RAA as it is portrayed in some Google image search results is more of a rhetorical move than a logical one, as formally it leads to explosion.

MT can be derived from MP and contraposition — Lionino

Ok, I'm going to assume you mean this proof (the one wikipedia lists as "Via contraposition"):

1
P→ Q (Given)
2
¬ Q (Given)
3
¬Q →¬P (Contraposition (1))
4
¬ P (Modus ponens (2,3))

This is the proof of modus tollens that you like - it proves modus tollens without assuming it, correct?


So it's pretty straight forward to use the same format, I'll take my previous argument which assumes Modus Tollens:

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

And reformat it to be in the style above, the proof that you like of modus tollens that doesn't assume modus tollens:

ρ→(φ^~φ) (premise)
~(φ^~φ) (law of non contradiction)
~(φ^~φ)→ρ (contraposition)
:. ~ρ (modus ponens)

Something that I read recently, very interesting, and I can't remember where, on the topic of logic, is that syllogisms can be said to be question begging (this is a point that has been made by philosophers in the past).
"All men are mortal; Socrates is a man; therefore, Socrates is mortal" is of no value, since we could not know that the premise, "All men are mortal" is true unless we already knew that Socrates is mortal. So we learn nothing from the syllogism.
https://philosophy.stackexchange.com/questions/104220/do-you-gain-further-truth-from-syllogisms

I didn't really talk about proving MT at any point, so I don't know why you are asking that.

didn't really talk about proving MT at any point, so I don't know why you are asking that. — Lionino

You don't know why I'm asking what?

What my favourite proof is.

you want a proof of ρ→(φ^~φ) , therefore ~ρ that doesn't assume modus tollens, if you give me your preferred proof of Modus tollens I can give that to you.

Why would you need a proof of X in order to find a proof of Y that doesn't use X? That makes zero sense.

if you give it to me I will show you how it makes perfect sense

I've already done it above, but I can do it again

You want a proof of some argument Y that doesn't assume modus tollens.

You presumably have a proof of Modus tollens that you like, that doesn't itself assume modus tollens.

Whatever format of argument that proof takes - that argument that doesn't assume modus tollens, but proves Modus tollens - I can use that exact same format of argument to prove Y similarly without assuming Modus tollens.

MP+contraposition is just MT. So you are not proving φ→(ψ^~ψ)⊢~φ without MT.

so you believe the MP+contraposition argument is circular? It's just using mt to prove mt?

Where did I say that? MP+contraposition is equivalent to MT. MP itself can be proven from more fundamental operations. If you are using those operations to prove something you are using MP, but with extra work.

If it's circular, fine, give me one that isn't circular. I assumed, perhaps wrongly, that when I asked you for a proof of mt that you like, that you wouldn't like an explicitly circular one.

Give me a proof of mt you like that isn't circular.

yes, using MP, not mt.

My proof did not assume mt, it did assume contraposition and MP.

If you believe that's the same as assuming mt, then that means the proof of mt that uses those two assumptions is circular.

The same logic applies to MT...
Assuming contraposition and MP is the same as assuming MT.

I didn't suppose ¬P. — Lionino

Sorry I misread a quote from above. You are right. You supposed S.

I know that S follows from the axioms of the theory. Not an assumption.
Conclusion: P. — Lionino

You know equally well that ¬P follows. Conclusion: ¬S.

You are importing "the axioms of the theory." They are nowhere to be found. They are background conditions, absent from your proof.

so there are more rules to the game then, apparently.

Rule 1. Don't assume mt.
Rule 2. Don't simultaneously assume contraposition and MP

Can I assume MP if I don't also assume contraposition? Can I assume contraposition if I don't also assume MP? Are there any more rules you haven't explicitly stated yet?

Are there any proofs of MT that obey the rules of the game we're playing? Obviously the one you've been talking about doesn't obey .

Banno asked a good question:

So, what is a "direct proof"? I gather you think using MT is direct, but RAA isn't? WHat's the distinction here? — Banno

(i.e. What is the difference between a direct proof like modus tollens and an indirect proof like reductio ad absurdum?)

I said:

Modus tollens requires no "and-elimination" step. Is that a good way to put it in your language? — Leontiskos

Put differently:

One is a statement in the meta-language and the other in the object language. They are different levels of statement. — TonesInDeepFreeze

A direct proof requires no recourse to the meta-language. When the reductio identifies a contradiction it is dipping into the meta-language. That exchange earlier with Tones was about whether the reductio is truth-functional. It turns out that you cannot represent a reductio in the object language.

Another way to put it is that in modus tollens we have two premises whereas in reductio ad absurdum we have a premise and a supposition, and the difference between a premise and a supposition only exists at the level of the meta-language.

