Were there any that were not from you? — Banno

Go back and see. Test your a priori thesis for once.

I mostly ignore users who run into a thread shitting on everyone in sight who is not a mod, and that's what I largely did when Tones entered.

I mostly ignore users who run into a thread shitting on everyone in sight who is not a mod, and that's what I largely did when Tones entered. — Leontiskos

Check the actual record of the posts.

Fair enough.

I mostly ignore users who run into a thread shitting on everyone in sight who is not a mod, and that's what I largely did when Tones entered. — Leontiskos

You ignored him for twenty-odd pages?

You ignored him for twenty-odd pages? — Banno

"When Tones entered." But his primary complaint has been that I ignore his posts (and not a few times have I logged in to find more than a dozen new posts from Tones alone, which are naturally ignored).

There may be something in what you are attempting to articulate. Perhaps a difference between Aristotelian logic and prop calculus could be shown in some interesting way. But quite a few of your comments were simply demonstrably incorrect. This thread was a lost opportunity for you.

There may be something in what you are attempting to articulate. Perhaps a difference between Aristotelian logic and prop calculus could be shown in some interesting way. But quite a few of your comments were simply demonstrably incorrect. This thread was a lsot opportunity for you. — Banno

I think you're just unwilling to consider a closer look at the logic machine. One can paper over the differences between RAA and other inferences and get along fine, but there are also interesting differences to be catalogued. And yes, RAA does present a very interesting juncture between ancient and modern logic. MP and MT are commensurable with ancient (and colloquial) logic in a way that RAA is not. RAA directly leverages the LEM in an entirely unique way. But none of my early posts were written with logicians like you or Tones in mind, so it does not surprise me that they did not resonate with you.

If one looks at the manner that a meta-logician justifies RAA it will quickly be seen that the justification is altogether different from the other rules of inference. Looking back at my phil logic text, the author's proof takes the form of mathematical induction applied to the various levels of RAA (i.e. the number of suppositions that an RAA utilizes).

I haven't found RAA to be the most interesting part of this thread, but it should be emphasized that the OP is not ideal material for RAA. RAA ideally requires a set of axioms, the scope of which is then extended over a new proposition. The OP is really not any such thing, and I maintain that the intuitive inference to ~A has more to do with MT than RAA.

(1) The worst thing about you is that you lie about me. And that you tried to wiggle out of that with a specious point about lying, to which I've responded to twice but which you ignore.

(2) The second worse thing about you is that you reply without understanding and, often, not even recognizing that your arguments based in ignorance and confusion had already been dispatched. How ludicrous that you kept asking "by what rule?" when over and over and over you had been told "by RAA".

You did it again in your latest post. You continue to complain that I posted many posts in a row when I've already answered: I was not posting in the the thread during a time when you were posting prolifically (including posts quoting me), so I caught up later, and at a mere fraction of the number of posts as yours. The fact that others weren't posting when I was catching up is not a fault of mine. There's no rule, and there should be no rule, that a poster can't later catch up in a thread. How hypocritical and juvenile of you to demand that you can go post a ton of stuff, including quoting another poster, but that the other poster bears fault for catching up later.

(3) The third worst thing about you is that you don't know Jackson Browne about this subject yet you roar your ignorant, over-opinionated confusions about it.

MP and MT are commensurable with ancient (and colloquial) logic in a way that RAA is not. — Leontiskos

RAA is derivable from MT, and MT is derivable from RAA. [Edit: To be precise, RAA is derivable from MT and LNC, and MT is derivable from RAA.]

Moreover, the next post:

You don't know Jack Kennedy about this subject.

Pythagoras's proof that the diagonal of a square is not commensurate with a side is a quintessentially famous example of reductio. And, if I recall correctly, so is Euclid's proof of the irrationality of the square root of 2.

