Salerno’s Knowledge

Yesterday was the day: Joe Salerno revealed the source of the knowability paradox! It’s normally called Fitch’s Proof, because Fred Fitch first published it in 1963.

But: Fitch says in a footnote that he owes the idea to an anonymous referee. Joe announced yesterday the results of his investigation.

Would it be fair now to add to Church’s Thesis and Church’s Theorem–Church’s proof?


Comments

Salerno’s Knowledge — 47 Comments

  1. Joe suggested that we should now call it the Fitch-Church Paradox (or was it the Church-Fitch Paradox?) because each made significant contributions to the way the paradox is now understood. I don’t know anything about it other than what I have learned from Joe, but his suggestion seemed exactly right to me.

  2. How clear is the attribution to Church? Is this is available records / correspondence or what? What’s the scoop Joe?

  3. Fritz, Joe can say more, but I have in my possession a copy of a referee’s report in Church’s meticulous and absolutely beautiful handwriting (signed by him as well) containing the proof. It has a slight mistake in it, if I remember correctly, but an innocuous one.

  4. Jon actually received from me copies of two referee reports from 1945. The *first* referee report, and the one where we find the first articulation of what Fitch later (in 1963) called Theorem 4, is not actually signed. But it is obvious from the handwriting that it is Church. The knowability paradox is the contrapositive of what Fitch called Theorem 5. Theorem 5 is a development, and a more provocative formulation, of Theorem 4. So importantly Fitch contributed to what today we call the knowability paradox. Moreover, the entire literature on the subject is inspired by Fitch’s publication. For these reasons I am inclined to call the result the Church-Fitch paradox of knowability. Besides (and with tongue somewhat in cheek), wouldn’t it be a bit anti-climactic to slip one of the very few Fitch legacies into the pocket of one who already has such an abundance of logical riches? 🙂

    I hope to have the first three sections of my paper online today or tomorrow (although I’m having trouble converting .dvi to .pdf). I’ll put it in the *in progress* section of my website ( http://pages.slu.edu/faculty/salernoj/Default.html ). The paper is called “Knowability Noir: 1945-1963”. Will continue to update it as sections become presentable. Comments are of course welcome. Moreover, if I receive copyright permissions, I’ll publish the full set of referee reports in my forthcoming volume of new essays on the paradox, and others can decide for themselves where credit is due.

    joe

  5. I would rather call the proof of

    (CT5) WVER –> SVER

    (in Williamson’s terminology) ‘The Church-Fitch result’. In fact, it is not clear whether the proof really constitutes a paradox. Kvanvig claims that it is one, other claim that it is not. ‘Result’ , or ‘Proof’ might be more neutral.

    Actually, Kvanvig offers two arguments for his paradxicality view: a) (CT5) clashes with our intuitions about the distinction between what is actual and what is merely possible; b) if true at all, WVER is necessarily so and, if true, SVER is at best possible so. But possible claims cannot be equivalent with necessary ones.

    Wrt to a), the difference between WVER and SVER is that the former is a modal thesis, whereas the latter is not. The difference is not: the former expresses a possible claim, whereas the latter expresses an actual one. So, it seems, Kvanvig owes us an argument to the effect that modal claims cannot imply non-modal ones (is that latter thesis true? I don’t think so).

    Wrt to b), both WVER and SVER are metaphysical claims. So they are both either necessarily true or necessarily false.

    The realist will think that they are both necessarily false. But still, one might think, there exist possible worlds in which all truths are known. This is false: the realist cannot (or at least will arguably not) endorse such a possibility.

    The antirealist will think that they are both necessarily true.

    So what’s paradoxical?

    The proof might be philosophically surprising, but I’m now convinced that (CT5) is definitely not a paradox.

    (Also, note that Church-Fitch proved a logical equivalence between two possible formalizations of the two notions of truth universally knowable and truth universally known. It might well be that the problem is formal and not, so to say, conceptual.)

    (Although to different formalizations correspond different concepts, of course.)

