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Reference: Peter Simons, Parts: A Study in Ontology (New York: Clarendon Press, 2000) Abbreviations
x«y x is a proper part of y x=y x and y designate the same individual x≈y x and y either designate the same individual or are both empty Ex x exists !x there is at most one x E!x there is exactly one x
Definitions
SD1 (part): x<y iff x«y or x=y
SD2 (overlap): x○y iff ∃z┌z<x or z<y┐^ (x,y have at least one common part)
SD3 (disjoint): x|y iff ~(x○y) (x,y, have no common part)
SD4 (product): x·y iff ιz∀w ┌w<z ≡ (w<x & w<y)┐ (a product is the common part of x,y such that any common part of x,y is a common part of it)
SD7 (sum): x+y ≈ ιz∀w ┌w○z ≡ (w○x or w○y)┐
SD9 (F-fusion): σx ┌Fx┐^ ≈ ιz∀y ┌x○y ≡ ∃z ┌Fz & z○y┐┐
SD10 (F-product): πx ┌Fx┐^ ≈ σx ┌∀y ┌Fy ⊃ x<y ┐┐
SD11 (difference): x–y ≈ σz ┌z<x and z|y┐ (a difference is the largest part of x that has a common part with y)
SD12 (universe): U = σx ┌x=x┐
SD13 (complement): x* ≈ U–x
Ordering Axioms These constrain the proper parthood relation.
SA1: x«y ⊃ ~(y«x) Proper parthood is not symmetric.
SA2: (x«y & y«z) ⊃ x«z Proper parthood is transitive.
Supplementation Principles These constrain how to supplement an individual's proper part in order to obtain a whole.
SF2: x«y ⊃ ∃z┌z«y & ~(z<x)┐ Any individual with a proper part has some other part that is not part of its proper part.
SA3: x«y ⊃ ∃z┌z«y & z|y┐ Weak Supplementation Principle (WSP) Any individual with a proper part has at least two disjoint proper parts.
SA4: (∃z┌z«x┐^ & ∀z┌z«x ⊃ z«y┐) ⊃ x<y Proper Part Principle (PPP) If an individual has a proper part and shares all of its proper parts with some other individual, then that latter individual is part of the former individual.
SA5: ~(x<y) ⊃ ∃z┌z<x & z|y┐ Strong Supplementation Principle (SSP) Anything that is not part of some individual has a part that is disjoint from that individual.
SA6: x○y ⊃ ∃z∀w ┌w<z ≡ (w<x & w<y)┐ Overlapping individuals have a unique product.
SF3: x○y ⊃ E!(x·y) Overlaps form a unique product.
SA24: ∃x Fx ⊃ ∃x ∀y ┌y○x ≡ ∃z ┌Fz & y○z┐┐ General Sum Principle (GSP) ∃x Fx ⊃ E! σx┌Fx┐^ (via SD9) There is a unique fusion of all Fs.
Theorems
SCT7: x=y ≡ ∀z ┌z<x ≡ z<y┐ Identical individuals have exactly the same parts.
SCT71: (∃z ┌z«x ┐& ∃z ┌z«y┐) ⊃ (∀z ┌z«x ≡ z«y┐^ ⊃ x=y) Mereological Extensionality Individuals with exactly the same proper parts are identical.
Counterexamples to Extensionality
F.C. Doepke, "Spatially Coinciding Objects," Ratio 24 (1982): 45-
A person and the person's body have exactly the same parts. A person exists only if their parts are undergoing certain processes. But the person's body can exists even if its parts are not undergoing certain processes. So a person and their body are distinct. Hence, individuals with the same parts need not be identical.
David Wiggins, "Mereological Essentialism: Asymmetrical Essential Dependence and the Nature of Continuants," in Ernest Sosa (ed.), Essays on the Philosophy of Roderick M. Chisholm (1979): 297-316.
Tibbles (cat) consists of Tib (cat body) and Tail (cat tail). Tibbles and Tib+Tail have exactly the same parts. Tibbles can lose its tail and continue to exist. Tib+Tail cannot lose its and continue to exist. [Tacit Principle: (◊Fx & ~◊Fy) ⊃ x≠y] So Tibbles and Tib+Tail are distinct. Hence, individuals with the same parts need not be identical.
on-Extensional Mereology
SD1* (part*): x≤y iff ∃z ┌z«x┐^ ⊃ ∀z ┌z«x ⊃ z«y┐^ or (~∃z ┌z«x┐^ ⊃ (x«y or x=y)
SD1* entails SA2 (transitivity of proper parthood).
