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Recipe

Take an English argument all the way to a verdict

Use when: you're handed an argument in ordinary English and asked whether its conclusion really follows

Goal: a verdict of valid or invalid, backed by a proof or by a countermodel read back into English

Taught in From English to verdict

Quick recipe

  1. Reconstruct in standard form
  2. Write a symbolization key
  3. Translate each line
  4. Test for a bad row
  5. Certify the verdict
  6. Read the result back into English
  1. 1

    Find the main conclusion, list the premises that support it, and drop everything that gives no support. Supply a missing premise only if the passage clearly relies on it.

    Example

    "The café will be busy, since it's Saturday and it always fills up on Saturdays" becomes: 1. It's Saturday. 2. The café fills up every Saturday. \therefore The café will be busy.

    Why: every later stage works on this list, so anything left out or added here changes the argument being judged.

  2. 2

    Give each basic claim its own capital letter. A claim and its denial share one letter.

    Example

    S: it is Saturday. B: the café is busy. "The café isn't busy" is \neg B, not a new letter.

    Why: one letter per claim is what lets the table see that two sentences are about the same thing.

  3. 3

    Symbolize each premise and the conclusion, using the chapter 10 patterns for "only if," "unless," "neither," and the rest.

    • the passage says "A only if B" — write A \to B.
    • the passage says "A unless B" — write A \vee B.

    Why: the test in the next step judges the formulas, so the formulas have to say what the English says.

  4. 4

    Look for a row where every premise is true and the conclusion is false: by full table when there are few letters, by the short method when there are many.

    Why: that row is what "invalid" means, so finding it or ruling it out settles the question.

  5. 5

    No bad row: write a formal proof of the conclusion from the premises. A bad row: state it as a countermodel, one truth value per letter.

    Why: a proof and a countermodel are the evidence a reader can check line by line without redoing your search.

  6. 6

    For a countermodel, describe the situation it stands for. For a proof, say in a sentence why the conclusion is forced.

    Why: the verdict is about the English argument, so it should end in English.

Common mistakes

  • Symbolizing "unless" as an exclusive or, which can turn an invalid argument valid.
  • Offering a row with a false premise as a countermodel.
  • Calling an argument invalid because a proof attempt stalled.
  • Treating "valid" as "the conclusion is true," which skips the question of whether the premises are.