Thursday, 24 February 2011

A Liar Paradox of Material Implication

Here I present a new objection to the material or "hook" analysis of indicative conditionals - the thesis that an indicative conditional 'If A then C' has the truth-conditions of the so-called material conditional - based on Liar-like reasoning. This objection seems invulnerable to any Grice-Lewis-Jackson-inspired pragmatic rejoinder.

(1) If (1) is true, (1) is false.

Let us call (1)'s antecedent 'A', and its consequent 'C'. I propose that the following sentence is intuitively true, or true based on intuitive and unproblematic reasoning:

(S) On the assumption that (1) is neither true nor false, A and C are false.

The reasoning is: Assume that (1) is neither true nor false. Then A is false, since it says that (1) is true, and C is false since it says that (1) is false.

If we accept the hook analysis, however, this reasoning is not secure. For if A and C are false, and the hook analysis is right, then (1) must be true. But if (1) is true, then A must be true (since it says that (1) is true), and C must be false (since it says that (1) is false). But then, by the hook analysis, (1) must be false, since it has true antecedent and false consequent. But if (1) is false, then A must be false since it says that (1) is true, but then by the hook analysis (1) must be true. But then...

The point is, the hook analysis treats (1) as a truth-functional compound, and this places it squarely in the Liar family, making our straightforward argument to (S) veer into paradox. Yet (S), and our argument for it, seem clearly correct. Therefore we should reject the hook analysis.

There are already plenty of intuitions around which seem to cast doubt on the hook analysis. This one has more bite, I submit, since it cannot be explained away with the customary sorts of pragmatic story. Take a case like 'If I die tonight, I will be alive tomorrow'. The typical proponent of the hook analysis will maintain that, given that I will not die tonight, this sentence is true but not assertable, since I should assert something stronger,1 or something robust with respect to the antecedent,2 etc. But (S) is an example of an intuitively true sentence which comes out as paradoxical (i.e. leads to paradox) if we apply the hook analysis to (1). It is hard to see how any Grice-Lewis-Jackson-inspired pragmatic story could account for our asserting, or treating as true, sentences which are "really" paradoxical.

That's the objection. While not exceedingly complicated, it is quite easy to misunderstand, so I shall conclude with a few clarifications. Firstly, the argument is not: when we apply the hook analysis to (1) we get Liar-like paradox, and since Liar-like paradox is undesirable, we should reject the hook analysis for (1). It is irrelevant to my objection whether Liar-like paradox is good, bad or indifferent. It is also irrelevant whether there is (or could be) a solution to these paradoxes. The point is simply that, intuitively, we do not get into Liar-like paradox with (1) and (S), and so the hook analysis seems to deliver the wrong answer on this point. For a truth-functional analysis of English conjunctions, on the other hand, generation of Liar-like paradox would be the intuitively right answer for certain sentences (e.g. 'This sentence is false and this sentence is false').

Secondly, I put (S) in the form 'On the assumption that X, Y', because if I had used the conditional form, the objection would have become messy through having to avoid begging the question against the hook analysis.

Finally, I am not maintaining that (1) is in no way paradoxical. It is paradoxical. To illustrate:

Suppose (1) is true. Then by (1) and modus ponens, it is false. Therefore, by conditional proof, if (1) is true then it is false. But that is just what (1) says, so it is true, but then by modus ponens (this time not within the scope of any assumption), it is false. Paradox.

What should be said about this and similar arguments, I regard as an open problem. Some thoughts: rejecting unrestricted conditional proof seems like a promising avenue, since several authors have done this for independent reasons.3 However, it is hard to shake the feeling that if (1) is true, it is false. Perhaps the object of this feeling could be accounted for as a 'Mackie conditional' or 'telescoped argument',4 and thus kept semantically distinguished from (1) read as an ordinary conditional. But if these telescoped arguments turn out to be truth-apt in some sense, and to sustain modus ponens, we would seem to be back where we started.

In sum, (1) does not appear to be a (full) member of the Liar family. An important difference can be expressed thus: while there are arguments involving the assumption that (1) is true which lead to paradox (and not just within the scope of the assumption), the bare assumptions that (1) false, or that it is neither true nor false, do not intuitively yield any such arguments (as they do with Liar-like sentences). Hence, we can and should accept (S) as straightforwardly true. And this means rejecting the hook analysis.

Tristan Haze
The University of Sydney


References

Beall and Murzi. draft. 'Two flavours of curry paradox'.
[draft available on the authors' websites, where it is listed as under review]

Bennett, Jonathan. 2003. A Philosophical Guide to Conditionals. Clarendon Press, Oxford.

