It is true that knowing the way many writers in logic have not found the subject silly helps to understand why it is of interest. — TonesInDeepFreeze

Argumentum ad populum, then? Such a weak response does not stand up to the arguments that are found in the OP and in this thread.

The proponent of the "Liar's paradox" wants to say that something like, "This sentence is false,"* represents something that is simultaneously true and false in the way that constitutes a formal contradiction. I have no idea what they purport to mean by this. I think they are confused. I challenge them to give a coherent explanation for their thesis.

* Or that, "I am lying," represents an utterance that is simultaneously a lie and a non-lie.

- You haven't given any arguments for your unfounded assertions. The only thing that got close were some poorly written sentences. Hence my reply:

Despite the fact that these sentences of yours are not grammatically correct, you are of course welcome to try to defend your assertions.

Here is the central sort of question you are avoiding:

is it possible for someone to speak, "I am lying," while simultaneously meaning that they are lying and that they are not-lying?
— Leontiskos — Leontiskos

You also seem to be committing a genetic fallacy. — wonderer1

How so? Try making a real argument, bud.

And I did engage the original question. And I've given good background and information about. — TonesInDeepFreeze

You <made some unfounded assertions> and then wrote 8 short non-committal replies with more unfounded assertions, all in response to posts that were written some five years ago. So no, I don't think you have.

Recently in another thread another poster took exception to certain senses of 'grammatical'. — TonesInDeepFreeze

See: Grammatically Correct (Collins). Here are the sentences in question:

And we may consider sentences that are displayed without implication (sic) that they have an implied or even hypothetical speaker. There (sic) instances in which we may consider display (sic) of a sentence so that we may consider it in and of itself. — TonesInDeepFreeze

The point here is that if you really think the cases that the OP has in mind, such as the "Liar's paradox," are coherent, then you are free to make such an argument. Thus far you have not done so, despite ample opportunity.

Do you think that being employed by an institution is somehow contradictory to being a man who seeks truth? — wonderer1

There is no more ubiquitously conflicting interest than the interest in truth. Consider the Fauci case:

1. Masks are effective against Covid-19.
2. If society knows this, then there may not be enough PPE for medical professionals.
3. Therefore, I must lie and say that masks are ineffective against Covid-19.

This is a perfectly standard expedient lie, and there may be nothing that humans are more adept at than the expedient lie. When science becomes fettered to an end that is separate from truth, conflicts of interest such as these inevitably arise. The sort of institutions that science has now become wed to all hold such heterogenous ends.

Science is a practice of bookkeeping guess work.

But the only way to make that bookkeeping guess work worthwhile is through honesty, or perhaps another virtue.

So, transcendentally: How is it possible to arrive at scientific truth? The only possible way is through honest bookkeeping. — Moliere

The only way to arrive at truth is to desire truth, and those who desire truth as a means to something else do not desire truth qua truth. Scientists were once lovers of truth, and because of that they were reliable. But now that science has become a means, scientists are no longer reliable. Their science (and its truth) is a means to some further end, and because of this the science has lost its credibility. When the scientist was a man who sought truth we believed him to be speaking truth, but now that the scientist is an employee of institutions, we believe him to be acting in the interests of those institutions.

Covid is a very good example. Fauci appealed to his scientific bona fides to inform us that masks are ineffective against Covid-19. We later learned that he was lying in order to ensure enough personal protective equipment (PPE) for medical professionals. We thought the scientist was speaking the truth, whereas in fact he was acting in the interests of his institution by speaking outright lies.

Now-a-days I'd say science is a profession tailored to the economy. I want to figure out how to tie it to Marx, duh, and so call it knowledge-production. — Moliere

I would want to say that the reason science is not knowledge-production is because it is tailored to the economy. Modern science is GDP-production, or arms-production, or health-production, and is only incidentally knowledge-production. This has been particularly true since the inception of the modern research university. Speculative knowledge has been more or less entirely eclipsed in our culture.