  6. PS

    Of course WVER and SVER have different senses, and of course we intuitively think so. But note that this is not to say that we have an intuition according to which WVER and SVER should not be logically equivalent. At least in Fregean semantics (as Cesare Cozzo has reminded me), logically equivalent claims *can* have different senses (this explains why, when we draw mathematical inferences, we learn and know something new).

  7. Julien, you say,

    Actually, Kvanvig offers two arguments for his paradxicality view: a) (CT5) clashes with our intuitions about the distinction between what is actual and what is merely possible; b) if true at all, WVER is necessarily so and, if true, SVER is at best possible so. But possible claims cannot be equivalent with necessary ones.

    Wrt to a), the difference between WVER and SVER is that the former is a modal thesis, whereas the latter is not. The difference is not: the former expresses a possible claim, whereas the latter expresses an actual one. So, it seems, Kvanvig owes us an argument to the effect that modal claims cannot imply non-modal ones (is that latter thesis true? I don’t think so).

    Wrt to b), both WVER and SVER are metaphysical claims. So they are both either necessarily true or necessarily false.

    I think neither of these arguments are my arguments. The second one I clearly rejected in the book, based on the distinctive S5 axiom. The first claim you cite is a bit closer to what I actually argued, but there was nothing in my argument that requires defending the claim that modal claims cannot imply non-modal claims.

    The argument I gave was this: there are two claims, one which is a claim about actual knowledge and the other about possible knowledge. These two claims, if the proof is granted, are logically equivalent. But in general, claims about what is actual and claims about what is possible are not logically equivalent, so we need an explanation here to free us from the air of paradox. Much of the book is an investigation of what sorts of explanations might be given here.

  8. I see. (Thanks much)

    But still, I think, the realist could insist that those two claims – one about what is actual, the other about what is possible – are both necessarily false, no matter what they are about (arguably, that’s what she *will* say). So she might have good reasons to welcome the logical equivalence proved by Church-Fitch.

    A corollary of the paradoxicality-thesis was that Church-Fitch raised a logical problem for both parties: the realist and her opponent. If the argument above is correct, apparently, such a corollary would be false.

    What do you think?

    (If correct at all, I submit, the argument might weaken a little bit the need for adopting a neo-Russellian theory of propositions).

  9. What’s crucial about the paradox is not that the two claims are true in all and only the same worlds, but that they are interderivable. It’s a proof-theoretic result, so the fact, if it is one, that both claims are necessarily false doesn’t alleviate concern about the the interderivability claim.

    Here’s a different example. It’s one thing to claim that 2+2=4 has the same modal status as the claim that water is H2O. It would be more troubling to find a proof that one could derive, from a first-order logic together with a knowledge operator and a possibility operator, that each claim could be derived from the other.

    So, yes, many realists will claim that both claims are necessarily false. But that doesn’t resolve the paradoxicality.

  10. Thanks much for the clarification, I think it’s helpful.

    Myself, I’m inclined to insist that Church-Fitch only proved that a certain formalization of the knowability claim collapses into strong verificationism, although I know Edgingtonian claims run the risk to be trivial sometimes…

  11. A quick point. 2+2=4 and water = H2O are indeed more ‘distant’ than WVER and SVER are. Moreover, I think it might be tendentious to say that the first claim is about possible knowledge whereas the second is about actual knowledge. Rather, they both seem to be claims about truth.

    Now the first one says that all truths are knowable, in the sense that

    (1) For every p true at some w, there is a world x relative to w at which it is known that p is known at x.

    Is that true? Church conjunctions, I think, prove that this claim is actually false. But I know this is to beg the question. The following argument might lead us to abandon (1) for reasons that are perfectly independent from the Church-Fitch result.

    Let’s substitute

    (2) The president of the United States is male

    for p in (1). For some world x, it seems, the president of the United States is not a male. So knowledge of (2) in different worlds could be knowledge of something different (the argument, if I represent it correctly, is Helge Rueckert’s).

    As long as we have independent reasons to abandon (1), it seems, one might wonder (i) whether (1) really expresses the anti-realist’s notion of knowability and (ii) why, after all, we should block the derivation of SVER from (1).

    Finally, if one already has independent reasons to think that (1) is false, it should not be surprising that (1) leads to implausible conclusions such as SVER.

