
\(\mathscr M, \mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \varphi \) iff \(\mathscr M, \mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \zeta v_1{\ldots } v_n\); and
-
(2)
\(\mathscr M^{\bigstar }, \pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\vDash \varphi \) iff \(\mathscr M^{\bigstar }, \pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\vDash \zeta v_1{\ldots } v_n\).
Now, (i) and Theorem 7 imply that:
\(\mathscr M, \mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \varphi \) iff \(\mathscr M^{\bigstar }, \pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\vDash \varphi \)
The conjunction of this result with (1) and (2) implies that:
\(\mathscr M, \mathscr t,\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n]\vDash \zeta v_1{\ldots } v_n\) iff \(\mathscr M^{\bigstar }, \pi (\mathscr t),\pi (\mathscr g[v_1/\mathscr x_1,\ldots ,v_n/\mathscr x_n])\vDash \zeta v_1{\ldots } v_n\)
But this contradicts the conjunction of (ii) and (iii). Hence, the reductio assumption is false. ζ is not definable in the E-theory solely in terms of <. This concludes the proof of Theorem 8. □
1.2 A.2 Undefinability of Timeless Parthood and Immediate Parthood
Here’s how we’ll appeal to Theorem 8 to show that timeless parthood and immediate parthood are not definable in the E-theory solely in terms of parthood. We will begin by characterising a frame \(\mathscr F=\langle \mathscr T,\mathscr P,\mathscr{d}_0,\kappa ,\leq \rangle \) and a model \(\mathscr M\) based on \(\mathscr F\). Then, we will define a bisimulation between \(\mathscr M\) and itself. Finally, we will show that there is a time \(\mathscr t\in \mathscr T\), a variable-assignment \(\mathscr g\) based on \(\mathscr F\) and objects \(\mathscr x\) and \(\mathscr y\) in \(\mathscr d_{\kappa ,\mathscr t}\) such that:
-
\(\mathscr M, \mathscr t,\mathscr g[x/\mathscr x,y/\mathscr y]\vDash x\triangleleft y\), yet,
-
\(\mathscr M, \pi (\mathscr t),\pi (\mathscr g[x/\mathscr x,y/\mathscr y])\not \vDash x\triangleleft y\),
and
-
\(\mathscr M, \mathscr t,\mathscr g[x/\mathscr x,y/\mathscr y]\vDash x\ll y\), yet,
-
\(\mathscr M, \pi (\mathscr t),\pi (\mathscr g[x/\mathscr x,y/\mathscr y])\not \vDash x\ll y\),
From Theorem 8 it will follow that none of timeless parthood and immediate parthood is definable in the E-theory solely in terms of parthood.
1.2.1 A.2.1 The Model
Our model will be based on the following frame:
Definition 45
Let \(\mathscr F=\langle \mathscr T,\mathscr P, \mathscr{d}_{0}, \kappa ,\leq \rangle \), where:
-
\(\mathscr T=\{1,2\}\);
-
1 ≤ 2;
-
\(\mathscr d_{0,1}=\mathscr d_{0,2}=\{\mathscr a,\mathscr b\}\);
-
\(\mathscr P=\{\mathscr s\}\);
-
\(\mathscr d_{0,1,\mathscr s}=\mathscr d_{0,2,\mathscr s}=\{\mathscr a,\mathscr b\}\);
-
κ = ω.
The model \(\mathscr M\) is defined as follows:
Definition 46
Let \(\mathscr M=\langle \mathscr T,\mathscr P,\mathscr d_{0}, \kappa ,\leq ,\mathscr v\rangle \) where:
-
\(\mathscr v(\sigma )=\mathscr a\), for every individual constant σ;
-
For every \(n\in \mathbb {N}\), every n-ary predicate ζ, and every \(\mathscr t\in \mathscr T\): \(\mathscr v(\zeta )=\emptyset \).
Other models based on \(\mathscr F\) would have been equally appropriate.
We will now distinguish a few elements definable in terms of frame \(\mathscr F\):
Definition 47
Let:
-
\({\mathscr{G}}^{\mathscr{a}}_{\mathscr{b}}\) and \({\mathscr{G}}^{\mathscr{b}}_{\mathscr{a}}\) be functions from \(\mathscr T\) to \(\wp (\mathscr B_0)\) such that:
-
\(\mathscr {G^{a}_{b}}(1)=\{\mathscr{a}\}\), \(\mathscr {G^{a}_{b}}(2)=\{\mathscr{b}\}\);
-
\(\mathscr {G^{b}_{a}}(1)=\{\mathscr{b}\}\), \(\mathscr {G^{b}_{a}}(2)=\{\mathscr{a}\}\).