From an interesting and pertinent article by the Harvard historian of science, Steven Shapin:

So, by the middle of the 20th century, the scientific community — in the United States and many other Western countries — had achieved a goal long wished for by many of its most vocal members: it had been woven into the fabric of ordinary social, economic, and political life. For many academic students of science — historians, sociologists, and, above all, philosophers — that part of science which was not an academic affair remained scarcely visible, but the reality was that most of science was now conducted within government and business, and much of the public approval of science was based on a sense of its external utilities — if indeed power and profit should be seen as goals external to scientific work. Moreover, insofar as academia can still be viewed as the natural home of science, universities, too, began to rebrand themselves as normal sorts of civic institutions. For at least half a century, universities have made it clear that they should not be thought of as Ivory Towers; they were not disengaged from civic concerns but actively engaged in furthering those concerns. They have come to speak less and less about Truth and more and more about Growing the Economy and increasing their graduates’ earning power. The audit culture imposed neoliberal market standards on the evaluation of academic inquiry, offering an additional sign that science properly belonged in the market, driven by market concerns and evaluated by market criteria. The entanglement of science with business and statecraft historically tracked the disentanglement of science from the institutions of religion. That, too, was celebrated by scientific spokespersons as a great victory, but the difference here was that science and religion in past centuries were both in the Truth Business.

When science becomes so extensively bonded with power and profit, its conditions of credibility look more and more like those of the institutions in which it has been enfolded. Its problems are their problems. Business is not in the business of Truth; it is in the business of business. So why should we expect the science embedded within business to have a straightforward entitlement to the notion of Truth? The same question applies to the science embedded in the State’s exercise of power. Knowledge speaks through institutions; it is embedded in the everyday practices of social life; and if the institutions and the everyday practices are in trouble, so too is their knowledge. Given the relationship between the order of knowledge and the order of society, it’s no surprise that the other Big Thing now widely said to be in Crisis is liberal democracy. The Hobbesian Cui bono? question (Who benefits?) is generally thought pertinent to statecraft and commerce, so why shouldn’t there be dispute over scientific deliverances emerging, and thought to emerge, from government, business, and institutions advertising their relationship to them? — Steve Shapin, Is There a Crisis of Truth?

The matter I addressed whether self-referential sentences all must be disqualified, not whether the liar sentence in particular must be disqualified. — TonesInDeepFreeze

What I am primarily interested in is the OP. I am sure Russell can speak for himself.

And we may consider sentences that are displayed without implication that they have an implied or even hypothetical speaker. There instances in which we may consider display of a sentence so that we may consider it in and of itself. — TonesInDeepFreeze

Despite the fact that these sentences of yours are not grammatically correct, you are of course welcome to try to defend your assertions.

Here is the central sort of question you are avoiding:

is it possible for someone to speak, "I am lying," while simultaneously meaning that they are lying and that they are not-lying? — Leontiskos

But, again, (1) It is a light year away from a "habit". — TonesInDeepFreeze

Perhaps you have a habit that you are not aware of. Someone wrote a single post in the whole thread and you managed to misquote that single, short post. I submit that what is at play is the strawmanning that you are often engaged in, for your idiosyncratic interpretations always harm the legitimacy of your interlocutor's position.

You wrote a falsehood, and apparently for effect. Instead of owning your own words, you speciously turn it back on me, to fault me for catching your lie. — TonesInDeepFreeze

There is a wonderful dovetailing between you and this thread with respect to conflating (purported) falsehoods with lies. But we've been over that already.

You skipped my examples that are not of that kind. — TonesInDeepFreeze

I addressed your example of word-counting.

Just now, you referenced the sentence "colourless green ideas sleep furiously" without there being an implied speaker other than a hypothetical one. — TonesInDeepFreeze

Where have I said that the implied speaker cannot be hypothetical?

And previously — TonesInDeepFreeze

I addressed this in my reply to that post.

