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:
- Suppose that we cannot know whether ‘this sentence is false’ is true or false. (This is the Dharmakīrtian thesis.)
- ‘This sentence is false’ implies that it’s not the case that ‘this sentence is false.’
- Say that P abbreviates ‘this sentence is false.’ By the means of classical logic, we can reason as follow:
- P → ~P formal representation of 2,
- (P → ~P) → ~P the contemptible consequence rule,
- ~P modus ponens from A. and B.
- 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 A is actually false or true. Importantly, ‘→’ is material implication. It means that A → B expressions are false if and only is A is true and B is false, and they are true otherwise. Not-A is true if and only if A is false.
Let’s look at the contemptible consequence rule again. If A is true, then not-A is false. If so, then A → not-A is false. Consequently, because not-A is false, (A → not-A) → not-A is true. Alternatively, if A is false, then not-A is true. If so, then A → not-A is true and because not-A 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.