Abstract
Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected (Varzi in Noûs 31:26–58, 1997). In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of empty parts, in delivering a balanced and bounded metaphysics of naive space.
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Notes
This project can be taken as an exercise in pure metaphysics, naive physics, or spatial representation (at least). By not committing to whether e.g. the problems raised are epistemic, or ontological, etc., our results may be useful to researchers in any of the above areas. Our aim is to work with a naive view of things like the Great Red Spot as continuous objects in continuous space, treating objects as regions of space, and spaces as objects.
This is to set aside other views and issues involving vagueness, the ‘problem of the many’, supersubstantivalism, and other dialectical alternatives (e.g. about what it means to be in contact, whether topological properties can change over time, whether or not there is a privileged class of ‘unified’ objects as in (Koslicki 2008)—required, if we are to have room to give a formalization of one (reasonably attractive) view that has yet to receive sufficient attention. For recent discussions of similar material, see Varzi (1997), Casati and Varzi (1999, p. 74), Dainton (2010, chap. 17), Hudson (2005, Chaps. 2, 3) , Priest (2006, Chaps. 11, 12, 15) and Arntzenius (2012, Chap. 4).
According to Putnam, citing a short unpublished note (Putnam 1994).
Thanks to referees for pressing this point.
If neither an object nor its complement are fully bounded, both are partly open and partly closed. As such, the object has a proper part that is closed, and therefore the complement’s bordering part must be open, and vice versa. This defers the problem to a smaller scale; see Casati and Varzi (1999, p. 80).
Interestingly, this is itself a continuity intuition—roughly, if \(a\) is very similar to \(b\), then \(\partial (a)\) is very similar to \(\partial (b)\), making the boundary operator like a continuous function.
Casati and Varzi try to relocate the asymmetry ((Casati and Varzi 1999, p. 80, 87)), but the fundamental forced choice remains: as ever, in cutting a continuous line in half, there is a single point that either falls on the left or on the right, but not both—an asymmetry.
The standard account of boundary in point-set topology reflects the Classical view, defended in (Bolzano 1950). The Glutty view seems to be found first in Peirce (1933), and then (Priest 2006). Varzi (1997) cites Leonardo as holding a Gappy view, with additional possible adherents being (Hestevold 1986) and (Sorensen 1986). The Coincidence view is in Brentano (1988) and Chisholm (984). Finally, the Eliminativist is most notably in Whitehead (1919). NB that the use of the term Classical does not indicate that all other approaches are based in a non-classical logic.
To be accurate, there are a number of different boundary-free theories. Lumping all such theories under the category of Eliminativism is simply for ease of explication, and because below we will prefer boundaried theories over boundariless ones.
The closest thing anyone has put forward is the bi-Heyting algebraic theory of Stell and Worboys (1997). This algebra is developed primarily to yield point-free models of the Region Connection calculus of Randell et al. (1992). This structure is importantly different from the one we propose below. In particular, it is based in classical logic. But a full examination of those differences will have to wait for another time.
See Smith (1997, p. 18).
Thanks to Marcus Rossberg for raising this point, which is reminiscent of the problem of higher-order vagueness.
There is a school of thought, owing to Poincaré, Weyl, and Whitehead, that the problems associated with boundaries are in fact problems about conceptualizing space in terms of points. This is an idea is preserved in point-free (or ‘pointless’) geometry; see the recent ((Arntzenius 2012, chap. 4)). Some in fact take mereotopology to be a recasting of geometry in terms of qualitative regions, rather than quantitative points (or sets of points) (Pratt-Harmon 2007), and so the move to mereology itself to be a solution to the puzzles.
Strictly speaking, HM1 is redundant, since the Boolean complement axiom HM4 implies the reflexivity of parthood. Proof: If \(\forall x (\exists y \ y \nleq x)\) is false, then \(\exists x \forall y \ y \leqslant x\). Hence, \(x\) is the universe, and \(x \leqslant x\). If \(\forall x (\exists y \ y \nleq x)\), then find the complement \(\overline{x}\). As above, \(x \upharpoonleft \!\!\downharpoonright \overline{x}\). By HM4, \(\forall y ( y \upharpoonleft \!\!\downharpoonright \overline{x} \supset y \leqslant x)\), so \(x \leqslant x\).
