# Category Archives: Reductio ad absurdum philosophy

In his book, The Two New Sciences Galileo Galilea gives several arguments meant to demonstrate that there can be no such thing as actual infinities or actual infinitesimals. One of his arguments can be reconstructed in the following way. Galileo proposes that we take as a premise that there is an actual infinity of natural numbers the natural numbers are the positive whole numbers from 1 on :.

He also proposes that we take as a premise that there is an actual infinity of the squares of the natural numbers. We can see this because we can see that there is a one-to-one correspondence between the two groups. If we can associate every natural number with one and only one square number, and if we can associate every square number with one and only one natural number, then these sets must be the same size.

But wait a moment, Galileo says. There are obviously very many more natural numbers than there are square numbers. That is, every square number is in the list of natural numbers, but many of the natural numbers are not in the list of square numbers. The following numbers are all in the list of natural numbers but not in the list of square numbers.

So, Galileo reasons, if there are many numbers in the group of natural numbers that are not in the group of the square numbers, and if there are no numbers in the group of the square numbers that are not in the naturals numbers, then the natural numbers is bigger than the square numbers. And if the group of the natural numbers is bigger than the group of the square numbers, then the natural numbers and the square numbers are not the same size.

We have reached two conclusions: the set of the natural numbers and the set of the square numbers are the same size; and, the set of the natural numbers and the set of the square numbers are not the same size.

Galileo argues that the reason we reached a contradiction is because we assumed that there are actual infinities. He concludes, therefore, that there are no actual infinities. Our logic is not yet strong enough to prove some valid arguments.

Consider the following argument as an example. This argument looks valid. By the first premise we know: if P were true, then so would Q v R be true. But then either Q or R or both would be true. And by the second and third premises we know: Q is false and R is false. So it cannot be that Q v R is true, and so it cannot be that P is true.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.

It is a style of reasoning that has been employed throughout the history of mathematics and philosophy from classical antiquity onwards. In its most general construal, reductio ad absurdum — reductio for short — is a process of refutation on grounds that absurd — and patently untenable consequences would ensue from accepting the item at issue. This takes three principal forms according as that untenable consequence is:. The first of these is reductio ad absurdum in its strictest construction and the other two cases involve a rather wider and looser sense of the term.

Some conditionals that instantiate this latter sort of situation are:. What we have here are consequences that are absurd in the sense of being obviously false and indeed even a bit ridiculous. Despite its departure from what is strictly speaking so construed — conditionals with self-contradictory — time to time conclusions — this sort of thing is also characterized as an attenuated mode of reductio. But while all three cases fall into the range of the term as it is commonly used, logicians and mathematicians generally have the first and strongest of them in view.

The usual explanations of reductio fail to acknowledge the full extent of its range of application. For at the very minimum such a refutation is a process that can be applied to. The task of the present discussion is to explain the modes of reasoning at issue with reductio and to illustrate the work range of its applications. But this view is idiosyncratic. Elsewhere the principle is almost universally viewed as a mode of argumentation rather than a specific thesis of propositional logic.

Accordingly, the above-indicated line of reasoning does not represent a postulated principle but a theorem that issues from subscription to various axioms and proof rules, as instanced in the just-presented derivation.

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The reasoning involved here provides the basis for what is called an indirect proof. This is a process of justificating argumentation that proceeds as follows when the object is to establish a certain conclusion p :. As this line of thought indicates, reductio argumentation is a special case of demonstrative reasoning.

What we deal with here is an argument of the pattern: From the situation. An example my help to clarify matters. Consider division by zero. A classic instance of reductio reasoning in Greek mathematics relates to the discovery by Pythagoras — disclosed to the chagrin of his associates by Hippasus of Metapontum in the fifth century BC — of the incommensurability of the diagonal of a square with its sides. The reasoning at issue runs as follows:. Let d be the length of the diagonal of a square and s the length of its sides.

If there were a common divisor, we could simply shift it into u. Now we know that. This means that n must be even, since only even integers have even squares. But this means that m must be even by the same reasoning as before.

