Not Every Truth Can Be Known:
at least, not all at once
According to the knowability thesis, every truth is knowable. Fitch's paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the "conjunctive knowability thesis") to the effect that for every truth pthere is a collection of truths such that(i) each of them is knowable and
(ii) their conjunction is equivalent to p.
I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on one other issue connecting knowledge and possibility. If some things are knowable but false, then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is quite weak.
Greg Restall
Comments
I am glad to be among the first persons to see it online. Now, let me make some comments about the manner the paradox is presented at the online encyclopedia:
Fitch's (1963) paradox challenges `common sense', as any paradox does. Still, there are ways to begin explicating or describing a paradox adequately. Surely, it is better to depart from ideas that everyone may intuitively agree to.
Everyone's intuition is that what is knowable is not necessarily known. In the same manner, not everything that is visible has been seen.
The way Fitch's paradox is presented in Stanford Encyclopedia is misleading for larger audiences. It says that the principle of knowability claims all truths are knowable ( (KP) p ? ?Kp). Then it concludes: "If one accepts the knowability principle, she must deny that there are unknown truths. (...) In sum, if all truths are knowable, then all truths are known. `
Well, if I accept (1) bellow:
(1) a. All Humans are mortal.
b. Greg Restall is human.
c. Thus, Greg Restall is mortal.
Should I deny that there are humans who are not dead? Should I conclude that Greg Restall is dead just because he is mortal? Should (1a) and (c) be understood as (2) and (3)?
(2) All humans are dead.
(3) Greg Restall is dead.
What may happen does not necessarily happens. One must make a difference between what is ` x`-able and what is already ` x`-ed.
I prefer the way you start presenting the paradox in your paper. The text at the Online Encyclopedia causes the wrong idea from the start, although it re-presents the paradox in the proper manner further on.
Tony Marmo at April 24, 2004 06:07 PM
Bravo! I love this paper. I'll still be thinking about it for a long time I'm sure, but initially, two comments. (i) In the middle block of paragraph 23 there are some p's which should beq's. Were you not taught as a lad to mind your p's and q's? Sorry, couldn't resist 8-) (ii) There's something weird about formalising `P is knowable' as \Diamond KP where \Diamond is an ordinary alethic modal operator. It's knowable that there is milk in the fridge (M)---go and look. It's not knowable that the colour of this banana is killer yellow---look at it and die. I would have thought it is not knowable that there is no milk in the fridge---because there is milk in the fridge! But \Diamond KM is true---there's an (alethically) accessible world in which you drink all the milk, and in which you (then) know there is no milk in the fridge. So to my ears, if P is not true, then not only is P not known, but P cannot be known, i.e. P is not knowable. In fact "You cannot know P unless P is true" is just the truth condition on knowledge (not "You do not know P unless P is true"). So if \Diamond KP is to mean that P is knowable, then the following must be true (at every world in every model): \Diamond KP --> P. This will then make (3) on p.7 not a truism, but a falsism! (For (\exists q)(\Diamond Kq\wedge \Diamond K\neg q) to be true, q\wedge\neg q (i.e. "q and not q" for the tex-illiterate) would have to be true.) But then your version of the knowability thesis does not follow from a truism (at least not this one, because it is a falsism) and so is not (yet shown to be) almost trivially true.
Nick Smith at April 24, 2004 09:57 PM
In Mathematics one works with constants, variables and incognitae, not only propositions. In Formal Semantics we know that we cannot account for linguistic phenomena without the notions of `contant' and `variable'. A Semantics without variables could yield many paradoxes and/or miss important data respecting human languages.
Now, I ask both of you this: hasn't the notion of incognita a place within Logic? You have [[x]] but you do not know the denotation of [[x]]. You can only know it after, let us say, you solve an equation or inequation, or a problem. I guess that, for instance, the correct Semantics of questions like:
(1) Who gave Cinderella a poisoned apple?
would imply that x in (2) is better understood as an incognita:
(2) Who is x, such as x gave Cinderella a poisoned apple?
In Cinderella's traditional story the answer is [[x]]=none, for none gave her a poisoned apple. You find the denotation of the incognita in the same manner as you find the solution of an equation.
Now, my question to both of you:
What if one tried to see the knowability paradox as an argument in favour of the notion of incognita within a Logic system?
