a curious proof

7/31/2019 A Curious Proof

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A Puzzle about Belief and the Limits of Knowledge

Heidi Howkins Lockwood

Yale University

May, 2008


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As we know,

there are known knowns.There are things we know we know.We also know there are known unknowns.That is to saywe know there are some things

we do not know.But there are also unknown unknowns,the ones

we dont know we dont know.

~ U.S. Defense Secretary Donald Rumsfeldat a Feb. 12, 2002 Department of Defense news briefing

in response to questions about U.S. intelligence onweapons of mass destruction in Iraq


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Overview

I. Introduction

II. Proof of the Possibility of Believing an Impossibility

Part 1A conditional proof

Part 2Discharging the Cartesian premise

III. The Limits of Knowledge (and other operators):A Lb-like incompleteness result

IV. Discussion

Diagonalizing on propositions


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Introduction Proof of Possibility of Believing an Impossibility: Part 1, Part 2 Structural Limits of Knowledge Discussion

Introduction

It is generally accepted that it is impossible to knowtheimpossible. To hold otherwise would require radical

revisions to the logic of knowability.

What is impossible is always false, what is false is nevertrue, and knowledge presupposes truth. It is thereforeimpossible to knowp ifp is impossible.


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But what about operators that dont presuppose truth operators such as imagining, representing, and believing?

Mere belief as opposed to true belief is a case in point.

Isit possible to believe that 2+3=6, or that a triangle has 5sides, or that there is a greatest even number, assuming fullcomprehension of concepts such as addition, greatest, even,

and so forth?


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Many philosophers have thought not.

Many theories of belief have presumed, either explicitly orimplicitly, that it is not possible to even merely believe alogical or conceptual impossibility.

This would be true, for example, of Bob Stalnakerspossible-worlds view of belief propositions, David Lewismap-like representational account of belief, and Ruth

Barcan Marcus traditional dispositionalism all of whichpresuppose externalist theories of belief.


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The thesis that it is impossible to believe an impossibilityalso appears at many points in the history of philosophy.

Berkeley links belief to imagination and denies that we canimagine impossibilities.

Logical positivists such as Schlick use the unthinkability ofcontradictions to distinguish between empiricalimpossibilities and logical impossibilities.

And so on.


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Lets dub the generic view that impossibilities or contradictionscannot possibly be the object of a possible beliefpossibilism.

In symbols:

p (Bp ~p)


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The proof proceeds in two parts.

The first part is a conditional proof of

the possibility of believing an impossibilitybased on the fixed point premisethat it is possible to believe that it is

possible that it is possible to believe an impossibility.

In the second part of the proof,

we will discard the reliance on the fixed point.


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Proof of the Possibility

of Believing an Impossibility

Part 1


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Suppose that I assert, contrary to possibilism, that I believe that itis in fact possible that it is possible to believe an impossibility.

In symbols:

B(p (Bp ~p))

Lets call this the fixed point premise. As we will see in amoment, it guarantees the truth of the proposition believed.

To make the proof more readable, lets use c to refer the formula

p (Bp ~p).


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Apart from calling me cognitively impaired and perhaps in thegrip of a Kripkean referential delusion, the possibilist will nodoubt assert this: it is impossiblethat it is possible to believe animpossibility.

And therein lies the catch.

For, if the is correct in asserting that the object of my beliefthe claim that it is possible to believe an impossibility is

impossible, then she is admitting that I have succeeded inbelieving an impossibility, and thereby refuting her own claim.


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Lets take a closer look at this argument in symbols.

Aside from the fixed point premise, the only assumptions arethose of classical logic and a standard S4 modal framework.

We will discard the fixed point premise in Part 2 of the proof.


