handout

Facts as figures Max Weiss

Goals I

The aim is to understand a system of three disjoint kinds: I

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facts, relations, and objects (FRO).

how do things of these kinds fit together? I I

what concept articulates best their interaction? articulates best means I I

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generates the genuine distinctions doesn’t generate specious ones

There is also a suspicion that three is a crowd: I

is one of the kinds of thing somehow definable by the others? I

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could a fact be specified by its constitution by objects in a relation?

that would be nice, though not sine qua non

Completion

A fact as a unification of constituents I I

I’ll assume that a fact uniquely consists of some objects standing to each other in a relation. More precisely, let’s assume the following Completion Principle I

This means that for any fact F . . . I I I

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there is a unique relation R, and there is a unique bunch OO of objects, such that F is a (bare) completion of R by OO.

let’s call R and OO the unifier and the constituency of F

So (following Russell and Fine) I have in mind a broadly Russellian conception of fact Also when I say ‘fact’ I just mean a (very broadly) possible fact I

an actual fact is a fact that obtains

Facts are completion-indefinable I I

What can we say about the structure of completion? distinct facts may 1. have the same constituency I

Mary is older than Susan; Mary is taller than Susan

2. have the same unifier I

Mary is older than Susan; Terry is older than Hoosen

3. have the same constituency and the same unifier I

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Mary is older than Susan; Susan is older than Mary

It would be nice if completion defined each fact: I

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as we’ve seen, some facts are completion-indiscernible but distinct completion doesn’t individuate facts and so doesn’t define them well, definability of facts is maybe unachievable anyway

Permutability I I

Let’s try to use completion to articulate other FRO-structure pretheoretically, some ‘dyadic’ relations like adjacency are ‘strictly symmetric’ I

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this not just that they happen always to be reciprocated but that reciprocation is not another fact can symmetry be articulated by the concept of completion? it looks as though if R is symmetric, then. . . I

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for no o, p is there more than one completion of R by o, p.

does that condition specify symmetry? I I

well, it’s unclear that the condition is sufficient some relations seem intuitively asymmetric, but satisfy the condition I

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perhaps one could deny that these are asymmetric I

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e.g., begetting, or preceding-in-the-numbers, or exemplifying that is, all asymmetric relations would be symmetric!

it might also be that metaphysical possibility isn’t the right concept here

Partial permutability I

more worrisome is that the analysis of symmetry doesn’t generalize I

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we could say that a relation is permutable it has at most one completion by any OO but then we seem unable adequately to characterize partial permutability consider the lifting relation, as in I

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this relation isn’t permutable, because we do need to distinguish I

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Alice lifts Barbara and the piano

just by counting completions, we have no way to distinguish I

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Alice and Barbara lift the piano

the indifference between lifters, from the difference between lifter and lifted

So, it looks like the concept of completion doesn’t articulate enough FRO-structure We need to find something more expressive

A traditional answer: sequential completion I I I

Recall that completion relates a fact F , a relation R, and a bunch OO of objects We might consider sequential completion which relates F and R with a sequence σ Instead of the Completion Principle we’d have the Sequential Completion Principle: I

for each fact F I I I I

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so we can define the constituency of a fact as the terms of any of its sequential-constituencies I

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there’s exactly one relation R, and there’s at least one sequence σ of objects, such that F is a sequential-completion of R by σ, and if both σ and τ are sequential-constituencies of F , then the terms of σ are the terms of τ .

then the completion principle is derivable

unlike with bare completion we can hope to identify facts by their unifier and sequential-constituency, that is: I

no two facts have the same unifier and sequential constituency

Sequential completion: expressiveness I I

Sequential-completion expresses more structure than does bare completion it distinguishes symmetry and asymmetry: I

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R is symmetric if every sequential completion of R by some (o, p) is also a sequential completion of R by (p, o) R is asymmetric if no sequentical completion of R by some (o, p) is a sequential completion of R by (p, o)

it also characterizes partial permutability: I

thus R would be permutable in some positions if no completion of R is changed by reordering on those positions I

for example, a completion of R by hAlice, Barb, the pianoi is also a completion of R by hBarb, Alice, the pianoi but not by hthe piano, Barb, Alicei.

Sequential completion: overgeneration I

recall that the basic goals of understanding FRO-structure are I I

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generating relevant distinctions not generating irrelevant ones

sequential completion draws just about as many distinctions as possible without props arguably it draws too many! I I

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suppose R is not symmetric, e.g. that of preceding then there’s some F such that F is the completion of R by o, p but not the completion of R by p, o now, intuitively R has a strict converse. I.e., I

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there’s an R 0 such that F is the completion of R 0 by p, o but not by o, p then R 6= R 0 , contradicting uniqueness of unifiers.

thus, a relation and its converse should have the same completions I

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but sequential-completion distinguishes a relation and its converse and this contradicts unifier uniqueness.

