The contemptible consequence problem

Hello again, this is Szymon, a PhD student researching the Dharmakīrtian approach to liar paradox. According to this approach—you can find more about it in my previous post—the liar sentence is ambiguous, unbelievable, and cannot express a warranted belief.

There’s a logical problem with the Dharmakīrtian approach I want to discuss today. In the next post, the last in the series, I will sketch some answers to this problem and share some general observations on the relationship between Dharmakīrtian epistemology and logic. 

The logical problem comes from Tom Tillemans and I call it the contemptible consequence problem. What is it? In classical logic, there’s a rule, the so-called contemptible consequence rule. It says that if a sentence implies its own negation, then this sentence has to be false: (A → ¬A) → ¬A. The problem is that we could seemingly know that the liar sentence is actually false if we were to reason according to this rule.

How this reasoning goes? Consider this argument:

  1. Suppose that we cannot know whether ‘this sentence is false’ is true or false. (This is the Dharmakīrtian thesis.)
  2. ‘This sentence is false’ implies that it’s not the case that ‘this sentence is false.’
  3. Say that abbreviates ‘this sentence is false.’ By the means of classical logic, we can reason as follow:
    1. P → ~P                                  formal representation of 2,
    2. (P → ~P) → ~                     the contemptible consequence rule,
    3. ~P                                          modus ponens from A. and B.
  4. We know that it’s not the case that ‘this sentence is false’ because it follows from (3). So, we know that ‘this sentence is false’ is false. Contradiction between 1. and 4.; so, 1. must be false.

Before we look into this argument in more details, let me explain some logic lurking in its background. 

The guiding idea behind the contemptible consequence problem is that the inference from a contradiction with own words—a sentence implying two impeding beliefs—to the truth of its negation is guaranteed by the contemptible consequence rule. Let me show you that, for any proposition A, (A → not-A) → not-A has to be true no matter whether is actually false or true. Importantly, ‘→’ is material implication. It means that A → expressions are false if and only is A is true and B is false, and they are true otherwise. Not-is true if and only if is false.

Let’s look at the contemptible consequence rule again. If is true, then not-is false. If so, then A → not-is false. Consequently, because not-is false, (A → not-A) → not-A is true. Alternatively, if is false, then not-is true. If so, then A → not-A is true and because not-is also true, (A → not-A) → not-A is true as well. Consequently, (A → not-A) → not-A is true no matter whether A is false or true. 

Now we see how the logic in the background of the problem works. However, that doesn’t answer the epistemological question how deferring to the contemptible consequence rule justifies the belief that the liar sentence is actually false. Let’s unpack the 1.-4. argument step-by-step how it answers this epistemological question.

The first step assumes the central thesis of the Dharmakīrtian approach to liar paradox. Given that the liar sentence is a contradiction with own words, we cannot know that it is true or that it is false. Generally, the whole argument is a reductio. It assumes something to be true, derives a contradiction from the assumption, and concludes that that the assumption has to be false.

The second step introduces a fact about the liar sentence’s meaning. For a Dharmakīrtian, the liar sentence implies two contradictory sentences just like ‘my mother is barren’ implies two sentences. The sentence ‘my mother is barren’ implies that ‘this person has a child’ and ‘this person cannot have children’. The liar sentence implies that ‘the liar sentence is true’ and that ‘the liar sentence is false’. This is indeed what Dharmakīrtian account says about the liar sentence’s meaning.

The third step is a logical argument. It starts with a formal representation of the fact about the liar sentence’s meaning described in the second step. It represents it as a material implication. Then, using the contemptible consequence rule and modus ponens, it derives the liar sentence’s falsity.

The fourth steps concludes that, given the logical argument in the steep three, we can know that the liar sentence is false, contrary to what the Dharmakīrtian approach says. We know it because any proposition that implies its negation has to be false no matter whether it’s actually true or false. Consequently, assuming the Dharmakīrtian approach to liar paradox leads to a contradiction and so the Dharmakīrtian approach is false.

You might not be bought into the contemptible consequence problem immediately, so let me motivate it a bit now.

Standardly, the liar sentence is a focal point of study of logic. As I’ve mentioned in my previous post, the liar paradox tells us something important about truth and logical rules. In contrast, for Dharmakīrtians, the liar sentence is primarily an epistemological phenomenon inviting a question what we can justifiably believe about it. If you are interested in how logic and epistemology interact, as I am, solving the contemptible consequence problem is a good way to illuminate this interaction.

I have several answers to the contemptible consequence problem. Do you have yours? Are you convinced by the 1.-4. argument? Let me know in the comments!

In my next post, I will sketch some of my answers to the problem and briefly discuss the general relationship between logic and Dharmakīrtian epistemology.

9 Replies to “The contemptible consequence problem”

  1. Hi Szymon, I think your project is very interesting, and have enjoyed reading your posts and comments.

    To answer your question, I don’t think Tilleman’s approach works, because it seems to me that we can also run the 1-4 argument for
    (not-A → A) → A as well. Then we are right back where we started. Is this right?