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And this means that m and nboth being even, will have common divisors namely 2contrary to the hypothesis that they do not. Accordingly, since that initial commensurability assumption engendered a contradiction, we have no alternative but to reject it.

The incommensurability thesis is accordingly established. As indicated above, this sort of proof of a thesis by reductio argumentation that derives a contradiction from its negation is characterized as an indirect proof in mathematics. On the historical background see T. The use of such reductio argumentation was common in Greek mathematics and was also used by philosophers in antiquity and beyond.

Aristotle employed it in the Prior Analytics to demonstrate the so-called imperfect syllogisms when it had already been used in dialectical contexts by Plato see Republic I, CA; Parmenides d.

The mathematical school of so-called intuitionism has taken a definite line regarding the limitation of reductio argumentation for the purposes of existence proofs.

The only valid way to establish existence, so they maintain, is by providing a concrete instance or example: general-principle argumentation is not acceptable here. Accordingly, intuitionists would not let us infer the existence of invertebrate ancestors of homo sapiens from the patent absurdity of the supposition that humans are vertebrates all the way back.Intuitionistic logic encompasses the general principles of logical reasoning which have been abstracted by logicians from intuitionistic mathematics, as developed by L.

Brouwer beginning in his [] and [].

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Because these principles also hold for Russian recursive mathematics and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics.

Although intuitionistic analysis conflicts with classical analysis, intuitionistic Heyting arithmetic is a subsystem of classical Peano arithmetic. It follows that intuitionistic propositional logic is a proper subsystem of classical propositional logic, and pure intuitionistic predicate logic is a proper subsystem of pure classical predicate logic.

Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing constructive reasoning about uncountable structures e. In his essay Intuitionism and Formalism Brouwer correctly predicted that any attempt to prove the consistency of complete induction on the natural numbers would lead to a vicious circle.

Brouwer rejected formalism per se but admitted the potential usefulness of formulating general logical principles expressing intuitionistically correct constructions, such as modus ponens. Formal systems for intuitionistic propositional and predicate logic and arithmetic were fully developed by Heyting [], Gentzen [] and Kleene [].

Beth [] and Kripke [] provided semantics with respect to which intuitionistic logic is correct and complete, although the completeness proofs for intuitionistic predicate logic require some classical reasoning.

Intuitionistic logic can be succinctly described as classical logic without the Aristotelian law of excluded middle:. Brouwer [] observed that LEM was abstracted from finite situations, then extended without justification to statements about infinite collections. One may object that these examples depend on the fact that the Twin Primes Conjecture has not yet been settled. Formalized intuitionistic logic is naturally motivated by the informal Brouwer-Heyting-Kolmogorov explanation of intuitionistic truth, outlined in the entries on intuitionism in the philosophy of mathematics and the development of intuitionistic logic.

Troelstra and van Dalen [] for intuitionistic first-order predicate logic. A proof is any finite sequence of formulas, each of which is an axiom or an immediate consequence, by a rule of inference, of one or two preceding formulas of the sequence. Any proof is said to prove its last formula, which is called a theorem or provable formula of first-order intuitionistic predicate logic. Thus the last two rules of inference and the last two axiom schemas are absent from the propositional subsystem.

If, in the given list of axiom schemas for intuitionistic propositional or first-order predicate logic, the law expressing ex falso sequitur quodlibet :. Since ex falso and the law of contradiction are classical theorems, intuitionistic logic is contained in classical logic.

In a sense, classical logic is also contained in intuitionistic logic; see Section 4. Decidability implies stability, but not conversely. The conjunction of stability and testability is equivalent to decidability. Here 1, 2 and 5 are axioms; 4 comes from 2 and 3 by modus ponens ; and 6 and 7 come from earlier lines by modus ponens ; so no variables have been varied. The first such calculus was defined by Gentzen [—5], cf. Kleene []. Section 4. These topics are treated in Kleene [] and Troelstra and Schwichtenberg []. While identity can of course be added to intuitionistic logic, for applications e. Each is capable of numeralwise expressing its own proof predicate.