-
-
\(\mathscr {g^{a}_{b}}\) and \(\mathscr {g^{b}_{a}}\) be a functions from \(\mathscr T\) to \(\mathscr B_0\) such that:
-
\(\mathscr {g^{a}_{b}}(1)={\mathscr{a}}\), \(\mathscr {g^{a}_{b}}(2)={\mathscr{b}}\);
-
\(\mathscr {g^{b}_{a}}(1)={\mathscr{b}}\), \(\mathscr {g^{b}_{a}}(2)={\mathscr{a}}\);
-
-
\(\mathscr e_1=\langle \mathscr {g^{a}_{b}},\mathscr {G^{a}_{b}}\rangle \);
-
\(\mathscr e_2=\langle \mathscr {g^{b}_{a}},\mathscr {G^{b}_{a}}\rangle \);
-
\(\mathscr R:\mathscr T\to \wp (\mathscr B_1\times \mathscr B_1\times \mathscr B_1)\):
-
\(\mathscr R(1)=\mathscr R(2)=\{\langle \mathscr a,\mathscr e_1,\mathscr e_2\rangle ,\langle \mathscr b,\mathscr e_1,\mathscr e_2\rangle \}\);
-
-
\(\mathscr e_3=\langle \lambda \mathscr t.\mathscr a,\lambda \mathscr t.\mathscr e_1, \lambda \mathscr t.\mathscr e_2,\mathscr R\rangle \);
-
\(\mathscr e_4=\langle \lambda \mathscr t.\mathscr b,\lambda \mathscr t.\mathscr e_1, \lambda \mathscr t.\mathscr e_2,\mathscr R\rangle \).
We observe without proof that the following holds of the elements just characterised:
Observation 1
1. \(\mathscr e_1\in \mathscr d_{1,1}\) | 7. \(\mathscr e_2\in \mathscr d_{1,1,\mathscr s}\) | |
2. \(\mathscr e_1\in \mathscr d_{1,2}\) | 8. \(\mathscr e_2\in \mathscr d_{1,2,\mathscr s}\) | 15. \(\mathscr e_3\in \mathscr d_{2,2,\mathscr s}\) |
3. \(\mathscr e_1\in \mathscr d_{1,1,\mathscr s}\) | 9. \(\mathscr R\in {D^{3}_{1}}\) | 14. \(\mathscr e_4\in \mathscr d_{2,1}\) |
4. \(\mathscr e_1\in \mathscr d_{1,2,\mathscr s}\) | 10. \(\mathscr e_3\in \mathscr d_{2,1}\) | 15. \(\mathscr e_4\in \mathscr d_{2,2}\) |
5. \(\mathscr e_2\in \mathscr d_{1,1}\) | 11. \(\mathscr e_3\in \mathscr d_{2,2}\) | 16. \(\mathscr e_4\in \mathscr d_{2,1,\mathscr s}\) |
6. \(\mathscr e_2\in \mathscr d_{1,2}\) | 12. \(\mathscr e_3\in \mathscr d_{2,1,\mathscr s}\) | 17. \(\mathscr e_4\in \mathscr d_{2,2,\mathscr s}\) |
Observation 2
\(\forall \mathscr x\in \mathscr B_{\kappa }\) , \(\forall \mathscr t\in \mathscr T\) : \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr e_3\) iff \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr e_4\) .
The embodiments \(\mathscr e_3\) and \(\mathscr e_4\) will play a crucial role in showing the undefinability of timeless parthood and of immediate parthood in terms of parthood. As will be shown, \(\mathscr M\) is <-bisimilar to itself via a <-bisimulation π that maps \(\mathscr a\) to itself and maps \(\mathscr e_3\) to \(\mathscr e_4\). Yet, \(\mathscr a\) is a timeless part of \(\mathscr e_3\), even though \(\mathscr a\) is not a timeless part of \(\mathscr e_4\). Similarly, \(\mathscr a\) is an immediate part of \(\mathscr e_3\), even though \(\mathscr a\) is not an immediate part of \(\mathscr e_4\). In conjunction with Theorem 8, this result shows that timeless parthood and immediate parthood are not definable in terms of parthood. We now turn to the definition of the <-bisimulation π.
1.2.2 A.2.2 The <-bisimulation
We begin by defining the set of all partial functions, \(\mathscr H_i\), from \(\mathscr T\) to \(\mathscr B_i\), for each i such that 0 ≤ i < κ:
Definition 48
For each i such that 0 ≤ i < κ, let:
-
\(\mathscr H_i=\{\mathscr h:\mathscr T\to \mathscr B_i\}\);
-
\(\mathscr H_{\kappa }=\bigcup _{i<\kappa }\mathscr H_{i}\).
The <-bisimulation π will be defined in terms of the following function between entities defined in terms of the frame \(\mathscr F\):
Definition 49 (𝜃-Function)
Let 𝜃 be a function with domain \(\mathscr B_{\kappa }\cup \mathscr D^{+}_{\kappa }\cup \mathscr H_{\kappa }\) simultaneously defined as follows:
-
1.
\(\forall x\in \mathscr B_{\kappa }\):
-
(a)
If \(\mathscr x\in \mathscr B_0\), then \(\theta (\mathscr x)=\mathscr x\);
-
(b)
If \(\mathscr x\not \in \mathscr B_0\), then:
-
(i)
if \(\mathscr x\neq \mathscr e_3\) and \(\mathscr x\neq \mathscr e_4\), then
\(\theta (\mathscr x)=\langle \theta (\mathscr m_1),\ldots ,\theta (\mathscr m_n),\theta (\mathscr X)\rangle \),
where \(\mathscr x=\langle \mathscr m_1,\ldots ,\mathscr m_n,\mathscr X\rangle \);
-
(ii)
if \(\mathscr x=\mathscr e_3\), then \(\theta (\mathscr x)=\mathscr e_4\);
-
(iii)
if \(\mathscr x=\mathscr e_4\), then \(\theta (\mathscr x)=\mathscr e_3\);
-
(i)
-
(a)
-
2.
