## Tuesday, 17 July 2012

### A List of Achievements of Analytic Metaphysics

Somewhat tongue in cheek. Motivated by some of the recent polemics against metaphysics.

A List of Achievements of Analytic Metaphysics

1. Leibniz’s Principle of Identity of Indiscernibles.
2. Theory of continuous quantities, from Leibniz to Robinson.
3. Frege’s analysis of cardinality.
4. Abstraction and abstraction principles (Frege, Dedekind).
5. Invention of quantification theory; predication; what variables are (Frege).
6. Existence not a predicate, rather a quantifer (Frege).
7. Concepts as functions (Frege).
8. Theory of infinity (Bolzano, Cantor).
9. Mereology (Lesniewski, et al.).
10. Theory of relations (Russell).
11. Non-classical logics.
12. Incompleteness of formal systems (Gödel).
13. Concept of a computable function (Gödel, Turing, Church, et al).
14. Rotating solutions of Einstein's equations with CTCs (Gödel).
15. Tarski’s theory of truth; object language/metalanguage; undefinability theorem.
16. Kripke models; possible worlds anaylsis (Kripke, Lewis, et al.).
17. Kripke’s fixed-point theory of truth; grounding.
18. Formal semantics & pragmatics.
20. Supervenience (Kim, et al.)
21. Representation theorems; applicability of analysis.
22. Field’s theory of applicability of mathematics; conservation theorems.
23. Properties of identity and indiscernibility.
24. Dependence?

Of course, many of the contributions to metaphysics listed here were made by individuals working in the intersection of mathematics, logic and philosophy.

1. I'm a bit confused why you would classify results like (5) and (15) as results in metaphysics when they seem so much more substantially results about language and/or logic. And is (1) 'analytic' in the relevant sense?

2. Slightly confused as well. Many of those results have something to do with metaphysics only if one believes in a (certain kind of) metaphysics. Very few of the contemporary critics of analytic metaphysics have anything against logic, maths, etc. (!).

3. Hi Colin, I'd not want to separate logic, language and metaphysics (particularly truth). I tend to think of metaphysics as (at least to a large degree) conceptual analysis or explication, so the overlap with work in logic & language is very significant.

I count work on truth as metaphysics, even if it's deflationary metaphysics; it's seems to be a metaphysical doctrine that there is no genuine property of being true, or that there is a truth predicate 'wahr', that exists solely for purpose P. (I think pretty much the opposite!) The thing is, it's metaphysics either way.

Yes, I want to classify Leibniz's PII as (successful) armchair analytic metaphysics. (Though Leibniz also has his theory of monads, that I can't quite make sense of.)

1. Hi Jeffrey,

To call something an achievement of analytic metaphysics is to suggest that: 1. the method used to arrive at the result is "analytic" in some sense and 2. that "analytic metaphysics" can appropriately take credit for having produced the result.

What textual justification is there for interpreting Leibniz as having arrived at PII through conceptual analysis?

(14) is a result in theoretical physics; if it counts as an achievement of analytic metaphysics, then what would prevent other results in theoretical physics from being similarly appropriated?

(8) and (12) are mathematical achievements by mathematicians who did not understand what they were doing as "analytic metaphysics." Thus, metaphysics cannot appropriately take credit for their achievements.

But you also said your list was somewhat tongue in cheek, so I'm not sure which of your assertions, if any, were not made seriously.

2. Hello Jamie, many thanks for the comments.

"But you also said your list was somewhat tongue in cheek, so I'm not sure which of your assertions, if any, were not made seriously."

All of them. I am deadly serious. There are many more.
I give this list in a tongue in cheek way merely as a polite way of responding to the demand that parts of logic, mathematics and the foundations of science be "discontinued".

4. Alexander Becker17 July 2012 at 17:45

This list seems to me to reflect one of the problems of much of current "metaphysics": it is split up in highly specialized fields of investigation which miss what used to be one of the marks of metaphysics - its capacity to give an overview, to integrate different areas of investigation. In this respect, one the most exciting and fruitful ideas of metaphysics from an analytic point of view is the idea of "logical form". Significantly, it is missing in your list.

5. Enzo, suppose we take analytic metaphysics to be the attempt to give analyses and explications of concepts such as existence, infinity, truth, relation, function, identity, individuation, property, vagueness, supervenience, parts, possibility, structure, counterfactual dependence, dependence, etc. This normally uses (often develops: e.g., Frege, Cantor, Russell, Tarski, Gödel) methods of logic and mathematics (e.g., Kripke used the theory of inductive definitions and monotone operators).

So, as an example, I've written on the topics of truth & deflationism, and identity & indiscernibility.

But is that logic or metaphysics? I'd say they're both.

6. Russell(&V Whitehead?) famously simpified Leibniz' theory of identity in PM. We may take a=b as b has all the properties of a. I.e. one does not need: b has all and only the properties of a.