    Very Best

    JM

  12. There’s nothing tendentious about the characterization in question, since nothing whatsoever hangs on the terminology. What is important is the presence of a possibility operator in one and its absence in the other. Claiming that both statements are statements about truth simply ignores this point.

    The argument involving (2) I can’t decipher. The fact that the Pres. is female in some worlds is irrelevant to the truth of (1). (1) only requires that there be some world where (2) is known to be true. That won’t be a world where the Pres. is female, of course, since knowledge requires truth.

    I also don’t grasp your last point. The fact that a given claim has a certain truth value doesn’t change it’s derivability relations. So even if (1) is false, that doesn’t change the fact that it is surprising when a claim about actual knowledge is interderivable with a claim about possible knowledge. The truth or falsity of the claims in question is irrelevant to what’s paradoxical here.

  13. As regards the Pres. argument, here’s Rückert’s own formulation. I quote from Rückert (2003: 369-70) – he’s considering the prospects for the schema (1).

    “Things look different when we are concerned with contingent truths (or necessary, but only a posteriori knowable truths), because then the knowing subjects have to know something (for example via experience) that might be different between the two possible worlds. An example: Take α to be “The President of the United States is male”. In our real world in order to know α one has to have had experiences that are somehow related to George Bush (to know α is knowing something of Bush), whereas in another possible world, in which Al Gore would have become president, one needs experiences that are somehow related to Al Gore in order to know α (to know α is to know something of Gore). That’s the deeper reason why we rejected schema (1) in the beginning (…): It is not at all clear that someone in one possible world having knowledge which he expresses by α and someone in another possible world having knowledge which he expresses by α, know ‘the same’. As the example above shows, their knowledge might be of different objects, a very good reason for assuming that they are not knowing ‘the same’.”

    It seems that you’re right: for the reasons you stated above, the argument does not seem to tell against (1). (Maybe Helge could add something on this.)

    Concerning the disappearing diamond in the equivalence licensed by Church-Fitch, here is another modal collapse (cf. Carrie Jenkins, The Mystery of the Disappearing Diamond, forthcoming in Joe’s book):

    (3) (p –> ◊p) (p –> p).

    Why should (LD) – the equivalence between WVER and SVER – be paradoxical and not (3)?

    You’ll probably say that the inter-derivability of two logical truths is not at all surprising, whereas the derivability of SVER from WVER *is* surprising (and, indeed, paradoxical).

    At this point I’m a bit puzzled and I don’t know much what to think.

    For, on the one hand, if logic tells us that diamonds can harmlessly (and proof-theoretically) disappear, one might wonder why we should find perplexing the fact that a diamond is missing within the non-logical right hand-side of (LD).

    On the other hand, it is true that, although our intuitions about the distinction between actuality and possibility basically just say that p cannot follow from ◊p, if asked whether the inference from ‘All xs are F-ble’ to ‘All xs are F’ is valid, competent speakers will probably tend to say “No, it is not”.

    So yes, in a sense, I might give up here. But it should be noted that (i) the inference above is only *intuitively* incorrect, (ii) the distinction between All xs are F-ble and All xs are F is not a paradigmatic case of the actual/possible distinction and (iii) (very) competent speakers such as Timothy Williamson and Carrie Jenkins don’t find the inference – or at least its exemplification (LD) – paradoxical at all.

    A minor point. You write:

    “The fact that a given claim has a certain truth value doesn’t change it’s derivability relations.”

    In a sense, this is true. However, if ex falso quodlibet holds good (we’re not Graham Priest!), the truth or falsity of a claim seem to have an import on its derivability relations.

    A final (quick) point. It seems that the naïve comprehension principle – which intuitively seems perfectly true – is logically equivalent to and inter-derivable with 0=1. Should we follow our intuitions and change our logic (or our semantics) to block the problematic inference? Or should we rather modify the intuitive formulation of the principle itself?

    So one might wonder whether Russell’s proof is a real paradox. The contradiction he derived might just be a clue (something like: Contradiction! There must be something wrong with the intuitive principle). Jenkins raises similar worrie. There might be an analogy with the Church-Fitch proof (maybe).

