Is Mereology Ontologically Innocent? Well, it Depends…

Abstract

Mereology, the theory of parts and wholes, is sometimes used as a framework for categorisation because it is regarded as ontologically innocent in the sense that the mereological fusion of some entities is nothing over and above the entities. In this paper it is argued that an adequate answer to the question of whether the thesis of the ontological innocence of mereology holds relies crucially on the underlying theory of reference. It is then shown that upholding the thesis comes at high costs, viz. at the cost of a quite strong logical background theory or at paradoxical ways of predicating and counting.

Technical Appendix

Every fusion of a finite number of individuals $a_1,\dots ,a_n$ is due to commutativity and distributivity of sum easily performable by an iterative application of the two-placed composition operation. For the infinite case one would just need to add a name-forming operator _Σx similar to the iota-operator for descriptive descriptions. One may then define possibly infinite fusions contextually by $_{{\Sigma }}x\varphi [x]=y\leftrightarrow \forall z(z\circ y\leftrightarrow \varphi [x])$.

In the following proofs we make use of Boolos’ result that PFO can be embedded into monadic second order logic (MSO) and vice versa with the following one-one mapping tr cf. (Linnebo 2003), p.74:

• ${tr}(x\sqsubset x\hspace {-0.22em}x)=X(x)$

• ${tr}(\sim \hspace {-0.25em}\varphi )=\sim \hspace {-0.25em} {tr}(\varphi )$, tr (φ & ψ) = tr (φ) & tr (ψ), …

• tr (∃xφ[x]) = ∃xtr (φ[x]), tr (∀xφ[x]) = ∀xtr (φ[x])

• tr (∃xxφ[xx]) = ∃Xtr (φ[xx]), tr (∀xxφ[xx]) = ∀Xtr (φ[xx])

To shorten MSO proofs we also make use of the result that MSO can be embedded into restricted Zermelo set theory, also called ‘monadic second order Zermelo set theory’, consisting of five axioms: extensionality, pairing, power set, union, infinity and separation (the latter is the only monadic formula: $\forall X\forall x\exists y\forall z(z\in y\leftrightarrow (z\in x~\&~X(z)))$). For details see (Pollard 2015, chpt.VII.2–4).

Proof

• Proof of T1: By definition PFO7 and the translation manual tr we get: ${tr}(\forall x\hspace {-0.22em}x\forall y\hspace {-0.22em}y\exists ^1_1z\hspace {-0.22em}z\forall z\hspace {-0.22em}z_1(z\hspace {-0.22em}z_1$$\square$$z\hspace {-0.22em}z\leftrightarrow (z\hspace {-0.22em}z_1$$\square$xx ∨ zz₁$\square$yy))) is logically equivalent to $\forall X\forall Y\exists ^1_1Z\forall Z_1(\exists x(Z_1(x)~\&~Z(x))\leftrightarrow \exists x(Z_1(x)~\&~(X(x)\vee Y(x))))$. The uniqueness claim ($\exists ^1Z\dots$) follows immediately by transitivity of the biconditional ($\leftrightarrow$). By restricted monadic second order comprehension (instantiation: $\exists Z\forall x(Z(x)\leftrightarrow (X(x)\vee Y(x)))$) we get $\exists _1Z\dots$.

• Proof of T2: By definitions PFO7 and PFO8 and tr one has to prove that $\forall Z_1(\exists x(Z_1(x)~\&~Z(x))\leftrightarrow \exists x(Z_1(x)~\&~(X(x)\vee Y(x))))$ is equivalent to $\forall x(Z(x)\leftrightarrow (X(x)\vee Y(x)))$. The step from the latter to the former is up to applying finally PFO2 for generalising Z₁ completely FO. A set-theoretical translation of the former would be: z₁ ∩ z ≠ ∅ iff z₁ ∩ (x ∪ y)≠∅, therefore z₁ ∩ z = ∅ iff z₁ ∩ (x ∪ y) = ∅ (for all z₁). Now, suppose z ≠ x ∪ y. Then there is an a ∈ z which is not in x ∪ y: a∉x ∪ y (or the other way round: a ∈ x ∪ y, but a∉z). Let z₁ = {a}. Then z₁ ∩ z = {a}, whereas z₁ ∩ (x ∪ y) = ∅ in contradiction to the equivalence above (similarly for the other case). Hence z = x ∪ y, which can be translated back to $\forall x(Z(x)\leftrightarrow (X(x)\vee Y(x)))$.

