at least some 1 in every 50 people would interpret it as no A without B — Lionino

If the idea here is, "It's not necessarily a good translation, but it's the best we have," then I would ask why it is better than the standard, "If A then B"?

I think A→B is better translated as, "If A then B."

¬(A→B) is suitably translated as, "A and not B."

The logical negation of A→B is different than an English negation, for the logical negation is more intuitively a negation of ¬A v B, which goes back to <this post>.

If one wants to make the negation translatable into English then "No A without B" is perhaps the best candidate, but it is not the best candidate apart from that single motive. Again, in propositional logic the negation of a conditional is never anything more than a counterexample, and this is the bug we are dealing with.

Would anyone interpret A→B as A implies B if they weren't taught about symbolic logic, like 99% of the world? — Lionino

Actually, yes, I think they would. People tend to understand that arrows signify directionality, in the sense of starting point → destination.

If you showed them the truth table of A→B, I can quite see it that at least some 1 in every 50 people would interpret it as no A without B. — Lionino

Sure: 2% of people might interpret it as, "No A without B," but that doesn't make for a very good translation.

Sorry, this whole Benjamin thing is too confusing for me to keep up. — Lionino

It is supposed to be simple: Has Benjamin agreed with Aaron? Has Caleb? Has Daniel?...
Or else: Aaron gives the condition, "Not A without B." Have the others fulfilled that condition or failed to fulfill it? The most obvious fulfillment would be, "A with B."

My point is that even the 2% who interpret it as, "No A without B," don't quite know what they mean by that. The real translation in those terms is something like, "No A without B in the domain of A-B pairs." Things like 'C' or '¬B' give no A without B, but they fail because they are not in the form of A-B pairs. Things like '¬A' or 'B' succeed when they are implicitly placed into the context of A-B pairs.

The point here is that if we sit down and think about what "No A without B" means in English, without assuming ahead of time that it means A→B, then we will recognize that it does not mean the same thing that A→B means. In some ways it does and in some ways it doesn't. My counterexample that began this whole tangent shows one of the ways that it doesn't.

I didn't really understand the Taleb-Nephlim dialogue but Daniel is just saying A but without saying anything about the value of B. — Lionino

Sure, and in English is to say A without saying anything about the value of B to say A without B? It would seem so.

Would anyone interpret "Not A without B" as A→B unless they knew ahead of time that they were supposed to interpret it that way? It seems highly doubtful.

¬(A∧¬B) is also no A without B. It says that A=1, B=0 is false. — Lionino

The technical problem here is that the English "Not A without B" in no way circumscribes the domain as ((¬)A, (¬)B) pairs. Neither Benjamin, Caleb, Ephraim, or Gregory are saying A without B, and yet only Benjamin and Caleb's answers entail A→B.

For example, the only way to claim that Gregory's answer does not entail "Not A without B" while Benjamin's does, is to beg the question and assume that "Not A without B" is equivalent to A→B. Without that assumption there is no reason to think it is correct that ¬A ⊢ «Not A without B» and incorrect that C ⊢ «Not A without B».

I said 'allowed' there to simply mean true no matter the truth value of the other variable. If ¬A is not disallowed, it means it is true. ¬A is simply A is false or 0. Not A without B means that A=1,B=0 is false, therefore every other combination of the values of the variables gives us true. Since A=0 in the case that ¬A, not A without B is true, and so is A→B. — Lionino

Ah, okay, I see where you are coming from now. It seems like a strange interpretation:

• Aaron: "Not A without B"
• Benjamin: "Not A"
• Caleb: "B"
• Daniel: "A"
• Ephraim: "Not B"
• Gregory: "C"
• Frank: "It looks like everyone is in perfect agreement with Aaron, except for Daniel."

In English it is usually different to say, "Not A without B," and, "Anything which is not A without B is true."