Cf. Remark 4 below. Standard axiomatizations of mereology usually include a supplementation axiom—the usual candidate is Strong Supplementation: \(\forall x \forall y (y \nleq x \supset \exists z( z \leqslant y \wedge z \upharpoonleft \!\!\downharpoonright x))\). This stipulates a ‘remainder’ in cases where one object is not part of another. One instance of \(y \nleq x\) is the case where \(x < y\). In that case, supplementation is used to guarantee that if \(y\) has a proper part, then it must have another. Contraposing gives the crucial direction of this equivalence. See Hovda (2009, Part 4) for how our axiom set is equivalent to ones with e.g. supplementation.
Assume for reductio that there are at least two objects \(0\) and \(1\), and \(\forall x \ \mathbb {0} \leqslant x\). Note, \(1\!\!1 \nleq \mathbb {0}\): if it were, \(1\!\!1 \leqslant \mathbb {0} \wedge \mathbb {0} \leqslant 1\!\!1\) which by HM2 implies \(\mathbb {0}=1\!\!1\). But that is contrary to our supposition. So by e.g. Strong Supplementation, \(\exists z (z \leqslant 1\!\!1 \wedge z \upharpoonleft \!\!\downharpoonright \mathbb {0})\). But nothing is disjoint from \(\mathbb {0}\), since it is part of everything. Contradiction.
This diagram and others below are intended to be illustrative only — i.e. not presented as a consistency argument for the theory. For such an argument, see Appendix 2.
Why? Well, since we have more than one object around, HM6 gives \(\lnot \exists x \forall y \, x \leqslant y\), which is just equivalent to \(\forall x \exists y \, x \nleq y\). But \(a\) is a witness for \(y\) when \(x\) is instantiated by \(b\).
Proving a proposition inside a paraconsistent theory means, as elsewhere, that it is true in that theory. (In the model of Appendix 2, it means for the proposition to have value ‘t’ or ‘b’.) This is paraconsistent logic: a theorem may be taken to be proved, without a guarantee that its negation fails.
Specifically, \(t = \mathsf {lub}[x : w=a \vee w=a \mapsto x \leqslant w]\) iff
$$\begin{aligned} \qquad \forall x ((\forall z( z\!=\!a \vee z\!=\!b \!\mapsto \! x \leqslant z) \!\mapsto \! x \!\leqslant \! t) \wedge \forall w \forall x (\forall z ((z\!=\!a \vee z\!=\!b \mapsto x \leqslant z) \mapsto x \leqslant w) \mapsto t \leqslant w) \end{aligned}$$i.e. if \(x\) is a lower bound of \(a\) and \(b\), then \(t\) is an upper bound of \(x\), and any upper bound of lower bounds of \(a\) and \(b\) is above \(t\). The second conjunct entails that \(t\) is a lower bound of \(a\) and \(b\), i.e. \(\forall x((x=a \vee x=b) \mapsto t \leqslant x)\). Uniform variable relabeling of the first conjunct gives: \(\forall w \forall x ((x=a \vee x=b \mapsto w \leqslant x) \mapsto w \leqslant t)\). Conjoining (in a different order): \(\forall x((x=a \vee x=b) \mapsto t \leqslant x) \wedge \forall w \forall x ((x\!=\!a \vee x\!=\!b \mapsto w \leqslant x) \mapsto w \leqslant t)\) iff \(t \!=\! \mathsf {glb}[w : w\!=\!a \vee w\!=\!b]\) by definition of \(\mathsf {glb}\).
Cf. ((Casati and Varzi 1999, p. 46)). In some cases, the defining condition may not be met, but we will ignore this complication in what follows.
Indeed, that the following fails is instructive: \(x \leqslant y\) does not imply that \(x \not \leqslant \overline{y}\). The proof would go by reductio, arguing that if \(x \leqslant y \wedge x \leqslant \overline{y}\) then \(y \bullet \overline{y}\), which is impossible—but it is not impossible here. And even if it were, we would only have it that \(x \not \leqslant y \vee x \not \leqslant \overline{y}\), which goes nowhere without disjunctive syllogism. The model in Appendix 2 does not satisfy \(x \leqslant y \rightarrow x \not \leqslant \overline{y}\).