Edit: Indeed, this is instructive given that the unique <modus tollens> we are considering also <uniquely requires recourse to the meta-language>. No other modus tollens requires recourse to the meta-language. Nevertheless, the recourse that it requires is different from the recourse that a reductio requires. <If we avoid the meta-language we will only continue banging our heads against the wall>.

(@Lionino)

Assuming contraposition and MP is the same as assuming MT. — Lionino

This of course makes the argument you brought up for MT circular. That's fine, we can move past that and find one that obeys the rules presumably.

meta-language — Leontiskos

Another interesting point goes to natural language. "A→(B∧¬B) means ¬A."

Compare:

1. (φ^~φ) means explosion
2. (φ^~φ) means reductio-rejecton
3. (φ^~φ) means false

Without recourse to the meta-language, there is no way to adjudicate. I think this goes back to 's point.

This sounds like the "Scandal of Deduction," and it actually holds not just for syllogisms but for all deterministic computation and deduction. From an information theoretic perspective, because the results/outputs of computation and deduction always occur with a probability equal to 100% it follows that they are not informative. Everything contained in the conclusion must be contained in the premise; we learn nothing from deduction. The premises must always assume the conclusion.

For some reason, philosophy has generally taken Hume's Problem of Induction more seriously than this problem, but they are equally intractable from a formal perspective. The entire early modern move to prefer deduction and "analyticity," and it's continuation in analytic philosophy, essentially just ignores that deduction is as undermined as induction.

I wrote an introduction on this a while back for 1,000 Word Philosophy, but they weren't interested in the topic.

https://medium.com/@tkbrown413/introducing-the-scandal-of-deduction-7ea893757f09

I do believe I have a solution here that goes beyond psychologism and grounds an answer for why deduction is indeed informative in physics and biology: https://medium.com/@tkbrown413/does-this-post-contain-any-information-3374612c1feb

The problem shows up because logicians, who tend to be the folks most interested in this problem, only look for formal solutions. But the issue is that "eternal implication," or "implication occuring outside time" is assumed. We can think of computation abstractly, but it remains defined by step-wise actions. Yet these abstractions are taken to be "real" as opposed to merely tools.

However, in the brain or in digital computers two things hold:

1. Computation always occurs over time.
2. Computation involves communication and can be thought of in terms of communication models (some very good work on this has been done and the two end up being almost the same thing, "information processing" indeed.)

Hence, deduction is informative because it involves communication. A message cannot be received before it is sent.

Floridi and others have tried very complex formal explanations and I think these just miss the point. What could be more "surface level" information than what color font a word is written in? We recognize color automatically, seemingly in "no time at all." The color "eternally" implies the word.

Except there is the Stroop Test. If you spell out the names of colors and put the font in a different color, people have a hard time reading off the color of the font. They take much longer. Their error rate goes from virtually always 0 to something fairly high if they are trying to do it quickly.

Why? Because deduction/computation, be it in computers or humans, always involves communication and must occur over some region of space-time, not "all at once and all in one place." Aristotle gets at this in his essentially processual conception of demonstration in the Posterior Analytics.

Again, the criticism I had of Wittgenstein for assuming implication is the "real deal," and causality is "superstition" applies here.

1. (φ^~φ) means explosion
2. (φ^~φ) means reductio-rejecton
3. (φ^~φ) means false — Leontiskos

It seems plausible that:

1. (φ^~φ) takes on the meaning of <explosion> as the antecedent of a modus ponens
2. (φ^~φ) takes on the meaning of <reductio-rejecton> as the penultimate step of a reductio
3. (φ^~φ) takes on the meaning of <false> as the consequent of a modus tollens

It's as if (φ^~φ) can be whatever we need it to be for our current purposes, and this should not be surprising.

Note:

• <Sense (2) is quite different from sense (3)>
• <Lionino's "reductio" seems to be ambiguous between senses (2) and (3)>
• <My term FALSE is meant to capture sense (3)>

This sounds like the "Scandal of Deduction," and it actually holds not just for syllogisms but for all deterministic computation and deduction. From an information theoretic perspective, because the results/outputs of computation and deduction always occur with a probability equal to 100% it follows that they are not informative. Everything contained in the conclusion must be contained in the premise; we learn nothing from deduction. — Count Timothy von Icarus

Yes, I was thinking about this as well.

The Scandal also has some implications for philosophy of mathematics.

Some might have it that 2+2 is just another name for 4. 2+2 is 4.

And in some sense it might be right to think of it this way, but not in terms of computation. Why?