Moreover, A→B does not follow from Ephraim or Gregory's answers in the way that «not A without B» does, and Daniel's answer seems to falsify «not A without B» without falsifying A→B.

By double negation ¬¬(A→B) is simply not A without B. — Lionino

I was thinking of ¬¬(A→B)↔¬(A∧¬B). This is not the same as your interpretation of "Not A without B."

I think they are there implicitly in "not A without B" as spoken. — bongo fury

This is why I would prefer "No A without B." The "parentheses" (however one wishes to depict them) become more important when you want to transform the proposition logically, or draw a modus tollens, etc.

Nah, I actually answered that line of thinking quite handily. — schopenhauer1

's "I never said's" confirm what I took from your response. It's like you were responding to a different post, and this seems to happen a lot in these antinatalism threads.

---

I think you have an established conclusion that you want to achieve no matter what. — Lionino

I think the responses were already in hand before the objections were read.

The subject that experiences the "eternal here". — Lionino

Hmm, okay.

To change a soul essentially would be to swap souls. We don't consider people to swap their consciousness, they are born with one and die with that same consciousness. — Lionino

Sure, but going back to my contention that this question is not adjudicable, if someone claims that an essential change like this has taken place, don't we just tell them, "We don't consider people to swap their consciousness, they are born with one and die with that same consciousness"? Are these theories and claims falsifiable?

Brain-washing or memory loss. — Lionino

So a larger amount of memory loss than being unable to recognize family members?

If you get cloned then die, you stop experiencing — Lionino

A number of folks seem to think that if you get cloned then die, you don't stop experiencing.

Some say you died, others say you kept living.
If it is the case that we die, we stop experiencing, and someone else with the same genes and memories as us keeps living.
If the soul is constantly annihilated and another one spawns in its place, the idea is that we are living only for a fraction of time, to then die and be replaced by a clone that will start living right after us, to then die again and be replaced too.
There is a difference between dying and keeping living, just like there is a difference between dying after being teleported or keep living. — Lionino

Let me put it this way:

1. We are not constantly being annihilated and recreated.
2. We are being constantly annihilated and recreated, but we don't know it given the efficacy of the reconstruction/recreation.
3. We are being constantly annihilated and recreated, and we do know it.

(3) is experientially/epistemically distinguishable from (1), but everyone accepts that (3) is false, so the contrast is moot. (2) is not experientially/epistemically distinguishable from (1), and therefore there is no practical difference between (1) and (2). So if we are left to choose between (2) and (3), it would seem that we get to choose between something that is otiose and something that is clearly false. This brings me back to this idea:

This gets to the separate argument that perdurance is the prima facie view, and that it should stand if there are no good objections. — Leontiskos

This whole thing is reminiscent of the Cartesian move that, "We of course have good reason to believe that X, but do we also have the fullness of certitude?" What standard of proof is being imposed, here? Are we trying to jump over the fence or over the moon?

Perhaps because, if there is no experience that happens at a point in time, but only experiences that happen through time, we cannot separate one experience from the other. And the continuity between those experiences is indeed the psychological continuity, which is allowed by the spatio-temporal continuity of brain states. — Lionino

That's a fair argument, but what about sleep? Usually when we sleep we lose consciousness, along with the experiential and psychological continuity.

Quotation mark!, "death" there stands for brain-death. I think the word 'death' itself is typically meant as brain-death (¿is there another kind?). Coma may be seen neurologically as a long and/or deep sleep. Dementia is a fast decrease of mental elements, leading ultimately to brain death — Lionino

Well as I understand it there are clearly documented cases of people coming back from brain death, which is why I distinguished it.

No evidence of consciousness after brain-death. — Lionino

How so? Is there evidence of non-consciousness after death? Is your definition of 'soul' necessarily embodied?

Your argument must be something like <The only (second-person) evidence of consciousness is bodily movement; after death there is no bodily movement; therefore after death there is no consciousness>. This sort of argument is only objectionable in the case where we have an extremely high standard of proof a la Descartes, which we perhaps do in this thread. This sort of argument is probable but not certain.

