Reductio ad absurdum
The following is a valid form of argument: “If P, then Q. But not-Q. So not-P.”
Some students initially have difficulty understanding why this is a valid form of
argument. Think of it this way: We know that if P, then Q. Now suppose for the
sake of argument that P is true. Then Q would have to be true, too, right?
Since if P, then Q. But we know that Q is not true!–this is one of our
premises. So our supposition that P is true must be wrong: it leads us to
something that we know is false. That is, it must be the case that not-P.
This kind of reasoning is known as reductio ad absurdum: you accept
some hypothesis for the sake of argument, and then you show that the hypothesis
leads to a contradiction, or to some other conclusion you know independently to
be false. Hence the hypothesis can’t be true. It has to be rejected.
It can be disorienting when you come across a philosopher employing a
reductio, if you misunderstand him as actually subscribing to the
contradiction he derives. You have to recognize that the philosopher who offers
a reductio does not endorse the contradiction himself. He’s arguing that the contradiction is something that follows from his
Here’s an example of a reductio. (I got this wonderful example from my colleague Tim Maudlin.)
A computer scientist announces that he’s constructed a computer
program that can play the perfect game of chess: he claims that this program is
guaranteed to win every game it plays, whether it plays black or white, with
never a loss or a draw, and against any opponent whatsoever. The computer scientist
claims to have a mathematical proof that his program will always win, but the proof
runs to 500 pages of dense mathematical symbols, and no one has yet been able
to verify it. Still, the program has just played 20 games against Gary Kasparov
and it won every game, 10 as white and 10 as black. Should you believe the
computer scientist’s claim that the program is so designed that it will always
win against every opponent?
No. Here’s why: Suppose for the sake of argument that a perfect chess program that always wins were possible. Then we could program two computers with that program and have them play each other. By hypothesis, the program is supposed to win every game it plays, no matter who the opponent is, and no matter whether it plays white or black. So when the program plays itself, both sides would have to win. But that’s impossible! In no chess game can both white and black be winners. So the supposition that a perfect chess program is possible leads to an absurd result. So that supposition must be false. A perfect chess program with the abilities the computer scientist claims must not be possible.
This is a reductio. We assumed some hypothesis for the sake of argument and showed that it leads to an absurd result. Hence the hypothesis must be false.
An equivocation is a bad form of argument where one of the key terms can be
understood in two ways, and the plausibility of the argument depends on reading
the term differently in different premises. For instance, consider the
- All politicians are snakes.
- No snake has legs.
- So no politician has legs.
There’s a metaphorical sense of the word “snake” in which premise 1 might have
some plausibility. But for premise 2 to be plausible, we have to understand the
word “snake” there in its literal sense. There’s no single sense of the word
“snake” which makes both premises plausible. So this argument does not
establish its conclusion: it equivocates on the word “snake.”
Here are some trickier examples of equivocation:
Nature is governed by fixed and unchangeable laws. But every law is the work of some legislator. Therefore, there is some legislator responsible for the governing of Nature.
It’s impossible for two objects to be separated by a vacuum. For if a vacuum
is to separate them then nothing can be between them. But if nothing is between them, then they obviously aren’t separated.
That dog over there is a father. In addition, that dog over there is yours. So that dog must be your father.
Begging the question
This does not mean “prompting or inviting
the question,” though you’ll sometimes see people (even prominent journalists)
misusing the expression that way.
To beg the question is to assume the very point at issue in attempting to argue for it. This is also sometimes called “circular reasoning.” Here is an example of an argument which begs the question:
We know that God exists, because it says so in the Bible. And we can trust the Bible on this matter because it’s the Word of God, and so must be correct.
This argument begs the question because one of its premises says that the Bible is the Word of God. Presumably, one would only accept this premise if one already believed that God exists. But that’s precisely what we’re supposed to be arguing for!
A good rule of thumb is the following: if an argument contains a premise or
step that would not be accepted by a reasonable person who is initially prone
to doubt the argument’s conclusion, then the argument begs the question.
We will seldom see obvious cases of begging the question in our readings. It’s the unobvious cases of begging the question which are really dangerous, because they’re so hard to spot.
Issues about the Burden of Proof
If no positive argument has been given for a claim P, then the following line
of reasoning is fallacious:
[BAD] P has not been shown to be false. So it must be true.
If however, P is some claim which seems intuitively to be true,
or if in our dispute or investigation there is some presumption that P is
true, then anyone who seeks to prove not-P bears what we call the burden
of proof. If he doesn’t succeed in proving not-P–if we can show that his
arguments that not-P are no good–then we’re entitled to go on believing P.
In such a case, we’re legitimately reasoning as follows:
[OK] There is some
presumption that P is true. And P has not been shown to be false. So we can reasonably continue to accept P.
Of course, this isn’t a deductive argument that P. There might be some reason why P is
in fact false–we just haven’t thought of it yet.
Here’s an example of this sort of argument:
The CIA carefully scrutinized Margaret Thatcher for years, and never found her guilty of any terrorist activities or conspiracies. Nor is she known to associate with any terrorist organizations. Hence, until we acquire evidence to the contrary, we can reasonably accept that Margaret Thatcher is not a terrorist.
