Wolfgang Schwarz

Atomism

Ordinary objects - persons, planets, rivers and tables - are unextended atoms. They occupy only one point of space at only one time at only one world.

At first sight, this might sound absurd. Don't ordinary things obviously exist at many different places, times and worlds? Isn't the Yangtze clean in Geladandong and dirty in Shanghai? Wasn't it clean in Shanghai in 1500? And isn't it clean in Shanghai now at some other possible world?

Fortunately, Atomism need not deny any of this. For even though the Yangtze is an unextended atom that strictly speaking only occupies a single point, it has many counterparts at other points. And these counterparts make all those statements true.

Let me illustrate how this works for a new language, L_a. Besides the usual logical machinery (quantifiers, variables, propositional operators), L_a has individual constants ("Yangtze", "Humphrey", etc.); predicates ("is an atom", "is dirty", etc.); and some index-shifting operators (on which more below).

All individual constants of L_a denote unextended atoms, and all predicates express properties of atoms. Thus "Yangtze is dirty" is true iff the atom denoted by "Yangtze" has the atom-property expressed by "is dirty". This is an extrinsic property: whether an atom satisfies "is dirty" depends on how it is surrounded by other atoms. Roughly, an atom satisfies "is dirty" iff it is part of something that's dirty. "is an atom", by contrast, expresses an intrinsic property, the property shared by all and only the atoms.

Index-shifting operators of L_a are "at world w", "at time t" and "at place p". It is here that we need counterparts. Suppose the atom denoted by "Yangtze", let's call it J, is located somewhere in Shanghai and satisfies "is dirty". Now if "p" denotes some place in Geladandong, "at [place] p, Yangtze is clean" is true in L_a, because J has a counterpart at p that satisfies "is clean". More precisely, the truth-conditions for utterances of L_a-sentences can be given as follows:

An utterance of an L_a-sentence is true iff the sentence is true relative to the place, time and world of the utterance.

An L_a-sentence A is true relative to a place p, a time t and a world w iff

• A has the form "B₁ and B₂" and both B₁ and B₂ are true relative to p, t and w;
• A has the form "not B" and B is not true relative to p, t and w;
• A has the form "for all x B(x)" and B(n) is true relative to p, t and w in all extensions of L_a in which the the new constant n denotes some atom located at p, t and w;
• A has the form "at place p', B" and B is true relative to p' (the place denoted by "p' "), t and w; likewise for "at time t' " and "at world w' ";
• A has the form "n is F" and there is an object denoted by "n" and this object has a counterpart located at p, t and w that satisfies "is F"; likewise for multi-place predicates.

Generalized index-shifting operators such as "possibly", "always" and "somewhere" are easily added. For instance, we can say that "possibly B" is true relative to p, t and w iff for some w', B is true relative to p, t and w'.

Counterparts can also be used to specify the semantics of certain extrinsic L_a-predicates. Thus an atom x satisfies "is round" iff the fusion of the x-counterparts located at the same time and place as x is round.

One more complication. Note that unembedded quantifiers in L_a are always restricted to things at the place, time and world of the utterance. This is reflected by the semantics of individual constants: in ordinary context, they denote atoms located at the place, time and world of the utterance -- or they do not denote at all. So in ordinary contexts, "Yangtze exists" is true only if uttered somewhere inside the Yangtze, where "Yangtze" denotes some Yangtze atom located at the place, time and world of the utterance. Hence in L_a, "Yangtze" behaves much like an indexical: its denotation systematically varies with the context of utterance. Just to give it a name, call the (partial) function from contexts to atoms determined by this variation the A-intension ("A" for Atomism, if you will) of "Yangtze"-

But L_a -- this is the complication -- also provides the means to speak from a 'God's eye' perspective. Then the quantifiers are not restricted to the here, now and actual, so that "there is" becomes roughly synonymous with "at some place, time and world, there is". In such a context, sentences with names for things that exist elsewhere can also be true. In this case, it is highly indeterminate which atom the name denotes. For instance, if I now say (in L_a, and in 'God's eye mode'), "Yangtze exists", my utterance of "Yangtze" is indeterminate between infinitely many atoms at various places, times and worlds, namely between all the atoms in the range of the A-intension of "Yangtze".

In between ordinary, strict mode and God's eye mode, we may allow further modes, where quantifiers are restricted, say, to actually existing or present things, and names are indeterminate only between those atoms in the relevant A-intension that exist actually and at present.

This completes my sketch of L_a. Atomism now says that English itself works pretty much like L_a. That is, the semantics I offered for L_a is a largely correct, albeit simplified, semantics for English. Persons, planets and rivers really are unextended atoms; they exist at different places, times and worlds only in virtue of their equally unextended counterparts. In suitable contexts, residents of Geladandong can truly say "the Yangtze is clean" and residents of Shanghai "the Yangtze is dirty" because "Yangtze" denotes different objects in the two cases. (More precisely, it is both times indeterminate what exactly "Yangtze" denotes, but the resolutions of the indeterminacy, the range of objects between which "Yangtze" is indeterminate, are disjoint in the two cases.)