Also, consider the following sentence that I am not asserting but merely displaying so that we can talk about it:

This sentence has five words. — TonesInDeepFreeze

Again:

Sure you do. When someone considers the claim, "Colourless green ideas sleep furiously," you will inform them that the statement they are considering is nonsensical. We could say that to consider a possible utterance is to speak it secundum quid, and what is not able to be spoken is not able to be considered. The objection to such a consideration is always something like, "No one in their right mind would ever speak such a thing." To consider an utterance that has no possible speaker is to consider a nonsensical utterance.

Bringing this back, then, to the OP, we should ask whether the "sentences" in question—along with their attributed meaning—have any possible speaker. For example, is it possible for someone to speak, "I am lying," while simultaneously meaning that they are lying and that they are not-lying? No, it is not. There is no possible speaker in such cases, and hence the "sentences" are nonsensical (even in the additional cases where they are thought to have an extrinsic object). — Leontiskos

"This sentence has five words," has a possible speaker, therefore it can be spoken, and therefore it can be spoken secundum quid (in the form of a consideration). When we consider a statement we say, "What would it be like to make a statement such as this? :chin:" When, "No one in their right mind would ever speak such a thing," then it cannot be legitimately considered.

You are lying that I "continue". — TonesInDeepFreeze

You are full of vapid nitpicking. But I am glad you corrected your mistake this time.

Not for me. I can consider a sentence for consideration without assuming an implied speaker, and certainly not an implied speaker who asserts it to be true. — TonesInDeepFreeze

Sure you do. When someone considers the claim, "Colourless green ideas sleep furiously," you will inform them that the statement they are considering is nonsensical. We could say that to consider a possible utterance is to speak it secundum quid, and what is not able to be spoken is not able to be considered. The objection to such a consideration is always something like, "No one in their right mind would ever speak such a thing." To consider an utterance that has no possible speaker is to consider a nonsensical utterance.

Bringing this back, then, to the OP, we should ask whether the "sentences" in question—along with their attributed meaning—have any possible speaker. For example, is it possible for someone to speak, "I am lying," while simultaneously meaning that they are lying and that they are not-lying? No, it is not. There is no possible speaker in such cases, and hence the "sentences" are nonsensical (even in the additional cases where they are thought to have an extrinsic object).

For example, in a math book may appear sentences that were typed by an author but are not considered to be specific to any one person. For example, I can display the sentence, "Harry Truman was a president" and that sentence can be discussed no matter that its just typed by me. — TonesInDeepFreeze

There is always an implicit or implied speaker. When you consider a claim like that you are implying the linguistic intentions of the average English speaker in order to infer meaning. A nonsensical statement fails in all of this, insofar as there is no true speaker and there is no implicit speaker. The only people who pretend that such nonsensical statements have meaning are, again, "philosophers."

The other poster said that sentences have truth value only if they refer to "the material world" and not themselves. — TonesInDeepFreeze

You continue your habit of falsely attributing quotes. He said nothing about the "material" world.

This sentence has five words.

Not true? — TonesInDeepFreeze

It is true that we can treat sentences as objects of predication, but the difference is that the number of words that a sentence contains is a material property, not a formal property. So could say, "Colourless green ideas sleep furiously," has five words, but he could not say, "Colourless green ideas sleep furiously," is true (or meaningful). Counting words and affirming a truth value are two different things.

There are formulations in which there is no speaker nor reference to "I' or things like that. — TonesInDeepFreeze

Many "philosophers" mistakenly hold that sentences have meaning apart from speakers, and when one reifies sentences in this way they have taken the first step towards this sort of self-confusion. They strangely believe that a sentence can self-negate itself because they have taken their eye off the ball: the speaker. — Leontiskos

Same goes with "This statement is false", not all statements that can be uttered in a language are meaningful, and I agree it's not much use to spend much time pondering about them — leo

it says nothing about anything, like saying “this statement is Fred”. — Fire Ologist

As the statement "colourless green ideas sleep furiously" expresses a nonsense proposition, then so does the statement "this statement is false". — RussellA

Yep. :up:

Or the so-called "Liar's paradox":

• "I am lying."
• "Lying about what? You haven't yet managed to construct a coherent sentence."

