An example used to prove that an if-then statement is false. For that counterexample, the hypothesis is true and the conclusion is false.
C&P page: 90 Definition: A counterexample to an argument pattern consists of a real-life argument which matches that pattern and yet has 100% true premises and a false conclusion. Comment: Obviously, such a counterexample demonstrates that the pattern in question is not valid-- i.e. can not be relied upon to generate valid arguments. "A) If John Lennon was trampled by a herd of elephants, he died. B) John Lennon was not trampled by a herd of elephants. Therefore © John Lennon did not die." This constitutes a counterexample to the (Jabberwock) argument pattern "If A then B. Not A. Therefore not B."
a.k.a.: Counter argument C&P page: 261 Definition: A counter-example to an argument (as opposed to one to an argument pattern) constitutes (broadly) a demonstration that the premises of that argument could be true under certain conditions where the conclusion would nevertheless be false. Comment: This demonstration will usually consist of adding a premise to the argument, that details a particular way in which the original premises could count as true, and under which it is at least not certain that the conclusion is true. "Little Freddie had turned a blotchy yellow-green by now, and was running a temperature of 107. A scratchy ak-ak-ak escaped from his dry and wrinkled throat." You might be tempted to draw from these premises the conclusion that Little Freddie is sick (to say the least!). But now add in the premise that Freddie is a Geezengolian from the planet Zenith. Are you so sure now what's sick and what's normal
an example of a conditional statement being false
an example that refutes a claim about some subject-matter
an example that undermines the reasoning by showing that even if the reasons are true, they don't necessarily support the conclusion
a positive example not covered by T or a negative example covered by at least one clause in T
a specific example, rooted in what is possible, that makes the reasons in a CPA true and the conclusion false
a structure (which assigns truth-values to all the sentence letters in the sequent) which makes all the formulae (if any) before the turnstile true, and the formula (if any) after it false
an example which contradicts some statement or argument (ex. a counterexample to the statement "All fifteen year-olds have blue hair" would be a fifteen-year-old without blue hair); for an argument, a counterexample would be a situation in which the premises of the argument are true and the conclusion is false; counterexamples show statements to be false and arguments to be invalid.
An example given to establish the invalidity of a CPA. It involves specifying a situation in which the reasons are true but the conclusion is false. This demonstrates that it is not the case that the conclusion must be true if the reasons are.
An example that proves a statement wrong (Lesson 2.3).
An example of a conditional statement in which the hypothesis is true and the conclusion is false.
An example to show that a rule is not true for all numbers. Example:Show by counterexample that the commutative property does not work -- 4-5 does not equal 5-4.
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification (a "for all" statement).
