Show: There are infinitely many prime numbers. If so, this is not a contradiction. What should we plug into ? This is a supplement to sections 1.6 and 1.7 of the Logic Text. Conditional Derivation is a powerful tool! Given that we invaded, it follows from your assumption that we brought stability. Socrates: Haven’t we also said that the gods quarrel and differ with one another, and that there’s mutual hostility among them? So the prime factors of P are not on the list. You can use it whenever the show line is a conditional. Indirect proof is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true. You should read those before you read this. Can you see how this derivation fits that pattern? In an indirect geometric proof, you assume the opposite of what needs to be proven is true.

This is tricky. To prove p, assume ¬p and derive a contradiction such as p ∧ ¬p. Other articles where Indirect proof is discussed: reductio ad absurdum: …ad absurdum argument, known as indirect proof or reductio ad impossibile, is one that proves a proposition by showing that its denial conjoined with other propositions previously proved or accepted leads to a contradiction. How do we represent Dove’s line of reasoning? Dove says: Assume you are right, and it is true that if we invaded, then we brought stability. Here is another famous argument by reductio: Euclid’s proof that there are infinitely many prime numbers. As a first step, plug Euthphryo’s proposal, P, into .

[…] Assume for the sake of argument, that the wolves are hunted to extinction. Euthyphro proposes a definition or account of piety: P: What is loved by the gods is pious, and what is not loved by the gods is impious. Some other Republican could be President. On lines (3) and (4), Dove brings down the premises. Conditional Derivation (CD): box and cancel by deriving the consequent of the conditional on the show line. 2. The sentence on line (2) is not the antecedent of the conditional on the Show line, and the sentence on line (4) is not the consequent. So, assuming the premises are true, Hawk’s claim is false. To derive the conditional by CD, you need to show that, if we assume the antecedent, we can derive the consequent. But (7) contradicts (3). And from that, and our third premise, it follows that forest fires will become common. […] Direct Derivation (DD): box and cancel by deriving the sentence that is on the show line. If there are only finitely many primes, then for some finite. Neither will R. We don’t have what we need to apply MP or MT. In common speech the term reductio ad absurdum refers to anything pushed to absurd extremes. Ex 2.1.2 The sum of an even number and an odd number is odd. This is a contradiction. Section 1.7 introduces Indirect Derivation (ID). We can represent this same line of reasoning as a derivation. In official notation, it looks like this. The gods often disagree and quarrel (premise). ¥Use logical reasoning to deduce other facts. Definition Of Indirect Proof. You are also allowed to assume the negation of the sentence on the show line.1. If we were allowed to do this, we could infer from the premise. Video Examples: Introduction to Indirect Proof Socrates wants to show that this is wrong. Here is an example drawn from Plato’s dialogue, The Euthyphro: Euthyphro: What is loved by the gods is pious, and what is not loved by the gods is impious.