\(\forall \mathscr X\in \mathscr D_{\kappa }, \mathscr t\in \mathscr T\):
\(\theta (\mathscr X)(\mathscr t)=\{\langle \theta (\mathscr x_1),\ldots ,\theta (\mathscr x_n)\rangle :\langle \mathscr x_1,\ldots ,\mathscr x_n\rangle \in \mathscr X(\mathscr t)\}\);
-
3.
\(\forall \mathscr h\in \mathscr H_{\kappa }\): \(\forall \mathscr t\in \mathscr T\), \(\theta (\mathscr h)(t)=\theta (\mathscr h(\mathscr t))\).
We now show that relevant restrictions of 𝜃 turn out to be bijections:
Lemma 15 (Bijections)
For everyi < κand\(\mathscr t\in \mathscr T\)and\(n\in \mathbb {N}\):
-
1.
\(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection from \(\mathscr d_{i,\mathscr t}\) to \(\mathscr d_{i,\mathscr t}\) ;
-
2.
\(\theta \restriction _{\mathscr d_{i,\mathscr t,\mathscr s}}\) is a bijection from \(\mathscr d_{i,\mathscr t,\mathscr s}\) to \(\mathscr d_{i,\mathscr t,\mathscr s}\) ;
-
3.
\(\theta \restriction _{\mathscr {D^{n}_{i}}}\) is a bijection from \(\mathscr {D^{n}_{i}}\) to \(\mathscr {D^{n}_{i}}\) ;
-
4.
\(\theta \restriction _{\mathscr H_{i}}\) is a bijection from \(\mathscr H_{i}\) to \(\mathscr H_{i}\) .
Proof of Lemma A.2.2
-
1.(a)
(\(\forall \mathscr x\in \mathscr d_{i,\mathscr t}:\theta (\mathscr x)\in \mathscr d_{i,\mathscr t}\)): Suppose, for reductio, that there is a least ordinal i < κ such that \(\theta (\mathscr x)\not \in \mathscr d_{i,\mathscr t}\), for some \(\mathscr t\in \mathscr T\) and \(\mathscr x\in \mathscr d_{i,\mathscr t}\). Clearly, \(\mathscr x\neq \mathscr a\), \(\mathscr x\neq \mathscr b\), \(\mathscr x\neq \mathscr e_3\) and \(\mathscr x\neq \mathscr e_4\). So, i is a successor ordinal j + 1, \(\mathscr x=\langle \mathscr m_1,\ldots ,\mathscr m_n,\mathscr X\rangle \), by Definition 7, and \(\theta (\mathscr x)=\langle \theta (\mathscr m_1),\ldots ,\theta (\mathscr m_n),\theta (\mathscr X)\rangle \), by Definition 49.
Suppose \(\mathscr x\) is an n-ary rigid embodiment. Then: (i) \(\theta (\mathscr m_l)\) is a constant function from \(\mathscr T\) to \(\mathscr B_{j}\), for each 1 ≤ l ≤ n, by Definitions 4, 49 and the reductio assumption; (ii) \(\theta (\mathscr X)\in \mathscr {D^{n}_{j}}\), by Definitions 4, 49 and the reductio assumption; and (iii) \(\langle \mathscr m_1(\mathscr t),\ldots ,\mathscr m_n(\mathscr t)\rangle \in \mathscr X(\mathscr t)\) if and only if \(\langle \theta (\mathscr m_1)(\mathscr t),\ldots ,\theta (\mathscr m_n)(\mathscr t)\rangle \in \theta (\mathscr X)(\mathscr t)\), by Definition 49. But \(\langle \mathscr m_1(\mathscr t),\ldots ,\mathscr m_n(\mathscr t)\rangle \in \mathscr X(\mathscr t)\), since \(\mathscr x\in \mathscr d_{j + 1,\mathscr t}\), by Definition 7. So, \(\langle \theta (\mathscr m_1)(\mathscr t),\ldots ,\theta (\mathscr m_n)(\mathscr t)\rangle \in \theta (\mathscr X)(\mathscr t)\). So, \(\theta (\mathscr x)\) is an n-ary rigid embodiment in \(\mathscr d_{j + 1,\mathscr t}\), by Definitions 4 and 7. But this contradicts the reductio assumption.
Suppose instead that \(\mathscr x\) is a variable embodiment. Then: (i) \(\theta (\mathscr X)\) is an individual concept in \({\mathscr{D}}^{1}_{j}\), by Definitions 6, 49 and the reductio assumption; (ii) \(\theta (\mathscr m_1)\in \mathscr H_{j}\), by Definition 6, Definition 49 and the reductio assumption; (iii) \(\theta (\mathscr X)(\mathscr t)=\{\theta (\mathscr m_1)(\mathscr t)\}\) if \(\theta (\mathscr m_1)(\mathscr t)\) is defined and otherwise \(\theta (\mathscr X)(\mathscr t)=\emptyset \), for every \(\mathscr t\in \mathscr T\), by Definition 6, Definition 49 and the reductio assumption; (iv) \(\mathscr m_1(\mathscr t)\in \mathscr X(\mathscr t)\) if and only if \(\theta (\mathscr m_1)(\mathscr t)\in \theta (\mathscr X)(\mathscr t)\), by Definition 49. But \(\mathscr m_1(\mathscr t)\in \mathscr X(\mathscr t)\), since \(\mathscr x\in \mathscr d_{j + 1,\mathscr t}\), by Definition 7. So, \(\theta (\mathscr m_1)(\mathscr t)\in \theta (\mathscr X)(\mathscr t)\). So, \(\theta (\mathscr x)\) is a variable embodiment in \(\mathscr d_{j + 1,\mathscr t}\), by Definitions 6 and 7. But this contradicts the reductio assumption.