Hints for calculating: While both \(x - a = b\) and \(x - b = a\) imply that \(x = a \sqcup b\), the converse does not hold, by dint of failure of disjunctive syllogism, which here is \(a \sqcap (\overline{a} \sqcup b) \leqslant b\). Indeed, \(x = a \sqcup b\) leaves irreducible traces, in that it implies only \(x - a = \mathbf{0_a} \sqcup b\) and \(x - b = a \sqcup \mathbf{0_b}\). Note that in general cancellation property, \(x \sqcap y = x \sqcap z \rightarrow y=z\), is not valid, e.g. \(\mathbf{0_x} = \mathbf{0_y}\) does not imply \(x=y\).
In the classical case, adding CA1–CA4 to HM0–HM6 would result in the mereotopological theory called \(GEMTC\) in ((Casati and Varzi 1999, p.59)). For related structures, see for example, the De Morgan algebras with interior and closure operators as given in (Zhang and Yao 2007) developed for quite different applications, and based in classical logic.
Additionally, objects may be regular open, meaning that \(x = i(c(x))\) or regular closed, meaning that \(x = c(i(x))\), though this is not always the case.
A null individual is proposed by Carnap as the designatum of an empty name, where he says that the null thing “corresponds to the null class of space-time points” ((Carnap 1947, p. 36)).
((Casati and Varzi 1999, p. 45)). Arntzenius uses a bottom element with only parenthetical comment ((Arntzenius 2012, p. 136)). Roeper ((Roeper 1997, p. 253)) invokes one in order to use the Stone Representation Thereom. Forrest ((Forrest 2002, p. 81)) utilizes a null object, which he dismisses as ‘fictitious’.
Properly understood, anyway. Recall Geach’s objection at the end of Sect. 4.1.
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Acknowledgments
Thanks to audiences at the 25 Years In Contradiction Conference at Glasgow University, the Metaphysical Indeterminacy Worskshop VI at the University of Leeds, the 2009 Mereology Workshop at the University of St Andrews, the Australian Metaphysics Conference at Kioloa, the Pukeko Logic Group, and the University of Otago for comments and discussion on various versions of these ideas. Thanks especially to Ralf Bader, Guillermo Badia, Elizabeth Barnes, Can Baskent, Jc Beall, Franz Berto, Ben Burgis, Ross Cameron, Filippo Casati, Katherine Hawley, Cory Johnson, Manuel Lechthaler, Hitoshi Omori, Graham Priest, Adam Reiger, David Ripley, Marcus Rossberg, Lionel Shapiro, Stewart Shapiro, Reed Solomon, Diego Tajer, Jason Turner, Gabriel Uzquiano, Achille Varzi, Alan Weir, Robbie Williams, and Gareth Young. Research on this paper was supported by a grant from the Marsden Fund, Royal Society of New Zealand.
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Appendices
Appendix 1: Logic
The logic for our paraconsistent mereology is the weak relevant logic DKQ, previously applied to paraconsistent set theory; see ((Priest 2006, p.251)), (Weber 2012). Set theory in DKQ is non-trivial—it has a relative ‘consistency’ proof—because it has a classical model in which some sentence is not satisfied (Brady 2006).
The language is that of first order logic with identity and membership. The usual shorthand is used: \(A \vee B\) for \( \lnot (\lnot A \wedge \lnot B)\); \(A \leftrightarrow B\) for \((A \rightarrow B) \wedge (B \rightarrow A)\); \(\exists \) is \(\lnot \forall \lnot \).