    Sorry for the long post!

  14. Carrie’s example (3) is quite interesting, but I think in only a limited way. You claim,

    You’ll probably say that the inter-derivability of two logical truths is not at all surprising, whereas the derivability of SVER from WVER *is* surprising (and, indeed, paradoxical).

    In a way, that’s part of what I’d say. They are trivial logical truths, provable in one step each. So the equivalence is not surprising, in part because of the triviality of the logical truths in question and in part because neither truth is needed to prove the other.

    The equivalence is surprising, though, in the case of Fitch’s proof. You claim Williamson doesn’t think so, but that’s not so. He thinks that the proof is surprising, but that it is valid nonetheless. We’ve talked about this at some length. Whether Carrie thinks it is surprising, I don’t know, but nothing in her discussion hinges on denying the point. Moreover, it would be an amazing discovery if the interderivability weren’t surprising, since then we’d have to question the command of English possessed by anti-realists who have incautiously endorsed the knowability claim. Surely their problem is not a linguistic competence one.

    On the point about paradigmatic instances of the actual/possible distinction, I think the book is very clear about exactly what distinction is in question, and the choice of terminology here is purely stipulative. There is a possibility operator in one claim that is not in the other; nothing more is intended or needed.

  15. I think both Carrie and Williamson believe that, however surprising, the proof is definitely not paradoxical (although the distinction is a vague one, as you say in the book).

    At least, that’s what Carrie says and Williamson writes (true, in his book he’s only considering the ‘old puzzle’. However, when I presented him last year with the new one, he said that many first order logic equivalences were surprising as well – I should have his examples somewhere in Italy).

    Anyway, I don’t know whether this really matters. The thought would be: since the fundamental paradoxicality Kvanvig detects in (LD) rests on our deepest modal intuitions, one might wonder why some competent speakers that apparently share such intuitions don’t find (LD) paradoxical. (Do they have different intuitions?) I should give the word to them, however.

    Best

    JM

  16. Hello all!

    Julien has informed me about the discussion going on here, so I’ll join it (even if I don’t have a lot of time and shoudn’t think about Fitch, or better Church-Fitch, at the moment). Just a few brief comments:

    1) Concerning the name of the paradox: If it has been found out (and it looks like that!) that the referee from whom Fitch heard about the derivation which leads to Fitch’s paradox was Church, then we should give him the credit and call the paradox the Church-Fitch-paradox in the future (even if it is a bit difficult to pay attention to this every time the paradox is mentioned at first).

    2) What is the paradox? I don’t think that one should call a certain theorem or a derivation in a formal system a paradox. These are just what they are: technical results in formal systems.
    On the other hand, a paradox is some argument which leads from premisses and assumptions which all seem to be plausible to an implausible conclusion. So, in the Church-Fitch-case the paradox is not just the theorem or the derivation, but we only get a paradox if we assume that the formal system that is used is the adequate tool to formalise thoughts about truth, knowledge, metaphysical possibility etc. (looked at it in isolation it is just a formal system, nothing else), if we assume that the relevant formulas are good translations of the corresponding philosophical theses, and if we assume that the anti-realistic thesis taht every truth might possibly be known is plausible and strong verificationism is not (or at least, that we hold it plausible that the anti-realistic thesis, even if false, is a weaker claim than strong verificationism).
    Now, I think that Church-Fitch’s proof etc. is perfectly valid (within that formal system!, there cannot be any doubt about that) and that the corresponding formal translation of the anti-realistic thesis is perfectly legitimate one. On the other hand, I think that the anti-realistic thesis has another reading (as it often happens with formulations in natural languages) according to which it doesn’t collapse to strong verificationism. This relates to the kind of knowledge involved…