Consider a program P that solves Hamiltonian path problems like the "traveling salesman," through brute force. Then consider an input I for that program with an absolutely massive number of nodes. Then consider O as the output of the program, which figures out the shortest path between all the nodes.

In a certain sense, we might be inclined to say that P(I) = O in the same way we would like to say 2+2 just is 4. However, if I includes enough nodes then all of the world's super computers running P(I) until the heat death of the universe still won't have been able to actually compute O yet.

So then, in a very important functional sense P(I) is not "the same thing as O." If we let abstraction get taken for reality we end up with some weird "problems."

This shows up with non-constructive descriptions as well. "The first number that violates the Goldbach Conjecture," is a rigid designator (if such a thing exists). However, there is loads it doesn't tell us, like what digit the number would start with. Discovering this would seemingly require some sort of Herculean computational effort (given a simple search, and given it exists) even though we have a rigid designator description of what we are looking for.

Well, what to make of this? Perhaps mathematics is better thought of in terms of signs and relations instead of identity. This will all be explained in my forthcoming magisterial book introducing Hegelian-Semiotic-Process-Thomism, also to be known as "The Correct Philosophy." :cool:

Metabasis eis allo genos is a complicated topic. I expressed it this way originally:

Every time we make an inference on the basis of a contradiction a metabasis eis allo genos occurs (i.e. the sphere of discourse shifts in such a way that the demonstrative validity of the inference is precluded). — Leontiskos

Note that this is a sufficient condition and not a necessary condition. The same thing can be expressed in terms of the "meta-language":

One is a statement in the meta-language and the other in the object language. They are different levels of statement. — TonesInDeepFreeze

Whenever some logical move requires recourse to the meta-language, we are involved in metabasis. <The three senses> of interpreting a contradiction that I set out are all utilized in the service of a metabasis. This sort of ambiguity always attends metabasis. Sorting out the ambiguity requires us to go beyond the object language at hand.

It seems plausible that:

(φ^~φ) takes on the meaning of <explosion> as the antecedent of a modus ponens
(φ^~φ) takes on the meaning of <reductio-rejecton> as the penultimate step of a reductio
(φ^~φ) takes on the meaning of <false> as the consequent of a modus tollens

Weren't you mocking me earlier in the thread for relating this side of logic to the principle of explosion? Or was that someone else?

Something that I read recently, very interesting, and I can't remember where, on the topic of logic, is that syllogisms can be said to be question begging (this is a point that has been made by philosophers in the past).
"All men are mortal; Socrates is a man; therefore, Socrates is mortal" is of no value, since we could not know that the premise, "All men are mortal" is true unless we already knew that Socrates is mortal. So we learn nothing from the syllogism. — Lionino

This is really the problem of knowledge as expressed in places like the Meno:

I know what you want to say, Meno. Do you realize what a debater's argument you are bringing up, that a man cannot search either for what he knows or for what he does not know? He cannot search for what he knows—since he knows it, there is no need to search—nor for what he does not know, for he does not know what to look for. — Meno, 80e, (tr. Grube)

Aristotle applies his notions of act and potency to basically say that in knowing something partially we can come to know it more fully. When the mind engages in argument this is what it is doing, according to Aristotle. We are unfolding implications previously unseen.

The problem shows up because logicians, who tend to be the folks most interested in this problem, only look for formal solutions. But the issue is that "eternal implication," or "implication occuring outside time" is assumed. We can think of computation abstractly, but it remains defined by step-wise actions. Yet these abstractions are taken to be "real" as opposed to merely tools.

However, in the brain or in digital computers two things hold:

1. Computation always occurs over time.
2. Computation involves communication and can be thought of in terms of communication models (some very good work on this has been done and the two end up being almost the same thing, "information processing" indeed.) — Count Timothy von Icarus

Right, and this is related to my claim:

If this is right then (b∧¬b) introduces instances of formal equivalence that are not provable. — Leontiskos

I believe that given the way formalized logic works, there can be sentences which are formally equivalent and yet underivable from one another. According to Sime one implication of this can be seen in terms of Peano arithmetic (link).

Why? Because deduction/computation, be it in computers or humans, always involves communication and must occur over some region of space-time, not "all at once and all in one place." Aristotle gets at this in his essentially processual conception of demonstration in the Posterior Analytics. — Count Timothy von Icarus

Going back to Meno, if argument was not temporal then we could presumably never gain new knowledge. The other interesting question is how to account for forms of non-temporal knowledge.

So then, in a very important functional sense P(I) is not "the same thing as O." — Count Timothy von Icarus

But probably only because it is NP-complete. When P is not NP-complete it is a more difficult question whether P(I) is the same thing as O. P(I)=O and 2+2=4 are very different in that sense.