Counter-examples: How do you show that an argument is
invalid or weak?
Remember that an argument is valid if it’s impossible for the premises
to be true and the conclusion false, and it is strong if it’s very unlikely
that the premises are true and the conclusion false.
So to show that an argument is invalid, you only need to find a case in
which the premises are true and the conclusion false, whereas to
show that it is weak, you have to show that it is quite possible for the
premises to be true and the conclusion false. To do so, you will
construct counter-examples.

Definition: A counter-example to an argument is a situation which
shows that the argument can have true premises and a false
conclusion.
If the argument being evaluated is deductive, then we can show it to
be invalid and, therefore, bad if we can describe a counter-example.
Recall this lovely example:
Everybody loves a winner, so nobody loves me.
In standard form, the argument looks like this – with the suppressed
premise:
P1Everybody loves a winner.P2[I am not a
winner.]Therefore,CNobody loves me.
Here’s a counter-example to this argument:
Suppose that everybody loves all winners and that I am not
a winner (so both premises are true.) Still, the conclusion
can be false if one of the people out there who love all the
winners also loves the occasional non-winner, including
me. We can imagine such a person saying: “I love all
winners, but I love you too, even though you’re not a
winner.”
Counter-examples and validity
If there are no counter-examples to a particular argument, then it is
valid, as it is then impossible to find a situation in which the premises
of the argument are true and the conclusion is false. That means that
in every situation in which the premises are true, then the conclusion
is also true, and this is what we need to know to conclude that an
argument is valid.
This is an important link between the concepts of validity and counterexample:

An argument is valid if and only if there are no counter-examples
to the argument.
Counter-examples for Non-deductive Arguments
Can a counter-example be used to show that a non-deductive
argument is weak? It can, but only if the counter-example itself
represents a plausible way things might have been.
Since a non-deductive argument acknowledges that the conclusion
might be false when the premises are true, but only in exceptional
circumstances, you need to find a counter-example that is not so
exceptional.
That is, you need to find a reasonable situation in which the premises
are all true and the conclusion false. If you want to attack a nondeductive argument with a counter-example and do real damage, your
counter-example must describe a situation which not only makes the
premises true and the conclusion false, but is also quite likely to come
about.
If all you can come up is too far fetched, then you’re probably
describing one of the exceptional cases that were excluded to begin
with, and so you would fail to provide a counter-example, and you
could conclude that the argument is strong.
Here’s an example of an argument, one that you may very well have
heard before:
Smoking marijuana is no more dangerous to your health or
to society than drinking alcohol is. And drinking alcohol is
legal. Therefore, smoking marijuana should probably
become legal.
In standard form, the argument looks like this:
P1Smoking marijuana is no more dangerous to your health
or tosociety than drinking alcohol is. P2Drinking alcohol is
legal.P3[If the smoking of marijuana and the drinking of
alcohol havesimilar impact on health and society, they
should have thesame legal status.]Therefore,
probably,CSmoking marijuana should become legal.
Notice that we have introduced what we considered to be an essential
suppressed premise in the standard form. Now, can we find a
plausible counter-example?
Yes we can, here’s one:
The negation of the conclusion is consistent with the
premises. Indeed, we might as well say that alcohol should
become illegal, precisely because the impact of drinking on
the society is similar to the impact of smoking marijuana –
maybe even worse!
This constitutes a counter-example, because it is possible for the
premises to be true and the conclusion false, and given the premises
provided, neither situation seems more plausible than the other.
Implausible counter-examples are counter-examples all the same, but
they can only show that a deductive argument is invalid. However,
only a plausible counter-example can show that a non-deductive
argument is weak.
Deductive or Non-deductive?
You might also think of the search for a counter-example as a method
that helps you determine whether an argument is deductive or nondeductive. If finding a counter-example to an argument makes us want
to say that argument is no good, then the argument must be
deductive, because in deductive arguments the premises are intended
to give conclusive support for the conclusion.
On the other hand, if generating a counter-example does not incline
us to give up the argument, then it is a non-deductive argument
because non-deductive arguments have conclusions which are only
meant to be strongly suggested by their premises, and leave it open
that the conclusion may be false.
Here’s another way of making the point. If the only counter-examples
you can find are far-fetched stories, then you may have an indication
to think that the argument is non-deductive.
Here’s an example:
Wolfgang robbed the safe. Wolfgang’s fingerprints were
found on the burgled safe. Lots of money, which was in the
safe, was found hidden in Wolfgang’s house. Wolfgang
was seen by several witnesses near the scene of the
burglary when it was committed.
Now, we can come up with counter-examples to the argument:
Wolfgang didn’t rob the safe. Sigmund did, but he was
perfectly disguised as Wolfgang, had a copy of Wolfgang’s
fingerprints that he put on the safe, and hid some of the
money in Wolfgang’s house.
Space aliens robbed the safe and engraved Wolfgang’s
fingerprints on the safe with lasers coming out of their eyes.
The safe was never robbed. The whole thing was invented
by the bank to get back at Wolfgang for switching banks.
They really don’t like it when they lose customers.
So it’s easy to come up with counter-examples, but much harder to
come up with plausible counter-examples. This indicates that the
argument is meant to be non-deductive.
© Patrick Girard, University of Auckland
https://www.futurelearn.com/courses/logical-and-criticalthinking/15/steps/822526
Accessed 7/6/21