• Proof of T3: By definitions PFO7 and PFO8 and tr one has to prove $(\exists ^n_nxX(x)~\&~\exists ^m_my(Y(y)~\&~\sim \hspace {-0.25em}X(y)))\leftrightarrow \exists Z\exists ^{n+m}_{n+m}z(Z(z)\leftrightarrow \forall Z_1(\exists w(Z_1(w)~\&~Z(w))\leftrightarrow \exists w(Z_1(w)~\&~(X(w)\vee Y(w)))))$ which can be translated further into the monadic second order Zermelo set theoretically valid cardinality claim: |x| = n and |y ∖ x| = m iff |z| = |x ∪ y| = n + m.

• Proof of T4: Follows immediately from T3 by the extra condition: For all x holds: x ∈ X ∪ Y.

• Correctness of ε in AP2 interpreted as $~_1\hspace {-0.35em}\subseteq _{1+}$ ($x~_1\hspace {-0.35em}\subseteq _{1+}y$ iff $x\subseteq y$ and |x| = 1): One just has to prove that the following statements as set-theoretically valid: for all x exists a y: $y~_1\hspace {-0.35em}\subseteq _{1+}x$; for all x and all y: If $x~_1\hspace {-0.35em}\subseteq _{1+}y$, then there exists a z such that $z~_1\hspace {-0.35em}\subseteq _{1+}x$, and for all z₁,z₂: If $z_1~_1\hspace {-0.35em}\subseteq _{1+}x$ and $z_2~_1\hspace {-0.35em}\subseteq _{1+}x$, then $z_1~_1\hspace {-0.35em}\subseteq _{1+}z_2$; furthermore: For all z: If $z~_1\hspace {-0.35em}\subseteq _{1+}x$, then $z~_1\hspace {-0.35em}\subseteq _{1+}y$. The existence of a singleton-subset is guaranteed by the power set axiom. Regarding the equivalence: (⇒) Assume $x~_1\hspace {-0.35em}\subseteq _{1+}y$. Then we know |x| = 1 and by this $x~_1\hspace {-0.35em}\subseteq _{1+}x$, hence: There is a z: $z~_1\hspace {-0.35em}\subseteq _{1+}x$; since |x| = 1 we also get by extensionality that the element(s) of a singleton-subset of x are/is unique. And by transitivity of $~_1\hspace {-0.35em}\subseteq _{1+}$ we arrive at the inclusion condition. (⇐): Assume (i) there is a z such that $z~_1\hspace {-0.35em}\subseteq _{1+}x$, (ii) for all z₁,z₂: If $z_1~_1\hspace {-0.35em}\subseteq _{1+}x$ and $z_2~_1\hspace {-0.35em}\subseteq _{1+}x$, then $z_1~_1\hspace {-0.35em}\subseteq _{1+}z_2$, (iii) for all z: If $z~_1\hspace {-0.35em}\subseteq _{1+}x$, then $z~_1\hspace {-0.35em}\subseteq _{1+}y$. Now assume that $x~_1\hspace {-0.35em}\not \subseteq _{1+}y$. Then there are three cases to be considered: $x\not \subseteq y$ or |x|≠ 1 or y = ∅. But by (i) we know that |x|≥ 1 and hence by (iii) we get: y ≠ ∅. Furthermore, take x to be $\{x_1,x_2,\dots \}$. By (ii) we get: x₁ = x₂, x₁ = x₃, …. Hence |x| = 1. But now we also get by (iii) that $x~_1\hspace {-0.35em}\subseteq _{1+}y$, hence $x\subseteq y$. So $x~_1\hspace {-0.35em}\subseteq _{1+} y$ in general. (Correctness for AP0 is straightforward: $z~_1\hspace {-0.35em}\subseteq _{1+} x$ exactly when $z~_1\hspace {-0.35em}\subseteq _{1+}y$ for any z implies $x~_1\hspace {-0.35em}\subseteq _{1+}z$ exactly when $y~_1\hspace {-0.35em}\subseteq _{1+}z$ for any z and vice versa.)

