How to work with this page

This page is the workshop for Lesson 8 of the Logic unit — the unit’s last lesson. You work through it alone, in about 45 minutes, with your Lesson 8 handout beside you. It has one job: to make sure that when you say a logic “gives something up”, you can name which law it gives up and what it buys with it — and that you never again read “0.7 true” as “70% sure”.

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 classical laws, the four systems that decline one of them, the price each pays, 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 7 from memory

Before anything new: eight questions on Lesson 7, without your notes. Guess first: how many of the eight will you get? …… of 8.

Stage 1 — The case first

Between 1932 and 1968 the Chisso Corporation discharged methylmercury into Minamata Bay. Thousands of residents who ate the fish suffered neurological injury, from severe Hunter–Russell syndrome at one end of the spectrum to subtle sensory disturbance at the other. In March 1973 the Kumamoto District Court ruled against Chisso and ordered damages: the company had the foreseeability and the means to prevent the discharge. That settled one question for the plaintiffs in front of the court. It did not settle who counts as a Minamata-disease patient entitled to compensation. The 1977 certification criteria required a combination of several severe symptoms and left thousands of partially affected residents uncertified; the 1995 settlement, the 2004 Supreme Court ruling against the state, and the 2009 Special Measures Law each redrew the line. Fifty years were spent contesting which side of it a person falls on.

The certification board’s problem, laid out:

1. Exposure produced a continuous spectrum of harm, from severe to barely measurable.
2. Certification is a yes-or-no status: a person is a Minamata-disease patient, or is not.
3. Therefore, wherever the line is drawn, someone just below it is uncertified while a marginally worse neighbour is certified.

Stage 2 · Part A — Three laws, four systems

A1. The three classical laws

Classical logic — Aristotle’s, formalised by Frege and Russell — rests on principles that feel like bedrock. The lesson names three:

Two of these are easy to confuse, so hold them apart from the start. Non-contradiction forbids both: P and not-P cannot both be true. Excluded middle forbids neither: one of P and not-P must be true. A logic can drop one and keep the other.

A2. Four systems, each named by the law it declines

Dialetheism is paraconsistency pushed one step further. A paraconsistent logic tolerates a contradiction in a system without letting it spread; the dialetheist says some contradictions are not merely tolerable but true. You need the first to hold the second without collapse; you can hold the first without the second.

A3. Degrees are not probabilities

Zadeh’s proposal pays for the Sorites by abandoning bivalence: a truth value is any real number in [0, 1]. “John is tall” can be 0.7 true; “these 100 grains are a heap” can be 0.3 true. The number measures how far the predicate applies, not how sure you are. You may be completely certain that John is 0.7 tall.

Zadeh’s three rules for combining degrees:

Part B — Explosion and the constructive refusal

B1. Explosion, and why paraconsistency is closer to practice

In classical logic a single contradiction is fatal to the whole system: from “P and not-P” every proposition follows — that the defendant owes ¥937 million, and that he owes nothing, and that the moon is cheese. This is the principle of explosion. It is why classical mathematicians fear an inconsistency above everything: one is enough to make the system prove anything.

A paraconsistent logic refuses that step. A contradiction stays where it is; it licenses nothing beyond itself. That is closer to how legal codes, databases and working physics actually behave — and it is only that. Paraconsistency does not say the contradiction is true, does not say contradictions are fine, and does not let you derive what you like. It says: the fire does not spread.

B2. The intuitionist’s one-way street

To the intuitionist, to assert a proposition is to have constructed it — a proof, or an object that witnesses it. “Not-P” means: P leads to absurdity. So “P or not-P” may be asserted only when you can assert one of the two, and for a conjecture that is neither proved nor refuted, neither half is yet available. That is what giving up excluded middle means: not that a third value exists, but that neither of the two has been reached.

Recall from Lesson 2: modus ponens and modus tollens are both valid in constructive mathematics, and contraposition runs in one direction only — from “if P then Q” to “if not-Q then not-P”. The return journey needs double-negation elimination (from “not-not-P” to “P”), and that is the move the intuitionist refuses. The same refusal blocks the classical proof by contradiction that concludes P: assume not-P, derive absurdity, and you have shown not-not-P, which for the intuitionist is not yet P.

Part C — The four corners

C1. Nāgārjuna’s scheme

The Buddhist catuṣkoṭi (“four corners”), deployed systematically by the 2nd-century Madhyamaka philosopher Nāgārjuna, asks of any proposition P whether

1. P,
2. not-P,
3. both P and not-P,
4. neither P nor not-P.

The third corner is what non-contradiction forbids; the fourth is what excluded middle forbids. Classical logic keeps only the first two. Nāgārjuna’s own central use is to argue that the four corners exhaust the available positions on a question such as dependent origination — and that ultimate reality (paramārtha) cannot be fixed by any of them. Whether that makes the catuṣkoṭi an early dialetheist logic (Priest’s reading), a paraconsistent move (Garfield), or a non-logical device for breaking attachment to any position (the Madhyamaka self-understanding) is contested — and Stage 3 will ask you what turns on it.

Part D — Spot the error

D1. How arguments go wrong in this lesson

Six 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 (Position A: keep the binary predicate and move the line; Position B: go degree-valued).
2. Your reason — and say which law it keeps or gives up: bivalence, kept because a right is had or not had; or bivalence, dropped because the harm is continuous. Say whether your reason is about this domain or about logic everywhere.
3. The price your side pays. A: the plaintiff whose symptoms fall just below the threshold while a marginally worse neighbour is certified — and the answer you owe them. B: everything else that must then become degree-valued — guilt in criminal law? legal personhood? — and the point at which the graded instrument stops being the right one.

Then check your own paragraph:

• Did you keep “the binary is painful here” apart from “bivalence is wrong everywhere”? The first is a claim about a domain; the second is a claim about logic, and it needs its own argument.
• Did your reason name a law, not an outcome (the uncertified patients, the fifty years)? The outcome is evidence for dropping or keeping the law; the paragraph has to say so.
• Did you carry your Vasa commitment across from museum to courtroom — or revise it? A revision is allowed; an unnoticed one is not.
• Did you say the price in your own words, not the sheet’s?
• Did you notice that no logic wins this on the merits — that your choice of instrument is grounded in something that is not itself a logical result?

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

Done?

You can now:

• name the three classical laws and keep excluded middle apart from non-contradiction;
• say, for fuzzy, paraconsistent, intuitionistic and dialetheist logic, which law each gives up and what it buys;
• read “0.7 true” as a degree, never as a probability;
• say what explosion derives from a contradiction and what paraconsistency refuses;
• say which classical steps the intuitionist refuses, and why contraposition runs one way only;
• place a sentence in one of the four corners — and see when it is in none of them;
• catch “anything goes” and “scepticism” being used for paraconsistency and intuitionism — in your own essays first;
• and, looking back across the unit: say where each case lived — at a premise (Surgisphere, Loomis, Tuskegee), at a base rate (Sally Clark), at a system’s own specification (MCAS), at a threshold (RBS, the Vasa, Minamata) — and see that this lesson’s question, what grounds the rules?, is Carroll’s Tortoise from Lesson 1, come back around.

Go back to your sheet.