We can apply DNI to one of the premises, but that won’t get us anywhere. We might symbolize the pair as P and Q. We brought stability to the Middle East (1 2 by Modus Ponens). You can’t do it the other way around. ABD ACD Logic of Indirect Proof Proof: How to Write an Indirect Proof 1. This is one is unclear. “Assume temporarily” that the conclusion is not true. If a Republican is President, then Romney is. So. I urge you to go read the Euthyphro yourself, and see whether or not you think that Socrates’s reasoning is valid. Can the same person be both happy and unhappy at the same time? This is not a contradiction: both sentences can be true together. If the vegetation is overgrazed, forest fires will become more common. But that is not a valid inference! reduction and indirect reduction or reductio ad impossibile, Aristotle was able to reduce all syllogisms to those of the first figure. A lively and informative new podcast for kids that the whole family will enjoy. By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. Euthyphro: Apparently. The first step, after entering the show line, is to make the assumption, for the sake of argument, that the antecedent true: Here, ‘ASS CD’ is short for ‘Assume, for the sake of a Conditional Derivation’. Let’s begin by considering two real examples of arguments by reductio, one from philosophy and one from mathematics, and one toy example. And from that, and our second premise, it follows that the vegetation will be overgrazed. If we try to construct a direct derivation, we meet a dead end: What moves can we make on line (5)? From that assumption, and our first premise, it follows, by MP, that the deer population will explode. This shows that if (1) is granted, then (5) follows, and so establishes the truth of the conditional. Proofs by Contradiction and by Mathematical Induction Direct Proofs At this point, we have seen a few examples of mathematical)proofs.nThese have the following structure: ¥Start with the given fact(s). It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards. The answers are provided at the end of the chapter, so you can check your work. Since we have shown that ¬p →F is true , it follows that the contrapositive T→p also holds. So your assumption leads to a contradiction, and must be rejected. Each of these forms of derivation models a natural form of reasoning. Example #3 Write an indirect proof in paragraph form: Given: m X m Y Prove: X and Y are not both right angles. Our system of derivation includes three different kinds of derivation—that is, three different ways of boxing and canceling. If the deer population explodes, the vegetation will be overgrazed. Since q2 is an integer and p2 = 2q2, we have that p2 is even. You are also allowed to assume the antecedent of the conditional on the show line. ID models a powerful form of reasoning, common in mathematics and philosophy, that is often called reductio ad absurdum (Latin for “reduce to absurdity”). So, on the assumption that* wolves are hunted to extinction, forest fires become common. No, this is not a contradiction. Ex 2.1.1 The sum of two even numbers is even. Try fitting it into the pattern we used to define contradictions. Can you make it fit the scheme for ID?

The first sentence is in informal notation. Here is an example of the sort of reasoning we can model using Conditional Derivation: Before reading on, try, on a piece of paper, to symbolize each sentence of the argument, using this scheme.

This is a contradiction. Euthyphro: It seems that way. …ad absurdum argument, known as indirect proof or reductio ad impossibile, is one that proves a proposition by showing that its denial conjoined with other propositions previously proved or accepted leads to a contradiction. If the show line is a negation, you can assume its unnegation instead.↩, If Romney is President, then a Republican is President. Suppose, for reductio, that there are only finitely many primes. Strategic Advice: If the show line is a conditional, try to construct a conditional derivation. And that is what we are trying to show: W → R. So we box and cancel by Conditional Derivation (CD), pointing to line 8, where we successfully derived the consequent: So that is what a Conditional Derivation looks like. If you get wrong answer and don’t understand why it is wrong, reread section 1.6 and this supplement with an eye to figuring out what you missed. Practice questions Use the following figure to answer the questions regarding this indirect proof. Reductio ad absurdum is a mode of argumentation that seeks to establish a contention by deriving an absurdity from its denial, thus arguing that a thesis must be accepted because its rejection would be untenable. We did not bring stability to the Middle East (premise). If we invaded Iraq, then we brought stability to the Middle East. We can then use that assumption, together with our premises, to derive the consequent, R: This derivation demonstrates that if W is assumed, then R follows.

But we didn’t bring stability. Show: It is not the case that if we invaded Iraq, then we brought stability to the Middle East. If the wolves are hunted to extinction, the deer population will explode. On line (5), Dove derives S from (2) and (3). Go back up to the top of this page, and the schematic presentation of CD. We need two sentences, one the negation of the other, that together capture line (5) of the argument. Again, which of these pairs of symbolic sentences are contradictions and which are not? (an indirect form of proof). Try plugging W into the box and R into the circle. Proof: By contradiction; assume √2is rational. Reductio ad Absurdum. But if being unhappy just means “not being happy”, then this is a contradiction. Since p / q = √2 and q ≠ 0, we have p = √2q, so p2 = 2q2.

Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. Example: Prove that if you pick 22 days from the calendar, at least […] But here we are just interested in the structure of the reasoning.