Internet Encyclopedia Of Philosophy (if it is correct):

"As indicated above, this sort of proof [Pythagoreas's 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. L. Heath, A History of Greek Mathematics [Oxford, Clarendon Press, 1921].)

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, 338C-343A; Parmenides 128d). "

/

Stanford Encyclopedia Of Philosophy (if it is correct):

"Both Zeno of Elea (born c. 490 BCE) and Socrates (470–399) were famous for the ways in which they refuted an opponent’s view. Their methods display similarities with reductio ad absurdum, but neither of them seems to have theorized about their logical procedures. Zeno produced arguments (logoi) that manifest variations of the pattern ‘this (i.e. the opponent’s view) only if that. But that is impossible. So this is impossible’."

/

Wikipedia (though I don't always trust Wikipedia):

"This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate."

and

"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. 570 – c. 475 BCE).[8] Criticizing Homer's attribution of human faults to the gods, Xenophanes states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and ox bodies.[9] 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 using reductio ad absurdum. Euclid of Alexandria (mid-4th – mid-3rd centuries BCE) and Archimedes of Syracuse (c. 287 – c. 212 BCE) are two very early examples.[10]

The earlier dialogues of Plato (424–348 BCE), relating the discourses of Socrates, raised the use of reductio arguments to a formal dialectical method (elenchus), also called the Socratic method.[11] Typically, Socrates' opponent would make what would seem to be an innocuous assertion. 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.[6]

The technique was also a focus of the work of Aristotle (384–322 BCE), particularly in his Prior Analytics where he referred to it as demonstration to the impossible (Greek: ἡ εἰς τὸ ἀδύνατον ἀπόδειξις, lit. "demonstration to the impossible", 62b).[4]"

See ( ) how what you are suggesting doesn't square with what is the case in prop logic? Deriving RAA from MT, and MT from RAA are common introductory exercises.

RAA directly leverages the LEM in an entirely unique way. — Leontiskos

Can you show this using Prop logic? If not, then why can't it be dismissed as an artefact of the limitations of Aristotelian logic?

The IEP article is surprisingly interesting.

RAA directly leverages the LEM in an entirely unique way. — Leontiskos

That was addressed long ago in this thread.

If Gu{P} |- Q & ~Q, then G |- ~P

makes no use of LEM.

However

If Gu{~P} |- Q & ~Q, then G |- P

does require LEM.

IEP gives this as the form of the reductio:
If p ⊢ ~p, then ⊢ ~p

And then

Suppose (1) p ⊢ ~p

(2) ⊢p → ~p from (1)

(3) ⊢p → (p & ~p) from (2) since p →p

(4) ⊢ ~(p & ~p) → ~p from (3) by contraposition

(5) ⊢ ~(p & ~p) by the Law of Contradiction

(6) ⊢ ~p from (4), (5) by modus ponens

@Leontiskos, is there anything in the article that corresponds to what you have been attempting to articulate?

Deriving RAA from MT, and MT from RAA are common introductory exercises. — Banno

It would be hard to dispatch Tones' army of strawmen. I think they are infinite.

Within the paradigm of classical propositional logic there is a certain parity between RAA and other rules of inference (although, as noted, there are also significant differences). But the way that a different paradigm conceives of reductio vis-a-vis direct inferences will not be the same as classical propositional logic. I almost guarantee that Aristotle will see a reductio as a metabasis eis allo genos (and part of the difficulty here is that an absurdity and a contradiction are not synonyms in the historical senses of reductio ad absurdum. Metaphysical and logical absurdities are both utilized historically under that name.).

Now RAA can certainly be used to derive other classical inferences, but a large part of our discussion in this thread revolved around the question of whether RAA can be derived from MT, and this is not at all apparent. This is the question that I was most interested in, because I think the inference to ~A is based in MT. On my view there is merely an analogy between RAA and MT, such that RAA is not an instance of MT. Again, this question was dealt with at some length in the middle of the thread, and the reason we ended by talking about RAA is because neither you nor Tones were comfortable arguing directly from MT.

