Thomas Bradwardine’s Solution to the Liar

I recently interviewed Stephen Read on Thomas Bradwardine’s solution to the Liar and Read rehearsed Bradwardine’s argument step-by-step. Since the specifics of the argument can be hard to follow (but also rather ingenious), I decided to try writing out a simplified version of the argument. I kept in a tiny bit of symbolism, for ease of presentation, but am not assuming familiarity with formal logic. I include reference and links to logical principles where they are invoked unless they are fairly obvious and intuitive (eg, double negation elimination). My intention was to make the argument as short and accessible as possible without sacrificing accuracy.

You can, of course, also find more thorough reconstructions of the argument at the Stanford Encyclopedia of Philosophy, in Stephen Read’s paper, and in Bradwardine’s book itself.

It should be said: Bradwardine assumes that a statement is true if and only if everything it says is the case.

The crucial move uses Bradwardine’s principle of “closure”: that all statements implicitly say everything that follows from the supposition that what they say is the case.

~ means “not” and T<L> means “L is true”.

Case 1: Suppose the Liar (L) says only that it itself is not true.

says [~T<L>]
Suppose L is not true (as it says). Then not everything it says is the case:
~~T<L>
T<L>
says [T<L>] (closure)

It is therefore not the case that L says only that it itself is not true (because if that were so, it would also mean that it’s true, contradicting the premise that it only says that it’s not true).…

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Stephen Read: Liar Paradox | WSB #41

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Stephen Read

“This sentence is false”. Is that sentence true or false? If it’s true, then what it says must hold; but what it says is that it’s false, so it must be false. But if it’s false, then what it says must not hold; but what it says is that it’s false, so it must not be false. But if it’s not false, it must be true. So if the sentence is true, it is false, and if it is false, it is true. The sentence, therefore, seems to be both true and false, which seems absurd.

Philosopher and logician Stephen Read is one of the preeminent scholars on this “liar paradox”. He is known, in large part, for rediscovering and defending a long forgotten solution to the paradox first proposed by the medieval philosopher Thomas Bradwardine. In this first half of our conversation, Read covers the paradox’s rich and influential history. It was first discovered, in its full form, in the 4th century BCE by Eubulides (who also first set down the sorites paradox). It became a central problem in the 20th century via its association with Russell’s Paradox, a major problem in the foundations of mathematics. Later in the century, two thinkers – Alfred Tarski and Saul Kripke – proposed monumentally influential theories of language and truth motivated, largely, by the paradox. But even after their contributions, the consensus is that the paradox remains unsolved. …

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Dialetheism: From Language to Reality

A physical contradiction?

I recently published a post in defense of dialetheism. I argued that in the case of statements about “man-made” states of affairs, it is obvious that some contradictions are true. For example, the law can easily contradict itself in such a way that a statement about what is legally mandated be a true contradiction. I invented “Timmy the Square Circle” to show that, similarly, there can be true contradictions about fictional characters. If this doesn’t seem intuitively obvious, read that post before this one.

The concluding paragraph included this teaser:

It is perhaps now tempting to draw a sharp line: the world of man-made ideas allows for true contradictions, reality doesn’t. However, this line is not so sharp.

If we grant that there are true contradictions about what is made up, does this tell us anything about whether there are true contradictions about objective reality? To say there are is a stronger, and intuitively harder to swallow, version of dialetheism. As we’ll see, however, there is no way to say anything about anything without talking, in part, about the man-made. This inescapable fact leaves open the possibility of true contradiction in claims about the physical world, even if it’s the case that the physical world itself, independent of our descriptions of it, cannot be contradictory.

Conceptual reality: Liar and Sorites paradoxes

We first need to establish that there are different “levels” of objective reality, and accepting a contradiction in one level may be much more counterintuitive than in another level.…

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