Therefore, the reductio assumption is false. Hence, for every i < κ and \(\mathscr t\in \mathscr T\): \(\theta (\mathscr x)\in \mathscr d_{i,\mathscr t}\), for every \(\mathscr x\in \mathscr d_{i,\mathscr t}\);
-
1.(b)
(\(\forall \mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t}\): \(\theta (\mathscr x)=\theta (\mathscr y)\) ⇒ \(\mathscr x=\mathscr y\)): Suppose, for reductio, that there is a least ordinal i < κ such that \(\theta (\mathscr x)=\theta (\mathscr y)\) and \(\mathscr x\neq \mathscr y\), for some \(\mathscr x\), \(\mathscr y\) in \(\mathscr d_{i,\mathscr t}\) and \(\mathscr t\in \mathscr T\). Clearly, i is a successor ordinal j + 1, where j ≥ 2. So, by the reductio assumption, and Definitions 7 and 49, there are \(\mathscr x=\langle \mathscr m^{\mathscr x}_1,\ldots ,\mathscr m^{\mathscr x}_n,\mathscr X\rangle \in \mathscr d_{j + 1,\mathscr t}\) and \(\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr X\rangle \in \mathscr d_{j + 1,\mathscr t}\) such that \(\mathscr x\neq \mathscr y\) and \(\theta (\mathscr x)=\langle \theta (\mathscr m^{\mathscr x}_1),\ldots ,\theta (\mathscr m^{\mathscr x}_n),\theta (\mathscr X)\rangle =\theta (\mathscr y)=\langle \theta (\mathscr m^{\mathscr y}_1),\ldots ,\theta (\mathscr m^{\mathscr y}_n),\theta (\mathscr Y)\rangle \). By the reductio assumption and Definition 49, \(\theta (\mathscr m^{\mathscr x}_{l})=\theta (\mathscr m^{\mathscr y}_{l})\), for every l such that 1 ≤ l ≤ n, and \(\theta (\mathscr X)=\theta (\mathscr Y)\). But then, \(\theta (\mathscr x)=\theta (\mathscr y)\). Yet, this contradicts the reductio assumption. So, for every ordinal i < κ and \(\mathscr t\in \mathscr T\): \(\forall \mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t}\): \(\theta (\mathscr x)=\theta (\mathscr y)\) ⇒ \(\mathscr x=\mathscr y\);
-
1.(c)
(\(\forall \mathscr y\in \mathscr d_{i,\mathscr t}\exists \mathscr x\in \mathscr d_{i,\mathscr t}\): \(\theta (\mathscr x)=\mathscr y\)):
Suppose, for reductio, that there is a least ordinal i < κ and a \(\mathscr y\in \mathscr d_{i,\mathscr t}\) such that for every \(\mathscr x\in \mathscr d_{i,\mathscr t}\), \(\theta (\mathscr x)\neq \mathscr y\). Clearly, i is a successor ordinal j + 1, where j ≥ 2. So, \(\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \), by Definition 7. Now, by the reductio assumption, there are \(\mathscr m^{\mathscr x}_{1}\), …, \(\mathscr m^{\mathscr x}_{n}\) and \(\mathscr X\) such that \(\mathscr m^{\mathscr y}_1=\theta (\mathscr m^{\mathscr x}_1)\), …, \(\mathscr m^{\mathscr y}_n=\theta (\mathscr m^{\mathscr x}_n)\) and \(\mathscr Y=\theta (\mathscr X)\). Let \(\mathscr x=\langle \mathscr m^{\mathscr x}_{1},\ldots ,\mathscr m^{\mathscr x}_{n},\mathscr X\rangle \). Suppose \(\mathscr y\) is a rigid embodiment in \(\mathscr d_{j + 1,\mathscr t}\). Then, for every l such that 1 ≤ l ≤ n, \(\mathscr m^{\mathscr x}_l\) is a constant function from \(\mathscr T\) to \(\mathscr B_{j}\), \(\mathscr X\in {\mathscr{D}^{n}_{j}}\) and \(\langle \mathscr m^{\mathscr x}_{1}(\mathscr t),\ldots ,\mathscr m^{\mathscr x}_{n}(\mathscr t)\rangle \in \mathscr X(\mathscr t)\), by the reductio assumption and Definitions 4, 7 and 49. So, \(\mathscr x\) is a rigid embodiment in \(\mathscr d_{j + 1,\mathscr t}\), by Definitions 4 and 7. Suppose instead that \(\mathscr y\) is a variable embodiment in \(\mathscr d_{j + 1,\mathscr t}\). Then, \(\mathscr X\) is an individual concept in \({\mathscr{D}}^{1}_{j}\), \(\mathscr m^{\mathscr x}_1\in \mathscr H_j\), \(\forall \mathscr t\in \mathscr T\): \(\mathscr m_1(\mathscr t)\) is defined if and only if \(\mathscr X(\mathscr t)=\{\mathscr m_1(\mathscr t)\}\), and \(\mathscr m^{\mathscr x}_1(\mathscr t)\in \mathscr X(\mathscr t)\), by the reductio assumption and Definitions 6, 7 and 49. So, \(\mathscr x\) is a variable embodiment in \(\mathscr d_{j + 1,\mathscr t}\). Either way, \(\theta (\mathscr x)=\mathscr y\), where \(\mathscr x\in \mathscr d_{j + 1,\mathscr t}\). But this contradicts the reductio assumption. So, for every ordinal i < κ and \(\mathscr t\in \mathscr T\), \(\forall \mathscr y\in \mathscr d_{i,\mathscr t}\exists \mathscr x\in \mathscr d_{i,\mathscr t}\): \(\theta (\mathscr x)=\mathscr y\).