Tony Marmo at April 25, 2004 05:57 PM
A question: Can "Every truth is conjunctively knowable" be read as form of logical pluralism? That is, if there is "conjunctively knowable" might there also be "foo knowable"?
If there is more than one way of knowing, then we might enquire whether Fitch meant K as all ways of knowing, or just one way. Does the paradox remain if K meant all ways?
RdR at April 30, 2004 06:30 AM
I am not a Logician though, I think I can answer your question:
If there is more than one sort of mental process that you call `to know', then you actually have more than one predicate.
Suppose you have two ways of `knowing', represented by K and G. Then if there is a proposition p and we do not know it, you can write:
notK(p).
To write that we know we do not know p, it could be:
G(notK(p))
In such case, I guess it is more difficult to build a similar paradox, but one could try it. I hope the others correct me if I said anything wrong.
Tony Marmo at April 30, 2004 07:53 AM
Yes, if we allow multiple ways of knowing, we might know in one way that we (don't) know in another way [your G(notK(p))].
But my question was (twofold) whether Fitch distinguished such multiple ways of knowing (which in Greg's paper would mean multiple ways of deriving logical consequences), and whether the paradox would hold if Fitch's interpretation of knowing was "for all ways of knowing".
That is, should we read the knowability thesis as (p)(K) (p -> <>Kp) ? And can we derive from that the paradox (p)(K) (<>K(p & -Kp)), given that that derivation also uses a particular K.
RdR at April 30, 2004 09:58 AM
As far as I understand Fitch's paradox, it assumes that there is only one kind of `known' predicate (K).
But one has to take Philosophy into account and, yeah, to know that something exists may be considered one way of knowing. To know what such thing is, to be able to define or describe it is another way or part of knowing it.
As I said before, the issue about assuming that there is p such that we do not know p is that actually we know or assume that it exists, but we cannot say what is its denotation or describe it, etc.
In many sciences it is often the case that in order to explain certain facts one has to relate them to some unknown or yet to be discovered entity, whose existence is already accepted by scientists, but which seem to lack a denotatum or cannot be defined or characterised.
Tony Marmo at April 30, 2004 11:01 AM
I think multiple ways of knowing might be a way around the paradox. Kp & K-Gp is not necessarily a paradox. For example, let's say K is "knowing theoretically" and G is "knowing empirically". I may theoretically know p and theoretically know that I don't empirically know p.
However, are multiple ways of knowing necessary?
Let's say that there is only one way of knowing. According to <>K( p & -Kp) such knowing is allowed to be higher order. But then other paradoxes seem possible. If numerals are names of sentences and K operates on sentences or names of sentences and 1 is the name of (K-1), then 1 seems paradoxical.
So, maybe we don't allow K to be higher-order. But that would imply that the first K in <>K( p & -Kp) is different (in extension at least) than the second K. So, multiple ways of knowing seem necessary.
RdR at April 30, 2004 02:43 PM
By the way, in Romance Languages we have two verbs for the English `to know':
[1] Portuguese `saber', French `savoir'
and
[2] Pt `conhecer', Fr `connaitre'.
The first has to do with the word `wisdom' in English and also with the kind of knowledge that enables someone to do something, i.e., the idea of `know-how'.
The second has to with the idea of `knowledge' in English and also refers to a kind of experience that is part of a collective, ie, to get a share of a common knowledge or to share knowledge.
Some people feel that the first implies `deeper knowing'. On the other hand, you may use the second to express the idea that you met someone or found something.
Tony Marmo at April 30, 2004 03:38 PM
PS:
Some French examples to illustrate the case:
(1) Je sais qu'il y a des choses inconnues. I know that there are unknown things.
(2) Je sais qu'il y a des choses que je ne connais pas. I know there are things I do not know. (The first verb is `savoir', the second is `connaitre').
Tony Marmo at April 30, 2004 04:08 PM
I'm loving the comments! What fun to have such interested and interesting readers.
A few responses. Clearly Nick is right, there's a reading of knowable according to which what is knowable is true. I'm tempted to say that this reading is a confused scope issue, resulting from the following reasoning: necessarily, what is known is true, therefore, if you can know something, it is true. But that doesn't follow at all. It's just follows that if you can know something it can be true.
