Posts on: Properties
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.
Gómez Sánchez (2023) asks an important and, in my view, unsolved question: what kinds of properties may figure in the laws of "special science" (chemistry, genetics, etc.)?
For the most part, the patterns captured in special science laws are not entailed by the fundamental laws of physics, nor by the intrinsic powers and dispositions of the relevant objects. Some kind of best-systems account looks appealing: the Weber-Fechner law, the laws of population dynamics, the laws of folk psychology etc. are useful summaries of pervasive and robust regularities in their respective domains. They are the "best systematisation" of the relevant facts, in terms of desiderata like simplicity and strength.
I've long been puzzled by the nature of quantities, but I've never really
followed the literature. Now I've read Jo Wolff's splendid monograph on the
topic. I'm still puzzled, but at least my puzzlement is a little better
informed.
The basic puzzle is simple and probably familiar. On the one hand, being 2m high
or having a mass of 2kg appear to be paradigm examples of simple, intrinsic
properties. On the other hand, these properties seem to stand in mysterious
relationships to other properties of the same kind. First, there's an exclusion
relationship: nothing can have a mass of both 2kg and 3kg. Second, there are
non-arbitrary orderings and numerical comparisons: one thing may be four times
as massive as another; the mass difference between x and y may be twice that
between z and w. If 2kg and 8kg are primitive properties, why couldn't an object
have both, and where does their quasi-numerical order and structure come from?
It is tempting to think that there is nothing more to physical
quantities than their nomic role: that to have a certain mass just is
to behave in such-and-such a way under such-and-such conditions.
But it is also tempting to think that the "Galilean equivalence" of
inertial mass and gravitational mass is a true identity; i.e.,
that
Inertial mass = gravitational mass.
However, the role associated with "inertial mass" is completely
different from the role associated with "gravitational mass". So if
having such-and-such inertial mass is having the relevant
dispositions associated with "inertial mass", and likewise for
gravitational mass, then the Galilean equivalence could not be an
identity. It would rather state an empirical law, according to which
two distinct quantities always have the same value.
In "Ramseyan
Humility", Lewis argues for a thesis he calls "Humility". He never
quite says what that thesis is, but its core seems to be the claim
that our evidence can never rule out worlds that differ from actuality
merely by swapping around fundamental properties. Lewis's argument, on
pp.205-207, is perhaps the most puzzling argument he ever gave.
Lewis begins with some terminology.
Sometimes, a property A entails a property B while B does not entail A, and yet there seems to be no interesting property C that is the remainder of A minus B. For instance, being red entails being coloured, but there is no interesting property C such that being red could be analysed as: being coloured & being C. In particular, there seems to be no such property C that doesn't itself entail being coloured.
This fact has occasionally been used to justify the claim that various other properties A entail a property B without being decomposable into B and something else. I will try to raise doubts about a certain class of such cases.
As a principle of plentitude, Recombination for Individuals is far too weak. If there happens to be nothing that is both red and dodecagonal, the recombination principle for individuals gives us no world where anything is. Likewise, if it happens that no red thing is on top of a blue thing, the principle gives us no world where this is different. But combinatorial reasoning seems to give us such worlds.
Let F be a fundamental property, understood as a maximal class of possible things that are perfectly similar in one respect. (This is one of Lewis's four proposed definitions of fundamental properties, and I think the best one.) And suppose I have F. What would it take to know that I have F?
Given that F is some class { Wo, Fred, ... }, and given that having F means being a member of F, it might seem puzzling how I can be ignorant about whether or not I'm F: how could I fail to know that I am a member of { I, Fred, ... }? But here we are substituting corefering expressions in a (hyper)intensional context, which is illegitimate. If I knew that F = { I, Fred, ... }, then I probably ought to know that I am F. So if I don't know that I am F, that's because I don't know that F = { I, Fred, ... }.