Therefore, for every ordinal i < κ and \(\mathscr t\in \mathscr T\), \(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection from \(\mathscr d_{i,\mathscr t}\) to \(\mathscr d_{i,\mathscr t}\).
-
2.(a)
(\(\forall \mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)\in \mathscr d_{i,\mathscr t,\mathscr s}\)): Suppose, for reductio, that there is a least ordinal i such that \(\theta (\mathscr x)\not \in \mathscr d_{i,\mathscr t,\mathscr s}\), for some \(\mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\) and \(\mathscr t\in \mathscr T\). Clearly, \(\mathscr x\neq \mathscr a\), \(\mathscr x\neq \mathscr b\), \(\mathscr x\neq \mathscr e_3\) and \(\mathscr x\neq \mathscr e_4\). So, i is a successor ordinal j + 1, \(\mathscr x=\langle \mathscr m_1,\ldots ,\mathscr m_n,\mathscr X\rangle \), by Definition 8, and \(\theta (\mathscr x)=\langle \theta (\mathscr m_1),\ldots ,\theta (\mathscr m_n),\theta (\mathscr X)\rangle \), by Definition 49. Since \(\mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\), there is some \(\mathscr m_l\) such that \(\mathscr m_l(\mathscr t)\in \mathscr d_{j,\mathscr t,\mathscr s}\), by Definition 8. So, by the reductio assumption and Definition 49, there is some l such that 1 ≤ l ≤ n and \(\theta (m_l(\mathscr t))=\theta (m_l)(\mathscr t)\in \mathscr d_{j,\mathscr t,\mathscr s}\). But then, \(\theta (\mathscr x)\in \mathscr d_{j + 1,\mathscr t,\mathscr s}=\mathscr d_{i,\mathscr t,\mathscr s}\), by Definition 8. This contradicts the reductio assumption. Therefore, the reductio assumption is false: for every i < κ and \(\mathscr t\in \mathscr T\), \(\forall \mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)\in \mathscr d_{i,\mathscr t,\mathscr s}\).
-
2.(b)
(\(\forall \mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)=\theta (\mathscr y)\) ⇒ \(\mathscr x=\mathscr y\)): Suppose, for reductio, that there is a least ordinal i < κ and such that \(\theta (\mathscr x)=\theta (\mathscr y)\) and yet \(\mathscr x\neq \mathscr y\), where \(\mathscr x\) and \(\mathscr y\) belong to \(\mathscr d_{i,\mathscr t,\mathscr s}\). Hence, there are \(\mathscr x\) and \(\mathscr y\in \mathscr d_{i,\mathscr t}\) such that \(\theta (\mathscr x)=\theta (\mathscr y)\), by Definition 8 and yet \(\mathscr x\neq \mathscr y\). But this contradicts the claim that \(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection from \(\mathscr d_{i,\mathscr t}\) to \(\mathscr d_{i,\mathscr t}\). Contradiction. So, the reductio assumption is false: for every ordinal i < κ and \(\mathscr t\in \mathscr T\), \(\forall \mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)=\theta (\mathscr y)\) ⇒ \(\mathscr x=\mathscr y\).