Anyway, I'll add a section to the paper indicating what one can say about conjunctive knowability in the case where we say that only truths are knowable.
As to incognita , I'm not sure about what to think about this. I s'pose I'm not sure what is the best way to talk about objects of intentional attitudes in the case of complete ignorance.
Different kinds of knowledge? I think that Fitch's paradox is restatable if we talk about being known-in-any-which-way. Let's say that something is known-in-any-which-way, if it's known in way-1 or way-2 or way-3, etc. Or so suspect anyway.
Greg Restall at April 30, 2004 04:49 PM
I left a message to Kai von Fintel and friends in his weblog, sort of inviting them all to come and speak their minds. I hope he read all of this and may say something.
In one of Kai von Fintel's courses in Intentional Semantics, he explains that `to know' in English has to with a reflexive world accessibility relation:
(0) wRw
It is the only `opaque' verb that has this character in English. Thus, this should explain a sentence like (1) is odd, but (2) is ok:
(1) #Bob knows that John Howard is the Prime Minister of France.
(2) Bob assumes that John Howard is the Prime Minister of France.
You cannot fix (1) as (1'), although (2') is an option:
(1') #*Bob WRONGLY knows that JH is the PM of France.
(2') Bob WRONGLY assumes that JH is the PM of France.
But, I wished Kai von Fintel could give us more details of such analysis in the way he uses it.
Tony Marmo at April 30, 2004 05:10 PM
About knowing and linguistics: I think Germanic languages make similar distinctions. In Dutch there is "weten" and "kennen". I can say, "Ik ken Jan" (I know John) and "Ik weet logica" (I know logic), but not "Ik weet Jan". Similarly, "Ik ken logica" would sound provincial. To me "kennen" is more like "knowing by acquaintance" and "weten" is more like "knowing by learning" (some skill or fact).
About knowing-anywhich-way: Don't we still run into reflexive paradoxes? It would mean the second K in <>K(p & -Kp) would include the same knowing as the first K. That is, reflexive knowing about the sentence that asserts the knowledge. So, paradoxes like "I know I don't know this sentence" would be possible.
RdR at May 2, 2004 11:35 AM
Richard, on knowing-any-which-way, I think I misunderstood the original motivation for the distinction. I'm happy with having knowledge tout court as a disjunction of knowing1, knowing2, knowing3, etc., because I think that the knowability paradox isn't a real problem. I reckon that we non-omniscient knowers should just bite the bullet and say that some truths are not knowable in any way at all.
If you want to distinguish between different kinds of knowing as a way of saving yourself from Fitch's paradox, then you'll need to resist that conclusion, of course.
Greg Restall at May 3, 2004 04:14 PM
Oh, and some news for everyone. I've managed to get a bit further with the paper on the basis of Nick's comment that on some views of knowability, to be knowable is to be true. (So <>K pentails p.)
It turns out that if to be knowable is to be true, then (modulo some sane choices about how propositional quantification works) conjunctive knowability is inconsistent with the S4 axiom for possibility: that <><> pentails <> p. That came out of the blue for me: I didn't see that coming at all , so thanks, Nick, for pointing me in the direction of thinking about this.
If we're happy for possibility to be weak (below S4) and knowledge to be weak (also below S4), then it turns out that there are models in which at every point every proposition is conjunctively knowable, and at every point not all truths are known, so we don't have a collapse into omniscience.
I'll write this up in gory detail soon.
Greg Restall at May 3, 2004 04:23 PM
Thanks Greg, that's agreeable: I would refute Fitch's paradox on the grounds that K can't be reflexively higher-order (on pain of a version of the liar's paradox). However, each kind of knowing might be known by a different kind of knowing (higher level), which allows us to talk about knowability but escapes Fitch's paradox.
RdR at May 3, 2004 05:04 PM
The paper's been updated. The current file is version 0.85, and it contains the gory detail I mentioned above. Check paragraphs 3.11 and 5.1 to 5.8 for the really new bits.
The proof of the failure of transitivity is not as nice as I'd like it to be, but it will do for now. If I can come up with anything nicer, I'll update it again.
It needs some nice pictures, too. But that will come later. I need some sleep.
Greg Restall at May 4, 2004 12:40 AM
Yeah, version 8.5 has more pages and is clearer.
