Stephen Read’s `Unity of Fact’, section 4.2: Combinatorialsm

In the first half of section 4, Read has argued that modal realism fails.
However, there’s another way of saving the possible world approach to modality from circularity.
It’s David Armstrong’s Combinatorialism.
Read points out that combinatorialism presupposes atomic, i.e. logically independent proposition s, and turns to argue against them.
Certainly, and Read doesn’t even explicate this first step, allegedly atomic propositions do not contain any logical constants but are of the form R^n a_1, … a_n, that is, represent facts which consist of some n-placed relation and n particulars.
The problem is, however, that every property (and relation) can be determined further. Read’s example is red and its different shades crimson and scarlet.
Since nothing can be both crimson and scarlet (all over), the propositions `crimson(a)’ and `scarlet(a)’ aren’t independent, hence not atomic.
Further, the incompatibility of different ways of specification cannot be resolved by analysis but repeats at any `lower’ level
This must be so, since otherwise the analysis would have failed since it does not capture the essential differences, Read points out [p. 338].

Read considers how Armstrong claims to solve this problem. I’m sorry to say that I don’t quite get Armstrong’s solution, at least not from Read’s exposition. Perhaps I should have a look into the book (A Combinatorial Theory of Possibility). Anyway, Read dismisses it [p. 340], and so for the time being I shall turn to his diagnosis of why the combinatorialist variant of the possible-world approach to modality has failed as well.

As it was to be expected from the preceding sections, Read thinks that combinatorialism contradicts to the unity of fact. States of affairs cannot be taken apart, since it is only them obtaining which ensures their unity. At this point I cannot put it more concisely than Read himself:
`[...] once one has taken a fact apart, there is no way to put it back together again’ [p. 340].
Therefore, combinatorialism is flawed at its basis, or so Read argues.
The unity of fact excludes atoms of which non-actual states of affairs can be construed by recombination: `What combinations there are, are all there are’ [ibid.].

Now, in the very last paragraph of this section, Read takes a surprising move: He identifies Wittgenstein’s bipolarity account of truth and falsity as deficient. Two poles do not suffice to account for all the different facts which render a proposition true or false. The one proposition that a is crimson is made false by the fact that a is scarlet as well as by a’s being vermilion, ultramarine, cyan or of some other (shade of a) colour.

I’m curious now what Read is going to make out of this in his conclusion…

Read’s `Unity of Fact’, Section 4.1: `Modal Realism’

(This continues my reading of Stephen Read’s `The Unity of Fact’)
On the basis of the Wittgensteinian explanation of the unity of fact which he has just excavated,
Read now turns to criticize possible-worlds accounts of modality.
As a starting point, he considers that such accounts appear circular unless `possible world’ is explicated in non-modal terms.
Then, he turns to criticize two approaches of doing this the first of which is modal realism (henceforth `MR’). Read spends half a page explaining modal realism, I’ll skip this.

The first objection which Read raises is insisting on the actualist claim that `real’ and `actual’ are co-extensive, and everything (widest scope) falls under both (they’re `blanket terms’) [p. 335].
However, Read turns to specify his criticism to each of the two, and in fact very different modal realist approaches. On one hand, there’s MR which explains a’s possibly being F by a’s being F at some world, and therefore buys into trans-world-identity. On the other hand, there is David Lewis’ MR which rejects trans-world identity and instead endorses counterpart theory such that a is possibly F iff some counterpart of a is F.

Read’s objection against trans-world identity MR is: It implies non-actual, not-obtaining states of affairs whose unity remains obscure.
Assume that a and b don’t stand in the relation R. There’s no fact aRb. However, it is possible that aRb.
According to trans-world identity, therefore, there’s a state of affairs aRb, however, not actually but merely at some possible world. These non-actual state of affairs, however, cannot be united by R since a and b don’t stand in the relation R.

Counterpart MR avoids these difficulties. Not a and b, but their counterparts a’ and b’ are related by R. The state of affairs a’Rb’ obtains, and the unity of this fact can be explained by R.

However, Read raises another objection which, however, I have problems to understand. Let’s see how far I get.
Counterpart MR requires an account of how an individual is represented by its counterparts.
Here, it is this representation which is supposed to constitute the non-actual fact.
David Lewis explains this as de re representation.
a’ represents de re a, and likewise b’ b. Therefore, a’Rb’ constitutes that (merely) possibly, aRb.