Axioms. All instances of the following schemata are theorems:
\(\begin{array}{l@{\quad }l@{\quad }l} {\textit{I}} &{} A \rightarrow A &{} \\ {\textit{IIa}} &{} A \wedge B \rightarrow A &{} \\ {\textit{IIb}} &{} A \wedge B \rightarrow B &{} \\ {\textit{III}} &{} A \wedge (B \vee C) \rightarrow (A \wedge B) \vee (A \wedge C) &{} (\textit{distribution})\\ {\textit{IV}} &{} (A \rightarrow B) \wedge (B \rightarrow C) \rightarrow (A \rightarrow C) &{} (\textit{conjunctive syllogism}) \\ {\textit{V}} &{} (A \rightarrow B) \wedge (A \rightarrow C) \rightarrow (A \rightarrow B \wedge C) &{} \\ {\textit{VI}} &{} (A \rightarrow B) \leftrightarrow (\lnot B \rightarrow \lnot A) &{} (\textit{contraposition}) \\ {\textit{VII}} &{} A \leftrightarrow \lnot \lnot A &{} (\textit{double negation elimination}) \\ {\textit{VIII}} &{} A \vee \lnot A &{} (\text{ excluded } \text{ middle })\\ {\textit{IX}} &{} (\forall x)A \rightarrow A(a/x) &{} \\ {\textit{X}} &{} (\forall x)(A \rightarrow B) \rightarrow (A \rightarrow (\forall x)B) &{} (\hbox {with } x \hbox { not free in } A)\\ {\textit{XI}} &{} (\forall x)(A \vee B) \rightarrow A \vee (\forall x)B &{} (\hbox {with } x \hbox { not free in } A) \end{array}\)
The law of excluded middle implies the law of non-contradiction, \(\lnot (A \wedge \lnot A)\).
Rules. The following rules are valid:
\(\begin{array}{l@{\quad }l@{\quad }l} {\textit{I}} &{} A, B \vdash A \wedge B &{} (\textit{adjunction}) \\ {\textit{II}} &{} A, A \rightarrow B \vdash B &{} (\textit{modus}\;\textit{ponens}) \\ {\textit{III}} &{} A \vdash (\forall x) A &{} (\textit{universal}\;\textit{generalization}) \\ {\textit{IV}} &{} A \rightarrow B, C \rightarrow D \vdash (B \rightarrow C) \rightarrow (A \rightarrow D) &{} (\textit{hypothetical}\;\textit{syllogism}) \\ V &{} x = y \vdash A(x) \leftrightarrow A(y) &{} (\textit{substitution}) \end{array}\)
Structural Rules. The following meta-rules preserve validity:
This validates argument by cases, and, excluded middle, reductio:
Enthymematic conditional. Add a \(t\)-constant that obeys the two-way rule
The \(t\)-constant may be thought of as the conjunction of all truths. Then define \(A \mapsto B := A \wedge t \rightarrow B\). This defined operator has the property of weakening:
However, it does not contrapose: \(A \mapsto B \not \vdash \lnot B \mapsto \lnot A\).
Inconsistency and absurdity. For ease of presentation, we adopt an axiom asserting the truth of at least one contradiction:
for some fixed \(p\). After all, the axiom set PM is no more than a classical set re-written with paraconsistent conditionals. So there is no reason to expect that it generates a contradiction on its own. Within the language of PM so defined, then, there is at least one \(\mathbf {0_x}\), which is required for the theorems above. Otherwise they would simply read that if \(\mathbf {0_x}\) exists then..., which is classically true too, by ex falso quodlibet! By the same token, to ensure that there is at least one unacceptable sentence, a constant \(\bot \) is added to the language, and a final axiom
asserts that \(\bot \) is never designated.
Appendix 2: Non-triviality
Here we present an interpretation \(\mathcal {M}\) of the formal theory, showing that even if it is inconsistent it is not trivial. The presentation and proof that \(\mathcal {M}\) is a non-trivial model are conducted in standard classical logic, following standard practice in the relevant logic and mathematics literature (e.g. ((Priest 2006, Chap. 17))); the model is suggested by Meyer and Mortensen’s two-element model for relevant arithmetic. This model is not presented as the intended model of paraconsistent mereology. The axiomatization PM is not complete with respect to the model. Our only concern is soundness, which in the present context is non-triviality.
Fact. Paraconsistent mereology has a model.
Proof
To interpret the logic, we use truth tables for the paraconsistent logic RM3:
The designated values of RM3 are t and b. RM3 validity is preservation of designated values. An argument is invalid if there is a valuation making all the premises designated, but not the conclusion. The conditional of DK does not have a truth table, but DK is sound with respect to RM3. Anything invalid by RM3 truth tables is DK invalid.
The enthymematic conditional, \(A \mapsto B\), should be thought of as having an antecedent that is no more than ‘b’, since the connective \(t\) is the conjunction of all truths, and some of these are false (Beall et al. 2011). So the conditional never has an antecedent that is \(t\) only, and the only way it can get an \(f\) is if the consequent is \(f\).