Because there is nothing about these facts that would make us think we are actually dying in that moment if one doesn't subscribe to empty individualism. Meaning: if we are closed individualists in a substance metaphysics, choosing those scenarios as the moment of the death of a consciousness is arbitrary and perhaps straight up wrong. — Lionino

I don't see why you think it is arbitrary. You define the soul in terms of consciousness, and in those cases a dramatic and permanent change in consciousness occurs.

Well, in a way you could say Descartes' substance is defined as something to perdure. The matter then is whether that substance (1) exists or a substance (2) that has the definition of a substance (1) except perdurance. — Lionino

I suppose for me the way that Descartes was confronting forms of Pyrrhonism inflects all of these discussions surrounding his positions. Do we have the highest degree of certitude that the soul perdures, such that it could overcome the most extreme version of Pyrrhonism? No, I don't think so. But I also don't really see it as a useful exercise to engage that form of Pyrrhonism.

I have no evidence upon which to found this, but I think my life has had much more suffering than the average pessimist's; and yet, somehow, I think life is awesome.

In fact, it is the people who actually went through great hardships and actual suffering that seem to have the most positive outlook on life. The "always kinda-depressed but not really" type seems to be an existence that occurs almost exclusively in upper middle-class urban settings. There is almost a role-play element to it:
"Oh no, my crush is sleeping with another guy! There are children in Africa starving! Time to read another Dostoyevsky novel." — Lionino

I think this is correct and well put. :up:

If ¬A is true, not A without B is true — Lionino

I think this is simply incorrect.

Everything else is allowed. That everything else includes ¬A. — Lionino

Again:

The question is not whether ¬A is allowed, but whether ¬A ⊢ A→B. — Leontiskos

<"Not A without B" does not preclude ¬A> is a different proposition than <If ¬A is true, "Not A without B" is true>.

1. ¬A ⊢ A→B
2. ¬A ⊢ "Not A without B"

(1) is true. (2) is false. It is false for you to claim that the consistency of ¬A and "Not A without B" justifies (2). (2) requires more than consistency. It requires more than that ¬A is allowed. "¬A is allowed, therefore (2) is true," is an invalid claim.

Put differently, we can know from «not A without B» that ¬A is not disallowed, but we cannot know that the statement is made true by ¬A.

For something of a disambiguation, see:

Forms relating to ¬¬(A→B):

"Not(A without B)"
"Not A without B"
"No A without B" — Leontiskos

Going straight to the point, I would not say that loss of some information, be a memory or else, implies that someone's soul has been swapped. Their mind/brain has changed accidentally to a small extent (in losing that information, I am not talking about the demented condition as a whole), but essentially it is the same. — Lionino

So if someone can no longer recognize their family you would say that their "[soul] has changed accidentally to a small extent." What then would be an example of a soul that has changed non-accidentally, and to a large extent?

But then you see how it doesn't make sense for them not to be distinct? If our consciousness is being annihilated and created every time, aren't we then dying and a copy of us with the same memories being created each time in an empty-individualism fashion? I think that is starkly distinct from our conscious experience persisting. — Lionino

If we are aware of the annihilation-recreation then the experience is different. If not, it is not. But given that we are obviously not aware of such a thing, the thesis must be posited as something that we are not aware of. The objector is presumably saying, "What if, without your knowing it, your soul is being annihilated and recreated at each moment?"

To answer all questions and statements in your posts: yes. But it does not triviliase the proposal because we have two different options for the soul: process or substance. We must choose one. Is it findable in a snapshot of time and space? Choosing substance leads to the problem aforementioned; choosing process seems not to. — Lionino

So in the English-speaking tradition Descartes' dualism and philosophy is distinguished from what came before it. An Aristotelian substance could almost be defined as something which is known to perdure, in the sense that it self-subsists. As this thread shows, Descartes' "substance" cannot be known to perdure and is explicitly claimed not to self-subsist, and is therefore not a substance in the classical sense.