A fundamental fact about intuitionistic logic is that it has the same consistency strength as classical logic. For propositional logic this was first proved by Glivenko []. The negative translation of classical into intuitionistic number theory is even simpler, since prime formulas of intuitionistic arithmetic are stable.

The negative translation of any instance of mathematical induction is another instance of mathematical induction, and the other nonlogical axioms of arithmetic are their own negative translations, so. Direct attempts to extend the negative interpretation to analysis fail because the negative translation of the countable axiom of choice is not a theorem of intuitionistic analysis.

Section 6. Gentzen [] established the disjunction property for closed formulas of intuitionistic predicate logic. Kleene [, ] proved that intuitionistic first-order number theory also has the related cf.

Friedman [] existence property :.Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence.

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Is Miles right. Or is Frank's argument reasonable by making an opposing argument to contrast how absurd Miles's logic is? Reductio ad absurdum is not a fallacy. Rather, RAA is correct reasoning that exposes a fallacy. From the Logically Fallacious page for it :. The fallacy is in the argument that could be reduced to absurdity -- so in essence, reductio ad absurdum is a technique to expose the fallacy. Your example is not a valid case of Reductio ad Absurdum. It's just an example of an absurd argument.

Miles: "If someone created a piece of art, they have full rights to allow or prohibit its reproduction". Frank: "Oh, so when I take a selfie in the city, I need to obtain permission from all the architects?

This is a perfectly valid counterpoint, and Miles needs to resolve it by, e. It is an argument from analogy, which is not deductive reasoning and needs to be evaluated differently. Reductio ad adsurbum requires that there be a valid chain of reasoning that leads from the initial premise to an impossible or unacceptable conclusion.

Your example is not RAA because Frank's response does not describe an actual consequence of the Miles's statement. There may be some similarity between the two situations in Frank's mind, but Miles argues that this is tenuous. Since the two situations are not comparable, we can't reasonably say that one leads to the other.

Since the conclusion doesn't follow logically from the premise, the absurdity of the conclusion can't be used to refute the premise. Frank's argument-as-stated is a non-sequitur. Reductio ad absurdum is when a system of logic is applied to arrive at an inconsistency absurdity. Doing this demonstrates that the system of logic is inconsistent broken.

Reductio ad absurdum can be used to attack a specific argument when it's done using only that specific argument and well-accepted premises. Then when the overall system is shown to fail, the target argument can be faulted as the other arguments are assumed to be faultless.

This can be taken to imply that A is absurd only if all of the others are eliminated as possible sources of error. In this scenario, Frank has successfully demonstrated that whatever system of logic he used to arrive at his conclusion is broken.

So, as stated, Frank's argument is a non-sequitur, as the conclusion does not follow from the stated premises. Dismiss Frank's argument as a non-sequitur. As stated, Frank's argument is a non-sequitur. If Miles doesn't care to help develop it, Miles can just disengage. Ask for Frank to explicitly lay out his arguments. As stated, Frank's argument is a non-sequitur and thus invalid — but, this could be fixed if Frank lays out all of his arguments.

If he does so, then the new statement of Frank's argument could be assessed. If Miles can show that one of Frank's other arguments isn't solid, then Miles can demonstrate that the absurdity doesn't demonstrate the inconsistency of Miles's own argument. If Frank can successfully argue his position using only arguments that Miles agrees with, then presumably Miles ought to accept that Frank has demonstrated an inconsistency in Miles's beliefs.

Since reductio ad absurdum isn't an attack on a specific premise, someone making a reductio ad absurdum is putting forth an attack on the collection of all arguments that they've used in their argument, as this is strictly necessary for their conclusion that the contested premise is faulty to follow.

However, in common social situations, not everyone seems to get this.In logicreductio ad absurdum Latin for '"reduction to absurdity"'also known as argumentum ad absurdum Latin for "argument to absurdity"apagogical arguments, negation introduction or the appeal to extremesis the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.