Grice, Herbert Paul. 1975. Logic and Conversation, in The Logic of Grammar, D. Davidson and G. Harman (eds.), Encino, California, Dickenson, pp. 64-75. Reprinted in Grice (1989).

Jackson, Frank. 1979. 'On assertion and indicative conditionals.' in The Philosophical Review 88, 565-589.

King, Peter. 2004. "Peter Abelard" in The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2008/entries/abelard/>.

Lewis, David. 1976. 'Probabilities of conditionals and conditional probabilities.' in Philosophical Review, 85(3):297315. Reprinted with Postscript in Philosophical
Papers, Volume II, pp. 133-152.

Mackie, J.L. 1962. Counterfactuals and causal laws, in R.J. Butler, (ed.), Analytical Philosophy, 1st series, Blackwell.

Thomason, Richmond H. 1970. 'A Fitch-style formulation of conditional logic' in Logique et Analyse, 52:397412.

1 cf. Grice (1975), Lewis (1976).
2 cf. Jackson (1979).
3 According to King (2004), Abelard rejected something like conditional proof. More recently, cf. Thomason (1970), Bennett (2003), and Beall and Murzi (draft).
4 cf. Mackie (1962).

41 comments:

  1. Thanks to Nick J.J. Smith, Graham Priest and Hezki Symonds for helpful comments on drafts of this article.

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  2. I tentatively prefer the Kripke Strong Kleene least-fixed-point account of truth, so I feel I must object here on several points. First, (S) is not true; it is neither true nor false. The argument goes as follows:

    T("S") will always have the same truth value as S.
    (1) is neither true nor false.
    (A) is just T("(1)"), so A is neither true nor false.
    (C) is just the negation of that; in the strong kleene valuation, then, it is also neither true nor false.
    The conjunction is then neither true nor false, also by the properties of the valuation.

    Speaking in broader terms, it is far from safe to assume that an assertion which hinges on the truth values of meaningless statements is meaningful.

    Second, it is argued that pragmatic accounts will not be able to give a good reason for asserting, or treating as "true", sentences which are actually paradoxical. Tim Maudlin gives such an account in his book about truth (which advocates the theory I employed above). The argument is simple: although (A) is neither true nor false, asserting it seems to be helpful. (This also is supposed to explain how it is that we can assert that (1) is neither true nor false; our claim there is neither true nor false, but is nonetheless assertible.)

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  3. Thanks for commenting! It is not clear to me what it comes to (at the very least in this context) to prefer the Kripke Strong Kleene least-fixed-point account of truth. Here's why I'm struggling:

    Kripke's main result here - the fixed point theorem - shows the possibility of defining a truth predicate (in a precise technical sense of that term) in a non-bivalent formal language. My argument was carried out in English. The fixed point theorem does bear on this in any obvious way (or so it seems to me). Now, presumably the object of your tentative preference is not simply this theorem itself, but rather the view that Kripke's theory somehow holds of natural languages and the notion of truth, or at least has some kind of explanatory, descriptive or revisionary implications for our thinking on these things.

    In lieu of light on this, for now I shall respond to your points on their merits as visible to me now.

    The first premise of your counterargument is: 'T("S") will always have the same truth value as S.'

    (I assume that when 'S' is a sentence, 'T("S")' is a sentence predicating truth of 'S'.)

    As a principle about English sentences, this seems simply false.

    Surely, there are meaningless sentences. Surely, meaningless sentences are not true or false. Surely, therefore, it is false to say that a meaningless sentence is true. So there must be some (meaningless) sentence MS such that T("MS") does not have the same truth value as it. So your first premise can't be right as a principle about English sentences, in which case your argument is not sound.

    Later, you write: 'Speaking in broader terms, it is far from safe to assume that an assertion which hinges on the truth values of meaningless statements is meaningful.'

    I think I agree with you about this, although perhaps not for the same reasons as you. But I don't think I make such an assumption. (I should say at this point that I have not committed myself on whether any of my specimen sentences is meaningful or not. Furthermore, I regard the notion of meaningfulness as quite indefinite, multifarious and open.)

    One worry: you talk of an assertion hinging on the truth-values of meaningless statements. But what is hinging? One natural suggestion would be that an assertion hinges on something iff it can be true or false depending on how that thing is. But then any statement which hinges on anything would presumably be meaningful.

    Certainly, a statement can appear to hinge (in the just-defined sense) on something without really hinging on it. And this appearing-to-hinge is compatible with meaninglessness, perhaps. If we allow this, in a more liberal sense of 'hinge', I am happy to agree with you that is it 'far from safe to assume that an assertion which hinges on the truth values of meaningless statements is meaningful'.