Whitehead in his process thinking was going behind Descartes in order to get beyond him. He was trying to go back to Plato and Aristotle. I think it is a false premise to associate Cartesian dualism with hylemorphism, or Cartesian substances with the classical notion of substance. Pre-moderns and post-moderns both tend to reject Descartes, at least in the English-speaking world. There is no need to choose between the Cartesian soul and a process view, for the classically Aristotelian view of the soul is different from both, and does not posit that the soul is "findable in a snapshot of time and space."

For example:

Descartes is not confusing anything, he is using 'substance' in the metaphysical sense then telling us what substances there are — the mind and the body. — Lionino

Classically the soul and the body are not two substances, as Descartes makes them.

Well, we know from experience that wood burns. We don't know from experience that the soul lasts, as we are very much philosophising about the subject that experiences. — Lionino

Well, you haven't nailed down what you mean by 'soul'. We have candidates: self, mind, consciousness, and memory. Whichever one you want to pick, I have more experience with its perdurance than with the combustibility of wood. So I don't see how a claim that the soul perdures is dogmatic but a claim that wood burns is not.

True. I think I address that point in a previous post: — Lionino

Consciousness then (or the soul etc) would start at birth or whenever we wanna say we first become conscious (mirror test?) and ends in death. — Lionino

I don't follow the middle term of these sorts of arguments in this thread (and there are many). For example, if the self or soul is, "a chain of experienced patterns that emerge subjective experience," then how does this tell us that the experienced pattern at birth is connected to the same chain as the experienced pattern at death? Why not say that it ends at dementia, or coma, or brain-death? Why not say that it goes beyond death? Why not, for that matter, say that it ends at a stroke that turns out not to be deadly? Or the day you have a religious experience? Or trip on LSD? What is the concrete argument for the continuity?

We can define a sentential constant 'f' (read as 'falsum'):

s be the first sentential constant:

f <-> (s & ~s)

That is not "gibberish". — TonesInDeepFreeze

A month ago I was talking about the implications of interpreting (B∧¬B) as 'FALSE' and all I received were superficial objections from the very people who were interpreting it in this manner but had not yet recognized it. I'm guessing those posts will have a very long shelf life.

- Eh... Let's try this:

"Not A without B" translates A→B into English. — A claim I attribute to Lionino

• Part of the meaning of A→B is (1)
• No part of the meaning of "Not A without B" is (1)
• Therefore, "Not A without B" does not translate A→B into English

I don't think a scientist needs to want to understand the natural world as a whole — Moliere

Sure, and that's not what I was saying. A scientist need not be interested in the whole of the natural world to be interested in the natural world.

No definition picks out anything in particular...

...

Definitions don't pick things out at all... — Moliere

I'm not really sure where to start with these sorts of claims. Do words pick out anything at all? It would seem that we are back to Aristotle's defense of the PNC in Metaphysics IV.

If "science" doesn't mean anything at all then we obviously can know nothing about science. If "science" does not pick out anything in particular, then it would seem that we can't use the word meaningfully. You seem to almost be doubling-down on your circular definitions in claiming that definitions don't pick out anything.

Okay, so we have:

1. "¬A being true means A→B is true"
2. "Not A without B"

What I am saying is that knowledge of (2) does not give us knowledge of (1), and yet everyone who knows what A→B means has knowledge of (1). Therefore (2) does not give us complete knowledge of A→B. (2) does not fully represent A→B.

(Edit: I am pointing to a problem with your claim that we can translate A→B into English as "Not A without B.")

- The question is not whether ¬A is allowed, but whether ¬A ⊢ A→B.

