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Recipe

Build a counterexample by substitution

Use when: you suspect a deductive argument is invalid and want a demonstration anyone can check, instead of a story about its own subject

Goal: a new argument with the same pattern word for word, whose premises are obviously true and whose conclusion is obviously false

Taught in Proving invalidity with counterexamples

Quick recipe

  1. Put the argument in standard form
  2. Blank out the content, keep the structure words
  3. Refill the blanks with familiar content
  4. Aim for true premises and a false conclusion
  5. Run the three checks
  1. 1

    Number the premises and put the conclusion last, after "so" or the therefore sign.

    Example

    "Every company that went viral had a funny account; we have one; so we'll go viral" becomes: (1) Every company that went viral had a funny account. (2) We have a funny account. So we'll go viral.

    Why: you can't copy a pattern until you can see which claim plays which role.

  2. 2

    Replace each repeated topic phrase with a bracketed blank. Use the same blank each time the same phrase appears. Keep all, some, no, not, or, if … then, and is/are.

    Example

    All [group 1] are [group 2]. [This one] is [group 2]. So [this one] is [group 1].

    Why: the pattern is what carries any guarantee, so it's the thing you have to keep fixed.

  3. 3

    Pick content everyone knows about. Fill the same blank with the same thing every time; give different blanks different content.

    Example

    [group 1] = lottery winners, [group 2] = ticket buyers, [this one] = me.

    • the blanks name groups — try animals, numbers, or places, where membership is common knowledge.
    • the blanks stand for whole claims — try claims about well-known geography or arithmetic.

    Why: the goal is a case where nobody can dispute which claims are true and which are false.

  4. 4

    Adjust your fillings until every premise is obviously true and the conclusion is obviously false.

    Example

    All lottery winners are ticket buyers (true). I am a ticket buyer (true). So I am a lottery winner (false).

    Why: that one combination is what a valid argument can never have.

  5. 5

    Compare your argument with the original, blank by blank. Then confirm every premise is true and the conclusion is false.

    Why: a counterexample that changes the pattern, or has a doubtful premise, tests nothing about the original.

Common mistakes

  • Changing a structure word, such as "all" to "some," while swapping content.
  • Filling one repeated blank with two different things, or two different blanks with the same thing.
  • Choosing content whose truth is debatable, so the premises aren't clearly true.
  • Ending with a conclusion that is true in the real world, which shows nothing.