To define a model \(\mathcal {M} = \langle D, V \rangle \), let the domain \(D = \{0, 1\}\) and \(V(\leqslant )\) be as follows:
That is, \(1 \leqslant 0\) is just false, \(0 \leqslant 1\) is just true, and it is both true and false that things are parts of themselves. For identity,
Because the domain is finite, quantifiers \(\forall \) and \(\exists \) can be understood as conjunctions and disjunctions, respectively.
Now it is just a matter of verifying the axioms, showing that they have designated values.
For PM1, \(V(0 \leqslant 0) = V(1 \leqslant 1) = b\), so \(V(0 \leqslant 0 \wedge 1 \leqslant 1) = b\) and therefore \(V(\forall x(x \leqslant x)) =b\).
For PM2, anti-symmetry, we consider all the possible values of \(x\) and \(y\):
In all cases, the required biconditional holds.
For PM3, transitivity, the only case where the consequent fails is \(V(1 \leqslant 0) = f\). Then the antecedent \(V(1 \leqslant y \wedge y \leqslant 0) = f\) whether \(y\) is \(1\) or \(0\). There is no case where \(V(x \leqslant y \wedge y \leqslant x) = t\).
For PM4, we set \(1\) as the complement of \(0\), and vice versa. Well, \(V(1 \upharpoonleft \!\!\downharpoonright 0) = b\) because
and
so the conjunction of the two is \(b\). And to show \(\forall y((\lnot y \bullet 0 \rightarrow y \leqslant 1) \wedge (\lnot y \bullet 1 \rightarrow y \leqslant 0))\), we have
making the conjunction of these four b, designated. To show the converses of all these, note that they all work as \(\rightarrow \) except the first, which would be \(V(0 \leqslant 1\rightarrow 0 \upharpoonleft \!\!\downharpoonright 0) = f\) since the antecedent is t and the consequent is b. But the \(\mapsto \) rendering drops the antecedent down to b and the sentence is satisfied. The same reasoning applies to show that \(0\) is the complement of \(1\).
For PM5, we need a witness to play \(\mathsf {lub}\). Now, \(V(A)\) is t, b, or f. Then take the \(\mathsf {lub}\) of the \(A\)s to be \(0\) when \(A\) is \(f\), and the \(\mathsf {lub}\) of the \(A\)s is \(1\) otherwise. In more detail, consider the case where \(V(A)\) is t or b. Well, \(V(x \leqslant 1)\) is always designated, so \(V(A \mapsto x \leqslant 1)\) is always designated. And then \(1\) is least since a check of truth tables shows
Otherwise, \(V(A) = f\). Then \(V(A \mapsto 0 \leqslant 0) = V(A \mapsto 1 \leqslant 0) = t\). And to show that \(0\) is least here,
Thus PM5 is always designated.
For PM6, \(V(1 \not \leqslant 1) = b\) and \(V(0 \not \leqslant 0) = b\), so \(V(\exists x(x \not \leqslant 1))\) and \(V(\exists x( x \not \leqslant 0))\) are both designated; and then so is their conjunction.
For PM7, we have \(V(0 \bullet 0) = V(0 \bullet 1) = V(1 \bullet 0)= b\), and \(V(1 \bullet 1) = t\). Then the cases are all designated:
noting that the second case would not be designated with an \(\rightarrow \) as the main connective.
For the topology, we need to interpret Kuratowski’s closure axioms. This can be done very easily: let
Again, this overdoes it, but suffices to show the existence of a model. For CA1, this follows from PM1; for CA2, \(c(c(x)) = c(x)\) for both \(0\) and \(1\); and so forth. The interior operator, \(i(x)\), is then \(\overline{c(\overline{x}}) = \overline{\overline{x}} = x\), which satisfies the dual interior properties.
Fact. Not every sentence of the language of PM is provable.
Proof
The sentence \(\forall x \forall y (x \leqslant y)\) gets value f in the model. All we need is one; but there are many, e.g. \(\forall x \forall y(x \bullet y \rightarrow x \leqslant y)\) gets f, too.
This completes the proof. \(\square \)
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Weber, Z., Cotnoir, A.J. Inconsistent boundaries. Synthese 192, 1267–1294 (2015). https://doi.org/10.1007/s11229-014-0614-z
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DOI: https://doi.org/10.1007/s11229-014-0614-z