But it does. If we understand A→B as «not A without B», and we have ¬A, it is within the scenarios that «not A without B» precludes, because it only precludes A, ¬B, it doesn't preclude ¬A ever. — Lionino

I think you may have mixed up a bit of the verbiage there, but I think you are saying that «not A without B» prescinds from whether or not ¬A justifies the conditional, and that is precisely my point. «not A without B» does not capture the fact that ¬A makes the conditional to be automatically true.

Or in other words, I can say, "¬A, therefore A→B," and clearly «not A without B» does not justify such a move. If all we knew about A→B was «not A without B», then we would not know that such a move is valid.

If we begin with Merriam-Webster, as you've done, then "Science is what scientists do as scientists" is filled out by our common-sense understanding of these terms. — Moliere

The single word "science" is equally "filled out" by our understanding of the term. A definition presupposes that one does not understand the term.

I've said more than just the statement of a theory, though: Good bookkeeping, communication of results over time, humans being coming together to create knowledge, the marriage to economic activity, and a basic sense of honesty — Moliere

Sure, but none of these pick out science in particular. For example, this describes an honest law firm as much as it describes anything else.

We generally know what we mean by the word, and generally know who is included — Moliere

I want to say that if we generally know what we mean by a term then we will be able to give a definition, and if we can't give a definition then we probably don't know what we mean by a term.

I've also said there are two explicit things I'd like a theory of science to accomplish: the demystification of process so that science is not perceived as magical, and a pedagogical simplification not for the purposes of identifying science, but for the purposes of learning how to do science: in some sense my definition of "science" is serviceable enough for those tasks, and we needn't begin at The Meaning of Being in order to say good an interesting things about the subject at hand. — Moliere

It seems to me that your "accounts" of science are very much in line with the magical thinking of the culture. We don't really know what science is, but we revere it and generate overly robust or overly simplistic conceptions of it. Is it demystifying to say that scientists do bookkeeping? Hmm, I don't know. "Bookkeeping" is a very ambiguous term, and the ambiguity lends credence to the idea that you might be saying something very substantial. "Scientists type in computers," "Scientists read literature," "Scientists test theories." These are all true facts about scientists, but we don't know which of them is telling us anything that is actually connected with science in itself without a definition.

And if someone doesn't have a definition of science then I'm not sure how they could be trusted to teach others how to do science. A pedagogue must understand what he is teaching.

The brainstorming process itself, though, is more about arriving at a thesis to defend, if there indeed be such a thing in the firstplace, or even a sharing of different perspectives on how we understand the beast science -- whereas for me I'm thinking about it from the perspective of what to do in order to be valuable to the scientific project as it presently stands... — Moliere

I want to say that a scientist is ultimately interested in understanding the natural world, and he does things that achieve that end. I think science is just the study of the natural world and the ordered body of knowledge that this generates.

As a philosopher I would agree with Feser and say that many "scientists" do not understand science, and because of this it is wrong to define science in terms of their work. For example, we could inquire about a cathedral like Chartres and its architecture. Someone might say, "The architecture of Chartres is what the builders did." Except this is confused, for what the builders ultimately did was take orders from the head architect, and when the head architect died the selection process was very protracted: you don't just entrust Chartres to any old builder.

That is I'm taking up a historical-empirical lens to the question -- the philosophical theory is "Science is what scientists do", which, of course, is defined only ostensively and so doesn't have some criteria for inclusion. — Moliere

We know that it has implicit criteria for inclusion given the fact that you qualify it every time it produces a false conclusion, such as in the case of Fauci or in the case of scientists who are not properly "acting as scientists."

Aristotelian definition in the broad sense is not something you can do without. If you say, "Grass is whatever grows here in the next five years," and a tree grows there in the next five years, then that tree is grass, which is absurd. In light of the tree someone might emend their claim and say, "Grass is whatever grows here in the next five years, such that it looks and behaves like grass does." But this is the vacuous, viciously circular sort of claim. "Any grass that grows here in the next five years is grass."