• Correctness of ε in AP3 interpreted as $~_{1+}\hspace {-0.35em}\subseteq _{1+}$ ($x~_{1+}\hspace {-0.35em}\subseteq _{1+} y$ iff $x\subseteq y$ and x ≠ ∅): Proof analogous to the correctness proof above (without making assumption (ii)).

• Correctness of ε in AP4 interpreted as ∩_≠o̸ (x ∩_≠o̸y iff there is a z such that z ∩ x ≠ ∅ and z ∩ y ≠ ∅): One just has to prove that: There is a z z ∩ x ≠ ∅ and z ∩ y ≠ ∅ iff there is a z such that there is a z₁z₁ ∩ z ≠ ∅ and z₁ ∩ x ≠ ∅, and there is a z₂ such that z₂ ∩ z ≠ ∅ and z₂ ∩ y ≠ ∅. (⇒): Follows immediately by iteration and piecewise existential generalisation. (⇒): Assume that x and y are connected via z₁ and z₂ respectively to z. Then x and y are connected directly via z₁ ∪ z₂ and hence there is a z connecting x and y (x ∩ (z₁ ∪ z₂)≠∅ ≠ y ∩ (z₁ ∪ z₂)). (Correctness regarding AP0 is due to the symmetry of the connectable-relation.)

• Proof of T5–T9: Straightforward FO (regarding AP1: set-theoretical).

• Proof of T10: Since the only primitive expressions are ε and (later on in Section 5) sum—all the other notions =, ≼, ≺ etc. are introduced by definitions alone—for proving the indiscernibility of identicals we have to consider only formulas that contain ε and sum as non-logical expressions. We do so by induction on the complexity of formulas: Assume x = y is valid in the theory, i.e. by definition DI1: $\forall z(z\varepsilon x\leftrightarrow z\varepsilon y)$ is valid in the theory. Then we have to consider the following inductive basis (degree of complexity of φ is 0): φ[x] is valid and of the form:

  • 1. z₁εz₂: In this case the substitution is idle, hence φ[x] = φ[x/y] and hence $\varphi [x]\leftrightarrow \varphi [x/y]$ is valid.

  • 2. xεz₁: By AP0 we get $x\varepsilon z_1\leftrightarrow y\varepsilon z_1$ and by this also yεz₁.

  • 3. z₁εx: With DI1 we get immediately z₁εy.

  • 4. xεx: With DI1 we get $y\varepsilon y\leftrightarrow y\varepsilon x$; with AP0 we get $x\varepsilon x\leftrightarrow y\varepsilon x$, and hence yεy.

  • 5. xεy or yεx: Cf. case 4 (xεy[x/y] = yεy = yεx[x/y]).

  • 6. sum(x, z₁)εz₂: By AM1 and DM3 we get $z\varepsilon z_2\leftrightarrow \forall z_3(\exists z_4(z_4\varepsilon z_3~\&~z_4\varepsilon z)\leftrightarrow \exists z_4(z_4\varepsilon z_3~\&~(z_4\varepsilon x\vee z_4\varepsilon z_1)))$; by the result on case 3 we get $z\varepsilon z_2\leftrightarrow \forall z_3(\exists z_4(z_4\varepsilon z_3~\&~z_4\varepsilon z)\leftrightarrow \exists z_4(z_4\varepsilon z_3~\&~(z_4\varepsilon y\vee z_4\varepsilon z_1)))$ and hence by DM3 and AM1 sum(y, z₁)εz₂.

  • 7. All the other cases of sum in φ are either analogous to 6 or follow immediately from basic features of sum (commutativity, distributivity, selfcomposition: T23-T25).

  The induction step is straightforward FO.

• Proof of T11-T25: Straightforward FO.

• Proof of T26: Take the following model: x = 1,y = 2. Then, by AP1 and AM1, we get z = sum(x, y) = {1, 2}. According to OII z = sum(x, y) iff x ∈ z, y ∈ z and for all z₁ ∈ z: z₁ ∈ x or z₁ ∈ y. But, although x ∈ sum(x, y), neither x ∈ x nor x ∈ y (if it were the case that x ∈ y, then sum(x, y) = y and by this it wouldn’t be the case that x ∈ sum(x, y)). W.l.o.g. this holds for any model whose domain contains at least two objects. So OII and AP1 are incompatible in case that there are at least two objects.