    I do however, incidentally, maintain that it is meaningful (false) to predicate truth and falsity of meaningless sentences. I have no theoretical reason for this. It just seems clearly right, and I have not seen any reason to suppose it isn't.

    There is something quite unsatisfying about these replies: they don't engage with Kripke's theory of truth. Still, I think they are valid replies, since I don't think it has been made plausible or clear that Kripke's theory of truth has any definite bearing on my argument.

    So, in my capacity as a defender of my argument, I'm shifting the burden back to you and saying that it's not yet sufficiently clear that you've given me anything to worry about. However, I do think you raise an inherently interesting and confusing issue about how Kripke's theory of truth might bear on natural language (and on truth itself!).

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  4. Correction. In the above comment, I said: 'I have not committed myself on whether any of my specimen sentences is meaningful or not'.

    That's not right, because I propose that (S) is true. I should have said: I haven't committed myself on (1).

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  5. 'The fixed point theorem does bear on this in any obvious way' -> 'The fixed point theorem does NOT bear on this in any obvious way'

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  6. For simplicity, I suppose I should not discuss my own (ambiguous) view and simply discuss Tim Maudlin's as I understand him.

    Keep in mind, then, that the following is not precisely my view.

    Maudlin believes that the best way to resolve the paradoxes of English is to interpret it with a Kripke-style 3-valued system. Maudlin offers the rule I mentioned, "S has the same truth value as T("S")," as a version of Tarski's T-schema which holds in this system. (He argues that it captures the intention of the T-schema.)

    Concerning the term "hinges", I'd say that in a 3-valued system, it's quite literal to use a definition along the lines of your first suggestion: the truth value has a potential to change.

    Since Maudlin's truth tables agree with the hook analysis for the classical 2 truth values, I take it to be a "hook theory" in the relevant sense. So, for that particular hook theory, the argument does not go through.

    Again speaking more broadly, I don't have any feeling that your argument would go through in other systems which resolve the paradoxes of truth-- though I haven't explicitly examined that. In any case, it seems clear to me that whether the argument goes through or not depends quite critically on one's preference concerning the resolution of the paradoxes, since it's an argument based on a self-referential sentence of the sort which gives rise to the paradoxes.

    What do you think?

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  7. OK, I concede (and thank you for pointing out) that my argument does not go through on a theory (such as Maudlin's, or your take on Maudlin's) which holds that, in English, a sentence predicating truth of a sentence always has the same truth value as the original sentence. (Let us call this 'the principle of unrestricted disquotational equivalence'.)

    However, for the simple(-minded) reasons given in my previous comment, I think this principle seems clearly wrong. (My hope is that enough people agree with me on this that my argument has a claim to wide acceptability barring certain minority positions. Probably it is vain to hope for any more than this with philosophical arguments.)

    Leaving my original argument to one side, since I admit it is blocked on Maudlin's theory as you describe it, here are a couple of inconclusive thoughts against that theory.

    I was about to start talking of the 'intuition' that the principle of unrestricted disquotational equivalence is false. But on second thoughts, 'intuition' is not the right word. It is not as if I have some vague feeling that the thing is false. It's that there seem to be very simple and clear reasons for rejecting the principle. More accurate, then, to speak instead of a 'reasoned belief' in the falsity of the principle.

    Now: to propose that we override the reasoned belief in the falsity of the principle of unrestricted disquotational equivalence, in order to obtain a "solution" to the Liar paradox, seems to me a strange form of wishful thinking. (I should say, I am ignorant of Maudlin's work, and don't mean to impute any of this to it.)

    I say 'strange', because it's not clear that there is anything undesirable about the Liar paradox's arising in natural language. One sort of approach to "solving the Liar paradox" is to try to account for it, to make sense of it, to obtain a clear view/description of the situation. For example: what is the semantic status of the Liar sentence? It doesn't seem to be true, false, or meaningless, in any ordinary sense. One might propose that it is 'unevaluable' - a new, special category - but then revenge problems arise. One may propose that unevaluable sentences have 'honorary truth-values', but then further revenge problems arise, leading to something like a type-hierarchy, with paradox reappearing in statements about all levels. (This happened to me!)

    Anyway, the theory under discussion doesn't appear to be an instance of this (broad) sort of 'accounting' approach. Rather, it seems to be like this: if the theory is right, there is no Liar paradox in English - the appearance of one must come from a lack of understanding of its semantics or something. But again, if this requires us to accept that a sentence predicating truth of a non-true sentence isn't false, I should think it would be very hard to accommodate. And it's hard to see the epistemic point.

