Is reduction to the absurdum to be considered a syllogism?

Question

I was reading this article called Aristotle on the fourth figure. Thea author while exposing Aristotle theory of syllogism, identify 3 kinds of syllogisms which believes Aristotle to discuss:

• Ostensive (Categorical?) Syllogism
• Hypothetical Syllogism
• Reduction to the Absurdum

Now in my understanding the Categorical Syllogism is the central one and the main discussed. It has 3 figures (or four depending on interpretation) and it can be perfect or imperfect.
Hypothetical syllogism is the one (citing Boethius) in which conditional statements appear. It can be divided in simple and complex, but Aristotle never discussed it except for some brief comments on its existence. It was developed by his students and,improved and organized by Boethius.
Concerning the role of reduction to the absurdum, maybe I did not really get its role but I'm quite skeptic of its inclusion in the above list. Even though my researches brought up some similiar views like this one:

Reductio ad absurdum. A 'reducing to absurdity' to show the falsity of an argument or position. One might say, for instance that the more sleep one gets the healthier one is, and then, by the logical reductio ad absurdum process, someone would be sure to point out that, on such a premise, one who has sleeping sickness and sleeps for months on end is really in the best of health. The term also refers to a type of reductive-deductive syllogism.

I'm looking for some in depth clarification, can we consider the reduction to absurdum a syllogism?
If yes, in which contexts? in which case? why?

PS:
This question is related to this one I asked, but being the questions both broad I decided to separate them, even though the understanding of one is linked to that of the other.

Answer 1

No; reduction to absurdum is not expressible with a syllogistic form of argument.

Reductio is a more "basic" argument: a propositional one.

We can express it either as a "law" or axiom:

⊢ (¬ϕ → ¬ψ) → ((¬ϕ → ψ) → ϕ)

or as a rule:

if we have the derivation of a contradiction from ¬ϕ, we may infer ϕ.

In more formal way:

if Γ, ¬ϕ ⊢ ψ and Γ, ¬ϕ ⊢ ¬ψ, then Γ ⊢ ϕ.

Aristotle uses reductio in his theory:

he “reduces” (anagein) each case to one of the perfect forms and that they are thereby “completed” or “perfected”. These completions are either probative (deiktikos: a modern translation might be “direct”) or through the impossible (dia to adunaton).

In other terms, he uses it to derive the otehr fugures from the first one, considered “perfect” or “complete” (teleios) i.e. that needs no otehr argument to show that the conclusion necessarily results from the premises.

We may say that he uses the perfect form as axiom and reductio is a rule of inference in his system.

---

In modern first order logic, we can prove Barbara from "basic" axioms and rules:

1) ∀x (Px → Qx) --- premise

2) ∀x (Qx → Rx) --- premise

3) Pa → Qa --- from Quantifier axiom: ∀x α(x) →α (t/x) and 1) by Modus Ponens

4) Qa → Ra --- --- from Quantifier axiom and 2) by Modus Ponens

5) Pa --- (temporary) assumption [a]

6) Qa --- from 5) and 3) by MP

7) Ra --- from 6) and 4) by MP

8) Pa → Ra --- from 5) and 7) by Deduction Theorem, discharging temporary assumption [a]

9) ∀x (Px → Rx) --- from 8) by Generalization Theorem (or Gen rule).

Thus, 1)-9) shows:

∀x (Px → Qx), ∀x (Qx → Rx) ⊢ ∀x (Px → Rx)

i.e. we have proved Barbara from propositional and quantifiers axioms and inference rules.

Answer 2

Islamic philosophy, which is based on Aristotelian philosophy, has the following 5 syllogisms:

Categorical

Conditional

Conjunctive

Disjunctive

Contradictory

The last one is a reductio ad absurdum syllogism.

This is well-explained here: https://www.wwnorton.com/college/phil/logic3/ch13/reductio.htm

The second technique we're going to consider is called proof by reductio ad absurdum (RA). The Latin phrase means a reduction to absurdity or contradiction; the technique is to show that if we accept the premises of an argument, but deny the conclusion, we contradict ourselves. This is a good way to establish that an argument is valid.

The reductio ad absurdum technique is to take the negation of the conclusion, add it as an assumption (just as in conditional proof), and then from the premises and that assumption, derive a contradiction--a statement of the form p ~ p.

There is also an example and a proof, but some of the characters do not come across with a copy and paste.

Answer 3

Reductio ad absurdum is a rule of inference. In this rule of inference one denies the conclusion of an argument and then checks to see if a contradiction necessarily occurs in the proof. If a contradiction occurs then the original conclusion must be true to begin with.

A syllogism is a type of argument that mathematics must translate. Mathematics doesn't deal with syllogisms in context or deal with what some words imply which lead to other concepts. A syllogism is a linguistic entity, whereas mathematical logic is symbol manipulation. A syllogism has rules of how to correctly create a valid syllogism that are not transferred over with symbol manipulation. One can reach the same conclusion from the translation from words to symbols but much of the power is lost. Sometimes it is not what we say that makes an impact or influence other people, but "how we say what we say" that makes a difference. This is how math teaching logic differs from philosophy teaching logic.

Answer 4

I would say it is possible to make a syllogism using Reductio ad absurdum.

The starting syllogism: One is not allowed to pick flowers to prevent the field being empty of flowers. The plural flowers indicates that it is forbidden to pick two or more flowers. One is allowed to pick one flower and the field will not get empty.

The reductio ad absurdum counterpunch:

One is not allowed to pick flowers to prevent the field being empty of flowers.

Everyone is allowed to pick one flower.

Everyone is allowed to pick one flower and the field will not get empty.

The first syllogism is created based on an example of a Reductio ad absurdum and goes something like this: Next to a field of flowers there is a sign saying not to pick the flowers. A little boy says to his mother that he is allowed to pick one flower. His mother denies saying "No, because if everyone is picking one flower, there will be no flower left either."

The creation of this syllogism was directed by: https://www.wwnorton.com/college/phil/logic3/ch13/reductio.htm, as the problem was how to create a premise that is opposite to the conclusion.