Read rejects this explanation. Only propositions can represent [p. 335] (I hope I’m not getting him wrong on this, comments welcome).
Therefore, a’Rb’ does not explain that possibly, aRb.

Read concludes that MR fails to explain sufficiently modality.

Questions
Have I understand correctly Read’s final objection against counterpart MR?

Read’s `Unity of Fact’, third section: Bipolarity

After having dismissed identity theories because they cannot explain the unity of fact, Read now turns to argue that Wittgenstein’s picture theory does better on this.

He starts with a diagnosis of the failure of the identity theories of truth: They miss `[...] the distinction between the level of representation and the level of what is represented’ [p. 326].
Wittgenstein appears to have given an account of these levels by his picture theory.
Read turns to criticize the `P-theory’ interpretations of it:
`Situation’ is the most generic category. Among the situations, the actual ones (both atomic and non-atomic) are called `facts’, and the atomic ones (both actual and non-actual) `states of affairs’.

Read considers Bradley’s interpretation of this ontology [p. 327], but rejects it since it does not explain the sense of false propositions.
At first glance, these seem to require situations that do not obtain (a Meinongian picture theory).
However, Read claims, this answer overlooks `[...] the problem of [...] the unity of the fact’ [p. 328]
He then turns to Russell’s multiple-relation theory of propositional attitudes and Wittgenstein’s criticism of it. As this seems to me somewhat an excursion, I omit rephrasing it. If you guys think I’m overlooking something, tell me.

I only summarize Read’s conclusion:
Wittgenstein’s theory did not, on the interpretation which Read favours, allow for non-actual situations.
Instead, all propositions represent facts, i.e. states of affairs which obtain. These make up the entire furniture of the world (`Die Welt ist die Gesamtheit der Tatsachen, nicht der Dinge’, TLP §1.1).
However, for any proposition and its negation there is only one fact, namely the actual state of affairs.
To denote the connection between propositions and facts Read here uses `correspond’. For example, both the proposition that Frege is a great logician and that it’s not the case that Frege is a great logician, correspond to the fact that Frege is a great logician.
Thus, there is no need to account for the unity of alleged non-obtaining facts. A fact aRb is unified by the terms (particulars) a and b standing in the relation R (instantiating the universal R).

The trick is, or so Read argues, that on Wittgenstein’s account every proposition `[...] has two poles, what makes it true and what makes it false [...]‘ [p. 330].
Thus, p and ¬p do not correspond to different facts,
one which obtains and the other which doesn’t,
but are differently polarized.
The same fact makes p true and ¬p false. In the example, the fact that Frege is a great logician makes true the proposition that Frege is a great logician but renders false the proposition that Frege isn’t one.
However, if the truth and the falsity of a proposition are two symmetrical sides of the same thing, there seems no way to distinguish between them, i.e. to define truth (and falsity).
Thus, the bipolarity of propositions again leads back to the problematic claim of the identity theorist, namely that truth is indefinable.
Wittgenstein’s solution to this difficulty Read relates to his account of tautologies. However, his brief explication of this crucial point [p. 331] leaves me puzzled.
I’d be glad if people could tell me how they understand this bit.

Meanwhile, however, I skip to p. 332
where Read, as a conclusion, turns to clarify Wittgenstein’s `Sachverhalt’-`Tatsache’-`Sachlage’ terminology.
Read’s interpretation is based on one of Wittgenstein’s letters to Russell (August 1919) and Ramsey’s 1925 lecture notes on Wittgenstein.
All three words, Read claims, are used for facts:
` ‘Sachverhalt’ = ‘atomic fact’ corresponds to an atomic or elementary proposition. ‘Tatsache’ = ‘fact’ is a conjunction of atomic facts. ‘Sachlage’ = ‘state of affairs’ is what is represented by any proposition such as p ∨ q’

Questions

1. I’ve skipped Read’s exposition of W’s criticism of Russell’s multiple-relation-theory. If you guys think I’m overlooking something, tell me.
2. Read’s brief explication of how W solves the difficulty that truth becomes indefinable [p. 331] leaves me puzzled. I’d be glad if people could tell me how they understand this bit.
   text

Read’s `Unity of Fact’, second section

After having looked at the introduction, I now go right into the article. Read starts by exposing the problems of indentity theories of truth.