    Another possibility, perhaps, is the proposal that, although the Liar paradox arises in English, we should revise our language so that it accords with theory proposed, principle of unrestricted disquotational equivalence included. That seems kind of crazy, and dubiously meaningful. Since I'm talking out of my hat (and possibly have been for some paragraphs), I'll leave it there.

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  8. I can understand why you think revising English would be "kind of crazy", but there is definitely an opposite side to that argument as well. English is a tool, after all; it should be more important that it be useful than that it accord with present habit.

    Of course, such a project would be irrelevant to your post, since you are arguing about standard English indicative conditionals.

    I suppose I think the most accurate view concerning English is that we lack strong norms about exactly how to reason when self-referential sentences are involved. People try to employ the rules which work for non-self-referential cases, but we find that these lead to paradox, and so we search for suitable restrictions/etc.

    If that's the starting position, then your argument about the hook interpretation is fairly weak, no? It should not be untenable that the hook interpretation leads to paradox for self-referential sentences, since other (more basic) principles of English also lead to paradox when self-referential sentences are involved. If the Hook truth-table were indeed a principle of English, it would not be surprising that it created trouble when combined with such sentences! Similar reasoning, after all, would have us reject the common-sense truth table of negation.

    (Hope you don't mind my harsh review!)

    So, you must be taking some *other* position about English. You think (1) English should not just be altered to avoid paradox, (2) there is nothing obviously undesirable about paradoxes arising in natural language, and (3) there are definite rules about how to handle self-referential sentences (at least, definite enough to allow your argument to go through). Is this right?

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  9. (Firstly, I don't mind the harsh review at all!)

    I don't think it's untenable that the hook interpretation leads to paradox. It clearly does. The difference between this and the case of, say, negation, is that I think paradox does intuitively arise with negations, and hence the truth-table analysis is plausible there. What I propose about conditionals like (1) on the other hand is that they do not intuitively lead to Liar-like paradox. So an analysis which predicts otherwise would on that score seem wrong.

    As for the three points at the end of your comment: I guess I endorse (1) and (2), but I'm not sure I need to for my argument, since (as you say) *it* concerns English as it is, and secondly, wouldn't be affected in any obvious way if one accepted the undesirability of Liar paradox.

    Regarding (3), I don't think I need anything quite so strong (and problematic-sounding!) as 'definite rules about how to handle self-referential sentences'. The 'rules' employed in my reasoning for (S) seem to be extremely basic ordinary rules or norms of reasoning. (After all, 'A sentence which predicates truth or falsity a sentence which is neither true nor false, will be false' is clearly an instance of a more general principle!)

    However, the applicability of fancier rules such as conditional proof, is not so clear, as indicated later in the article. But they aren't needed to get to (S).

    BTW, your 'lacking strong norms' idea regarding paradox in English strikes me as the right way to think about it. So we agree on something.

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  10. We agree on several things! :) After all, I also agree that the hook interpretation is bad.

    Where you said "this happened to me", well, me too! Here is an old blog post in which I put forward that kind of view:

    http://dragonlogic-ai.blogspot.com/2008/12/general-theory-well-i-have-been.html

    In any case, regarding (3), what I don't see is why I should think that the rules you need are more basic and obvious than other rules. If other rules might have exceptions in the case of Liar-type sentences, why not the ones you need? Liar-type paradoxes force us to make exceptions for extremely appealing intuitive rules no matter what, so pointing out that a rule is "clearly an instance of a more general principle" does no good in my eyes.

    If you agree that we lack strong norms for these cases, I don't see how you can think this is a strong case against hook.

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  11. Regarding your old post - it's good to know that wasn't a completely shamefully idiosyncratic episode from my intellectual past!

    Perhaps my remark about a rule being an instance of a more general principle was misplaced. However, the case in question is one where, with the rule on board, one gets (S) - and, crucial to my point, not Liar paradox - from considering (1).

    Now, the core of your objection (to my objection) seems to be this: if the rule in question needs to be rejected, in light of Liar-like paradox elsewhere in the language, then it will no longer be available for my reasoning toward (S). Is that right? (I haven't worked out a response yet.)

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  12. Yes, that is the core of my objection. Perhaps it's best to rephrase it-- it's more like the rule might have exceptions, not that they'd be rejected totally. For normal arguments, I wouldn't question obvious rules; but for cases involving self-referential sentences, they might not hold! If we're talking about common use, they are "unstable"... people may decide to object at different places (or perhaps be unsure where to object, but want to object somewhere).