7. Thanks, Alexander - I agree. (I missed vagueness too, and no doubt other things that count as developments and even achievements in analytic metaphysics.) Logical form is something I'm quite interested in too.

Many of the topics listed are pretty general ones in metaphysics, like "what is a function?", "what is a relation?", "is existence a predicate?", "what is a structure?", "how do we give a definition of truth?".

8. Frode, many thanks. Yes, I know. See p.1 of this RSL article, "Identity and Indiscernibility",

1. But this does not attribute the simplification to Russell or Whitehead.

2. Yes, you're right, I didn't.
I meant I know the simplification, which I why I formulated it in the simplified manner. I'd thought it might be well-known when I wrote that article in 2005. It's mentioned in Boolos & Jeffrey somewhere, and there it's attributed to R&W.

3. Definition 13.01 of PM.

4. Thanks - from memory, I think it's mentioned in the chapter "Second-order logic" in B&J, 3rd edition, 1989.

The RW formula AX(Xy -> Xz) defines identity in any full second-order model; but in a Henkin model, and it doesn't not always succeed in defining identity (a Henkin model needn't contain all units sets).

9. These are all towering, highly meaningful achievements. Ergo, by definition, they have nothing to do with metaphysics.

10. Sorry to parade my ignorance, but can you elaborate on 24?

11. There's older work (e.g., Yablo 1985; and going back to Kripke and Herzberger), and then some recent, very interesting, work, on notions of dependence and grounding. Some by Hannes Leitgeb.
Hannes' original work on this concerns a notion of semantic dependence, e.g., this important 2005 paper, "What Truth Depends On",

http://www.jstor.org/discover/10.2307/30226836?uid=3737864&uid=2129&uid=2&uid=70&uid=4&sid=56313725323

but the method is more widely applicable (it can be applied to set theoretic membership, necessity and identity, and Hannes has done this elsewhere).
Dennis Bonnay and Toby Meadows have worked on this too (no doubt many others ...). Here are some notes on this topics by Jo ̈nne Speck (who was visiting MCMP a few days ago),

http://www.bbk.ac.uk/philosophy/our-research/ppp/leitgebhandout.pdf

I don't know this work in sufficient detail to make a confident assessment, hence the question mark. But I think this topic will develop in the next few years.

12. Thanks Jeffery, I will check those out.

13. One worry would be that if (21) counts; why doesn't any mathematical result?

14. Topics (21) and (22) are very close, concerning the (metaphysical) problem of the applicability of mathematics. By "representation theorems" in (21), I mean the theory of how measurement scales (normally into real numbers) arise from qualitative characterizations of a system of things. Field's approach in Science without Numbers invokes representation theorems, adding the conservation theorem bit to justify the instrumentalism.
So, if (22) counts, then (21) ought to too; but maybe they should be squeezed into one item.

15. Jeffrey,

A quick question: could you say a bit more about (9)? In what sense is Mereology an "achievement" of analytic Metaphysics? There have been theories of Mereology long before Analytic Metaphysics came on the scene, Aristotle's Mereology being obvious example. Could you elaborate more on this?

16. Andres, thanks for the question.
"There have been theories of Mereology long before Analytic Metaphysics came on the scene, Aristotle's Mereology being obvious example."

I agree, yes, mereology goes back to Plato and Aristotle, and forward through mediaeval metaphysics. There is a nice book on ancient mereology by Verity Harte ("Plato on Parts and Wholes"). However, this stuff strikes me as lacking in precision.
So, while I appreciate its history, I take the first major intellectual achievement here to be the formal working out of it by an analytic philosopher, Lesniewski, who hoped it would provide a surrogate for set theory. Lesniewski's student, Tarski worked on this area too. In this context, being a part of is intimately related to the subset relation on sets, to Boolean algebras in general, and to topology (that is, a certain collection of subsets of a base set X, satisfying closure conditions).
There are various applications of this sort of thing: Leonard, Goodman, Quine, Lewis and others.

As an application, consider the finite sequences (words) over an alphabet {'a',‘b‘}, and consider the string token

aaaabbbbbaaaabbbbbaaaaa

I.e., I mean some token itself; imagine it is printed in ink. Call this t.
Then t is a token of the abstract type T

('a','a','a','a','b','b','b','b','b','a','a','a','a','b','b','b','b','b','a','a','a','a','a')

This type T is a word over {'a‘,‘b‘}. I.e., a finite sequence - a function f: {0, 1, ..., 23} -> {'a',‘b‘}.

No one has worked out the precise details of the analysis of "t is token of type T" (but Linda Wetzel's "Types and Tokens" (2009) is very good). But it seems to be right to start with the suggestion that t is a token of T because the mereological structure of the token (relative to the properties of being a token of 'a' and being a token of 'b') is isomorphic to the sequence f.

So, one can apply mereology to account for how mereologically structured expression tokens are instances of expression types.