Some properties are inherited from wholes to their parts: if x is (completely) made of steel, then its parts are also (completely) made of steel; if x is in the top drawer, then its parts are also in the top drawer. Other properties are upwards inherited from parts to wholes: if a part of x contains steel, then x contains steel; if a part of x touches the ground, then x touches the ground. Yet other properties are not inherited either way: if x is a hand, then x usually has non-hands as parts and is part of non-hands.
I believe that there is such a property as being two meters away. -- Not two meters away from me, or from somebody else, or two meters away from something or other. Just two meters away.
Admittedly, there is a sense in which something can be two meters away only relative to some point of reference. But compare properties like being empty and being bent. There is a sense in which, strictly speaking, persisting things can be empty or bent only relative to a time: this cup here is empty at the present time while it was full 5 minutes ago. Likewise, at least prima facie many things are empty or bent only relative to worlds: the cup is empty at the actual world, but full at other possible worlds. That's why properties are often modeled as something like functions from worlds and times to sets of objects.
Is it metaphysically necessary that like charges repel? One might think so: one might think that "charge" is partly defined by its theoretical role, so that this claim comes out analytic. Or one might think that science reveals to us the essence of properties, and that it is part of this essence of charge that like charges repel.
If that law about charges is metaphysically necessary, one might suspect that quite generally, nomological necessity coincides with metaphysical necessity (though see below for an argument against this suspicion):
In their contributions to Lewisian Themes, Rae Langton and Jonathan Schaffer both argue that quidditism -- the claim that possible worlds may differ only in which intrinsic properties play which causal/nomological roles -- does not entail skepticism about intrinsic natures because standard replies to skepticism about the external world carry over to skepticism about intrinsic natures.
But it seems to me that there is an important difference: if quidditism is true, we not only lack knowledge about intrinsic natures, but also any beliefs about them.
The fundamental properties provide a minimal basis for all intrinsic qualities of things. That is, whenever two things are not perfect qualitative duplicates, they differ in the distribution of fundamental properties over their parts; whenever two things do not differ by that distribution, they are perfect qualitative duplicates. It follows that all fundamental properties are intrinsic. But not all intrinsic properties are fundamental: the fundamental properties provide a minimal basis for all qualitaties. Hence there is no fundamental property of having a mass of either 1g or 2g, because instantiation of that property is already determined by the distribution of mass 1g and mass 2g. For the same reason, there is no fundamental property of being the fusion of a round thing and a distinct rectangular thing. By and large, fundamental properties are never logically complex (like A or B) and never structural (determined by the distribution of properties over the parts of their instances).
Roughly, the principle of recombination says that anything can coexist and
fail to coexist with anything else. But that's too strong: things do
have essential extrinsic properties; if Kripke's origin is essential
to Kripke, Kripke cannot fail to coexist with his ancestors. However, a
perfect intrinsic duplicate of Kripke could fail to coexist with Kripke's ancestors.
So less roughly, the principle of recombination goes somehow like this:
For any things in any possible world there is a world
which contains any number of perfect intrinsic duplicates of all those
things and nothing else (i.e. nothing distinct from all these
duplicates).
What is a perfect intrinsic duplicate? Something that has exactly
the same intrinsic properties as the original. What
is an intrinsic property? A property that belongs to objects
independently of what exists and goes on around them. The
instantiation of an intrinsic property in some region of a world is
independent of the instantiation of intrinsic properties in other
regions.
I believe that the so-called problems of intrinsic change and accidental intrinsic properties are real problems. But I believe that their names are misleading, and that they have nothing to do with whether or not we construe properties as sets of things or as functions from worlds and times to sets of things.
Suppose we do the latter, and we also endorse counterpart theory and temporal parts theory. The property of being bent is a function that maps world-time pairs to sets of things. These things are temporal parts of world-bound individuals, ordinary fusions of particle segments, just like us, except that they are smaller along the time axis and all bent. This is a perfectly reasonable and common-sensical view, I believe (but of course I'm biased), and I don't think Lewis has any reason to reject it as turning properties into relations. There is after all a simple equivalence between being bent construed as a function and being bent construed as a Lewisian set: the set is the union of the range of the function; the function indexes all members of the set by their world and time.