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  13. This is good - as you say, the rule in question may not have to be rejected totally. And that's the key to my response; I think the necessary restrictions can be made in a way which leaves my argument standing.

    Here's the unrestricted rule, which arguably fails in light of Liar-family paradoxes:

    Predication rule: A sentence which ascribes truth or falsity to a sentence which is neither true nor false, is itself false.

    I propose we tack on a restriction: '...unless the sentence in question is self-referential, part of a reference-loop, or part of a Yablo series'.

    Now, what I wish to point out is that this restricted predication rule is all I need for my reasoning to (S). For when we evaluate (1)'s antecedent and consequent, we take each of them as a sentence by itself, and thus the restricted predication rule is applicable (because neither the antecedent nor the consequent of (1) is self-referential, part of a reference-loop, or a Yablo series).

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  14. Now wait a minute! Aren't A and C both part of the referential loop of (1)? (I realise that there have been arguments for the view you need to say no, but I don't like them too well.)

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  15. To make my argument in a more reasoned manner:

    Either A (and C) considered "as a sentence by itself" has the same truth value as when we consider it within the context of (1), or it doesn't.

    If it does, then A and C are still part of a referential loop, so your reasoning to (S) is blocked by your own restriction.

    If it does not, then A and C considered by themselves may get (S), but are irrelevant to the hook analysis: your subsequent argument depends on the truth values of A and C considered in the context of (1), which you have not securely established.

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  16. I confess I don't understand - don't see any problem here.

    I don't understand this distinction between 'considering within the context of (1)' vs. considering it by itself. Of course, it refers to (1), and it is also a component of (1). When I said 'by itself' I really just meant: as a sentence (rather than some subsentential expression.)

    As I see it, A and C are not part of a reference-loop, although they are part of a self-referential sentence. (For simplicity let us say that a self-referential sentence is a special case of 'being part of a reference-loop', i.e. where the loop has one node.)

    A and C both refer to (1) - which is a different sentence from both A and C (it contains them, after all), and (1) refers to itself. So (1) is part of a reference-loop, but A and C stand outside, pointing into the loop, as entry-points. Since (1) does not refer to A or C, or any sentence besides itself, the loop does not encompass them.

    At the end of your comment, you write: 'your subsequent argument depends on the truth values of A and C considered in the context of (1), which you have not securely established'.

    Reading this literally, it seems to go right past my argument. For I don't need to securely establish the truth values of A and C. All I need to do is establish is that *on the assumption* that (1) is neither true nor false, A and C are false.

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  17. Ah! Sorry. I thought you were trying to make a very different argument. I have heard a point of view which claims that if I speak aloud:

    "What Abram is now saying is not true."

    And, at the same moment, you happen to speak the same thing, what I've said is paradoxical or meaningless or some such, but what you've said is true. I don't like this very well, because we've both said the same sentence, and we haven't included any indexicals like "I", so it seems they really should be the same sentence. I thought you were relying on that sort of view.

    I'll try and respond to the argument you did make.

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  18. The crux of my objection is still that A and C are part of a referential loop. Here is a drawing of (1):

    (1): A->C

    (where -> may or may not be hook.)

    Now, it seems evident to me that (1) only refers to itself by virtue of A and C referring to 1. In other words, there are two loops: (1)...(A)...(1), (1)...(B)...(1). Your analysis points out that the "..." in those loops cannot always be filled in with "reference", since (1) does not refer to A and C; rather, it uses them directly. If anything, I find this makes their inclusion in the referential loop *more* solid!

    In other words: If it were the case that A and C did not refer to (1), (1) would not be self-referential. Thus, they are essential to the loop.

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  19. What is strange about (1) is that if A is true and C is false, (1) is still true, but this violates the common accepted intuition on the literature that conditionals with true antecedents and false consequents are false.

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  20. Matheus,

    It seems to me that saying such a thing is similar to saying:

    "Consider two sentences:

    (L): (L) is not true.
    (A): (L) is true.

    What's strange about L is that if (A) is true, (L) is, but this violates the common accepted intuition that the negation of a true sentence is false."

    (I mean for L to be the negation of A; that is, ~A.)

    This argument seems to have a bad form.

    Do you think the two arguments are relevantly different in form?

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  21. Abram,

    I think that maybe there is one difference: in your examples we have two sentences that refer indirectly to one other, but my point is that conditional (1), which is one sentence, is true even when its antecedent is true and its consequent is false, which is bizarre. I'm familiar with the idea that our intuitions are not so important in cases involving semantic paradoxes because we have no habit of using paradoxical sentences, so we have no firm beliefs that we tend to accept about paradoxes. But my point is that in this case we have one independent reason to exclude this case of self-reference: conditionals with true antecedents and false consequents are always false.