I agree it is not a statement, meaning it is not about anything. — Fire Ologist

Yes, it seems to me that this is just another case of "philosophers" confusing themselves.

However you slice it, the intent of the "liar" determines whether he is lying. He either is or he is not. There is no both-and. The same goes for someone who claims to be speaking falsely rather than lying. Either they intend to speak falsely or they do not. Many "philosophers" mistakenly hold that sentences have meaning apart from speakers, and when one reifies sentences in this way they have taken the first step towards this sort of self-confusion. They strangely believe that a sentence can self-negate itself because they have taken their eye off the ball: the speaker.

For example, I can't conceive of anything as being other than it is, because as soon as I conceive it, it is what it is, and not something else. I cannot imagine something as being otherwise. This reminds of the law of identity, and it just might be. — Lionino

This is very close to the way that Aristotle defends the PNC in Metaphysics IV. Much of this is just a question of what we mean by 'logic'.

in "This statement is false" we're never saying what we're referring to. — leo

Right. The so-called "Liar's paradox" seems quite silly, akin to something a third grader thought up at recess.

I agree it's not much use to spend much time pondering about them — leo

Me too. :up:

I am thinking of what SEP calls, "Aristotle’s Challenge to the Opponent to Signify Some One Thing."

More:

The Aristotelian can counter that without those qualifications the dialetheist has not said anything meaningful at all. — SEP | 11. Dialetheism, Paraconsistency, and Aristotle

To which the dialetheist may simply say "so much for Aristotle". — Banno

I would suggest actually reading Metaphysics IV.

But what is second-order rules of discourse? — Lionino

The examples I gave were:

Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself). — Leontiskos

Note, though, that, "You are contradicting yourself," or, "This is a contradiction," is a different genus, and deviates from first-order discourse, moving into the meta-language. — Leontiskos

So an example of a second-order rule of discourse is, "Thou shalt not contradict thyself."

It's a mistake to think that there are laws of logic that have complete generality - and must be obeyed in all circumstances.

...

Logic sets up systems in which some things can be said and others are ruled out, but natural language is far broader than that, allowing for the breach of any such rule. — Banno

Yet if what Aristotle does in Metaphysics IV is correct, then there is a logical law that cannot be breached, namely the law of non-contradiction. Or in other words, "logic" is not a purely formal exercise. It was created for a reason and that reason has implications for reality/metaphysics.

Elaborate. — Lionino

For example:

The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this. — Leontiskos

"If the Baltic sea is salty, then the Eiffel Tower stands." According to material implication this is a perfectly good statement, but according to English it is foolish. There is nothing which surpasses this sort of statement according to material implication: the antecedent is true, the consequent is true, and therefore the implication is true. What more could we ask? But for the natural speaker what is lacking is a relation between the two things. What is lacking is a relation between the saltiness of the Baltic Sea and the standing-ness of the Eiffel Tower.

Passing to another kind? What kind? — Lionino

Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself). — Leontiskos

In the example I gave, "First-order claims and second-order rules of discourse."

A first order claim in propositional logic is something like, "P is true," or, "Q is false." Sentences consist of propositional affirmation, negation, and logical operators. Note, though, that, "You are contradicting yourself," or, "This is a contradiction," is a different genus, and deviates from first-order discourse, moving into the meta-language.

The key is that in English we prescind from many things that material implication does not prescind from — Leontiskos

For example, one can assert the material implication (P→Q) for three reasons:

1. P is true and Q is true
2. P is false (and Q is true)
3. P is false (and Q is false)

In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. One way to see this is to note that an English speaker will be chastised if they use the phrase to represent a correlation that is neither causative nor indicative, but in the logic of material implication there is nothing at all wrong with this.