For many things, there is no set that contains just those things. There is no set of all sets, no set of all non-self-members, no set of all non-cats, no set of all things, no set of pairs (x,y) such that x is identical to y, no set of (x,y) with x part of y, no set of (x,y) with x member of y.
If Lewis is right and there are proper-class many possibilia, there is also no set of possible philosophers, no set of possible dragons and no set of possible red things. However, if Lewis is right and there are proper classes, there will be proper classes of all these things. But there will still not be a class of all classes, a class of all non-self-members, a class of all non-cats, etc.
This is a follow-up to yesterday's entry.
Andy Egan argues that functions from worlds and times to sets of things are ideally suited as semantic values of predicates, even better than mere sets of things.
I agree, and so would Lewis. In fact, Lewis would say that functions from worlds and times are still too simple to do the job of semantic values. There are more intensional operators in our language than temporal and modal operators. Among others, there are also spatial operators and precision operators ("strictly speaking"). So our semantic values for predicates should be functions from a
world, a time, a place, a precision standard and various other 'index coordinates' to sets of objects. This is more or less what Lewis assigns to common nouns in "General Semantics" (see in particular §III). Other predicates like "is green" that do not belong to any basic syntactic category get assigned more complicated semantic values: functions from functions from indices to things to functions from indices to truth values. In later papers, Lewis argues that we may need several of the world and time coordinates and, more
importantly, a further mapping that accounts for context-dependence
(and to deliver the kind of truth-conditions needed in his theory of
linguistic conventions). Thus for predicates, we get something like a
function from centered worlds to functions from functions from possibly several worlds, times, places, precision standards, etc. to functions from such worlds, times etc. to truth values. (Alternatively, if we go for the 'moderate external strategy' (Plurality) and reserve "semantic value" for 'simple, but variable semantic values' ("Index, Context and Content"), we can say that the semantic value of a predicate in a given context is the value of the function just mentioned for that context.)
Andy Egan, in "Second-Order Predication and the Metaphysics of
Properties", argues that there is a bug in Lewis' theory of
properties which can be fixed by identifying properties
not just with sets but with functions from worlds (and times) to
sets. I disagree: there is no bug. But there are some interesting
questions about Lewisian properties nearby.
Here's the alleged bug. Consider the second-order property being
somebody's favourite property. This property belongs to
Green. So on Lewis' account, Green is a member of the
set being somebody's favourite property. But at another
possible world, Green is nobody's favourite property. So it is not a member of that set. Contradiction. In the parallel case of accidental properties of individuals, Lewis resorts to counterpart theory: If Graham Greene is a writer in our world and not in another world, that's not because Greene both is and isn't a member of the set writer, but because Greene is a member while one of his counterparts isn't. However, this solution doesn't work for Green because properties don't have counterparts.
Warning: another pointless exercise in conceptual geography.
Can intrinsic properties have their causal/nomic role essentially? It seems not. Suppose something x is P. If P essentially occupies a
certain causal role, say being such that all its instances attract one
another, we can infer from x's being P that either there are no other
P-things in x's surrounding or x and the other things will (ceteris
paribus) move towards one another. But if we can infer from x's being
P what happens in x's surrounding, P cannot be intrinsic. Being
intrinsic means belonging to things independently of what goes on
in their neighbourhood.
Fundamental (or 'perfectly natural') properties are properties on whose distribution in a world all qualitative truths about that world supervene. That is, whenever two worlds are not perfect qualitative duplicates, they differ in the distribution of fundamental properties.
This is not the only job discription for fundamental properties. If it were, far too many classes of properties could play that role. For instance, all qualtiative truths trivially supervene on the distribution of all properties, or on the distribution of all intrinisic properties, or (for what it's worth) on the distribution of all extrinsic properties. (That's because no two things, whether duplicates or not, ever agree in all extrinsic properties.)