1. You say:

"t is a token of T because the mereological structure of the token".

But I don't understand what is specifically mereological about this. It seems to be a general, if trivial, fact about the free concatenation algebra with two generators. And if you really wanted to reduce it to something else, the set-theoretic account seems just as good, perhaps better, than the mereological one.

2. You mean the type T (yes, it belongs free concatenation algebra over 'A' and 'B' ).
But I mean the token - i.e., the physical thing. Isn't the relation of its subtokens to the token parthood? So, e.g., take a token of 'ABA', say inscribed on a whiteboard. The token has parts which are tokens of 'A' and 'B': so the token t has three non-non-overlapping parts u1, u2, and u3, such that t is the fusion of all three, and u1 and u2 are tokens of 'A', and u2 is a token of 'B' (u2 also lies physically between u1 and u3.)

But what is the set-theoretic account? Can one do this without referring to parts of the token? (I.e., the parts of the token that are tokens of primitive symbols.) I guess the question is whether one can define "t is a token of T" without using the notion "part of".

(I spent a bit of time on creating an RSS feed.)

3. PS - I now think the infinitary sentence I mentioned in the eliminating relata post below doesn't work, but I'm not sure why.

4. Why do you think that?

5. Aldo, on the PS?
After you mentioned Scott's isomorphism theorem, I had a look at the details: it's limited to countable structures in the language $L_{\omega_1, \omega}$, with finitely long sequences of quantifiers. And Bell's article on infinitary logic on SEP has the same restrictions, and he says, "we mention one further result which generalizes nicely to $L_{\omega_1,\omega}$ but to no other infinitary language", adding "the restriction to countable structures is essential because countability cannot in general be expressed by an $L_{\omega_1,\omega\}$-sentence."

So that suggests that one doesn't get the general result I want, a sort of generalized Scott sentence - unless one introduces second-order variables. (Which is ok for my purposes.) It sounds reasonable, but I'm still not sure why though.

Or do you know if we can get a Scott-like sentence of this kind I'm after without second-order variables?

6. No, I don't know right off. But I see that Bell's result applies to $L_{\omega_1,\omega}$, which would rule out infinite strings of quantifiers (which you make use of, if I remember correctly).

7. Yes, I was thinking of saying when I wrote the post that I probably needed some set theory to show that the relevant infinitary formulas are well-defined.
What makes me thinks it's wrong (or, at least, not quite right) is that the basic construction I give is quite simple; so if it were right, it would be an immediate result in infinitary logic folklore. Given that it doesn't seem to be, there must be something that goes wrong - with the set of inequations (i.e., the literals $v_i \neq v_j$, with $i \neq j$) when A is an uncountable structure.

So, I'm now guessing that if $A$ is an uncountable set with $|A| = \kappa$, and we have a constant $\underline{a}$ for each $a \in A$, then given the set $\{\underline{a} \neq \underline{b} \mid a,b \in A \& a \neq b\}$ of inequations over distinct elements of $A$, then the infinitary conjunction $\bigwedge X$ along with the quantified infinitary disjunction $\forall x \bigvee_{a \in A}(x = \underline{a})$ doesn't fix the cardinality $\kappa$ uniquely.

17. In the Prolegomena, Kant said something like we are just as likely to stop doing metaphysics as we are likely to stop breathing. Interpret this in the most charitable manner as possible, and I think it expresses an important truth. Even if metaphysics---whatever it is---hasn't been put on the "secure path of a science" or made any achievements whatsoever, there is still the question of whether or not it is something we even could abandon. I'm imagining an idea here that echoes Quine on the indispensability of quantification over mathematical objects to [or for] our best scientific theories. If we think that certain philosophical domains [among others domains] are important or worth doing or lead to achievements or are needed for certain sciences, and if---a big if---metaphysical theories or metaphysical theorizing is indispensable to our best philosophical theories [or whatever], then won't we get the result that metaphysics cannot be abandoned without abandoning the whole lot [or the whole lot that it is indispensable to]? I think this is important because even if we do think we ought to abandon metaphysics, that is no argument that we can even achieve that end.

18. Most of these achievements is better classified as logic, instead metaphysics. Some were made in a metaphysical environment or discussion or motivation and in this sense they are also metaphysic. However, we do not need metaphysics in order to see the logical relevance of these achievements. Metaphysics works sometimes as motivation, but the logical achievements subsist by themselves.

19. I would definitely also include the vast area of nonmonotonic reasoning/logic which has been investigated in applied logic and knowledge representation for 35 years. It may be called non-classical, but it is rather different from what is usually running under this heading (multivalued, paraconsistent, relevance, ... logics.). It includes inference notions where the addition of new premises may force the withdrawal of previous conclusions. It is a characteristic of plausible or autoepistemic inference. The loss of monotonicity can be partly compensated by imposing other principles, like cumulativity.

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