SCT72: x<y ⊃ x«y
The converse of SCT72-- x«y ⊃ x<y --holds only with SA4.
Coincidence: x≤≥y iff x≤y & y≤x (x coincides with y) x,y are coincident iff x,y have exactly the same parts
CTD5: x≤≥ty iff x≤ty & y≤tx (x coincides at time t with y)
CTT22: x≤≥ty ≡ Exta & Extb & ∀x ┌x<ta ≡ x<tb┐
CTD5+SA2 ├ CTT
Superposition: x supt y iff x and y occupy exactly the same place at time t
Coincident individuals are perceptually indistinguishable. Coincident individuals are superposed. If x≤≥ty, then x supt y.
Without SA4, one cannot prove:
SF12: (x<y & y≤x) ⊃ x=y SF13: ∀z ┌z≤x ≡ z≤y┐^ ⊃ x=y
This shows that SA4 amounts to mereological extensionality.
Without SA4,
- parthood* (≤) is not antisymmetric
- parthood* (≤) remains reflexive and transitive
- coincidence (≤≥) is not identity (=)
SA4 ├ coincidence only if identity
Objections to the Possibility of Superposition
One-Many View
- a plurality of objects is not identical to a single object.
Problem: a ring and its gold, and a person and its body, are not cases involving a one-many relation
Relative Identity
- see (e), (f), (g) criticism from the Flux argument
Dichronic View
- when the ring comes into existence, the gold ceases to exist and is replaced by the ring, so neither exists at the same time as the other
- This rejects the view that a substratum exists that survives change; instead, change is the replacement of one individual by another.
Problem: On this view, it is hard to explain continuity of properties from, say, gold to a ring.
Reductivism
- the only real objects are the ultimate constituents of continuents; all else is a logical construction
Problem: Parts are not always logically prior to their wholes. For example, integrated wholes (e.g., organisms) have properties and obey laws that are relatively independent of their constituents, because they can survive and sustain these properties despite flux in their parts.
Concluding Remarks from Simons
There are four minimal constraints on the proper parthood relation:
Definition 'Part': x<y ≡ (x«y & (E!x & x≈y))
Definition 'Overlap': x○y ≡ ∃z ┌z«y & ~(z≈x) & ~(z≈y) & ~(z≈x)┐^ & ∃w ┌w«z & w<<x┐
Falsehood: x«y ⊃ (E!x & E!y)
Asymmetry: x«y ⊃ ~(y«x)
Transitivity: (x«y & y«z) ⊃ x«z
Weak Supplementation: x«y ⊃ ∃z ┌z«y & ~(z○x)┐
The basis for stronger mereological principles is the nature of the objects to which the part-relation applies.
- This results in local mereological systems with the above four global mereological properties.
Ontological Dependence
Weak Foundation: NEC(E!a ⊃ E!b) (NEC(x) : it is necessarily the case that x)
Problem: This entails that everything is ontologically dependent on necessarily existent individuals (such as numbers, God)
Problem: This allows self-dependence (reflexivity of the dependence relation).
DD1: x WRD on y ≡ ~(x=y) & ~NEC(E!y) & NEC(E!x ⊃ E!y) (Weak Rigid Dependence) x WRD y : x is weakly rigidly dependent on y
- This avoids the two problems with Weak Foundation.
x notionally depends on y : x cannot be described as x unless y exists
NEC ∀x NEC (Hx ⊃ ∃y ┌Wy & ~(x=y)┐
Notional dependence entails ontological dependence (weak rigid dependence) only for Fs that are essentially Fs -- that is, only if:
NEX (Fx ⊃ NEC(E!x ⊃ Fx))
Ontological dependence entails notional dependence. (Obvious.)
Symbols: « ≤ ≥ ○ · ≈ ≡ σ π « – ≅ ⊃ ∃ ∀ ┌^ ┐^ ι ├