A structural property is a property that belongs to things in virtue of their constituents' properties and interrelations. For instance, the property being a methane molecule necessarily belongs to all and only things consisting of suitably connected carbon and hydrogen atoms.
There is two-way dependence: Necessarily, if something instantiates a structural property, then it has proper parts that instantiate certain other properties; conversely, if the proper parts of a thing instantiate those other properties then, necessarily, the thing itself instantiates the structural property.
Sometimes the best argument for a certain assumption is that it proves fruitful in various theoretical contexts: Why believe in a plurality of worlds? Because the hypothesis is serviceable in semantics, decision theory, theories of intentional content, the interpretation of modalities, the definition of supervenience, etc. -- and that is a reason to believe that it is true. Another example, again by Lewis, is the argument for universals, or at least for a fundamental distinction between natural and unnatural properties: the assumption is serviceable to account for objective similarity, the determinacy of meaning and translation, the interpretation of some quantified sentences, the analysis of natural laws, etc. Similar arguments can be put forward for the existence of temporal parts, states of affairs, events and numbers.
These arguments presuppose that it is really the very same assumption, rather than a diverse family of similar sounding assumptions, that does all the work it is supposed to do. The case for numbers would be much worse if lots of different arithmetics were 'indispensable' in different branches of science. The problem is quite obvious for events: the events employed in relativity theory can hardly do as the events used in Davidsonian interpretations of English adverbs.
Re dense worlds, Dave Chalmers asks in what sense worlds that differ only in which intrinsic properties play which roles are indistinguishable. That's a very good question, and I'm afraid I don't have a good answer. He notes that those worlds differ in lots of respects, including their laws and quite probably the perceptions of their inhabitants.
What I want to say is that the worlds are somehow 'structurally alike', or 'isomorphic'. But that's hard to cash out. Is every Ramsey sentence that is true of one of them also true of the others? Then I would first have to restrict the 'old' terms of the Ramsey sentence. But that's a minor problem. What's worse is that this doesn't take care of more complicated rearrangements, where different parts of roles are played by different properties. Here the quantifiers of the Ramsey sentence would have to range over very gerrymandered (though intrinsic) properties. And given that gerrymandered properties are generally supposed to be causally inefficacious this is dubious. And finally, even if the Ramsey sentence account would work, I would still have to say why worlds that cannot be distinguished by Ramsey-sentences (or are otherwise 'structurally alike') are in any reasonable sense indistinguishable.
My attempts to get a copy of 'Ramseyan Humility' were unsuccessful, so I searched the web in the hope that somewhere somebody might have said something about what Lewis says in that paper. This is how I came across Paul Mainwood's BPhil Thesis Properties, Permutations and Physics (PDF). It's a very good thesis and contains (in section 4) an extended discussion of some of the problems I'm struggling with.
Assume that all facts in our world are determined by the distribution of basic intrinsic properties at space-time points. Some of the space-time points in our world might be empty, that is, no basic intrinsic property might be instantiated there (either by some particle or by the point itself). If so, consider another world which is exactly like ours except that at all these empty points some basic intrinsic property is instantiated (say, the basic intrinsic property that plays the role of a certain mass in our world -- "some mass", for short) which however has no effect at all on what goes on in the world. (So if that property is some mass, the laws of nature at this world must be different from the laws at our world since our laws don't accept masses that have no effects.) By the definition of "intrinsic" and a rather weak principle of recombination, such a "dense" world is possible. And obviously, it is in principle indistinguishable from our world.
One of my problems with Lewis is that he published so little on issues where he thought he had nothing new to say. Sometimes it's tricky to figure out what his views on these issues might have been. Knowing people who knew him personally, or having access to some of his communications would probably help. Have there already been efforts to collect his letters, or even to make some of his unpublished writings available somehow? (If this is really Lewis' computer, the data on it definitely should be backed up soon before it completely turns to dust...)