    • Hey Boram, thanks! It’s really nice to know that you enjoy reading my posts!

      I think you are totally right and I’ve been think the same thing. If we can know that the liar sentence is false because if it is true, then it is false, then, we can also know that the liar sentence is true because if it is false, then it is true. That would be the admirable consequence problem, after the so-called admirable consequence rule you mention: (not-A → A) → A.

      And I’m very interested in what you mean when you that ‘we are right back when we started.’ The way I think about it is this. If the contemptible and the admirable consequence arguments showed that we can justifiably believe that the liar sentence is true and that the liar sentence is false, then these two beliefs would be equally supported by evidence. Consequently, it seems that we haven’t learned anything about the liar sentence’s actually truth value.

      • Hi Szymon,

        When I wrote the above comment, I believed that Tillemans’s approach just brings us back to square one. Now I have a better opinion of his approach. But first let me explain why I thought it got us nowhere.

        I don’t know any Dharmakirtian epistemology, but I have some rudimentary grasp of Naiyayika epistemology, so that’s what I am working with.

        Now, with respect to the liar sentence, I don’t know whether it is true or false. So I have a doubt (saMzaya) of the form: A or not-A.

        Then Tillemans tells us that we can resolve this doubt. We can know that A is false, and thus know that not-A. The pramANa for this is the inference (i.e., the contemptible consequence rule):
        (A → not-A) → not-A

        Here, in Naiyayika terminology, I guess the subject (pakSa) of this inference is the liar sentence, the reason is “(A → not-A)”, and the probandum (sAdhya) is “not-A”.

        And the problem is that, given the inference (i.e., the admirable consequence rule),
        (not-A → A) → A,
        the reason is defective (hetvAbhAsa). The defective reason is either contradictory (viruddha) or counterbalanced (satpratipakSa), depending on how finely we individuate the consequence rules.

        In terms of their logical form, I think the contemptible consequence rule and the admirable consequence rule state one and the same rule, since A is the negation of not-A just as much as not-A is the negation of A. In that case, it is formally the same reason that allows us to infer both not-A and A. So this reason will be contradictory (viruddha).

        In terms of their specific content, the contemptible and the admirable consequence rules are different. In that case, the reason put forward in the contemptible consequence rule is counterbalanced (satpratipakSa) by the reason put forward in the admirable consequence rule.

        On either reading, the reason is defective in establishing the probandum (sAdhya) because there is another inference that establishes the opposite (sAdhyAbhava). So, we are still in a state of doubt of the form: A or not-A. Hence back to square one.

      • Here I will try to explain why I have a better opinion of Tillemans’s approach now.

        I don’t know what Naiyayikas think about the liar sentence. But suppose they think we cannot know whether it is true or false (like your Dharmakirti). Then what could be their basis for thinking so?

        We start out with the doubt whether A or not-A. Then we draw out the implications of the liar sentence: if A then not-A, and if not-A then A. In short, we can only engage in hypothetical reasoning (tarka), but this cannot give us knowledge: no knowledge in, no knowledge out. That is to say, we must first know whether the antecedent is true or not, and only then can we detach the consequent as a piece of derived knowledge through inference.

        What Tillemans does, I think, is to suggest a clever way of detaching the consequent via the rule
        (not-A → A) → A
        and modus ponens. Even if we do not know whether A or not-A, we do know that not-A → A, and this knowledge allows us to infer A. “Not-A → A” is the knowledge in, and “A” is the knowledge out.

        Likewise, via the rule
        (A → not-A) → not-A
        and the knowledge that A → not-A, we can detach the consequent not-A through modus ponens.

        And perhaps it can be argued that what we get from these two detached consequents is not a doubt of the form A or not-A, but a dialethic cognition of the form A and not-A.

        As to how we can know the implications A → not-A and not-A → A, perhaps a Naiyayika can accept that we know them through mAnasa-pratyaksha, i.e., by reflection on our cognitions regarding the liar sentence.

        So, I think there is some merit to Tillemans’s approach, if it is understood as leading to dialetheism. Whether it is ultimately viable or not will turn on technical matters that are beyond my ken, so I will end my part of the discussion here. 🙂

        • Hey Boram, that is all very helpful and right in the centre of my interests, so thanks a million!

          I’m totally bought into your idea that the initial doubt about the liar sentence’s truth or falsity doesn’t disappear on the basis of contemptible argument. The argument has a visible tarka/prasaṅga vibe, so, exactly as you say, there’s no place for knowledge-in/knowledge-out. I think that this idea generalises and that hypothetical inferences deferring to logical rules—like the contemptible and the admirable arguments—cannot be a valid means of cognition because of their lack of actual, existential commitments. That can have interesting implications for our understating of the role of logic in knowledge acquisition in general. That’s something a tad bit beyond my paygrade now but I would like to explore it further in the future, with Buddhist and Nyāya epistemologies in sight.

          BTW, I vaguely remember that they are people who think that tarka is a pramāṇa. Do you know anything about it?

