THE LOGIC NOTES

Counter-example Glossary

Definition

A counter-example to an argument form is an argument of that form with actually true premises and an actually false conclusion.

Comments

A counter-example shows the argument form in question to be invalid, since a valid argument with true premises would have a true conclusion.

You cannot show a particular argument in natural language to be invalid in this way unless the counter-example is that argument itself. This is for three reasons: firstly, the argument may not really be of the suggested form; secondly, one argument may be an instance of several forms, and so a counter-example to one of those forms leaves open the possibility that another one may be valid; thirdly, the argument may be valid for reasons that are not captured in any of the forms available in our logic.

Examples

  1. A counter-example to the claim
              ALLxSOMEyRxy   ⊢   SOMEyALLxRxy
    might be
              Every number except 1 has a prime factor;
              So there is a prime which is a factor of every number except 1.
  2. Sequent:   ALLx(Fx OR Gx)   ⊢   ALLxFx OR ALLxGx
    Counter-example: Every key on the piano is either black or white, so either all the keys are black or else all the keys are white.
  3. The parent says: "If you don't tidy your room you can't watch the movie". A little later, the child says: "Well, I tidied my room, so now I can watch the movie."
    "No!" says the parent, "That's not valid reasoning."
    "Yes it is," says the child, "NOTp IMP NOTq, p  ⊢  q."
    "Ah, no," replies the parent. "You might as well argue:
              If Socrates wasn't a biped, then he wasn't a footballer;
              Socrates was a biped;
              Therefore Socrates was a footballer."
    The child advises the parent to read Grice's essay on conversational implicature, and watches the movie anyway.

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