Here at Humboldt University, there's a reading group about analytic
philosophy (Sam already mentioned it). The flyer
advertising this group describes analytic philosophy as a sort of new and
fascinating kind of philosophy characterised by its perspicuity and
ignorance of philosophical tradition. The funny thing is that the
organisers of the reading group decided that we'll be discussing David
Wiggins' Sameness and Substance Renewed. I don't want to know
how much Hegel one has to read to find Wiggins perspicuous (and
ignorant of philosophical tradition).
This is a problem that cropped up several times in my thesis on Lewis,
but which I never seriously discussed.
Lewis argues, or rather, stipulates, that all fundamental ("perfectly
natural") properties are intrinsic. I agree that fundamental extrinsic
properties would be strange. For if a thing x's being F depends on the
existence and the properties of other things, it seems that F-hood should
be reducible to intrinsic properties (and relations) of all the things
involved. Moreover, fundamental properties are supposed to be the basis for
intrinsic similarity between things, and they could hardly be if they were themselves extrinsic.
What, in general, does it mean that something A satisfies a predicate 'F'?
Traditionally, there are three candidates:
1) 'A is F' means that A is F. That' all. Simple predications can't be analysed.
2) 'A is F' means that A instantiates the property F. Except in some special cases,
in particular the case where 'F' is 'instantiates'.
3) 'A is F' always means that A instantiates the property F.
It is not entirely obvious how to locate Lewis here. In some places, when
discussing Armstrong's request for analyses (or truthmakers) for
predication, he sounds like he favours (1): "the statement that A has F
is true because A has F. It's so because it's so. It just is." ("A world
of truthmakers", p.219 in Papers)
Brian Weatherson tells me that Lewis does mention Goodman's 'New Riddle' as
a task for natural properties in "Meaning without use: Reply to Hawthorne".
Lewis says here that we should not be scared off by "Kripkenstein's
challenge (formerly Goodman's challenge)" to find a distinction between
natural and unnatural extrapolation (p.150 in Papers in Ethics and
Social Philosophy, similar remarks can be found in the introduction to
Papers in Metaphysics and Epistemology). So the first suggestion
is very probably right.
(Reading Brian's comments it now seems to me when I argued that natural
properties can't solve the New Riddle I've been confusing it with the Old
Riddle. All the New Riddle requires is an objective distinction between
good and bad extrapolations. That induction based on good extrapolations
might nevertheless yield systematically false predictions ("not work") is the
Old Riddle.)
I think these conditions match the dot-matrix test better than the ones I
proposed earlier. They are more complicated, but closer to the matrices
and not too unnatural:
A property F is natural to the extent that the following conditions are
satisfied, where (1), (3) and (5) weigh heavier than (2), (4) and (6).
1) The Fs resemble each other intrinsically.
2) The Fs resemble each other extrinsically.
3) Anything that exactly resembles an F intrinsically is itself F.
4) Anything that exactly resembles an F extrinsically is itself F.
5) There are few intrinsic F-gaps.
6) There are few extrinsic F-gaps.
Something y is an intrinsic (extrinsic) F-gap if it isn't F and there are
Fs x and z such that y intrinsically (extrinsically) resembles both of them
more closely than x intrinsically (extrinsically) resembles z.
RL satisfies all conditions except (3), whereas R only properly satisfies
(1).
In section 6 of "Redefining 'Intrinsic'" (Philosophy and Phenomenological
Research 62, 2001), Lewis introduces an interesting test for comparative
naturalness of properties. The test is based on two-dimensional dot-matrix
pictures, where distance along the horizontal dimension measures intrinsic
dissimilarity, and distance along the vertical dimension extrinsic
dissimilarity. Roughly (p.385), a natural property demarcates a regular
region in the dot-matrix. Less roughly (p.391), two aspects of the region
are important for naturalness: spread and scatter.
My logfiles indicate that people are more interested in silly
logic puzzles than in pointless remarks on footnotes in the
metaphysical writings of David Lewis. Let's see if I can get my readership
down to zero with this one.
Besides perfectly natural properties, Lewis also needs somewhat less
natural properties in his philosophy of language and elsewhere. What
determines how natural a property is? Lewis gives three different
answers, in four different places, none of them longer than two sentences.