    3) As Julien has refered to an example of mine (and I think his presentation was at least a bit too sketchy, mislaeding and out of context), I should say some more about it (to do so I quote from a recent abstract of mine):
    “Here, a short sketch of my argumentative strategy to defend a version of soft anti-realism:
     One has to distinguish two aspects of knowledge: (1) How knowledge is (or might be) expressed by sentences or propositions. This aspect is captured by what I call knowledge de dicto. (2) The subject matter which the knowledge is about. This aspect is captured by what I call knowledge de re. If in our world someone knows de dicto that George W. Bush is male, and another one knows de dicto that the president of the U.S. is male, their knowledge de dicto differs but they nevertheless have the same knowledge de re because they know of the same person to be male. On the other hand, when we consider two knowing subjects who both know de dicto that the president of the U.S. is male, but live in different possible worlds (say one lives in our world and the other one in a world in which John Kerry had won the elections), then despite having the same knowledge de dicto they differ with respect to their knowledge de re because they know something about two different persons.
     Standard modal epistemic logic, which is used in the derivation of Fitch’s Paradox, is biased towards the first aspect of knowledge and the second aspect is neglected. Therefore, I propose to make use of the idea of subjunctive markers which has been developed by Wehmeier (2003) for modal logic for epistemic logic, too: If knowledge de re is at stake, the formula in the scope of the knowledge operator has to be in the indicative mood and if knowledge de dicto is at stake, the formula in the scope of the knowledge operator has to be in the subjunctive mood.
     The version of anti-realism I propose claims that what is at stake in (ART) is not knowledge de dicto, but knowledge de re. (Thus, one might say that in a certain sense I am defending a ‘realistic anti-realism’). It is easy to show that by formalising this idea adequately the derivation of Fitch’s Paradox can be blocked.”
    To sum up: I think the anti-realistic thesis which says that every truth might possibly be known has (at least) two readings (for illustrative purposes I use an instance):
    1) If it is true that the president of the US is male, then there is a possible world and there is someone in that world who knows the proposition “The president of the US is male”
    The corresponding general thesis is false (as shown by Church-Fitch!)
    2) If it is true that the president of the US is male, then there is a possible world and there is someone in that world who knows about the president of the US (namely Bush!)that he is male. This knowledge can be haved without knowing the proposition “The president of the US is male”, and in lots of possible worlds (namely in all worlds in which someone else, and not Bush, is the president of the US!) knowing this proposition would not even do the job!
    The general thesis according to reading 2) is the one, I think, that an anti-realist should endorse and defend.

    Much more to say, but I have already written too much…

    cheers
    Helge

  17. Hi Helge, in post 14 I quoted your example at lenght. Kvanvig’s point was that although in different possible worlds knowers may know something different when they know that the president of the US is a male, still this seems to be irrelevant to the truth of ART (for the uninitiated reader: ART is WVER, or (1) in post 12). After all, as long as there are possible words (relative to ours) in which Bush won the presidential elections and

    (3) The president of the US is male

    is known, the fact that Al Gore (or whoever) could have won the election does not seem to have any import on the truth of standard interpretation of the knowability claim. To repeat, what is required for the truth of this latter claim (if p at w, then, at x, Kp), is that there be some possible world x at which p is known and, if you want, knowledge of p in x and knowledge of p in w are knowledge of ‘the same’. But indeed, as far as (3) is concerned, there exist such worlds. So I agree that the way I first mentioned your argument was too sketchy, but I think Kvanvig reply was correct nevertheless.

    Concerning your views about what is a paradox, I totally agree with you.

    Best

    J

  18. I just thought I’d add a comment on the point I make (in my paper in Joe’s collection) that the collapse of
    1. p –> ◊p
    and
    2. p –> p
    is not paradoxical but (by Jon’s lights) it should be for the same reason he thinks the Church-Fitch result is.

    Jon’s response above seems to be that this is disanalogous to the Church-Fitch case, as the the inter-derivability of two logical truths is not surprising and that neither of 1 and 2 is needed to derive the other.

    I just wanted to flag that I anticipate this kind of response in my paper, and discuss another case which it does not work against, namely the interderivability of:
    3. A -> (AvB)
    and
    4. A -> B
    These are not logical truths and each is needed in order to derive the other.