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  22. "but my point is that conditional (1), which is one sentence, is true even when its antecedent is true and its consequent is false, which is bizarre."

    So, what makes you so sure of that assertion? What makes that more secure than the assertion that the Liar sentence (L) is an example of a negation which is true even when the negated part is true?

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  23. Ok, I concede! My argument doesn’t work.

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  24. Glad to see someone else becoming embroiled in Demskian controversy!

    BTW Abram: now it seems we agree that A and C are not part of a reference-loop, at least in *my* sense of reference-loop, where the connections in question actually have to be reference connections. (BTW less direct reference is included in this - Quine's ' "yields falsity when preceded by its own quotation" yields falsity when preceded by its own quotation'. )

    Now, what I'd be interested in is if you could come up with a clear counterexample to the Predication rule *as restricted* in my last post.

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  25. Oh! Ok. I didn't realise that was your definition. It seems like an unnatural definition to me, but I'm not in a state of mind to give you a good counterexample right now. I'll let you know when I think of one!

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  26. What about the example I gave? Re-labelling my sentences slightly (just to make sure they don't have the same names as yours):

    (L): ~T(L)
    (LL): T(L)

    According to your definition, (LL) is not part of the referential loop, right? What's wrong with the following reasoning:

    (L), being the Liar, is neither true nor false. So (LL), which asserts the truth of the Liar, is false (by your restricted predication rule). But then (L), which is just the negation of (LL), is true. Contradiction.

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  27. Tristan,

    Someone could reply that you are just begging the question: unless we already assumed that the hook analysis is wrong, we can't attribute truth values to A and C when we suppose that (1) is neither true nor false. If the hook analysis is right and (1) has the same truth conditions of a material conditional, the truth value of a material conditional is a function of the truth values of its parts. So when we suppose that A e C are neither true nor false, we also should suppose that (1) are neither true nor false, and vice-versa.

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  28. Correction: So when we suppose that A e C are neither true nor false, we also should suppose that (1) IS neither true nor false, and vice-versa.

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  29. Another problem is that the assumption that (1) is neither true nor false, is false, because a material conditional always has a truth value.

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  30. Matheus,

    Thanks for your thoughts. Firstly, on the issue you raise, about question-begging: the argument is meant to suggest that, intuitively and for simple reasons, we *can* (and should) - within the scope of the assumption that (1) is neither true nor false - ascribe truth-values to A and C. This cannot be squared with the hook analysis, but seems to be true. This, I think, gives us reason to reject the hook analysis (which itself does not have the same kind of simple plausibility as (S)).

    Regarding your second problem: To begin with, how is it a problem for my argument if the assumption that (1) is neither true nor false is false? OK, so maybe it's a false assumption (although I personally don't think so), but the crucial point is: On that assumption, A and C are false.

    Furthermore, you say this assumption is false 'because a material conditional always has a truth value'. But how can you assume that (1) is a material conditional? This would be to beg the question against my argument.

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  31. It’s my pleasure,

    My first objection is that the assumption itself that (1) is neither true nor false cannot be squared with the hook analysis. If we are under the assumption that (1) is neither true nor false we already are under the assumption that the hook analysis is false. But in that case we don’t need to ascribe truth-values to A and C and present a counter-example to the hook analysis, because we already are assuming that the hook analysis is false. The same happens in the question of plausibility: if the hook analysis does not have the same plausibility of the assumption that (1) is neither true nor false we don’t need your counter-example anymore: we can just defend that we have good reasons to believe that conditionals like (1) are neither true nor false and in that case the hook analysis is false.


    I presented my second objection in a hurry. The idea is that a defender of the hook analysis could object with the following modus tollens:
    If it’s true that (1) is neither true nor false, then the hook analysis is false.
    The hook analysis is true.
    Hence, it’s not true that (1) is neither true nor false

    In that case we will turn our attention to discuss what are the reasons to believe or in the hook analysis or in the assumption that (1) is neither true nor false.

    I’m not sure, but I have the impression that the best counter-examples to the hook analysis always concede, in a first moment, that the conditionals in the counter-examples have the same truth conditions of the material conditional to then prove, in a second moment, that this assumption results in consequences that we don’t accept. But your counter-example can’t concede in a first moment the conditional in question has the same truth conditions of a material conditional.

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  32. Matheus,

    Thanks, this makes things clearer. However, I'm not sure what you take to be the reason for your statement that 'If we are under the assumption that (1) is neither true nor false we already are under the assumption that the hook analysis is false.'