So, what could one say about ¬(A→B) in English? — Lionino

As I alluded to in the other thread, material implication captures English usage only insofar as it guarantees that if the antecedent is true then the consequent will also be true. Similarly, the negation of a material implication says that if the antecedent is true then the consequent will be false, and this is vaguely similar to the denial of an implication in English except for the fact that the falsity of the consequent is not guaranteed in English.

The key is that in English we prescind from many things that material implication does not prescind from, such as the value of the consequent in that denial case. As another example, if an antecedent is false then the material implication is true, whereas this does not hold in English. At the end of the day the English sense of implication simply isn't truth functional. It is counterfactual in a way that material implication is not.

And what about the following formulas:

A→(B∧¬B);
A→¬(B∧¬B);
¬(A→(B∧¬B))? — Lionino

I think in examining these we are combining two confusing and non-translatable logical concepts: material implication and contradiction. Neither one translates well into English, and their combination translates especially badly.

Further, I am of the opinion that speech about contradictions is always a form of metabasis eis allo genos. Even in English when we say, "If you make that claim you will be contradicting yourself," we are shifting between two different registers: first-order claims and second-order rules of discourse (i.e. Thou shalt not contradict thyself).

...it just takes forever if the input is large. — Count Timothy von Icarus

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. — Count Timothy von Icarus

If we have a large number of nodes with infinite time then P(I)=O will produce the ideal solution, it will be unique, and in that case what you conclude no longer holds:

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

If we don't have infinite time then there is no sense in which P(I) is the same thing as O, but if we have infinite time then there is an important sense in which P(I) is the same thing as O.

But my point is that it isn't plausible to compare 2+2=4 to an NP-Complete problem and then claim that 2=2=4 suffers from the same limitations as the NP-Complete problem. 2+2 isn't NP-Complete. The basic objection to your argument would be, "Well I agree that P(I) is not the same thing as O, but it doesn't follow that 2+2 is not the same thing as 4."

The phrase «A does not imply a contradiction» really means specifically «A being true, it does not imply a contradiction». I think this meaning is indeed encapsulated in A→¬(B∧¬B), especially when it can be translated as «A implies True». — Lionino

I don't see it this way. I think the phrase, "A does not imply a contradiction," either means, "A implies something and that something is not a contradiction," or else, "Whatever is implied by A, it is not a contradiction." Both of these are examples of meta-language, and neither is represented by A→¬(B∧¬B). The first means, "A implies B and B is not ¬A." The second means that whether or not A implies anything, it does not imply ¬A.

In any case we have to distinguish these two propositions:

• A→(B∧¬B)
• A→¬A

They are found in "S". Or you can just replace "S" with axioms of the theory. Axioms are naturally assumed.

...

I don't. I know that S and ¬P can't coexist. I know that S, so ¬P can't be the case. ¬¬P is P. — Lionino

See:

<Lionino's "reductio" seems to be ambiguous between senses (2) and (3)> — Leontiskos

So:

(S∧¬P)→(B∧¬B)
S
∴ P — Lionino

What is happening is apparently:

1. (P∧Q)→R
2. ¬R
3. ∴ ¬(P∧Q)

As noted earlier in the thread, a reductio is not representable in the object language, and therefore what you present is not a reductio in any formal sense.

The first thing to note is that the conclusion is ¬(P∧Q). In the first place P and Q can both be false. But if we add an additional condition that they cannot both be false, ¬(¬P∧¬Q), then to stipulate that one is true or false automatically determines the other value, and yet there is no principled reason to stipulate such a thing apart from mere stipulation. Or, if we stipulate that one is true, as you did, then we must accept that the other is false, even without the additional condition. Still, we have no reason to stipulate P instead of Q. There is no theory here or set of axioms, except in a purely stipulative and imaginary way. P and Q are exactly on a par as far as the formalization goes.