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First, using c to refer to p (Bp ~p),we have the fixed point premise:

(1) Bc


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First, using c to refer to p (Bp ~p),we have the fixed point premise:

(1) Bc

From this and the principle that whatever is actual is possible

(pp), we get:

(2) Bc


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(1) Bc

(2) Bc

Now lets suppose (ad reductio):

(3) c


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(1) Bc

(2) Bc

Now lets suppose (ad reductio):

(3) c

From (3) and the principle that whatever is impossible is not

possibly possible (pp), we have:(4) c


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(1) Bc

(2) Bc

(3) c

(4) c

The conjunction of (2) and (4) gives us:

(5) Bc c


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(1) Bc

(2) Bc

(3) c

(4) c

The conjunction of (2) and (4) gives us:

(5) Bc c

And existential generalization on (5) produces:

(6) p(Bp ~p)


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

But (6), of course, is just:(7) c


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

Another application of the principle that the actual is possible

gives us:

(8) c


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c

And (8) contradicts (3). So we have:

(9) c


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c(9) c

Applying double negation to (9) of course gives us:

(10) c


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c(9) c

(10) c


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c(9) c

(10) c

This argument proves c assumingthe truth of the fixed point

premise, Bc.


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c(9) c

(10) c


premise, Bc.

In other words, the affirmation ofbelief in the possibility ofbelieving the impossible isinfallible because it is self-fulfilling.


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(1) Bc

(2) Bc

(3) c

(4) c

(5) Bc c

(6) p(Bp ~p)

(7) c

(8) c(9) c

(10) c


premise, Bc.

In other words, the affirmation ofbelief in the possibility ofbelieving the impossible isinfallible because it is self-fulfilling.

If Bp(Bp ~p),

then in fact p(Bp ~p).


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But the truth of Bc is not incontrovertible.

One might, for example, worry about the semantic stability

of the terms in question.

We turn now to the project of strengthening the proof byeliminating the dependence on the fixed point premise.


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Proof of the Possibility

of Believing an Impossibility

Part 2


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Recall that c refers to p (Bp ~p). In Part 1 of the proof wehave shown that (Bc c).

To improve the readability of the second half of the proof, lets

embed the redundant in a new constant. In other words, let c*refer to

p (Bp ~p)

So we have now shown that (Bc* c*), or, alternatively(weakening it slightly by disregarding the first line of the proof),that Bc* c* is provable. This gives us:

(11) Bc* c*


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(11) Bc* c*

For the remainder of the proof, it will be convenient to note

that believability, which we have been representing as B,

intuitively satisfies what are known as the Hilbert-Bernays-Lb(HBL) conditions in provability logic.


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The Hilbert-Bernays-Lb (HBL) conditions:

(i) ifp , then Bp(ii) B(pq) (BpBq)

(iii) BpBBp


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(i) if p , then Bp

Condition (i) says that ifp is a theorem (is provably true), then thefact that it is possible to believe thatp is also a theorem. Note that

this condition does notencounter the well-known troublesassociated with its counterpart for knowability, the problematicknowability thesis or strong verificationist thesis. In particular, (i) isnot susceptible to the woes associated with the family of Fitch-Church knowability paradoxes because belief and believability arenot factive.

It merely asserts that we can derive the believability ofp from thederivability ofp.


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(ii) B(pq) (BpBq)

Condition (ii) says that if it is possible to believe thatp entailsq, then if it is possible to believe thatp, it is possible to believethat q. Worries can certainly be raised about the counterpart of

(ii) for belief (as opposed to believability). We might, for example,argue that the belief thatp entails qdoes not entail that thebelief thatp entails the belief that q, on the grounds that acertain sort of simultaneity of the belief thatp entails qand thebelief thatp is required for the formation of the belief that q.

But condition (ii) is a claim about the metaphysical orconceptual landscape, not an assertion about epistemicentailment. It merely asserts that if it is believablethatp entails q,then if it is believablethatp, it is believablethat q.


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(iii) BpBBp

Condition (iii) says that if it is possible to believethatp, then it is possible to believe that it is

possible to believe thatp. Again, the modality takesthe bite out of this claim.

Exhibiting the derivability of the believability ofpsuffices to show that the believability ofp is

believable.