          I will just keep going because there’s so much interesting stuff in your comments, but just two more things! Sorry!

          I also think that Tom’s argument is clever in that it starts with (A → not-A), and not with A or not-A. However, (spoiler alert!) I don’t think that material implication adequately represents the fact that the liar sentence implies its falsity or truth. I have a chapter about it, and I’ll be very happy to share it when it’s gonna get out.

          I haven’t heard about satpratipakṣa before reading your comments, so many thanks for mentioning it. Where can I learn more? It’s a very interesting idea for me for a slightly independent reasons I will just mention briefly. I’ve been working on the epistemology of peer disagreement and developed an account that in situations of equal evidential support, one should believe a contradiction rather than suspend judgment. The satpratipakṣa seems to describe a situation when p and not-p are supported to the same degree by the available evidence. From what you say, one way to respond this epistemic situation is to acknowledge that, as far as we know, both p and not-p, but, also, we don’t know whether p or not-p. I think this attitude is a good doxastic response to the liar sentence and to a host of other phenomena. And, to advertise my view a bit more, it’s not a standard dialetheism because dialetheists seem to think that they know that the liar sentence is both true and false.

          Boram, again, thanks a lot for all your comments! They help me a lot in my work and they are pure fun to engage with.

          • Szymon, thanks for indulging in my speculative suggestions. As I’ve said, I am enjoying your posts and looking forward for more here.

            To make a few replies:
            (1) It looks like there were some in the Dvaita, Vishishtadvaita, and Jaina schools that accepted tarka as an independent source of knowledge. I haven’t read the book, but it is discussed in Bagchi’s _Inductive Reasoning: A Study of Tarka…_ (https://archive.org/details/inductivereasoningastudyoftarkaanditsroleinindianlogicsitansusekharbagchi_202003_174_F)

            (2) Though logic is not my forte and it makes my head hurt, I will be happy to read your work on the inadequacy of material implication when it’s ready. My email should be accessible through my comments here.

            (3) The satpratipakSa-hetu (the doSa itself is called pratipakSa) is explained in NyAya primers like Tarka-bhASA and Tarka-saMgraha, under the topic of hetvAbhAsa. In Tarka-bhASA, it is also referred to as “prakaraNasama”, where it is mentioned that the defective reason in question is counterbalanced by a reason establishing the absence of the probandum in another equally strong inference (samAnabalam anumAnAntaram).

            The NyAya-sUtra itself does not mention satpratipakSa as a hetvAbhAsa, mentioning only prakaraNasama. Some understand it to mean the defect of indecision (e.g., Matilal), and others understand it as begging the question. So it’s unclear whether prakaraNasama originally meant what satpratipakSa hetvAbhAsa comes to mean later.

            I think it’s more likely that Naiyayikas will take Tillemans’ inference to involve the hetvAbhAsa of asiddhi. Perhaps the pakSa (liar sentence) here will be seen as defective, somewhat like the “sky lotus”. Because the liar sentence involves self-reference, it might be seen as involving the doSa of AtmAzraya. For the two-sentence version of the liar, the doSa may be itaretara/anonya-Azraya.

  2. Hi Boram, not quite where you started: rather at the fourth place, where true and false are both admitted. And as Szymon nicely put it, this now applies to the *meaning of the sentence, which was not the case for the previous simple evaluation of the reference.

    A little before Dharmskirti, Dinnaga distinguished between using a sign to make reference (sensuous intuition), and inference established between signs (mental intuition). And with that in hand, you come to the schema of judgement known as the *hetuchakra, and much debated in Indian logic.

    There, Szymon, you have a semiotic formalism that comprises possibilities of judgement and the fourth truth-value. At the expense, apparently, of a soft dualism of reference and sense, as Frege put it. I’m interested to see how your work might relate to that.

    The interest then carries to modern logic and science: Boole’s calculus of probability has to be invertible (exchanging 1 and 0 as true and false) which is tricky in an infinite universe, or on a continuum. In the thermodynamics from that time, ranging the temperature over infinity takes you strangely *below absolute zero, and experiment now reaches there, by manipulating magnetic spins.

    So there, unexpectedly, you have a physical model for the “space of meanings”, and strangely like some traditional views of consciousness in meditation, related to mesmerism or “animal magnetism”. All now easily dismissed as New Age blather, but preferable to Ken Wilber just taking Advaita Vedanta to also carry off the true essence (svabhava) of Buddhism.

    • Hey Orwin, good to hear from you again.

      I fully agree that there’s something suspicious in thinking that ‘the liar sentence implies that it is false’ equals ‘the liar sentence *materially* implies that it is false’. That made me study Buddhist philosophy of language and semantics of conditionals.

      A big chunk of my research concerns exactly what you ask about: how different Buddhist theories of valid inference could account for arguments deferring to logical rules. I will say a bit about it in my next post, so stay tuned.

  3. Pingback: Answering the contemptible consequence problem – The Indian Philosophy Blog

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