    Is this because of a feature of (1)? Or would the same hold for the corresponding assumption about any indicative conditional?

    The second course, at any rate, is easily averted on a natural reading of the hook analysis; think of the hook analysis as embodied by the truth-table for hook. This can be thought of as giving, for each of the truth-possibilities for the atoms, the value of the compound. So, while such an analysis does imply that when the atoms have truth-values T or F, so will the compound, I don't see it as ruling out that a compound might be neither true nor false. In such a case, if the analysis is right, at least one of the components will also have to fail of truth-value. (And this is just what can't happen on the assumption that (1) is neither true nor false, hence my objection to the hook analysis.) The hook analysis, on this reading, gives truth conditions, and falsity conditions too, but doesn't rule out (or say anything about) the case of neither truth nor falsity.

    Again, I am not saying that the assumption that (1) is neither true nor false is plausible. I sort of think it is, but it's not an official part of my objection. I don't need it to be true. The point is: it is plausible that *on this assumption*, A and C are false.

    Now, if the assumption in question *amounted to* the assumption of the falsity of the hook analysis, my argument might be a bad one (though this is actually not obvious to me). However, I don't think it does, in light of the reading indicated above. What do you think?

    By the way, to address later stuff in your comment: I definitely like that fact that you're scheming against the hook analysis, but I'm not sure about this talk of conceding the truth-conditions to the hook analysis in the 'first moment'. I don't think one should concede that at any point. An assumption in an argument is not any kind of concession. (Perhaps this is merely pedantic - I'm not sure.) Perhaps you're right, however, that reductio ad absurdum is a good strategy for arguing against the hook analysis.

    My argument could be framed as a reductio, incidentally: Assume the hook analysis. Then we cannot affirm (S) without getting into paradox. But (S) is true and affirmable without paradox. Contradiction. Thus, by reductio, our assumption was wrong.

    I don't find this very natural, but I give it in response to your statement that my 'counter-example can’t concede in a first moment [that] the conditional in question has the same truth conditions of a material conditional' (in order 'to then prove, in a second moment, that this assumption results in consequences that we don’t accept').

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  33. If the hook analysis is true, indicative conditionals have the same truth-conditions of a material conditional.
    (1) is an indicative conditional that doens’t has the same truth-conditions of a material conditional.
    Hence, the hook analysis is false.

    The first premise is uncontroversial. The second premise is true if it is true that (1) is neither true nor false. If we admit the two premises, we have to conclude that the hook analysis is false, and that is why the assumption that (1) is neither true nor false cannot be squared with the hook analysis.

    I don't buy your reading of the truth-table of the material conditional. The material conditional is a truth-functional operator of a bivalent truth-functional logic in which an atom always has a truth-value and a compound always has a truth-value which it's determined by its atoms. The case of neither truth nor falsity is a case of truth-conditions: if a compound in classical logic could be neither truth nor false this would appear in its truth-table.

    But even if a compound might be neither true nor false (in the case one of its components fail of truth-value) this is still a problem for your counter-example: on the assumption that (1) is neither true nor false, A or C fail of truth-value. But if A or C fail of truth-value, they can't both be false. Again, the assumption that (1) is neither true nor false is not very helpful and you face a dilemma:

    If the hook analysis forbids this assumption, you can use it only if you are already presupposing that the hook analysis is false
    If the hook analysis don't forbid this assumption, A or C (maybe both) fail to have truth-value, and in this case you counter-example doesn’t work either

    I'm not sure if your counter-example can be framed as a reductio or not. I surely think that it doesn't work. But working or not, is not a bad argument. On the contrary, it’s a very clever argument.

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  34. The argument at the beginning of your comment isn't really how I see mine as going, but I don't know if that's important. Another thing: later on, you seem to assert that 'on the assumption that (1) is neither true nor false, A or C fail of truth-value'. I don't see why that should be.

    Anyway, rather than get into these things, the main thing I want to respond to is this: You said you didn't buy my interpretation of the truth-table for hook (as an analysis). Now, all I said is that it was one natural interpretation, so I'll proceed as if your main point is that another, more restricted interpretation is possible. And I thank you for making this apparent.

    Putting the table aside for clarity's sake, we can formulate two relevant theories about indicative conditionals:

    (Theory 1) 'If A then C' is false iff 'A' is true and 'B' is false, true otherwise.

    (Theory 2) 'If A then C', if it is true or false, is false iff 'A' is true and 'B' is false, true otherwise.