1. (P∧Q)→R
2. ¬R
3. ∴ ¬(P∧Q)
4. __Suppose P
5. __∴ ¬Q

From the supposition we learn (P→¬Q), at which point P must be further asserted beyond supposition if we are to actually arrive at ¬Q:

• (P→¬Q)
• P
• ∴ ¬Q

(To suppose P and to assert P are here two different things)

..But it always goes back to the question of why we preferred P in (4) rather than Q.

Note too how this is different from a reductio:

(S∧¬P)→(B∧¬B)
S
∴ P — Lionino

...insofar as your supposition produces no contradiction at all, and the thing you suppose is never rejected. This sort of underlines that you are merely supposing that something is true without any reason. This is what confused me in my initial reply. When you called your proof a reductio I assumed your supposition was being rejected.

What makes the Hamiltonian Path problem intractable is precisely the extremely large number of operations and this can be true for any program provided it has enough steps. — Count Timothy von Icarus

An NP-Complete problem is, among other things, one that has no known polynomial time algorithm/solution. The point being that your P(I) = O is an approximate solution, not a deterministic solution. If O is only an approximation of a solution then of course it deviates from the ideal solution, and from an isomorphic relation.

Yup. We agree there, and that's basically what I mean with the story. It's just an introduction to a thought with a funny conclusion, not an argument or anything of that sort. — Moliere

Okay, fair enough. :up:

A more current but exactly the same example is Chidi from The Good Place :D — Moliere

I was told to watch it by all sorts of people but never did. :grimace:

Thanks that's very high praise :) -- It's all just me working out my own thoughts that I'm willing to share, though, and it's part of what I consider to be in fair trade: I like to read others' thoughts, and so share in kind. — Moliere

Makes sense.

Parables are hard anywhere I think. What makes them difficult is what also makes them attractive. I'm very much attracted to stories, though, because I think they set out nuances better thanwell even though the difficulty is that the nuances aren't specified and there's a certain amount of interpretation that has to go into them. — Moliere

Yep, and probably also because it is impossible to express all the nuance of certain things. In that case to even try is to show that you don't understand what you're dealing with.

Though maybe the distinction is between the sublime and the humorous? — Moliere

Yes, and that line can get fuzzy, too.

I never thought to interpret Balaam's Ass like that, though, which adds an interesting layer: "Get out of your head, dork!" is the kind of message I imagine which unites these. — Moliere

Haha - well the interesting thing about "the old book" is the presuppositions that are brought to it. I don't wish to reduce the value to those presuppositions, but when a text is approached as sacred or inspired it eo ipso comes to possess an unmatched power to express nuanced ideas, such as parables. This is something like Kierkegaard's idea that the believer measures himself against the infinite, and for that he stands taller.

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.

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.

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.

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.

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)>

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.

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)

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.

(S∧¬P), S does. — Lionino

And note that you supposed ¬P (which is the same as preferring S). Either way its a random pick for the second assumption. The point here is as I have said:

This is the formal conclusion, before the and-elimination step of the reductio takes hold:

A→(B∧¬B) {Assumption}
A {Assumption}
∴ (¬A ∨ ¬(A→(B∧¬B)))

...which is the same as, "∴ (A ∨ (A→(B∧¬B)))" We are merely picking an assumption to be true or false, for no reason.

Whether we call (1) a supposition or (2) a supposition is arbitrary, and purely stipulative. There is no formal reason to draw one of the disjuncts of (3) rather than another. — Leontiskos

We do because S fully follows from the axioms of a theory. — Lionino

Which premise do you think provides us with such information?

(S∧¬P) does not favor S over ¬P in the case of a contradiction. If a contradiction follows from (S∧¬P) it is no more rational to reject ¬P than to reject S.

I think you are referring to background conditions that are not formally present, hence my point. If we really had a set of axioms and a theory instead of a single assumption, then you could say that it follows from a theory. In this case we do not have that.