Reading off semi-lattice diagram

Question

I'm reading a chapter about mereology (in a handbook of linguistics), and I have some questions. A prepring is available here (p. 519). The symbol $\leq$ is to be interpreted as non-strict parthood, and $<$ is a strict parthood.

Consider a structure $(L,\oplus)$ where $L$ is a set and $\oplus $ is a two-place operation of join on this set that is associative, commutative, idempotent. Suppose that unique separartion holds: $$x < y \to \exists! z(x\oplus z = y\land \neg (x \circ z ))$$ where $x\circ y=^{def} \exists z(z \leq x \land z \leq y)$. After this the author says "Given unique separation, if we define parthood in terms of join by $x\leq y\iff x \oplus y = y$, then reflexivity, transitivity, and antisymmetry will follow as theorems".

First question: If at the beginning we only had this structure $(L,\oplus)$, how can we impose the axiom of unique separation if we don't yet know what how $< $ is defined on $L$? My only guess is that it is automatically presupposed that whenever we have $(L,\oplus)$ with the properties as above, it is assumed that $x \leq y$ is defined by $x\leq y\iff x \oplus y = y$ (and then $x < y$ is defined as $x \leq y$ and $x\ne y$). But the author doesn't discuss "$x\leq y\iff x \oplus y = y$" until after he has imposed the axiom of unique separation ...

Then the following diagram is presented: click. "The models in Figure show the parthood relation exhaustively. The elements that form the leaves of these models — $a,b,c$ in Structure (a), for example — are atoms, and they are pairwise disjoint." It is said that (a) does not satisfty unique separation because it has "too many parts" and (b) and (c) do not satisfy unique separation because they have "too few parts".

Second question: For (a), how do I see that $a\oplus b = d$ (and $a\oplus c = d$)? This must follow from the picture somehow, I might be misunderstanding how to read it. And similarly, for (b) and (c), why is it the case that there is no $x$ such that $\neg (x\circ b)$ and $b\oplus x = c$?

Update: If I may add a third question, how exactly distibutivity is used in full lattices to exclude cases like (a), (b), (c)? "The role that unique separation plays in eliminating Structures (a), (b), and (c) is played by distributivity in full lattices, that is, lattices with a bottom element."

Answer 1

The editors of the book The Cambridge Handbook of Formal Semantics remark in the preface that

This handbook is intended for everyone interested in the understanding of meaning. It presents a broad view of the semantics and logic of natural language and, as a helpful tool, of the logical languages employed.

So, we do not expect much exactness in formal details. No doubt, the authors could have been more attentive to giving a systematic exposition without such leaps.

We have

• $x$ and $y$ overlap: $$x\circ y\overset{def}{=}\exists z(z\leq x\wedge z\leq y)$$
• "$x$ is a sum of (the things $z$ in) $P$": $$sum(x, P)\overset{def}{=}\forall y(y\circ x\leftrightarrow\exists z(y\circ z\wedge P(z)))$$
• $z$ is composed exactly of $x$ and $y$ ('$\iota$' denotes the logical constant, definite description operator; originally, the symbol should be inverted iota): $$x\oplus y\overset{def}{=}\iota z\,sum(z, \{x, y\})$$

Working out our way backwards, renaming variables duly to avoid collisions, we can express the binary sum operation in terms of improper (reflexive) parthood, it would not be quite conspicuous to make out what it states, though.

For the order-theoretic perspective, the authors employs reflexive (improper) parthood, $\leq$, as the primitive relation. They could have chosen irreflexive (proper) parthood, too. It is up to the discipline applied whether something of substance turns on either choice by possible consequences. In metaphysics, I think, it does, and refer those interested to Stephen Kearns' paper Can a Thing Be Part of Itself? (free to read online).

As for the lattice-theoretic (algebraic) perspective, the alternative parthood relation, $<$, is taken as the primitive relation. Since, in this case, a part $x$, there should be a unique corresponding part $y$ that completes $x$ to the whole, the authors appeal to the principle of unique separation (as indicated by the quantifier $\exists !$) which is conceptually connected to the axiom schema of separation (Aussonderungsaxiom) in set theory. Accordingly, the lattice

fails in unique separation, for, as the binary summations $a\oplus b=d$ and $a\oplus c=d$ show, the parts do not have unique complementary parts.

In the lattice

the part $c$ comprises only $b$ lacking a complementary part (likewise, the lattice (b)).

I shall illustrate how the distributive law in set-theoretic terms plays the role of unique separation over the lattice (a) above. By distributivity, we should have

$$a\cup (b\cap c) = (a\cup b)\cap (a\cup c)$$

On the left-hand side, we get

$$a\cup (b\cap c) = a\cup\emptyset = a$$

that is, the parts do not "meet". On the right-hand side, we get

$$(a\cup b)\cap (a\cup c) = d\cap d = d$$

Hence, distributivity fails; generally, in cases unique separation fails.

For a nice overview of mereology addressing also formal aspects, I recommend Achille Varzi 's article Mereology.