David Lewis offers a lot of work for natural properties in his semantics,
his theory of mental content, materialism, supervenience, causation, laws
of nature, etc. Strikingly missing in this list (as opposed to the list of
Anthony Quinton, "Properties and Classes") is the solution of Goodman's New
Riddle of Induction. I don't know why Lewis never mentions this. Two
suggestions:
1) He thought it was just too obvious, and he disliked repeating arguments
of other philosophers (none of the items on Quinton's list occurs on
Lewis').
Brian Weatherson now says that 'the world exists' is exactly as natural as
'there is a G', where G applies to worlds that are exactly like this one.
I agree. But this only makes things worse, because the class G denotes
seems very natural: It contains our world and all its exact intrinsic
duplicates. Is this a gruesome gerrymander? We still need a
further restriction on best theories apart from naturalness.
Intuitively, some objects are more natural than others. For example, cats
are more natural than mereological fusions of cats and elephants. I think
that ultimately, naturalness of things should be definable in terms of
naturalness of the properties the things instantiate. I'm not quite sure
how exactly this is to be done, so for now I'll stick with the intuitive
notion of naturalness. Intuitively natural things are spatiotemporally
connected, constitute a causal unity, contrast with their surroundings,
etc. The world, that is, the mereological fusion of everything that exists
at any spacetime distance from us, does fairly well here: As far as I know,
it is perfectly connected, causally united (indeed, causally closed) and
contrasts clearly with everything outside of it (such as numbers or other
worlds, if such there be). Why then does Brian Weatherson think that the
world is gruesome?
I see two ways to exclude 'the world exists' as the best theory of
everything. The first is the one I already mentioned: to state that a good
theory must imply interesting truths a priori. The second is to
stipulate that a theory must not contain individual constants. I have some
sympathy with such a stipulation, though it may stipulate away haecceitism.
In her paper 'Logical
Parts', forthcoming in the december issue of Nous, L.A. Paul presents a nice
theory of objects according to which things are mereologically composed of
their properties. Here are a couple of potential problems.
First, the theory seems to conflict with Unrestricted Composition and
incompatible properties. For suppose that P and Q are incompatible
properties, like being square and being round. By Unrestricted
Composition, there is a fusion of P and Q (or, if you prefer, of P and Q
and Paul's red cup). This fusion has both P and Q as parts, hence, on
Paul's theory, it is both P and Q. But if P and Q are incompatible, nothing
can be both P and Q.
One of the many drawbacks of studying philosophy in Germany is that nobody kicks you when you spend years on your M.A. thesis. So hereby I publicly kick myself. As a consequence, this blog might become even more occupied with David Lewis than it already is.
To start this trend, here is a question about Lewis' functionalism. When a property P is multiply realizable, we cannot identify it with its realizers because then we would identify the realizers with each other, too. All we can do is locally identify P-in-k with its realizer in k, where k is a world or species or individual. Now what is P itself? In Lewis' papers on mind, he usually says that 'P' is systematically ambiguous or indeterminate, denoting the contextually salient realizer. (At least this is what I take him to say. He is not particularly explicit here.) The alternative would be to identify P with the diagonal property of being a P-realizer. The main reason why Lewis rejects this option (e.g. in 'Reduction of Mind') is that it is difficult to see how this diagonal property can occupy the causal role associated with 'P'. Difficult, but not impossible: In 'Finkish Dispositions' he proposes a solution, and consequently prefers identifying fragility not with the contextually salient realizer but rather with the diagonal property. Since then he apparently hasn't written anything more on the issue, so I would like to know if he has changed his mind on mental states (and heat, etc.) as well. (There may still be problems if the theoretical role is not entirely causal, so that even if the diagonal property can do the causal work, it might not be able to do the rest.) Or has he completely endorsed the third alternative – to simply leave the question unanswered: 'The folk well might have left this subtle ambiguity unresolved' ('Void and Object'). Can anyone help?