There is some presumption that Margaret Thatcher is not a terrorist. So unless a convincing proof that she is a terrorist turns up, it’s reasonable to believe that she’s not a terrorist. The burden of proof is on the person who wants us to believe that she is a terrorist.
As you can imagine, philosophers often seek to establish that it’s their
opponents, and not they themselves, who bear the burden of proof.
Where the burden of proof lies will sometimes depend on the dialectical situation. For example, contrast these two situations:
- Eric is a committed believer in God who is trying to convince Matt that God exists. Matt is not convinced by Eric’s arguments, and raises many doubts, which Eric attempts to answer. Matt is not an atheist. He is agnostic. Here Eric has the burden of proof. Matt only needs to examine and criticize Eric’s arguments. He is not obliged to argue that God does not exist.
- Karl is a committed atheist, who is arguing that God does not exist. Eric is a committed believer in God and he is trying to convince Karl that God does exist. Each person is trying to refute the other. Here both philosophers have the burden of establishing their position.
Arguments by Analogy
These sorts of arguments often raise issues about the burden of proof,
because they are hostage to the discovery of unnoticed disanalogies.
For example, here’s a common argument against the death penalty. Suppose Lefty argues:
Imposing the death penalty for murder is hypocritical and inconsistent. You only punish people for murder because you believe killing to be wrong. But then the death penalty itself must be wrong, because it too involves killing someone. And two wrongs don’t make a right. So imposing the death penalty is just as bad as killing someone in cold blood.
Lefty is trying to convince us that we have to take the same view of murder and of capital punishment, else we’re being inconsistent.
Now suppose Righty comes along, and criticizes Lefty’s argument as follows. (Again, credit for the example to Tim Maudlin.)
You say capital punishment is supposed to be analogous to murder. Well, then, you should also count other activities committed by the state as analogous to those same activities when committed by criminals. In particular, since kidnapping–confining someone against their will–is wrong when committed by criminals, so too must it be wrong for the state to confine people against their will (in jails). Hence, if your argument that capital punishment is inconsistent is successful, then by the same reasoning, it would also be inconsistent to jail kidnappers. That is clearly an unacceptable result. So there must be something wrong with your analogy. Murder and capital punishment are similar in some respects. But there are important differences between them, too. And these differences are morally important.
Of course, Righty hasn’t established here that the death penalty is morally acceptable; he’s only criticized Lefty’s argument that the death penalty is unacceptable. There might be other arguments against the death penalty, which are better than Lefty’s.
In this exchange, we’ve seen an example of shifting the burden of
proof. Lefty pointed out an analogy between murder and capital punishment and urged that they be regarded similarly. This puts the burden of proof on Righty,
who wishes to regard the cases differently: Righty has to find some
disanalogy, or to argue that the cases aren’t genuinely analogous.
In our exchange, Righty argues that if Lefty’s analogy were good, then so too should
a second analogy be good, but the second analogy leads to clearly absurd
results. So Righty concludes that the original analogy must be bad too.
This shifts the burden of proof back on Lefty, who has to argue that
the cases really are analogous after all.
A dilemma is a form of reasoning that presents a choice between two alternatives. Here is an example: “If P then Q. And
if R then also Q. But either P or R. So in any event, Q.” Dilemmas are
perfectly respectable forms of argument.
In an argument of this sort, P and R are called “the horns” of the
dilemma. If you want to reject a dilemma, then you have several choices:
- You can “take the dilemma by one of its horns,” that is, accept one of the
options (P or R) and argue that that option doesn’t lead to the consequences
your opponent says it leads to. (Or, you might argue that it does lead to those
consequences, but that those consequences are not so bad or implausible as your
opponent makes them out to be.)
- Alternatively, you can try to “go between the horns of the dilemma,” that
is, show that the options you’re presented with do not exhaust the relevant
A dilemma where the options do not exhaust the relevant possibilities is called
a false dilemma. Here’s an example:
Caliph Omar ordered the destruction of the Library at Alexandria, proclaiming that the books either contained the same doctrines as the Koran, and so were
unnecessary, or else they contradicted the Koran, and so were pernicious. In
either case, they should be destroyed.
This isn’t a good argument that the books should be destroyed, because it
hasn’t considered all the possibilities: what if the books at Alexandria
talk about different things than the Koran, and so neither contain the
same doctrines nor contradict the Koran?
Here’s another example of a false dilemma. Can you explain why?
Should we allow the government to take total control of the software industry, or must we allow companies like Microsoft to be completely free of government regulation?
- You can “take the dilemma by one of its horns,” that is, accept one of the
Some more forms of argument are described in A Philosophical Glossary for Beginners, on this same web site.
The following site describes even more forms of bad argument:
- Stephen’s Guide to the Logical Fallacies: http://onegoodmove.org/fallacy/welcome.htm
But the kinds of arguments described above are the main ones you’ll encounter in philosophical discussion, and they are the only ones you need to familiarize yourself with right now.