Do you intend to compare your approach with Beall (2000)?
Tony Marmo at May 4, 2004 04:07 AM
Tony: I don't intend to compare my approach with JC's (2000) account, except for my throwaway lines in paragraph 1.4. Do you think I should do more than that?
My reasons for not are that I'm presupposing that the reasoning to Fitch's conclusion is valid, I'm then reflecting on what that should tell us about knowability. It has struck me that in the case of p& ~K p, the verificationist should say that both pand ~K pare knowable, and that might be enough to be getting on with, even if the conjunction isn't knowable. The paper is an exploration of what one can do with that move.
Therefore, the kinds of moves that try to contract the propositional logic of the situation -- to make it possible that ( p& ~K p) is known -- don't appeal to me.
Of course, JC is an extremely bright guy, and one with whom I enjoy collaborating . This isn't to say that we agree on everything.
Greg Restall at May 5, 2004 10:10 AM
Well, to be honest with you, in Linguistics people always ask you to write an overview of what the others have done before you advance your own thoughts. I think it is a choire to do it in every paper. I am not doing it anymore, because journals always impose page limits.
I just asked it because I began to search Beall's paper thru the internet.
Tony Marmo at May 5, 2004 09:08 PM
PS:
As I like to work with models/situations. I think that the following has a good solution to the paradox:
Edgington, (1985) "The Paradox of Knowability" Mind 94, 557-568.
Tony Marmo at May 6, 2004 06:16 PM
By the way, a new paper by John MacFarlane on knowledge attributions might be of interest.
I haven't read it yet. I've found it thru Kai von Fintel's blog.
Tony Marmo at May 12, 2004 09:16 PM
Yes, I've downloaded Macfarlane's paper. I have skimmed through the earlier draft, and I very much like the line (on the assessment relativity of knowledge ascriptions). I'm not sure whether admitting this would change the picture markedly, or keep it pretty much the same. I should have a think about this, but I'm inclined to say that a real consideration of that is for another time and place.
Greg Restall at May 12, 2004 09:44 PM
This sentence by Gere, who also won the Foot in Mouth award by the Plain English Campaign in 2002, is of much greater philosophical interest:
`I know who I am. No one else knows who I am. If I was a giraffe and somebody said I was a snake, I'd think "No, actually I am a giraffe."'
In his post Can Derrida be ever wrong? (september 29 2003), Mark Lieberman makes the claim that Derrida's sentences are nonsensical.
He talks about a game someone played with people who took Derrida seriously. The game consisted of picking one of the long sentences of any document by Derrida and substuting antonyms for its words in order to produce variants of that sentence. Later the original sentence and its variants were presented to people who were supposed to know Derrida's works and the question was: `which sentence is the original?' He claims that Derrida's admirers were unable to establish which one was the original sentence. According to Lieberman, this suggests that Derrida's rethoric is `a sophisticated form of White Noise'.
I confess that Derrida often makes no sense to me either, regardless of whether I read him in English or French. Maybe a Plain French Campaign is in order and French Intelectuals like him should try to express themselves in plain French. But I think the nonsensical aspects of their discourse are not mere instances of odd usage of natural languages. Would Derrida make sense if he wrote his thoughts in simple and direct French????
Tony Marmo at May 16, 2004 02:36 AM
TENTATIVE ANSWER#1
Assume that the answer is no: even if Derrida wrote in plain French his sentences would still mean nothing.
Now we have the following situation: Derrida's admirers think they know that Derrida's sentences mean something. After the game they get to know they do not know what Derrida's sentences mean. But they still assume that Derrida's sentences mean something, whereby they claim they know the sentences mean something but they do not know what their meaning is.
This is a kind of situation where claim that
(1) they know they do not know p.
But (1) is false and yet the falsity of (1) does not entail they know p.
Moreover, if on one hand (1) is false, in natural languages (2) is not the negation of (1):
(2) They do not know they do not know p.
So we go to stage after the aforesaid game was played and Derrida presents his sentences in plain French and everyone finds out that his sentences mean nothing. Now they know that (1) is false. How would one express this `change of knowledge' in a natural language? Would it be (3)?
(3) Now they know that they do not know that they do not know p.