Note too that real definitions are actually falsifiable. Someone can say, "Ah, but Leontiskos, computer scientists are scientists who do not study the natural world." At such a point I would not be allowed to give a non-answer about how my definition was not intended to be a definition. (And there is a healthy debate about whether "computer science" is aptly named.) The centrality of falsifiability is something that science and philosophy share, albeit in different ways.

If the knowledge or insight is worth money, yes. However there's still a lot of academic and scientific studies that people, who have done them, would enjoy if their ideas would be picked up by others. — ssu

I think people want to share their ideas and they want to profit by their ideas, at the same time. If the latter is not possible then philosophy is only a hobby and not a career. If all philosophy were free then philosophers would make no money.

The traditional donor system helps address this problem, but only to a point.

Of course some of these overlap. For example, the multiple meanings of "without" make "Not A without B" ambiguous between a directional modus ponens and a non-directional ¬A∨B. — Leontiskos

I suppose it is worth asking whether these are the same two inferences, and whether the first is any more "directional" than the second:

• (A→B)
• A
• ∴ B

• ¬A∨B
• A
• ∴ B

I want to say that they are different mental inferences, which is why we think of them differently (in English). But this is part of the difficulty of the thread. "Rabbit holes."

I am starting to think that it is because the word "implies" has the idea of causality in it, while logic says nothing about causality. I reckon that it is better to think of a truth table as coexistence rather than causation. — Lionino

So I think you are overstating this idea. Conditionals have a directionality that partially mimics causality. Meta-logically, they are intended to support the inferences of modus ponens and modus tollens. These are directional, asymmetric inferences. When you think in terms of coexistence or when @bongo fury speaks about "Not A without B" or Venn diagrams, you both seem to be thinking primarily in symmetric, non-directional terms. For example, "coexistence" is not asymmetrical or directional like (A→B). It is symmetrical like (A^B).

Further, "A implies B" does not necessarily mean that A causes B. As I said on the first page, the relation can be indicative. For example, the antecedent can be a sign of the cause that is the consequent. "Warmth implies fire," does not mean that warmth causes fire. "Implies" can also be correlative, where two correlates are caused by a third thing, but this is a true case of coexistence, in which the relation is biconditional (and therefore symmetric) rather than merely conditional (and therefore asymmetric).

There are lots of legitimate ways to speak about (A→B) in English, and each is incomplete:

• Forms relating to ¬¬(A→B):
  
  • "Not(A without B)"
  • "Not A without B"
  • "No A without B"
  • "¬A and/or B"
• Forms relating to modus ponens:
  
  • "If A then B"
  • "A implies B"
  • "B follows from A"
  • "B from A"
• Forms relating to modus tollens:
  
  • "If ¬B then ¬A"
  • "¬B implies ¬A"
  • "¬A follows from ¬B"
  • "No ¬B without ¬A"
  • "Without B, no A"
• (I omit the forms relating to the idiosyncrasies of material implication)

There are also lots of legitimate ways to speak about ¬(A→B) in English, and each is incomplete:

• Forms relating to (A^¬B):
  
  • "A without B"
  • "A but not B"
  • "A and not B"
• Forms relating to the denial of modus ponens:
  
  • "Not(If A then B)"
  • "A does not imply B"
  • "B does not follow from A"
  • "No B from A"
• Forms relating to the denial of modus tollens:
  
  • "Not(If ¬B then ¬A)"
  • "¬B does not imply ¬A"
  • "¬A does not follow from ¬B"
  • "¬B without ¬A"
  • "B requires no A"
• (I omit the forms relating to the idiosyncrasies of material implication)

Of course some of these overlap. For example, the multiple meanings of "without" make "Not A without B" ambiguous between a directional modus ponens and a non-directional ¬A∨B.

