Logic 101 · Series

Reductio Ad Absurdum Arguments

Many logic textbooks are good at giving you the rules for symbolic logic, but you usually have to get another textbook on “practical” logic to learn how to actually object to arguments in plain language. An objection is a reason that you can give for denying the premises in an argument. Providing objections is how we fulfill the ‘D’ (Deny) in the R.E.A.D. Method.

One way that we can deny an argument is by using a special kind of conditional proof called the reductio ad absurdum argument. This Latin phrase carries over into the English as the “reduction to absurdity” argument. You’ll see exactly what that means in a moment.

In this post I’ll give you the logical rule for reductio ad absurdum arguments called the Negation Introduction (NI) rule, an explanation of the rule, and an extended example of how it works by talking about St. Anselm’s famous ontological argument for the existence of God.

Negation Introduction (NI)

P
[. . .]
~P

∴ ~P

Alternatively:

~P
[. . .]
P

∴ ~~P
∴ P

Explanation: Begin an argument or conditional proof (i.e., sub-argument) by assuming a claim (P) is true. If you run into the negation of that claim (~P) through your process of reasoning, you have encountered a contradiction (⊥) because then you have the premise P & ~P by the conjunction rule. If you encounter a contradiction, you can conclude that the negation of the starting claim is the case (~P). (The rule also works if you begin by assuming ~P. ~~P is logically equivalent to P.) That’s why it’s called the Negation Introduction rule.

By the way, some authors (like Paul Herrick) call this way of reasoning the “Indirect Proof” rule. That’s because you indirectly infer ~P by starting with P and eventually finding a contradiction; you don’t reason to ~P directly, in other words. I like this terminology, but I will stick to calling these types of arguments reductio ad absurdum (reductio for short) on this website.

St. Anselm’s Ontological Argument

In most introduction to philosophy courses that cover the existence of God, and in every introduction to the philosophy of religion course, St. Anselm’s (c. 1033-1109) ontological argument for God’s existence will come up. St. Anselm was the Archbishop of Canterbury from 1093-1109 A.D., and wanted to find an argument that would prove that God exists, and that God has all of the superlative attributes (omnipotence, omniscience, etc.) that are usually ascribed to him. He thought he discovered such an argument, and wrote about it his work called the Proslogion (1077-1078).

The basic idea is this: by understanding God as the greatest conceivable being (“that than which no greater can be conceived”), we can show that God exists from this concept alone. From reading the Proslogion, some scholars have suggested that Anselm was actually using a reductio ad absurdum argument to prove this. Here’s one way we can express his argument:

  1. Suppose God exists only in the mind as a mere idea.
  2. Existence in reality is greater than existence as a mere idea in the mind.
  3. We can conceive of a God who exists in reality.
  4. If (2) and (3) are true, then we can conceive of a God that is greater than one that exists in the mind as a mere idea.
  5. Therefore, we can conceive of a God that is greater than one that exists in the mind as a mere idea.
  6. If (5) is true, then we can conceive of a being greater than the greatest conceivable being.
  7. If we can conceive of a being greater than the greatest conceivable being, then God does not exist only in the mind as a mere idea.
  8. Therefore, God does not exist only in the mind as a mere idea.
  9. Therefore, God exists only in the mind as a mere idea and God does not exist only in the mind as a mere idea.
  10. Therefore, it is false that God exists only in the mind as a mere idea.
  11. If (10) is true, then God exists in reality.
  12. Therefore, God exists in reality.

I know that reading arguments that are placed in standard form like this typically aren’t exciting reading, but Anselm’s argument is a fun one to step through. Do you see how it follows the Negation Introduction rule and is therefore a reductio ad absurdum argument? It begins with the assumption that God exists only in the mind at premise (1), arrives at a contradiction in premise (9), and then concludes the negation of premise (1), which is why we get premise (10).

I’m not sharing this argument because I actually think it’s a sound one. I’m sharing it because it’s a clever reductio ad absurdum argument, regardless of whether the argument is sound or not. This argument will actually give us a springboard for discussing another way that we can deny arguments: by using a parody argument. I’ll dedicate an entire post to using parody arguments and counterexamples.

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