The "absurd" conclusion of a reductio ad absurdum argument can take a range of forms, as these examples show:. The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses. The second example is a mathematical proof by contradiction also known as an indirect proof which argues that the denial of the premise would result in a logical contradiction there is a "smallest" number and yet there is a number smaller than it.

Reductio ad absurdum was used throughout Greek philosophy. The earliest example of a reductio argument can be found in a satirical poem attributed to Xenophanes of Colophon c. But if horses and oxen could draw, they would draw the gods with horse and ox bodies. The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false.

Greek mathematicians proved fundamental propositions utilizing reductio ad absurdum. The earlier dialogues of Plato — BCErelating the discourses of Socratesraised the use of reductio arguments to a formal dialectical method elenchusalso called the Socratic method.

In response, Socrates, via a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion and adopt a position of aporia.

Calling Bullshit 10.2: Reductio Ad Absurdum

Much of Madhyamaka Buddhist philosophy centers on showing how various essentialist ideas have absurd conclusions through reductio ad absurdum arguments known as prasanga in Sanskrit. Aristotle clarified the connection between contradiction and falsity in his principle of non-contradictionwhich states that a proposition cannot be both true and false.

### Intuitionistic Logic

Therefore, if a proposition and its negation can both be derived logically from a premise, it can be concluded that the premise is false. This technique, known as indirect proof or proof by contradiction has formed the basis of reductio ad absurdum arguments in formal fields such as logic and mathematics. From Wikipedia, the free encyclopedia.

There is no smallest positive rational number because, if there were, then it could be divided by two to get a smaller one. Math Vault. Retrieved Encyclopedia Britannica.Reductio Ad Absurdum is disproving an argument by showing the absurdity of following it through to a logical conclusion.

Essentially, the argument is reduced to its absurdity. This works only if there is faulty logic in the argument to begin with. In a location where there is a sign saying not to pick the flowers, a small child says to his mother, "It's just one flower. You respond, "So, if it brings good luck, then I need to rub it so that my mom's cancer will go away, and my dad will get a new job, and our family will win the lottery. You are in trouble for skipping school, but you tell your father, "All of my friends were going!

He says, "Well, if all of your friends were going to jump off of a bridge, would you do that, too? I think it is agreed by all parties, that this prodigious number of children in the arms, or on the backs, or at the heels of their mothers, and frequently of their fathers, is in the present deplorable state of the kingdom Toggle navigation.

Reductio Ad Absurdum Examples. Reductio Ad Absurdum Reductio Ad Absurdum is disproving an argument by showing the absurdity of following it through to a logical conclusion.Philosophy Stack Exchange is a question and answer site for those interested in the study of the fundamental nature of knowledge, reality, and existence.

It only takes a minute to sign up. What is the exact difference between reductio ad absurdum and proof by contradiction? Wikipedia used to state that:. Reductio ad absurdum Latin: "reduction to the absurd" is a form of argument in which a proposition is disproven by following its implications logically to an absurd consequence. When I read this, I instantly thought, "Ah, that's proof by contradiction However, it continues like this:.

A common species of reductio ad absurdum is proof by contradiction also called indirect proofwhere a proposition is proved true by proving that it is impossible for it to be false. So, I suppose there is some subtle difference between them that isn't clearly explained in the article.

What exactly is that subtle difference? How are the two strategies related, and how do they differ? There is some instability in the terminology here.

More careful authors distinguish them, taking both RAA and indirect proof to be a species of proof by contradiction. I do wish for LaTeX! These two principles are very much worth distinguishing. Intuitionistic logic is one of the best studied and oldest non-classical logics, and one which plays a prominent role in many debates in metaphysics.

There are different types of absurd consequence absurdities. The two main ones being pairs of contradictory and contrary statements. Proof by contradiction meets the first type of absurdity. See the Square of Opposition from syllogistic logic for more on this. Sign up to join this community. The best answers are voted up and rise to the top.