Again, "No one transformation is more central to [the logical] 'meaning' than any other" (). Privileged meanings only emerge at the meta-logical level:

On the other hand, in English, or most European languages, nobody ever says "X implies false/true", that comes off as gibberish. The reason must be because the word 'implies' has the sense of (meta)physical causation, while logical implication is not (meta)physical causation; the latter starts with the antecedent being true, the former may have a false antecedent. — Lionino

1. If it rains, then the grass will be wet.
2. If Hitler was a military genius, then I'm a monkey's uncle.

These are equivalent at the first-order level, but not at the meta-logical level. At the first-order level they are both true and there is no difference between the truth of (1) and the truth of (2). At the meta-logical level, (1) partakes in the true purpose of a conditional whereas (2) does not (link). (2) is a consequence of the idiosyncrasies of the material conditional. This relates to my earlier point that the logical negation of a conditional is no longer a conditional, and in that case the modus ponens and modus tollens are no longer accessible, and because of this the directionality of the conditional dissipates.

If one does not make the meta-logical distinction between (1) and (2) then they will be tempted to claim that conditional logic cannot map asymmetrical or directional relations (including causation). This isn't right. A conditional can map an asymmetrical relation. Can it map something like causality? Yes and no: partially but not fully, because causation is not entirely truth-functional.

The key here is that propositional logic distinguishes (1) from (2) not in themselves, but extrinsically through modus ponens and modus tollens. Even though (2) is 'true', nevertheless it cannot be used to draw any substantial conclusion. Calling the conditional "true" is just a useful fiction which has no practical impact on the system. Or rather, it shouldn't. In the other thread we are seeing the havoc that meta-logical ignorance can wreak, for to permit standing contradictions gives the "dross" of the material conditional a potency it was never intended to have. It turns the useful fiction into non-fiction.

On the other hand, in English, or most European languages, nobody ever says "X implies false/true", that comes off as gibberish. — Lionino

If I am right then it is very likely gibberish in logic as well. It is at least clear that no one knows what it is supposed to really mean.

I think the matter of bringing logical propositions into English and vice-versa is still quite meaningful. — Lionino

I'm still not sure, but I would say, along the lines of my two previous posts, that an understanding of (1) does not necessarily provide any understanding of (2), whether that understanding has to do with logic or English:

1. A→C
2. A→(B∧¬B)

To think that (1) must provide an understanding of (2) is to think that (B∧¬B) is always substitutable for P, which it is not. And again, I think this throws the original derivation of A→(B∧¬B) into doubt in the first place.

NB: The riddle of the metabasis is the riddle of when we can take (B∧¬B) as P and when we cannot. If we don't know the answer to this question then none of the reasoning in this thread which treats (B∧¬B) as P is secure.

I am starting to think that it is because the word "implies" has the idea of causality in it, while logic says nothing about causality. I reckon that it is better to think of a truth table as coexistence rather than causation. — Lionino

Yep, I think this is right, and it's what I was trying to get at on the first page. I think my point about "denying without affirming a propositional negation" is also right, and Bongo developed that point. I wonder if the two can be brought together.

and use instead "not A without B", which is exactly understood in English as coexistence. — Lionino

In general I want to avoid thinking any English represents the logic, but I also I think this is a good point. But to give something of a counterexample, if A is false then we can say A→B, and yet your English does not capture this move. Thus:

Because of this the "meaning" of a logical sentence is merely what can be done with it, or what it can be transformed into, and no one transformation is more central to its "meaning" than any other. This is what I was trying to get at on the first page. — Leontiskos

Keeping to this counterexample, "not A without B" captures a truth-functional conditional, but it does not fully capture a material conditional. English involves "causation," but it additionally prescinds from the idiosyncrasies of the material conditional. I think Bongo's negation may have more to do with the materiality of the conditional than its lack of causation, although the two may well be related.