Positional completion

To recap. . .

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we tried barely-completing a relation with a bunch of objects I

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this gives too little structure

so we tried sequentially-completing a relation with a sequence of objects I

this gives too much structure

Less structure

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Let’s try something in between. How does sequential-completion run into a problem? I I

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A sequence is a labelling of objects by ordinals elements of an initial segment of the ordinals are structurally identifiable so, labelling of objects by ordinals yields reidentification of labels across different relations in turn, reidentification of labels implies distinguishability of converses

So let’s try choosing labels which aren’t so structured

positionalism I I

Each relation brings its own set of positions Now you get a positional-completion principle: I

for each fact F . . . I I

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there’s a unique relation R, and there’s an assignment t of objects to the positions of R, such that such that F is the completion of R by t.

As with sequential completion. . . I

we can also hope to identify a fact by positional-completion: I

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no two facts have the same unifier and same positional-constituency

And we can still maintain that no fact has two bare constituencies

Expressiveness of positional-completion

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Positional completion distinguishes symmetric and asymmetric relations I

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R is permutable on some positions if no completion is affected by prepending to the assignment a permutation of those positions R is asymmetric if no result of prepending a position-permutation to an assignment which has a completion also has a completion

Clearly it expresses partial-permutability too I

it’s permutable on some positions if no completion is affected by prepending to the assignment a permutation of those positions

positional completion: overgeneration? I

How does positional completion avoid the distinguishability of converses? I

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It looks as though given some permutation π of the positions of R, one could define an R 0 to be that relation such that F is a completion of R 0 by t iff F is a completion of R by t ◦ π Fine’s response: I

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“I doubt that there is any reasonable basis, under positionalism, for identifying an argument-place of one relation with an argument-place of another.”

Thus, Fine’s positionalist would avoid distinguishing converses by rejecting transrelational reidentification of positions The basis of the rejection is that positions are individuated only by their place in the labelling structure

positional completion: overgeneration, another try

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if R is symmetric, then F is a completion of R by t iff F is a completion of R by t ◦ π for all R-position permutations π I

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by the positional completion principle it follows that no completion of a symmetric relations has more than one constituent

So far as I can see, this objection doesn’t quite work I

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The positional-completion principle doesn’t imply that no two assignments yield the same relation In the case of sequential completion, we allowed that the same fact could be determined by two different sequences why not allow the same for positional completion? ???

substitution

a structuralist alternative

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Previously we tried to say ‘what’ a fact is I

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that is, to find a gadget uniquely specifying a fact by reference to a relation and some objects maybe that is too hard

let’s try just reporting how facts, relations and objects fit together

The concept of substitution

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Fine proposes to take as primitive the concept of substitution on objects in a fact for example, suppose again that F is the fact that Alice and Barb lift the piano I

you can simultaneously substitute Carol for Barb, and the car for the piano I

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you can also exchange Alice and Barb, or Barb and the piano I

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of course the result is not F

the second changes F but the first doesn’t.

pedantically, substitution is a mapping OO → F F from functions on the objects to functions on the facts

Substitution and and constituents

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Expressively this is a sea change from the completion-based approaches It’s not clear that, given a fact you can even identify its constituents I

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one idea is to say that o is a constituent of F if F is changed by some substitution which moves only o this would rule out a fact that such-and-such are all the objects

Substitution and unifiers

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It looks tricky because substitution is specified without mentioning relations but plausibly, substitution preserves the unifier I

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it doesn’t make sense to talk about substituting one relation for another e.g. if you substitute preceding for adjoining in a fact that something adjoins another, what do you get?

so one option is to recover unifiers by abstraction: I

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facts have the same unifier if they are connected by substitution-accessibility e.g., (a, b, b) ← (a, b, c) → (b, b, c)

Substitution: expressiveness

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since the notions of unifier and constituents became obscure, we need re-identify the explananda instead of symmetry of a relation we could talk about symmetry of objects with respect to a fact I

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for example, objects o, p might be said to figure symmetrically in F if F = F o,p p,o .

likewise partial permutability of a relation becomes permutability of objects in a fact

Generality of substitution

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the concept of substitution seems intuitively well-understood I

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although it’s complicated, Fine says our understanding of it is ‘general’

I’m worried that the apparent clarity of substitution on facts derives from clarity of substitution on strings I

substitution on strings is just composition of functions I I I

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a string is a function s from ordinals to characters suppose f is a function on characters then f determines a substitution which, applied to s, yields simply f ◦ s

so, substitution on strings is totally clear

Non-one-oneness

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It seems that substitution on facts is not clear I

if F is Bob loves Alice, then I I

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I guess I understand substituting Chris for Bob in F But I’m not sure I understand substituting Alice for Bob in F

Here Fine might reply: I

well, the sequentialist/positionalist is no better off since they allow repeating sequences or non-one-one assignments