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Full Validity Test Walkthrough

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State

Testing an argument by full truth table means tabling every premise and the conclusion together, then scanning every row for the one fatal pattern: all premises true while the conclusion is false.

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The argument: p horseshoe q, and tilde q, therefore tilde p. Two letters, four rows. Fill the reference columns. The first premise, p horseshoe q, reads true, false, true, true. The second premise, tilde q, is the reverse of q: false, true, false, true. The conclusion, tilde p, reverses p: false, false, true, true. Now scan. Row one: premises true and false — not both true, so this row cannot decide anything. Row two: false and true — again not both. Row three: true and false — no. Row four: both premises true, and the conclusion there is true as well. That is the only row where every premise holds, and the conclusion holds with it. No row anywhere shows all premises true beside a false conclusion. The verdict is valid, and now provably so rather than plausibly.

Watch for

Every row has to be checked, including the ones that look irrelevant at a glance. A single unexamined row can hold the fatal pattern, and an argument declared valid after a partial scan has been declared valid on the strength of the rows that happened to be looked at.

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