• Proof of T27: Since both, AP2 and OII are consequences of AP4 and AP4 is consistent (see below), it follows also that AP2 and OII are consistent.

• Proof of T28: Analogous to the proof of T27.

• Proof of T29: (That AP2 and AP3 follow from AP4 is straightforward FO.) The proof of OII by help of AP4 and M is a quite long, but can be verified, e.g., by an automatic FO-prover (used here: Prover9). Using all the definitions the problem reduces to:

  • From AP4: $\forall x\exists y~y\varepsilon x~\&~\forall x\forall y(x\varepsilon y\leftrightarrow \exists z(z\varepsilon x~\&~z\varepsilon y))$, and

  • the creative part of M (translation of T22): $\forall x\forall y\exists z\forall z_1(\exists z_2(z_2\varepsilon z_1~\&~z_2\varepsilon z)\leftrightarrow \exists z_2(z_2\varepsilon z_1~\&~(z_2\varepsilon x\vee z_2\varepsilon y)))$

  • follows OII—translated as: $\forall x\forall y\forall z((x\varepsilon z~\&~y\varepsilon z~\&~\forall z_1(z_1\varepsilon z\rightarrow (z_1\varepsilon x\vee z_1\varepsilon y)))\leftrightarrow \forall z_1(\exists z_2(z_2\varepsilon z_1~\&~z_2\varepsilon z)\leftrightarrow \exists z_2(z_2\varepsilon z_1~\&~(z_2\varepsilon x\vee z_2\varepsilon y))))$.

  Also AP4 and M are consistent which is easily verifiable (used here: Mace4). Here is the relevant input and output for Prover9 and Mace4 (for the proof the programme ran about 20min on an ordinary machine):

  
  

  %Assumptions: %AP4 ((all x(exists y(E(y,x))))&(all x(all y(E(x,y)<-> (exists z(E(z,x)&E(z,y))))))) & %DM3: O for overlapping (all x(all y(O(x,y)<->(exists z(E(z,x)&E(z,y)))))) & %DI1 (all x(all y(I(x,y)<->(all z(E(z,x)<->E(z,y)))))) & %AM1: S for sum (existence and uniqueness are built in) ((all x(all y(all z(S(x,y,z)<->(all z1(O(z1,z)<-> (O(z1,x)-O(z1,y))))))))&(all x(all y(exists z(S(x,y,z)& (all z1(S(x,y,z1)->I(z1,z)))))))) . % % %Goals: %OII (all x(all y(S(x,y,z)<->((E(x,z)&E(y,z))&(all z1 (E(z1,z)->(E(z1,x)-E(z1,y)))))))) .

• Proof of T30: Similar reduction and method as used above:

  • From OII—translated as above—, and

  • the creative part of M (translation of T22)

  • follows AP4.

  So, it turns out that under the assumption of unrestricted mereological composition OII and AP4 are equivalent, i.e. unrestricted composition and OII enforces gavagaian predication.

• Proof of T31–T35: Straightforward FO. Regarding T35 note that the not dissolved way of counting (by distinguishing only disjunct objects) is paradoxical but can be made coherent by adding to the basis case of the contextual definition for the condition that the entity under consideration is atomic. I.e.: For use $\exists x(A(x)~\&~\varphi [x]~\&~\forall y(\varphi [y]\rightarrow y=_px))$. But then one cannot claim any more that the “whole is the many counted as one”. Also the question why counting disjuncts and not, e.g., maternally identicals remains.

□

Some formulations of the assumptions and theorems in the language of Prover9 / Mace4:

%%%%%%%%%%%%%%%%%%%%%%%%% %AP0 (all x(all y((all z(E(z,x)<->E(z,y)))->(all z(E(x,z) <->E(y,z)))))). %AP2 ((all x(exists y(E(y,x))))&(all x(all y(E(x,y)<-> (((exists z(E(z,x))&(all z(E(z,x)->E(z,y))))&(all z1(all z2((E(z1,x)&E(z2,x)->E(z1,z2))))))))))). %AP3 ((all x(exists y(E(y,x))))&(all x(all y(E(x,y)<->((exists z(E(z,x)))&(all z(E(z,x)->E(z,y)))))))). %AP4 ((all x(exists y(E(y,x))))&(all x(all y(E(x,y)<->(exists z(E(z,x)&E(z,y))))))). %%%%%%%%%%%%%%%%%%%%%%%%% %DI1 (all x(all y(I(x,y)<->(all z(E(z,x)<->E(z,y)))))). %DM1: M for improper part (all x(all y(M(x,y)<->(all z(E(z,x)->E(z,y)))))). %DM2: P for proper part (all x(all y(P(x,y)<->(M(x,y)&-I(x,y))))). %DM3: O for overlapping (all x(all y(O(x,y)<->(exists z(E(z,x)&E(z,y)))))). %DM4: A for atom (all x(A(x)<->-(exists y(E(y,x)&-I(y,x))))). %DM5: U for universal (all x(U(x)<->(-A(x)&(all y(M(x,y)->x=y))))). %AM1: S for sum (existence and uniqueness are built in) ((all x(all y(all z(S(x,y,z)<->(all z1(O(z1,z)<-> (O(z1,x)-O(z1,y))))))))&(all x(all y(exists z(S(x,y,z)&(all z1(S(x,y,z1)->I(z1,z)))))))). %%%%%%%%%%%%%%%%%%%%%%%% %OII (all x(all y(S(x,y,z)<->((E(x,z)&E(y,z))&(all z1(E(z1,z) ->(E(z1,x)|E(z1,y)))))))) %OIC (exists x(exists y(exists z(((((((-I(x,y)&-I(x,z))&-I(y,z)) &A(x))&A(y))&A(z))&(all z1(A(z1)->((I(z1,x)-I(z1,y))-I (z1,z))))))))) %%%%%%%%%%%%%%%%%%%%%%%% %T5 (all x(E(x,x))) %T6 (all x(all y(all z((E(x,y)&E(y,z))->E(x,z))))) %T7 (all x(I(x,x))) %T8 (all x(all y(I(x,y)->I(y,x)))) %T9 (all x(all y(all z((I(x,y)&I(y,z))->I(x,z))))) %T11 (all x(all y((all z(E(x,z)<->E(y,z)))->I(x,y)))) %T12 (all x(all y(I(x,y)<->(all z(E(x,z)<->E(y,z)))))) %T13 (all x(all y(I(x,y)<->(E(x,y)&E(y,x))))) %T14ᅟ (all x(all y((I(x,y)&(exists z1(exists z2(exists z3(((((((-I (z1,z2)&-I(z1,z3))&-I(z2,z3))&E(z1,x))&E(z2,x))&E(z3,x)) &(all z4(E(z4,x)->((I(z4,z1)-I(z4,z2))-I(z4,z3)))))))))) ->(exists z1(exists z2(exists z3(((((((-I(z1,z2)&-I(z1,z3)) &-I(z2,z3))&E(z1,y))&E(z2,y))&E(z3,y))&(all z4(E(z4,y)- >((I(z4,z1)|I(z4,z2))-I(z4,z3)))))))))))) %T15 (all x(all y((exists z(E(z,x)&E(z,y)))<->(exists z(M(z,x) &M(z,y)))))) %T16 (all x(-(exists y(E(y,x)&-I(x,y)))<->-(exists y(P(y,x))))) %T17 (all x(-P(x,x))) %T18 (all x(all y(P(x,y)->-P(y,x)))) %T19 (all x(all y(all z((P(x,y)&P(y,z))->P(x,z))))) %T20 (all x(all y(E(x,y)->M(x,y)))) %T21 (all x(all y(M(x,y)->E(x,y)))) %T22 (all x(all y(exists z(all z1(O(z1,z)<->(O(z1,x)-O (z1,y))))))) %T23 (all x(all y(all z(S(x,y,z)<->S(y,x,z))))) %T24 (all x(all y(all z(all z1(all z2(all z3(((S(x,y,z1)& S(z1,z,z2))&S(y,z,z3))->S(x,z3,z2)))))))) %T25 (all x(S(x,x,x))) %T31 (all x(all y(M(x,y)<->(all z(O(z,x)->O(z,y)))))) %T32 (all x(all y(I(x,y)->(all z(O(z,x)<->O(z,y)))))) %T33 (all x(all y(((A(x)&A(y))&-I(x,y))->-O(x,y)))) %%%%%%%%%%%%%%%%%%%%%%%%