  19. Hi Carrie,

    yes, but in your latest example, there is no box (or diamond) which simply disappears.
    So, it would be good to have an uncontroversial example of formulas which are not logical truths, but which are derivable from each other, and the one differs from the other by a mere box (diamond).
    I’m not sure whether such examples exist (should exist).
    In the formalism which I prefer (modal epistemic logic with subjunctive markers), I can give a reconstruction of Church/Fitch’s proof, of course. But then, in this formalism, there is not just a diamond disappearing, but some things happen with the subjunctive markers, too.
    So, it might by interesting whether there are, in the standard formalism, uncontroversial disappearing boxes (diamonds) which are not more or less abviously vacuous and redundant.

    Helge

  20. @post 21 from Julien:
    According to the standard reading of ART worlds in which the president is not male, are pretty much irrelevant, of course (because, in such worlds nobody can know that the president is male because is isn’t). And, on the other hand, this special instance of ART is already verified by a world in which Kerry is president and somebody knows that the presindent(Kerry)is male. According to the standard conception it is completely irrelevant whether ‘the president of the US’ in the other worlds denotes Bush or not.
    But my point is, that there is a sensible reading of ART, and according to this reading it is irrelevant how the non-actual knower refers to Bush in his knowledge (whether by ‘the president of the US’ if possible, or something else). It is not even required that in this other world Bush (the actual president of the US) is the president of the US there. What is decisive according to this reading of ART is that there has to be a world in which somebody knows in the de re sense about Bush that he is male (in this world need not to be called ‘Bush’ either, of course).

    Helge

  21. Indeed, I see a diamond in post 23. It might just be a technical problem.

    Wrt to post 25, I think I don’t really understand your positive thesis.

    So here’s the quote from Helge (2003) in post 14:

    “Things look different when we are concerned with contingent truths (or necessary, but only a posteriori knowable truths), because then the knowing subjects have to know something (for example via experience) that might be different between the two possible worlds. An example: Take α to be “The President of the United States is male”. In our real world in order to know α one has to have had experiences that are somehow related to George Bush (to know α is knowing something of Bush), whereas in another possible world, in which Al Gore would have become president, one needs experiences that are somehow related to Al Gore in order to know α (to know α is to know something of Gore). That’s the deeper reason why we rejected schema (1) in the beginning (” … ): It is not at all clear that someone in one possible world having knowledge which he expresses by α and someone in another possible world having knowledge which he expresses by α, know ‘the same’. As the example above shows, their knowledge might be of different objects, a very good reason for assuming that they are not knowing ‘the same’.”

    It seems that the argument is presented as a potential concern for WVER. You write: “That’s the deeper reason why we rejected schema (1) in the beginning”, where (1) in Helge’s paper is WVER, the standard S5 (say) formalization of the Knowability Principle (Helge’s ART). Or am I missing something? I thought the argument was meant as an independent (from Church-Fitch) reason the anti-realist could have for preferring an alternative reading of the Knowability Principle (ART) to the standard one.

  22. It is important to specify *which* normal modal logic one is working under. This collapse would not occur in K, for instance. But here I assume we are discussing a result of S5. And since we know that ‘p only if p’ is valid w.r.t. the class of all reflexive, euclidean models, and that p only if p is a tautology (and therefore valid in all models), then both formulas are valid in the class of S5 models. There is nothing paradoxical about this. It is a fact about modal logic S5; indeed, it is a fact about the (weaker) logic KT. And it is not, to repeat, a fact about K.

    The ‘paradox’ is generated by the interpretation of the modal language: the valid formulas of K5 (or even KT, restricted to the example at hand) appear to clash with our extra-logical commitments about how belief works, or how belief should work. (Other ‘paradoxes’ include how even K is too coarse to properly model moral obligation, for instance, by ruling out the possibility of an agent having inconsistent moral obligations.)

    Framing the discussion in terms of a paradox might masks the nature of the problem (namely, that it is a problem of applied or philosophical logic, not a problem in modal logic) and to mask the sound reasons for the logic to behave this way.

  23. Drats. I’ve lost my diamond in the post. Try ‘p only if Dp’ is valid w.r.t. the class of reflexive, euclidean models.

  24. Hi Helge,

    “yes, but in your latest example, there is no box (or diamond) which simply disappears.”

    The point of this example is that if in the original case we should be worried about a ‘lost distinction’ between diamond-p and p, so in the new case we should be worried about a ‘lost distinction’ between AvB and A.