Read summarises Frege’s argument against the correspondence theory of truth: that it would lead into a regress.
He points out, however, that the regress itself does not yet make up an argument since operators such as `necessarily’ and `possibly’ yield similar regresses [p. 321].
Therefore, the regress argument doesn’t suffice as it stands but requires further reasoning as to why the truth-regress is vicious.

Read adopts Dodd’s distinction between modest and robust identity theories of truth.
On one hand,
modest identity theories say that what is true are the senses of sentences, thoughts, and that true thoughts are identical to facts.
Therefore, facts do not belong to the realm of reference, aren’t things.
Thus, the modest theory rejects that the furniture of the world includes states of affairs or propositions.

On the other hand,
a robust identity theory presumes that facts are states of affairs, and then identifies true propositions with facts, such that a true thought is some state of affairs.

Read now turns to ask how these families of theories would account for truth, and for falsehood [p. 324].
He points out that the robust theories imply a Meinongian account of erroneous beliefs: Since the object of propositional attitudes are states of affairs, somebody believing that Frege is a woman presupposes a `[...] non-actual state of affairs [...]‘ [ibid].
Thus, modest theories require non-existent things to exist, in some way.
Although this was Russell’s reason to reject robust identity theories of truth, however,
Read identifies another and more severe problem [p. 325].
Modest theories require facts whose unity remains obscure.
Take the fact that Frege is a woman. Certainly, there are its constituents, namely Frege and the property of being a woman. However, they aren’t unified, simply because Frege isn’t a woman. The robust theory thus cannot account for the unity of the facts it stipulates.

The modest theories, however, do better on this.
Read doesn’t spell out how since he is about to dismiss this solution anyway. Nonetheless, one explanation could look like this: If propositions belong to the third realm of sense, then one can believe that Frege is a woman without Frege and womanhood making up a state of affairs.

Read points out, however, that modest theories as well provide an only unsatisfactory account of the difference between true and false thoughts [pp. 325f].
Dodd takes this question to be sufficiently answered by the deflationist equivalence.
However, as Read points out, the equivalence is justified only by the regress arguments.
Since this he finds wanting (see above) Read rejects Dodd’s response.
Read concludes that both robust and modest identity theories fail to account for all the functions of propositions.

Reading Read, Introduction

This week I’m taking things relaxed. Actually, I should be revising my Magister thesis, but I’m procrastinating with Stephen Read’s `The Unity of Fact‘.

At a first glance, Stephen Read’s article seems to me of rather expository nature. In the introduction Read connects with the question, raised recently by Armstrong but going back to at least the Russell – Bradley – debate, what accounts for a states of affairs being one, if it cannot be the mereological sum of which things are involved in it. Read phrases this question as follows:
`The problem of the unity of the fact is that of reconstituting the fact from its constituents once they have been abstracted from it’
Read diagnoses that the question is ill posed. The relation (property) is not a constituent among others, such that their composition requires explanation. Instead, it is the relation (property) that makes up the fact.

It seems to me as if Read thus takes a position very close to the one Armstrong [1997 p. 30] ascribes to David Seargent.
Also, it seems to me related to the view which Dummett attributes to Frege, namely that concepts are formed by the analysis of propositions.
Now I have to look out for what distinguishes his view from these (there’s certainly something new about this).

… to be continued

Occham’s Razor, reverse-engineered

This is what it actually means, if you didn’t know it before:
“Occam’s Razor suggests that the simplest explanation is probably the best one.”
tells Joe Stewart, malware specialist and self-titled `reverse engineer’, the readers of the NY Times

Why it’s only the ascriber’s context which matters for the content of `knows’

Last Friday, I’ve summarized what a generic contextualism about `knows’ would be. The final formulation was:

GC”’.

The content of `S knows that P’ is a function of the character `has knowledge according to the contextually salient standards’ and the epistemic standards of the context.

However, one further issue requires clarification. When some speaker utters `S knows that P’ then at least two contexts can be discerned. On hand hand, there is the subject’s context, to whom knowledge is ascribed. On the other hand, however, there is also the context of the ascriber himself. Prima facie, GC”’ leaves open whether the subject’s or the ascriber’s context determines the salient standards.