-
2.(c)
(\(\forall \mathscr y\in \mathscr d_{i,\mathscr t,\mathscr s}\exists \mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)=\mathscr y\)): Suppose, for reductio, that there is a least i < κ and a \(\mathscr y\in \mathscr d_{i,\mathscr t,\mathscr s}\) such that, for every \(\mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\), \(\theta (\mathscr x)\neq \mathscr y\). Clearly, i is a successor ordinal j + 1, where j ≥ 2. So, \(\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \), by Definition 8. So, there is an \(\mathscr x=\langle \mathscr m^{\mathscr x}_1,\ldots ,\mathscr m^{\mathscr x}_n,\mathscr X\rangle \in \mathscr d_{j + 1,\mathscr t}\) such that \(\theta (\mathscr x)=\mathscr y\), since \(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection and \(\mathscr y\in \mathscr d_{j + 1,\mathscr t}\), by Definition 8. Now, from the reductio assumption and Definition 49 it follows that, for every l such that 1 ≤ l ≤ n, \(\mathscr m^{y}_l\in \mathscr{d}_{j,{\mathscr{t}},{\mathscr{s}}}\) if and only if \(\mathscr m^{x}_l\in {\mathscr{d}}_{j,{\mathscr{t}},{\mathscr{s}}}\). But there is some l such that \(\mathscr m^{y}_l(\mathscr t)\in {\mathscr{d}}_{j,{\mathscr{t}},{\mathscr{s}}}\), by Definition 8, since \(\mathscr y\in \mathscr d_{j + 1,\mathscr t,{\mathscr s}}\). Therefore, there is some l such that \(\mathscr m^{x}_l(\mathscr t) \in {\mathscr{d}}_{j,{\mathscr{t}},{\mathscr{s}}}\). But then, \(\mathscr{x} \in \mathscr d_{j + 1,\mathscr t,{\mathscr s}}\), by Definition 8. This contradicts the reductio assumption. So, for every ordinal i < κ, \(\mathscr t\in \mathscr T\)\(\forall \mathscr y\in \mathscr d_{i,\mathscr t,\mathscr s}\exists \mathscr x\in \mathscr d_{i,\mathscr t,\mathscr s}\): \(\theta (\mathscr x)=\mathscr y\).
Therefore, for every ordinal i < κ and \(\mathscr t\in \mathscr T\), \(\theta \restriction _{\mathscr d_{i,\mathscr t,\mathscr s}}\) is a bijection from \(\mathscr d_{i,\mathscr t,\mathscr s}\) to \(\mathscr d_{i,\mathscr t,\mathscr s}\).
-
3.
The claim that for every i < κ, \(\theta \restriction _{\mathscr {D^{n}_{i}}}\) is a bijection from \(\mathscr {D^{n}_{i}}\) to \(\mathscr {D^{n}_{i}}\) is a straightforward consequence of the fact that for every i < κ and \(\mathscr t\in \mathscr T\), \(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection from \(\mathscr d_{i,\mathscr t}\) to \(\mathscr d_{i,\mathscr t}\);
-
4.
The claim that for every i < κ, \(\theta \restriction _{\mathscr H_{i}}\) is a bijection from \(\mathscr H_{i}\) to \(\mathscr H_{i}\) is a straightforward consequence of the fact that for every i < κ and \(\mathscr t\in \mathscr T\), \(\theta \restriction _{\mathscr d_{i,\mathscr t}}\) is a bijection from \(\mathscr d_{i,\mathscr t}\) to \(\mathscr d_{i,\mathscr t}\).
□
Our final result about the 𝜃 function concerns the preservation of parthood under the 𝜃 function:
Lemma 16
For every \(\mathscr x,\mathscr y\in \mathscr d_{\kappa ,\mathscr t}\cup \mathscr D_{\kappa }\) , for every \(\mathscr t\in \mathscr T\) : \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\) if and only if \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\) .
Proof of Lemma 16
(⇒): Suppose, for reductio, that there are \(\mathscr x,\mathscr y\in \mathscr d_{\kappa ,\mathscr t}\cup \mathscr D_{\kappa }\) such that \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\) and \(\theta (\mathscr x)\not <^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\), for some \(\mathscr t\in \mathscr T\). Then, there is a least ordinal i such that \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\) and \(\theta (\mathscr x)\not <^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\), for some \(\mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t}\cup \mathscr \bigcup _{n\in \mathbb {N}}\bigcup _{n\in \mathbb {N}}{\mathscr D}^{n}_{i}\) and \(\mathscr t\in \mathscr T\). Now, i is a successor ordinal j + 1, where j > 2, since \(\mathscr e_3\) and \(\mathscr e_4\) share the same parts, by Observation 2, and \(\theta (\mathscr y)=\mathscr y\) for every \(\mathscr y\in \mathscr B_{2}\) such that \(\mathscr y\neq \mathscr e_3\) and \(\mathscr y\neq \mathscr e_4\).
So, \(\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \), by Definition 7. Now, either \(\mathscr x=\mathscr Y\), or else \(\mathscr x\leq ^{\mathscr t}_{\mathscr F}\mathscr m^{\mathscr y}_{l}(\mathscr t)\), for some l such that 1 ≤ l ≤ n. If \(\mathscr x=\mathscr Y\), then, \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\). But then, \(\theta (\mathscr x)=\theta (\mathscr Y)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)=\langle \theta (\mathscr m^{\mathscr y}_1),\ldots ,\theta (\mathscr m^{\mathscr y}_n),\theta (\mathscr Y)\rangle \). So, suppose that \(\mathscr x\leq ^{\mathscr t}_{\mathscr F}\mathscr m^{\mathscr y}_{l}(\mathscr t)\). If \(\mathscr x=\mathscr m^{\mathscr y}_{l}(\mathscr t)\), then \(\theta (\mathscr x)=\theta (\mathscr m^{\mathscr y}_{l}(\mathscr t))=\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)=\langle \theta (\mathscr m^{\mathscr y}_1),\ldots ,\theta (\mathscr m^{\mathscr y}_n),\theta (\mathscr Y)\rangle \). So, suppose instead that \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr m^{\mathscr y}_{l}(\mathscr t)\). Then, by the reductio assumption, \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)\). But \(\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)=\langle \theta (\mathscr m^{\mathscr y}_1),\ldots ,\theta (\mathscr m^{\mathscr y}_n),\theta (\mathscr Y)\rangle \). Hence, \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\), by the Definition of \(<^{\mathscr{t}}_{\mathscr{F}}\).