  25. So, to continue humming this tune, saying that it would be nice “to have an uncontroversial example of formulas which are not logical truths, but which are derivable from each other, and the one differs from the other by a mere box (diamond),”

    is unclear. p -> p is a propositional tautology, whereas p -> Dp is a theorem of some, but not all, systems of modal logic. Both are “logical truths” of the logic S5, however, in the sense that both are validities on S5 models; and both are theorems of S5, as one would expect.

    Also, I am not convinced that p -> p is not needed to *derive* p -> Dp, as remarked in [23]. I would like to see this proof. (Recall that propositional logic is built into propositional modal logic, so instantiating instances of T and K and manipulating those won’t establish the claim.)

  26. Jon – In my post 23, I lost the negations in front of the As! The equivalence I wanted to point out is of course between:
    not-A -> B
    and
    not-A -> AvB

  27. Carrie, yes, that’s an interesting example, and advances the discussion. Remember, though, that my concern is not simply about lost distinctions. It is about unexplained lost distinctions. The lost distinction you mention is easily explicable in terms of any model instancing the antecedent of the conditionals. So here the explanation of why the lost distinction isn’t troubling is a straightforward semantic one, just as with the S5 explanation of why diamond-box and box are interderivable. I’d be perfectly happy in the knowability case if there were a similar explanation!

  28. It seems that we have a (valid) proof, and that there are no models in which WVER is true ans SVER is false. I think Helge was right. We have a real paradox only if we think that our formalism correctly frames our intuitive notion of knowability. If other non-paradoxical interpetations of knowability are available, then the result shouldn’t be troubling. (But I know Carrie’s position is different.)

  29. Methinks too much protestation against the formalism here. We have two formulas and the question is what they say, in your preferred natural language. One of them surely *seems* to say that all truths are known and the other surely *seems* to say that all truths are knowable. If they don’t say that, then we deserve an account of what they do say and why the intuitive rendition is false. For if the intuitive rendition is true, then we have a very surprising logical equivalence, given the Fitch result, for which we are in need of an explanation of how it could be true.

  30. I take WVER to say something like: “All propositions, no matter at which world they are true, are knowable, no matter at which world they are known”. I think this is false, as Church-Fitch reminds us, much in the same way as “All sentences are correctly assertable” is — see Hand’s example “Je ne te tutois jamais”. After all, by stating that, if p at w, then it possible to know that p, I think we should be interested in possible knowledge of p at w — not of p at some x. Anyway, I think that’s the basic thought behind Edgington-like solutions of the problem. Truth is truth at a world, not truth tout-court. Your proposed invalidation of the proof, I think, is Edgingtonian in spirit. The same for Carrie’s and Berit & Joe’s recent (forthcoming) proposals. I’m convinced that Edgington correctly frames our notion of knowability (or, at least, the one that should be employed by anti-realists). So I don’t believe that Church-Fitch actually proved a logical equivalence between the two notions of truth universally knowable and truth universally known. Rather, I’m inclined to think that the proof just shows that a false claim, such as WVER, has bad consequences. I find Carrie’s psychological explanation in her mystery paper quite convincing. If we check Church-Fitch’s proof carefully, we should find it plausible. Carrie probably thinks that WVER *is* the (informal) Knowability Principle, and argues that the provable equivalence between the two notions of truth universally knowable and truth universally known is not and should not be paradoxical. I think she offers sharp arguments to this effect. Nevertheless, I am myself inclined to think that those two notions should *not* be provably equivalent. But it is just an intuition, whatever intuitions are.

    Anyway, here’s a question for you.

    Suppose Edgington’s proposal was not so problematic as it is commonly thought (this might be an impossible world, I concede that). Should we still consider the provable equivalence between WVER and SVER highly paradoxical, in your view?

    Very Best

    Julien

  31. Julien, you’re right that my proposal is very close to Edgington’s. I’m not happy with your gloss of WVER, though. I don’t think we should introduce concepts into the natural language characterization that don’t have any formal counterparts in WVER. So we are entitled to talk about possibility, knowledge, quantification, and the like, but not about worlds.

