Normativity of Logic and the Preface Paradox

normativity logic

I recently came across a surprising claim: that logic is normative. That is, it is in some sense wrong to deny logic. The claim isn’t surprising because it’s controversial; instead, it’s so obvious that it’s initially jarring to see it spelled out explicitly. What is controversial is the claim that followed: that logic’s normativity isn’t universal. In other words, that it is sometimes rational to accept a deductive argument as valid, accept all its premises as true, and yet still deny its conclusion. Let’s see why this might be and whether it holds up.

The Preface

A well-known paradox, the Preface, goes as follows: I assert each thing I state in this post. After all, if there were something here I did not wish to assert, I would not state it. However, I also assert that I’m wrong about at least one thing I say here. Write anything long enough, and chances are, no matter how thoroughly you check yourself, you’ll get at least one thing wrong. (This post isn’t very long, but as an amateur writing on a complex topic, the post needn’t be very long for me to feel confident that there’s at least one mistake in it.)

So far so good. Here’s the trouble. Let’s label my assertions in this post p0, p1, p2, … pn. I’m apparently asserting that each of those is true, but also denying that their conjunction – (p0 & p1 & p2 & … & pn) – is true.…

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A Quick Note on the “It’s Meaningless” Solution to the Liar Paradox

In an earlier post, I discuss one good reason to reject the solution to the Liar paradox that says that it’s meaningless: it means calling some other sentences, like “This sentence is in Japanese” and “This sentence has five words” meaningless, though they seem to be obviously meaningful.

Anyone who defends this solution has to bite the bullet on these sentences. It’s a tough bullet to bite, but at first it doesn’t seem implausible. Maybe those sentences only seem to be meaningful, though they aren’t really.

I came across three sentences today that convince me that rejecting all self-referential sentences is, in fact, utterly ridiculous. I found them in Tim Urban’s newest amazing article on Elon Musks’ newest mind-blowing venture, Neuralink. (By the way: go read it. Now. Elon Musk is turning humanity into the Starchild from the end of 2001 and you’re reading about loopy sentences? Get out of here!)

Here are the three sentences:

That’s why we still communicate using technology Bok invented, it’s why I’m typing this sentence at about a 20th of the speed that I’m thinking it, and it’s why brain-related ailments still leave so many lives badly impaired or lost altogether.

And

Right now, your eyes are making a specific set of horizontal movements that allow you to read this sentence.

And

None of this stuff will take any effort or thought—we’ll all get very good at it and it’ll feel as automatic and subconscious as moving your eyes to read this sentence does to you now.

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Vagueness: The Sorites Paradox

Imagine we have 0 grains of sand. Do we have a heap of sand? Of course not! Well, what if we add one grain? We obviously still do not have a heap. Okay, what if we add one more? One more after that?

No matter how many grains of sand we have, adding just one more will never turn a non-heap into a heap. This is called the “tolerance principle,” and it is the defining feature of vague properties. It says that a small enough change can never alter the applicability of a vague property.

Say you have a red shirt. Change the frequency by an imperceptible amount. Obviously, the shirt is still red. Take someone who is sober. One ml of beer will not make that person drunk.

A problem appears when we compound these small increments. Here’s a version of the argument:

1) 0 grains of sand is not a heap (premise)

2) 1 grain of sand is not a heap (by #1 & tolerance principle)

3) 2 grains of sand is not a heap (by #2 & tolerance principle)

10001) 10000 grains of sand is not a heap (by #10000 & tolerance principle)

Welcome to the sorites paradox (“sorites” = “heap” in Greek), the argument that allows us to prove that a 90-year-old woman is a child, a blade of grass is red, and Danny DeVito is tall. It was invented by Eubilides sometime in the 4th century BCE, when he also invented the Liar and a few other paradoxes.…

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