But this contradicts the reductio assumption. Therefore, for every \(\mathscr x,\mathscr y\in \mathscr d_{\kappa ,\mathscr t}\cup \mathscr D_{\kappa }\), for every \(\mathscr t\in \mathscr T\): \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\) only if \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\).
(⇐): Suppose, for reductio, that there are \(\mathscr x,\mathscr y\in \mathscr d_{\kappa ,\mathscr t}\cup \mathscr D_{\kappa }\) such that \(\mathscr x\not <^{\mathscr t}_{\mathscr F}\mathscr y\) and \(\theta (\mathscr x) <^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\), for some \(\mathscr t\in \mathscr T\). Then, there is a least ordinal i such that \(\mathscr x\not <^{\mathscr t}_{\mathscr F}\mathscr y\) and \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\), for some \(\mathscr x,\mathscr y\in \mathscr d_{i,\mathscr t}\cup \mathscr \bigcup _{n\in \mathbb {N}}\bigcup _{n\in \mathbb {N}}{\mathscr D}^{n}_{i}\) and \(\mathscr t\in \mathscr T\). Now, i is a successor ordinal j + 1, where j > 2, since \(\mathscr e_3\) and \(\mathscr e_4\) share the same parts, by Observation 2, and \(\theta (\mathscr y)=\mathscr y\) for every \(\mathscr y\in \mathscr B_{2}\) such that \(\mathscr y\neq \mathscr e_3\) and \(\mathscr y\neq \mathscr e_4\).
So, \(\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \), by Definition 7, and \(\theta (\mathscr y)=\langle \theta (\mathscr m^{\mathscr y}_1),\ldots ,\theta (\mathscr m^{\mathscr y}_n),\theta (\mathscr Y)\rangle \). Now, either \(\theta (\mathscr x)=\theta (\mathscr Y)\), or else \(\theta (\mathscr x)\leq ^{\mathscr t}_{\mathscr F}\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)=\theta (\mathscr m^{\mathscr y}_{l}(\mathscr t))\), for some l such that 1 ≤ l ≤ n. If \(\theta (\mathscr x)=\theta (\mathscr Y)\), then, \(\mathscr x=\mathscr Y\), since \(\theta \restriction _{\mathscr {D^{n}_{j}}}\) is a bijection between \(\mathscr {D^{n}_{j}}\) and \(\mathscr {D^{n}_{j}}\). But then, clearly, \(\mathscr x=\mathscr Y<^{\mathscr t}_{\mathscr F}\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \). Suppose \(\theta (\mathscr x)\leq ^{\mathscr t}_{\mathscr F}\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)\). If \(\theta (\mathscr x)=\theta (\mathscr m^{\mathscr y}_{l}(\mathscr t))\), then \(\mathscr x=\mathscr m^{\mathscr y}_{l}(\mathscr t)\), since \(\theta \restriction _{\mathscr d_{j,\mathscr t}}\) is a bijection between \(\mathscr d_{j,\mathscr t}\) and \(\mathscr d_{j,\mathscr t}\). So, \(\mathscr x=\mathscr m^{\mathscr y}_{l}(\mathscr t)<^{\mathscr t}_{\mathscr F}\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \). So, suppose that \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr m^{\mathscr y}_{l})(\mathscr t)\). Then, by the reductio assumption, \(\mathscr x<^{\mathscr t}_{\mathscr F} {\mathscr{m}}^{\mathscr{y}}_{l}(\mathscr{t})\). But \(\mathscr m^{\mathscr y}_{l}(\mathscr t)<^{\mathscr t}_{\mathscr F}\mathscr y=\langle \mathscr m^{\mathscr y}_1,\ldots ,\mathscr m^{\mathscr y}_n,\mathscr Y\rangle \). Hence, \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\), by the Definition of \(<^{\mathscr{t}}_{\mathscr{F}}\).
But this contradicts the reductio assumption. Therefore, for every \(\mathscr x,\mathscr y\in \mathscr d_{\kappa ,\mathscr t}\cup \mathscr D_{\kappa }\), for every \(\mathscr t\in \mathscr T\): \(\mathscr x<^{\mathscr t}_{\mathscr F}\mathscr y\) if \(\theta (\mathscr x)<^{\mathscr t}_{\mathscr F}\theta (\mathscr y)\). □
We are now in a position to define the <-bisimulation π:
Definition 50 (π Function)
Let π be a function with domain \(\mathscr B_{\kappa }\cup \bigcup _{n\in \mathbb {N}}\mathscr D^{n}_{\kappa }\cup \mathscr T\cup \mathscr P\) such that:
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\(\forall \mathscr x\in \mathscr B_{\kappa }\): \(\pi (\mathscr x)=\theta (\mathscr x)\);
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\(\forall \mathscr X\in \mathscr D^{n}_{\kappa }\): \(\pi (\mathscr X)=\theta (\mathscr X)\);
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\(\forall \mathscr t\in \mathscr T\): \(\pi (\mathscr t)=\mathscr t\);
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\(\forall \mathscr p\in \mathscr P\): \(\pi (\mathscr p)=\mathscr p\);
The following lemma states that π is indeed a <-bisimulation between \(\mathscr M\) and \(\mathscr M\):
Lemma 17
\(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M\) .