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Now, given that we have proved:

(11) Bc* c*

The technique of diagonalization, introduced by Gdel [1931],

gives us a formula asuch that a (Ba c):

(12) a (Ba c*)

Taking just the left-right direction of this biconditional gives us:

(13) a (Ba c*)


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

Clause (i) of the HBL conditions together with (13) gives us:

(14) B(a (Ba c*))

By clause (ii) of the HBL conditions, we know:

(15) B(a (Ba c*)) (BaB(Ba c*))


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))

And from (14) and (15) we get:

(16) BaB(Ba c*)

By clause (ii) again we know:

(17) B(Ba c*) (BBaBc*)


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))

(16) BaB(Ba c*)(17) B(Ba c*) (BBaBc*)

So by (16) and (17) we can infer:

(18) Ba (BBaBc*)


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

Clause (iii) of the HBL conditions gives us:

(19) BaBBa


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

From (18) and (19) we know:

(20) BaBc*


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

And (20), together with (11), gives us:

(21) Ba c*


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

By (12) and (21) we can now derive:

(22) a


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

By virtue of (22) and clause (i), we get:

(23) Ba


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

Finally, (23) together with (21) gives us:

(24) c*


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

(24) c*


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

(24) c*

Recall that c* is p(Bp ~p).

d f f b f b f d


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

(24) c*


We have now eliminated the initialdependence on the fixed pointpremise, and proven that it ispossible to believe an impossibility.

I d i P f f P ibili f B li i I ibili P 1 P 2 S l Li i f K l d Di i


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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc*)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

(24) c*


Note that this argument is not restricted

to belief.



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(11) Bc* c*

(12) a (Ba c*)

(13) a (Ba c*)

(14) B(a (Ba c*))

(15) B(a (Ba c*)) (BaB(Ba c*))


(18) Ba (BBaBc)

(19) BaBBa

(20) BaBc*

(21) Ba c*

(22) a

(23) Ba

(24) c*


It will work for any operator for which

the HBL conditions hold and it is

provable thatc c.



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Is there an analogous result for knowability?

For provability?For T?

I t d ti P f f P ibilit f B li i I ibilit P t 1 P t 2 St t l Li it f K l d Di i


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The first half of the proof relied only on an S4 modal frameworkand classical logic.

So we can extend the results of the first half to:

(KcK cK)(Bew(cBew) cBew)

(TcT cT)

where c is p(p ~p).

Intr d ti n Pr f f P ibilit f B li in n Imp ibilit P rt 1 P rt 2 Str t r l Limit f Kn l d Di i n


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For the second half of the proof, we need the HBL conditions:

(i) ifp , then p(ii) (pq) (pq)(iii) p(p)

These are standardly assumed to hold in the case in which the modal

operator is taken to be provability (Bew).

They also seem to hold for possible truth (T) and knowability(K),given an appropriate interpretation of .

Introduction Proof of Possibility of Believing an Impossibility: Part 1 Part 2 Structural Limits of Knowledge Discussion


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In other words, the proof we have just seencan be used to show:

p(Kp ~p)

p(Bew(p) ~p)

p(Tp ~p)



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The Structural Limits of Knowledge



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Introduction Proof of Possibility of Believing an Impossibility: Part 1, Part 2Structural Limits of Knowledge Discussion

Lets suppose for a moment that the HBL conditions hold forknowabilityi.e., lets suppose, for a factive knowledge operator

K, that K satisfies:

(i) ifp , then Kp

(ii) K(pq) (KpKq)(iii) KpKKp

(Notice that (i) is just the knowability principle.)



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Given this assumption (that the HBL conditionshold), then by Lbs Theorem [1955], we know:

K(Kpp) Kp



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It is worth pausing at this point to note that Gdels secondincompleteness theorem is an immediate consequence of Lbs

Theorem. This is easiest to see when the theorem is expressed as:

(pp) p

for if we express the inconsistency of a theory using then consistency is representable as:

~

which is equivalent to:



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In other words, withp, Lbs Theorem says that the consistencyof the theory is provable only if the theory is inconsistent:

()

which is of course just the second incompleteness theorem.



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Similarly, in the case of knowability, if we use the symbol torepresent a contradiction or impossibilityperhaps the proposition

expressed byp(Kp ~p)then the knowability of thepossibility of knowing an impossibility is representable as K andthe unknowability of the possibility of knowing an impossibility is

representable as ~K, which is equivalent to K.

So, byLbs Theorem, we have:

K(K) K

And this tells us that either it is possible to know an impossibility, orit is not possible to know that it is not possible to know animpossibility.



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So we now know that ifthe HBL conditions hold for knowabilityin particular, if the knowability thesis expressed by condition (i)holdsthen either it is possible to know a contradiction (knowledgeis inconsistent), or knowledge is incomplete.