    What I’m objecting to in the above discussion is the focus on the Fitch result that begins from anti-realism and moves to a formal characterization of it. Then the discussion revolves around whether the characterization is faithful to anti-realism. That’s backwards from the way the proof ought to be treated, I think. I think you start with the formal claims and the Fitch result and try to say in ordinary language what the formalization amounts to. At no point is faithfulness to anti-realist motivations an issue, when looked at from this point of view. Instead, what counts is whether the ordinary language rendition makes for paradox, and if it does, how it can be resolved. As I argue in the book, too much ink has been spilt thinking about the proof result from the parochial perspective of whether it causes a problem for anti-realism.

  32. Julien, I forgot to reply to your question. As I wrote above, my position isn’t that the equivalence is the problem. It’s only a problem if there’s no good explanation of it. So if Edgington’s proposal explained the equivalence, then even if surprising, anything paradoxical here would have been resolved. I want to spend some time looking at Carrie’s paper, which I haven’t done yet; perhaps what you term her psychological explanation will do the trick.

  33. Jon, in reply to your post 35, if that’s all it takes to count as having provided a ‘semantic’ explanation, I wonder why isn’t it enough in the Church-Fitch case to say that the two relevant semantic features of the knowledge predicate are that it is factive and distributive? From these features it follows that there is no model where K(p&not-Kp), and thence it follows that if p&not-Kp is true, some truths are not knowable (i.e. that if all truths are knowable no truth is of the form p&not-Kp).

  34. Carrie, the semantic features in question in 35 are simply truth values and truth-functional accounts of the connectives. The knowledge operator’s being factive and distributive are, I take it, proof-theoretic features of the operator.

    You are, of course, right that there is no model where K(p&~Kp): that’s provable on any view that makes K factive and distributive. The question is what follows from that. Your last claim assumes, as I’m sure you realize, a connection between the existence of a model and some claim about possibility such as an unrestricted rule of necessitation provides. And we should grant that rule here only if we can explain the surprising result in some way other than just pointing to the proof theory that generates the surprising result. I’d be happiest with a neat semantic theory that supports the result, but I don’t think that only a semantic story will do.

  35. Here’s another (psychological) argument for the non-paradoxicality of WVER iff SVER (maybe). Consider Fitch’s Theorem 5 of 1963:

    “If there is a true proposition which nobody knows (or has known or will know) to be true, then there is a true proposition which nobody can know to be true”.

    Since – on pain of contradiction – we cannot know our own ignorance of any given truth, the theorem seems quite plausible.

    Moreover, consider Fitch’s Theorem 4 (which is, by the way, Church’s):

    “For each agent that is not omniscient, there is a true proposition that that agent cannot know”.

    Formally, both theorems look quite similar. The second, I think, seems even more compelling than the first.

    Two concerns:

    1) If it is thought that *those* theorems are not paradoxical, then one might wonder why their contrapositives should be so.

    2) If it is thought that those theorems are intuitively valid, then one might wonder why they are both *invalid* once we go neo-Russellians and we accept the modal indexical package.

    Best (and thanks much for the answers above).

    Julien

  36. Julien, I don’t think either 4 or 5 are intuitively obvious, and of course they stand or fall with the proof of SVER from WVER. I don’t see any psychological facts here beyond what was already true of the original proof.

    What you say after quoting theorem 5 is true, but the theorem only follows from what you say given the rule of necessitation.

  37. Yes, I know… And your proposal requires a restriction of that rule. I just wanted to stress that both results strike me as quite acceptable. (This might be a psychological fact!)

    Suppose the war is over, but soldier S never stops fighting until he die. In fact, he’s in a little tiny island in the Pacific Ocean. Nobody bothered to tell him about the end of the war. The war ends. So there’s at least one true proposition S never knew, namely that the war is over.

    Was it possible, for him, to *know* that the war was over but he would never now that the war was over?

    Maybe, he’s been wondering for years whether the war was over. Actually, he probably feared all along that the war was over but he would never know.

    But he *couldn’t* know, I would say (as long as K is factive and distributes over conjunction).

    This is not intuitively obvious, as you say, but it is (at least) intuitively correct.

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