    My argument, I maintain, gives us good reason to reject (Theory 1). (Theory 2), however, can withstand my argument, provided the proponent is prepared to accept that (1) (i.e. 'If (1) is true, (1) is false') is neither true nor false. And this seems to me quite plausible. This having been clarified, I must concede that my argument cannot be expected to dissuade those who favour the (Theory 2)-version of the hook analysis. But I think it's worthwhile to see that (Theory 1) won't fly. For instance, not even the proponent of the hook analysis should, if told that an indicative conditional has a false antecedent, infer from that alone that it is true. They must know that the conditional is either true or false (and must also hold that sentences like (1) are neither true nor false).

    Thanks again for making me see this.

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  35. Tristan,

    That also appears to address my argument, since Theory 2 is the relevant one within the types of 3-valued systems I was thinking of.

    Still, I feel that the *reason* Theory 1 fails is that it bundles up some of the assumptions of the naive theory of truth...

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  36. Abram,

    Interesting. What would you say is the 'naive theory of truth', anyway? I've been a bit puzzled about something like this in other areas.

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  37. What I tend to mean is classical first-order logic along with a quotation, a truth predicate, and the T-schema. In this sense, "the naive theory" is taken to absolutely refer to something which is inconsistent it the worst sense (ie, we can infer anything we like).

    Some take the term to just refer to the unrestricted T-schema, so that the naive theory can be saved from paradox by restricting/modifying/abandoning classical logic.

    That sums up the number of ways I've heard the term used.

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  38. The idea that on the assumption that (1) is neither true nor false, A or C fail of truth-value came from a comment of yours (9 March 2011 01:07). You said that “while such an analysis does imply that when the atoms have truth-values T or F, so will the compound, I don't see it as ruling out that a compound might be neither true nor false. In such a case, if the analysis is right, at least one of the components will also have to fail of truth-value”. So, I considered this as taken for granted. But even if you drop this assumption I think I still can give some reasons.

    A function of truth returns truth values from truth values. If it doesn’t depart from truth values, the function stays mute. This is what a function is. What is curious about this is that the concept of function itself is conditional: if something, something. If not something, empty. A material conditional is a function that is only determined from its parts to the compound, but not vice-versa. This means that I can’t pass from de truth value of a conditional to infer the truth value of its parts, but I can pass from the truth values of its parts to the truth value of the conditional. We can say that a material conditional is neither true nor false, this doesn’t violate the definition – so I drop one of the arguments that I defended above: the assumption that (1) is neither true nor false is not, per se, incompatible with the hook analysis. But the rest remains the same: on the same classical definition the only circumstance in which a material conditional is neither true nor false is when its parts are neither true nor false, for if its parts were true or false, the conditional would be true or false.

    You said (a comment from 7 March 2011 06:05) that maybe the assumption that (1) is neither true nor false is false “(although I personally don't think so), but the crucial point is: On that assumption, A and C are false”. Know you can see the problem: you can’t sustain that (1) is neither true nor false without implying that A and C are neither true nor false – your counter-example is broken.

    The only difference between (Theory 1) and (Theory 2) is that (Theory 2) says literally something that is just implied in (Theory 1) – and that’s it. So if your counter-example doesn’t work against (Theory 2) it doesn’t work against (Theory 1) either. I didn’t understood your point when you said that “even the proponent of the hook analysis should, if told that an indicative conditional has a false antecedent, infer from that alone that it is true. They must know that the conditional is either true or false (and must also hold that sentences like (1) are neither true nor false)”. If he is a proponent of the hook analysis he must believe that the conditional is either true or false because that is what a proponent of the hook analysis believes!

    We can learn two important lessons from your counter-example. The first is that when we aim to present a counter-example to the applicability of a certain function we must be sure that we are not disregarding its mechanisms of functioning. The second one is that theories of conditionals must offer an explanation of self-referential conditionals like (1) that are plausibly neither true nor false. This last lesson deserves careful attention and research.

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  39. Matheus,

    I feel your new argument gives too little credit to Tristan's argument. I do think there is a distinction between Theory 1 and Theory 2... Thanks to the word "otherwise," Theory 1 entails that conditionals are always either true or false, whereas Theory 2 does not entail this.

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  40. Abram,

    but even if I concede this point the problem still remains: the only circumstance in which a material conditional is neither true nor false is when its parts are neither true nor false, and this stops the counter-example.

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  41. Matheus,

    I concede the point. Even if we accept (S) on the grounds that it has a false assumption and hence must be true (by either the hook analysis or the law of explosion if we accept that law), the rest of the argument shows nothing, since it makes use of the same false assumption and hence is just a proof that we can derive a contradiction from a false assumption.

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