However, this question resolves easily. Recall that the epistemic contextualist claims that `knows’ is an indexical. The content of indexicals, however, is determined by the context of the utterance. For example, the content of `I’ is the speaker of the utterance. Accordingly, in the case of `knows’ it is the ascriber’s context which determines the standards.

What’s cardinality, actually?

Wolfram gives no less than three different accounts, Wikipedia isn’t more definite. Is it that the notion is underspecified? How about this:

The cardinality of a set s is the set k of all sets s’ such that there is a bijection between s and s’.

This seems to be closest to the Tarski 1924 account, as explained by Wolfram.

Correction on Löwe’s and Müller’s Contextualism about Mathematics

Yesterday, I’ve presented Löwe and Müller’s 2008 proposal of a contextualist account of mathematical knowledge. But I haven’t been quite correct in my presentation. #’ is not their final proposal.

At the end of §5, they consider the objection that their talk of the subject’s `dispositional state of mind’ may be empirically void. Whether that’s a real problem or not, in any case they replace the phrase by a reference to `the Dreyfus-Dreyfus model of skill acquisition’ [p. 104], and formulate

S knows that P iff S’s mathematical skills suffice to produce the contextually required proof.

I have to say I’m not entirely clear about this reformulation. For one, it seems as if now, mathematical knowledge is not anymore relativized to the resources provided by the context, as in the previous account #’. If so, then the notion of skill is not merely a replacement for the problematic talk of dispositions (which was meant to specify the general talk of evidence in #) but also for the resources-clause.

Anyway, LM now write [p. 104]:
`Context determines the required form of proof or other justification, and context also sets the [...] required skill level’
So the skill is not so much the subject’s state of evidence but an additional requirement? I suppose I have to think through this again. I’ll try to come up with some explanation soon.

Löwe and Müller’s Contextualism about Mathematical Knowledge

The epistemic contextualism I’ve sketched elsewhere is meant to apply to all kinds of knowledge. In `S knows that P’, the variable `P’ ranges over all propositions. I now turn to a more specific variant of epistemic contextualism: Let `P’ henceforth range over mathematical propositions. The resulting position is a contextualism about mathematical knowledge.

Recently, Benedikt Löwe and Thomas Müller (henceforth `LM’) have proposed such a contextualism about mathematical knowledge [lm2008]. LM’s analysis has a two-fold basis. On one hand it presupposes a general analysis of mathematical knowledge in terms of proof (), on the other hand it rests on David Lewis’ variant of generic contextualism (#).

Without further motivation, LM presume that the following analysis of mathematical knowledge is generally accepted [p. 92]:

S knows P iff S could in principle generate a proof of P

.

Then, LM give several examples of mathematical knowledge ascriptions [§3, §4]. Together, these cases are meant to show that the meaning of `could in principle generate a proof’ varies across contexts. On LM’s view, this disproves any invariantist account of mathematical knowledge [p. 102]. Instead, mathematical knowledge requires a contextualist analysis. To formulate this, LM adopt the epistemic contextualism of David Lewis [1979, 1996].

A full exposition of Lewis’ account would go beyond the scope of this essay. LM as well confine themselves with a focused summary. They concentrate on his final analysis:

#

S knows that P iff S’s evidence eliminates every contextually salient possibility according to which not-P

[p. 102].

This analysis of knowledge is contextualist in so far as
the context of a knowledge ascription comes into play by ruling out certain possibilities. Thus, Lewis’ account allows the content of knowledge-ascriptions to vary drastically across contexts [p. 102].

LM now turn to combine # with the analysis of mathematical knowledge [p. 103]. First, the general `S’s evidence’ of # is specialized to the justification required for mathematical knowledge: The disposition of S to conduct a proof. Secondly, the context rules out respectively includes possibilities by determining which type of proof is appropriate for knowledge and which resources the subject is allowed to deploy. In sum, LM propose the following contextualism about mathematical knowledge:

#’

S knows that P iff S is disposed to produce the contextually required proof with the resources allowed by the context.

References
Lewis, David 1979. “Scorekeeping in a Language Game”, in Journal of Philosophical Logic v. 8, pp. 339-59.
- 1996. “Elusive Knowledge”, in Australasian Journal of Philosophy v. 74, pp. 549-67.
Löwe, Benedikt and Müller, Thomas 2008. `Mathematical Knowledge is Context Dependent’, Grazer Philosophische Studien
2008, V. 76, pp. 91 – 107