Proof of Lemma 17
The satisfaction of conditions 1-9 of Definition 43 is a straightforward consequence of Definitions 49 and 50 and Lemmas A.2.2 and 16. Therefore, \(\mathscr M\overset {\pi }{\leftrightarrows }\mathscr M\). □
The last result required for the application of Theorem 8 is the following:
Lemma 18
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1.
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(a)
\(\mathscr M,1,\mathscr g[x/\mathscr a,y/\mathscr e_3]\vDash x\ll y\)
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(b)
\(\mathscr M,\pi (1),\mathscr \pi (g[x/\mathscr a,y/\mathscr e_3])\not \vDash x\ll y\) ;
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(a)
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2.
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(a)
\(\mathscr M,1,\mathscr g[x/\mathscr a,y/\mathscr e_3]\vDash x\triangleleft y\) ;
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(b)
\(\mathscr M,\pi (1),\mathscr \pi (g[x/\mathscr a,y/\mathscr e_3])\not \vDash x\triangleleft y\) .
-
(a)
Proof of Lemma 18
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1.
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(a)
Clearly, \(\mathscr a\ll ^{1}_{\mathscr F} \mathscr e_{3}\), by Definition 9, since \(\lambda \mathscr t.\mathscr a\) is the first element of \(\mathscr e_3=\langle \lambda \mathscr t.\mathscr a,\lambda \mathscr t.\mathscr e_1,\lambda \mathscr t.\mathscr e_2,\mathscr R\rangle \) and \(\mathscr a=\lambda \mathscr t.\mathscr a(1)\) is such that \(\langle \mathscr a,\mathscr e_1,\mathscr e_2\rangle \in \mathscr R(1)\); So, \(\mathscr M,1,\mathscr g[x/\mathscr a,y/\mathscr e_3]\vDash x\ll y\);
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(b)
\(\pi (\mathscr a) \ll ^{\pi (1)}_{\mathscr{F}} \pi (\mathscr e_{3})\) if and only if \(\mathscr a\ll ^{1}_{\mathscr F}\mathscr e_4\). Now, \(\mathscr e_4=\langle \lambda \mathscr t.\mathscr b.\lambda \mathscr t.\mathscr e_1,\lambda \mathscr t.\mathscr e_2,\mathscr X\rangle \) and \( \mathscr a\neq \lambda \mathscr t.\mathscr b(1)=\mathscr b\), \(\mathscr a\neq \lambda \mathscr t.\mathscr e_1(1)=\mathscr e_1\) and \(\mathscr a\neq \lambda \mathscr t.\mathscr e_2(1)=\mathscr e_2\). So, \(\mathscr a\not \ll ^{1}_{\mathscr F}\mathscr e_4\), by Definition 9. Hence, \(\pi (\mathscr a) \not \ll ^{\pi (1)}_{\mathscr{F}} \pi (\mathscr e_3)\). Therefore, \(\mathscr M,\pi (1),\pi (\mathscr g[x/\mathscr a,y/\mathscr e_3])\not \vDash x\ll y\)
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(a)
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2.
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(a)
Clearly, \(\mathscr a \triangleleft ^{1}_{\mathscr{F}} \mathscr e_3\), by Definition 13, since \(\mathscr a \ll ^{1}_{\mathscr{F}} \mathscr e_3\) and \(\mathscr e_3\) is a rigid embodiment. So, \(\mathscr M,1,\mathscr g[x/\mathscr a,y/\mathscr e_3]\vDash x\triangleleft y\);
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(b)
\(\pi (\mathscr a) \triangleleft ^{\pi (1)}_{\mathscr{F}} \pi (\mathscr e_{3})\) if and only if \(\mathscr a \triangleleft ^{1}_{\mathscr{F}} \mathscr e_4\). The only immediate parthood sequence at time 1 linking \(\mathscr a\) to \(\mathscr e_4\) is \(\mathscr a \ll ^{1}_{\mathscr{F}} \mathscr e_1 \ll ^{1}_{\mathscr{F}} \mathscr{e}_4\). But \(\mathscr e_1\) is not a rigid embodiment. So, not: \(\mathscr{a} \triangleleft ^{1}_{\mathscr{F}} \mathscr{e}_{4}\). Hence, not: \(\pi (\mathscr{a})\triangleleft ^{\pi (1)}_{\mathscr{F}}\pi (\mathscr{e}_{3})\), by Definition 13. So, \(\mathscr M,\pi (1),\mathscr \pi (g[x/\mathscr a,y/\mathscr e_3])\not \vDash x\triangleleft y\).
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(a)
□
A straightforwardly corollary of the above results is the undefinability in the E-theory of timeless parthood and immediate parthood in terms of parthood. Theorem 2 is thus an immediate consequence of Theorem 8, Lemma 18 and Definitions 27 and 28.
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Jacinto, B., Cotnoir, A.J. Models for Hylomorphism. J Philos Logic 48, 909–955 (2019). https://doi.org/10.1007/s10992-019-09501-3
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DOI: https://doi.org/10.1007/s10992-019-09501-3