Can you show this using Prop logic? If not, then why can't it be dismissed as an artefact of the limitations of Aristotelian logic? — Banno

What I mean is that when logic becomes purely formal, abstracted entirely from natural language, then RAA and the suppositions that attend it take on a more central role. It becomes primarily a way to elaborate and extend a system.

IEP gives this as the form of the reductio:
If p ⊢ ~p, then ⊢ ~p — Banno

That is interesting and a bit mind-bending, but it goes to my point above that meta-logical justifications of RAA tend to be sui generis. IEP calls Whitehead and Russell's approach "idiosyncratic." I have no doubt that there are any number of creative attempts to justify reductio in classical propositional logic. It does not reduce as easily to the other rules of inference.

A more stark way to put the difference between a direct proof like MP and an indirect proof like RAA, is that in a dialogical context (which is my primary context) a MP cannot be rebuffed, but a reductio can. Laymen and logicians alike are on occasion apt to say, "An absurdity? A contradiction? So what? 'I contain multitudes'."

None of what you have claimed is novel, nor hopeful.

RAA is certainly a valid inference in classical logic.

...in a dialogical context (which is my primary context) a MP cannot be rebuffed, but a reductio can. — Leontiskos

I'll invite you to set out an example. It might be helpful.

Deriving RAA from MT [...] are common introductory exercises. — Banno

I'll invite you to derive RAA from MT as a way to engage with what I've already written.

My bad, I shouldn't have uncritically adopted your nomenclature. Laws of deduction are not usually derived from one another. But deriving equivalent schema to MT and RAA are exercises in basic logic. Here's one using MT:

ρ→(φ^~φ) (premise)
~(φ^~φ) (law of non contradiction)
:. ~ρ (modus tollens) — flannel jesus

And the conclusion is ρ→(φ^~φ)⊢~p, one of the variants of RAA.

Others have been offered.

and part of the difficulty here is that an absurdity and a contradiction are not synonyms in the historical senses of reductio ad absurdum. Metaphysical and logical absurdities are both utilized historically under that name.) — Leontiskos

Read the articles.

Look up Pythagoras to start.

whether RAA can be derived from MT, and this is not at all apparent. — Leontiskos

It is apparent that RAA can be derived from MT and LNC. (Among non-dialetheists, LNC should be uncontroversial):


RAA:
If Gu{P} |- Q & ~Q, then G |- ~P

MT:
If G |- A -> B and G |- ~B, then G |- ~A

An instance of MT:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P

LNC:
{} |- ~(Q & ~Q)


To derive RAA from MT and LNC:

Show:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P
and
{} |- ~(Q & ~Q)
implies
If Gu{P} |- Q & ~Q, then G |- ~P

Suppose:
If G |- P -> (Q & ~Q) and G |- ~(Q & ~Q), then G |- ~P
and
{} |- ~(Q & ~Q)

Show: If Gu{P} |- Q & ~Q, then G |- ~P

Suppose: Gu{P} |- Q & ~Q
So G |- P -> (Q & ~Q)
{} |- ~(Q & ~Q), so, a foritori G |- ~(Q & ~Q)
So G |- ~P


The other direction drives MT from RAA:

Show:
If Gu{P} |- Q & ~Q, then G |- ~P
implies
If G |- P -> Q and G |- ~Q, then G |- ~P

Suppose: If Gu{P} |- Q & ~Q, then G |- ~P

Show: If G |- P -> Q and G |- ~Q, then G |- ~P

Suppose: G |- P -> Q and G |- ~Q

{P} |- P
So Gu{P} |- Q & ~Q
So G |- ~P

It would be hard to dispatch Tones' army of strawmen. — Leontiskos

It's more than hard. It's impossible. Because it's impossible to dispatch what doesn't exist

I almost guarantee that Aristotle will see a reductio as a metabasis eis allo genos — Leontiskos

Your guarantees or "almost" guarantees are as worthless as a day pass to a rickety ride park closed for serial safety violations fifty years ago.

creative attempts to justify reductio in classical propositional logic. — Leontiskos

If Gu{P} |- Q & ~Q, then G |- ~P.