In other words, either the knowabilitythesis doesnt hold, in whichcase there are unknowable truths (knowledge is incomplete), or theknowability thesis does hold, in which case knowledge is eitherinconsistent or incomplete.



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y g p y , g

Those familiar with the family of Fitch-Church knowability paradoxmay detect a whiff of the knowability paradox in this result.

The Fitch-Church result is a surprising refutation of the thesis thatall truths are knowable based on three assumptions: (1) knowledge isfactive, (2) knowledge distributes across conjunction, and (3) there isat least one unknown truth.

It is easy to misinterpret the knowability paradox as a generalizationof Gdels first incompleteness theorem, which demonstrates that,for any consistent, sufficiently strong theoryTin the language of

arithmetic, there are truths unprovable in T. The problem with doingthis is that the Fitch-Church result rests on the existence of anunknown truth, which is arguably a contingent fact.



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y g p y , g

The proofs we have just completedfree the Fitch-Church result from the

arguably contingent assumption of theexistence of an unknown truth.



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y g p y , g

Discussion


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y g p y g

One way to understand the underlying tension betweenconsistency and completeness is as a symptom of a

cardinality mismatch.

A cardinality mismatch, that is, between the expressible andtherefore at most denumerablyinfinite number of-ables and

the non-denumerable number of potenial -ables that isgenerated through the iteration of.



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These thoughts raise a natural question:

What are we taking theobjects of knowledge (or belief) to be?



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These thoughts raise a natural question:

What are we taking theobjects of knowledge (or belief) to be?

Do the results presuppose any limits on theobjects of knowledge or belief?


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This presupposition enters the proof at

the point at which we diagonalized on theobjects of belief or knowledge.



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Propositions are (on most accounts) mind-independent extra-linguistic abstract entities. So how does diagonalization onpropositions work?

Well, briefly and without entering into the fray on propositions, lets

suppose that the objects of knowledge are propositions, and that aproposition is knowable only if it is expressible. (Propositions whichare ineffable, inexpressible, or otherwise non-assertable and non-communicable are not knowable, and therefore would not besuitable candidates for objects of knowledge.)

We know how to diagonalize on sentences and formulae. So inorder to diagonalize on propositions, it suffices to come up with asystematic method for mapping (via an injective function)expressible propositions onto sentences.



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Each proposition may in theory be expressed by many differentin fact, possibly infinitely manydifferent sentences.

In other words, for sentences S1 and S2 such that S1 S2 , given ameaning relationMwhich takes sentences to propositions, it maybe the case thatM(S1) =M(S2).

So we cant rely on an expression function to take propositions tosentences.

We can, however, rely on an expression relation,Ex, which takespropositions to equivalence classesof sentences:

Ex(p) = {S| Sexpressesp}



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Heres the key: since each equivalence class is a set of sentences,each of which has finite length, each equivalence class has a well-ordering and can be put into one-to-one correspondence with thenatural numbers.

In particular, since each member of each equivalence class is asentence of finite length, it can be assigned a unique Gdelnumber, call itg(S). We can then well-order the members ofEx(p)

by simply using the standard


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One final note:

The result we have discussed does not assume that propositions arethe objects of knowledge, but it does assume that it makes senseto assert that something is known or is knowableso at some level

it does assume that there are objects of knowledge. Theseobjects, however, are not restricted to potential truth-bearers suchas sentences or propositions. They could, for example, be someentity or feature of a possible world that is either a truth maker or aconstituent of a truth maker.

In the case of objects viewed as truth makers in the actual world,the knowledge could be knowledge by direct acquaintance, i.e.,knowledge that involves an unmediated relation between thesubject and the truth maker.



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In other words, the result is not restricted toknowledge that involves characterizing, representing,or conceptualizing the objects of knowledge. It does,

however, assume that those objects are what I will calldiscretely graspable, i.e., either finite or effectively finite

(expressible through a recursively or effectivelyenumerable string of symbols, or in principle

observable via deterministic effects).


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I am grateful to the participants in the

Yale spring 2008 works in progress seminar for helpful

comments and questions.

[email protected]

a curious proof

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