How to work with this page

This page is the workshop for Lesson 6 of the Logic unit. You work through it alone, in about 45 minutes, with your Lesson 6 handout beside you. It has one job: to make sure that when you meet a paradox you take it apart into its three pieces — premises, step, conclusion — and say which piece goes, instead of calling it “a contradiction” and walking away.

Each part is built the same way:

1. Explanation — short, with the lesson’s own examples.
2. Exercise — you type or choose, then press Check. Nobody sees your mistakes. The page counts your first try; that number is for you.

The page has three stages. Stage 1 puts the lesson’s case on trial and asks you to take a side before you have the tools. Stage 2 gives you the tools: the three parts of a paradox, the standard resolutions, the four responses to vagueness, the error-spotting. Stage 3 sends you back to your verdict with the tools in hand. Your sheet tells you when each stage is due.

Recall — Lesson 5 from memory

Before anything new: eight gaps on Lesson 5, without your notes. Guess first: how many of the eight will you get? …… of 8.

Stage 1 — The case first

The Swedish warship Vasa sank on its maiden voyage on 10 August 1628, about 1,300 metres into Stockholm harbour, capsized by a light gust because its ballast could not carry the upper gun deck and the carvings. It was salvaged in 1961 after 333 years on the seabed. From 1962 to 1979 its timbers were sprayed with polyethylene glycol to displace the water in the wood; the glycol has since been slowly transformed by sulphuric acid forming inside the timber from sea-derived iron and sulphur, and conservators spent the 2000s and 2010s neutralising and replacing the failing material. The museum says the ship is roughly 98% original wood — and every long fibre of that wood has been chemically infiltrated and partly substituted. Suppose a Swedish tribunal must now decide, for heritage and insurance purposes, whether the conserved object is continuous with the vessel that sank in 1628.

The argument the tribunal is handed, laid out:

1. The ship that sank on 10 August 1628 was the Vasa.
2. Treating or replacing one more fibre of a ship that is the Vasa does not turn it into a different ship.
3. Therefore, the object in the museum today, after every fibre has been treated, is the Vasa.

Stage 2 · Part A — What a paradox is

A1. The definition

A paradox is an argument: it proceeds from premises that look acceptable, by steps that look valid, to a conclusion that is absurd or contradictory. Three pieces — premises, step, conclusion — and the trouble is that all three cannot stand together. So one of the three must go. The work of a paradox is finding out which, and what it costs.

A paradox is not a contradiction. A contradiction — “it is raining and it is not raining” — is simply wrong; nothing follows except that you should stop saying it. A paradox is not a mere surprise either: a surprising truth (an infinite series adding up to exactly 1) is unexpected, but no absurdity follows from it. Only the paradox has the three-piece shape.

A2. The four paradoxes at a glance

Each conclusion is something no one accepts. Each argument looks valid. Each premise looks fine on its own. That is what makes them paradoxes rather than mistakes.

Part B — One of three must go

B1. The standard resolutions

If the premises look acceptable, the step looks valid, and the conclusion is unacceptable, then at least one of the three looks are deceiving. A resolution names which and pays the price. Three of the four paradoxes have a standard resolution; one does not.

Notice what none of the three standard resolutions does: reject the step. Achilles’ steps, the Liar’s if-then, Russell’s two horns are all left standing. What goes, each time, is a premise that looked too obvious to state.

B2. The Sorites and the four responses to vagueness

The Sorites arises for any vague predicate — heap, tall, bald, rich, young, red — because such a word has no sharp boundary. Remove one hair at a time from a full head: when exactly is the person bald? The lesson’s four responses each do something different with the boundary:

• Epistemicism (Timothy Williamson): there is a precise boundary; we just cannot know where it is.
• Supervaluationism: in borderline cases “John is bald” is neither true nor false — a truth-value gap.
• Fuzzy logic: truth comes in degrees; “John is bald” is 0.7 true for a borderline John.
• Contextualism: “heap” is context-sensitive; the boundary shifts with the conversational context.

B3. Russell’s set: the two horns

Call it R: the set of all sets that do not contain themselves. The set of all cats is not a cat, so it does not contain itself, so it belongs in R. The question is whether R belongs in R.

• Horn 1. Suppose R contains itself. Then R is a set that contains itself — so it does not meet the condition for being in R. Contradiction.
• Horn 2. Suppose R does not contain itself. Then R is a set that does not contain itself — so it does meet the condition, and belongs in R. Contradiction.

There is no third option, and neither horn is a bad step. What Zermelo–Fraenkel set theory does is refuse to let R be formed in the first place: the premise goes.

Part C — Spot the error

C1. How arguments go wrong in this lesson

Seven things students say, and the one mistake in each. Learn to name them: naming is what lets you find them in your own writing.

Stage 3 — Back to the case

Your verdict, with the tools

Take out what you wrote at Stage 1. Now rewrite it in three sentences:

1. Your side: the tribunal discovers the answer (Position A) or draws it (Position B).
2. Your reason — and say at which level it lives: indeterminacy in our judgements (a boundary we cannot find) or indeterminacy in the world (no boundary to find), and which of the four responses to vagueness you are leaning on.
3. The price your side pays. Position A: the borderline case decided wrongly — and the criterion you would apply (the keel? continuous registration? institutional intent?) with a word on why that one survives the Sorites turned on it. Position B: what stops Vasa jurisprudence collapsing into arbitrary consistency with the last ruling, once identity itself is stipulated.

Then check your own paragraph:

• Did you keep cannot know where the boundary is apart from there is no boundary? A paragraph that runs the two together is advising on a different case — that is the slide this lesson exists to catch.
• Did your reason name a premise or a level, not the outcome (the 98% figure, the museum’s claim)? The percentage is precise; the predicate is not; the number settles neither side.
• Did you say the price in your own words, not the sheet’s?

Only now: compare with the three model paragraphs — two good ones, one bad one.

Done?

You can now:

• say what a paradox is without using the word contradiction;
• take a paradox apart into premises, step, and conclusion;
• say which piece Cauchy, Tarski, and Zermelo–Fraenkel each gave up — and what it cost;
• name the four responses to vagueness and say what each does with the boundary;
• run Russell’s two horns and see that the resolution refuses the question rather than answering it;
• catch “we cannot know” being written as if it meant “there is nothing to know” — in your own essays first.

Go back to your sheet.