It's merely a matter of showing that if G along with P proves a contradiction, then there are no interpretations in which the all the members of G are true and P is true.

Proving that is actually rather routine and dull, hardly creative.

Laymen and logicians alike are on occasion apt to say, "An absurdity? A contradiction? So what? 'I contain multitudes'." — Leontiskos

What is supposed to be the point of that? Classical logic doesn't excuse contradiction.

I'll invite you to derive RAA from MT as a way to engage with what I've already written. — Leontiskos

RAA derived from MT and LNC. Done.

It becomes primarily a way to elaborate and extend a system. — Leontiskos

No, in a natural deduction system it is not a mere "elaboration" nor "extension". It is crucial for proving negations; it is key to having a system that deals with negation. Without such rules, the system would not be complete, in the sense that there would be validities not provable.

Laws of deduction are not usually derived from one another. — Banno

It's done frequently.

Ok. I'll take your word for it.

For example, proving the deduction theorem (thus deriving the rule of '-> introduction') for a Hilbert style system. That's one of the key topics in an early chapter of just about any intro text in mathematical logic that uses a Hilbert style system. Cf., e.g., Enderton's 'A Mathematical Introduction To Logic'.

Ok. Again, it seems to be so.

Actually, there've been other first insulters in this thread. — TonesInDeepFreeze

I think my insult was in fact in my thread, not on this one.

That last clause is wrong, obviously. (Maybe you corrected it subsequently.) — TonesInDeepFreeze

Υes, I did. In my thread, I explored the question "On the flip side, can the English meaning of "A does not imply B" be converted to first order logic formulas?". My conclusion thus far is no.

I referred to the last clause in this quote as it is still posted:

Now, the conclusion that I arrived at is that "A does not imply a contradiction" is not an accurate statement about ¬(A→(B and ¬B)), it would be a true statement about (A→¬(B and ¬B)) instead. — Lionino

"A does not imply a contradiction" is not a true statement about "(A→¬(B and ¬B))".

First, this is not a derivation of RAA. It is a putative modus tollens that looks a little bit like an RAA. As I said, there are analogical similarities.

Second, this is precisely what we argued about in the middle of the thread, and no one was willing to accept this sort of argument as a straightforward modus tollens. Again, the whole reason you have been so laser-focused on RAA is because the MT is so unconvincing. Recall that in order to run this "modus tollens" one must conceive of the contradiction as false (or 'FALSE') in a manner that is sui generis for a non-simple logical formula. Earlier in the thread you characteristically begged off from entering into that meta-logical dispute, and as a consequence refused to try to prove ~A via modus tollens.

-

Setting this out more clearly:

A corollary of what I have been arguing in this thread is the idea that reductio ad absurdum is a kind of black sheep in the logical family, or that there is a measure of discontinuity between RAA and the other inferences of classical propositional logic, such that there is no straightforward derivation of RAA from these other rules of inference.*

Now someone like yourself who is unwilling to engage in meta-logical discourse will naturally have a hard time seeing my thesis, and for this reason my thesis was directed towards people like Count Timothy rather than you or Tones. You say that I have made a number of well-documented errors in this thread. This is assertion and hot air which can in no way be substantiated, but there is a way for you to show that my corollary is mistaken. The corollary is mistaken if there is a straightforward derivation of RAA from the other rules of inference in classical propositional logic. All you need to do is provide such a derivation.


* What I have said more recently is that the more purely formal a system is, the less this discontinuity of reductio ad absurdum is able to be recognized.

All you [another poster] need to do is provide such a derivation. — Leontiskos

I did.

And his is okay too; all it needs is to be generalized as I did with G.