I do not think so. I think that at the stage former admirers of Derrida get at the same conclusion as Lieberman, i.e., they come know that Derrida's sentences mean nothing, the report that would accurately depict this fact is (4):
(4) They know that not p.
Or, if one wants to be more accurate, we have three stages:
(0) They assume they know p When Derrida's sentences seem to make sense.
(1') They assume they know they do not know p. When, after the game, they find out they do not understand Derrida's sentences.
(4') They assume they know that not p. When they find out that in plain French Derrida's sentences mean nothing.
Notes:
[1] Of course, here I adopt a three values system, where if one does not understand a sentence judges it undecidable and not false.
[2] In all of the aforesaid cases we have to consider that there are two propositional attitudes: `to know' and `to assume'.
TENTATIVE ANSWER#2
The second possible answer is yeas:
yeas, if Derrida re-writes his sentences in plain French they make sense.
Now we get these stages:
(0") They assume that they know p.
When they intially think they understand Derrida's sentences.
(1") They assume they know that they do not know p.
After the game has shown them they cannot understand Derrida's sentences.
(4"a) They assume they know that p.
When finally Derrida re-writes his sentences in plain French and show what they mean. But, assuming the KK thesis, (4"b) is the other possibility:
(4"b) They assume they know that they know p.
Tony Marmo at May 16, 2004 11:52 AM
OK, version 0.95 is uploaded now. This one has pictures . (And a nice new argument, too.)
Greg Restall at May 20, 2004 01:30 AM
Dear Greg,
I have too extra questions for you.
First, I am intrigued by the fact that nobody, who insists in defining opacity as failure to apply Leibniz' substitution of identicals, has tried to relate it to Fitch's paradox. I ask if you could tell me why.
Second, is it accurate that Leibniz' Substitutivity of Identicals principle is from his work `Discourse on Metaphysics'? I could not find it there.
Tony Marmo at May 28, 2004 07:28 PM
On Tony's two questions:
I am intrigued by the fact that nobody, who insists in defining opacity as failure to apply Leibniz' substitution of identicals, has tried to relate it to Fitch's paradox. I ask if you could tell me why.
I'm not really sure why, but here's a conjecture. Most people who are interested in formal treatments of Fitch's paradox do so thinking that the formalism of modal logic is the right way to think of the inferential properties of claims to knowledge. And here, even though we have opacity, we do have substitutivity of logical equivalents in this kind of epistemic contexts. I think that this is an idealisation, but an OK idealisation here. (In the paper I talk about reading Kp as pis a consequence of what is known, and this satisfies the substitutivity of logical equivalents.)
Second, is it accurate that Leibniz' Substitutivity of Identicals principle is from his work `Discourse on Metaphysics'? I could not find it there.
I have no idea! Does anyone around here know?
Greg Restall at May 29, 2004 10:46 PM
Thank you, Greg.
I was thinking of sentences like:
(1) Jimmy knows that Superman can fly.
(2) Superman is Clark Kent.
(3) (?)Jimmy knows that Clark Kent can fly.
If Fitch's paradox is considered:
(4) The Hulk does not know that Bruce Banner is smart.
(5) Bruce Banner is the Hulk.
(6) (?) Bruce Banner knows that the Hulk does not know he is smart. (7) (??) Bruce Banner knows that he does not know that he is smart.
What do you think?
Tony Marmo at May 30, 2004 12:17 AM
Greg and all,
I'm a little slow on commenting, but thought it might be worthwhile even at this late date. I've just completed a ms. on the paradox, following up on my `95 piece on it, and have a further piece, in progress, on my website. It argues that no one can be complacent about Fitch's paradox, simply taking the Fitch's proof as presenting a somewhat surprising result. I need to make some changes to it, but here's one way to put the point. "All truths are knowable" is, if true, necessarily true. "All truths are known" is false, at least when the domain in question is finite minds, but seemingly contingently false. And yet if Fitch's proof is sound, the two claims are logically equivalent. That ought to bother everybody...
jon kvanvig at June 16, 2004 08:03 AM
On Jon Kvanvig's comment:
Yes, that bothers me. I look forward to seeing the ms. to see how the bother dissipates. It should be fun.
Greg Restall at June 17, 2004 01:04 PM
Posted by Tony Marmo
at 01:32 BST
Updated: Friday, 25 June 2004 17:48 BST