You mean that saying "He is not beautiful" is not necessarily the same as saying "He is ¬beautiful"? — Lionino

Ha - that's an additional consideration that I was not thinking of (Diotima's point in the Symposium). Prescinding from this question and from the question of Buddhist logic, my point is primarily about conditionals or consequences, and can be set out in response to Bongo:

We wish to withdraw or deny an assertion without thereby committing to its negative. Deny it is the case there won't be a sea battle, without claiming there will. — bongo fury

Basically, but more precisely, I would say that we are denying an inference. In English we don't usually say, "You are wrong that there will be a sea battle tomorrow, and yet there may be a sea battle tomorrow."* Instead we say, "Your reasoning for why there will be a sea battle tomorrow is not sound, and yet there may be a sea battle tomorrow."

N: There is a wind from the north tonight
S: There will be a sea battle tomorrow

N→S

The denial in English would seem to be, "S does not follow from N." This doesn't mean that S will always be false whenever N is true. It only means that S need not be true when N is true. This seems to be evidence for Lionino's view that a causal connection is at play. Or as I said on the first page, "The English has to do with a relation between P and Q that transcends their discrete truth values."

then it would seem that we don't intuit negation in this case as a photographic negative of the Venn diagram, which is what logic would deliver.

...

So, not really negation. Not cancelling out the first. — bongo fury

These are good thoughts, but I think a kind of cancelling-out is taking place. It's just that the denial transcends the limitations of truth-functional logic.

To deny something requires understanding what is first being asserted, that it might be denied. If someone says, "Wet grass follows from rain," they are not asserting everything that is involved in the logical claim <[rain]→[wet grass]>, for they are not asserting the idiosyncrasies of the material conditional, such as the idea that ~[rain] justifies their claim. At the same time, they are asserting something more than the logical claim insofar as they do not believe that the falsity of their claim would mean that rain always produces dry grass. Something more subtle is being said and something more subtle is then in turn being denied, and these subtle affirmations and denials don't straightforwardly translate into the affirmations and denials of classical propositional logic.

Or going back to my earlier post and putting it in simpler terms, we can deny a conditional with a simple denial of the metaphysical relation, or else with a counter-conditional, or else with a counterexample. When classical propositional logic denies a conditional it is limited to doing so with a counterexample (e.g. N ^ ~S). This is something of a bug, for to deny the essence of a conditional is to deny its conditionality (e.g. "N does not ensure S"). English is capable of all three responses; propositional logic is only capable of one.


*I am changing the proposition to avoid confusing double-negatives.

If we've established an unreliability of the mind as to how it correlates with reality, I just don't see how you can call an end to that unreliability at a certain level and then feel safe to claim that what you know about your perceptions are accurate and not blurred, manipulated, altered, and corrupted by the mind. — Hanover

Right. When "science" undermines realism it undermines itself, and those who do not notice this live in an alternate reality where their perceptions are good enough when it comes to "science" and untrustworthy otherwise.* There is never a clear answer as to where the "science" ends and the "otherwise" begins.

* At times they even seem to labor under the idea that "science" makes no use of basic perceptions at all. "Trust the 'science', not your lying eyes!," as if science has no use for vision.

Introducing some of the insights from 's <thread>, I would say that the creators of classical propositional logic intended to create a system where conditionals are conditional:

A conditional, by its very name, signifies that which is not necessary.
[It is instead hypothetical] — Leontiskos

But when we place a contradiction in the consequent of a conditional it is no longer conditional (e.g. (A→(B∧¬B)). So if it is a meta-principle of classical propositional logic that all conditionals are conditional, then allowing the contradiction has upended this meta-principle.

Thus when draws ~A, he has implicitly accepted that the conditional is no longer a conditional. The objection to his move is to say that anything which undermines the meta-principle that all conditionals are conditional should not be allowed into propositional logic. Ergo: we should not accept and affirm formulae which are known to contain contradictions. This objection is of course not open to the truth-functionalist who has no recourse to meta-objections (despite the fact that these internal contradictions destroy the formal nature of validity).

I made a similar point earlier:

The most basic objection is that an argument with two conditional premises should not be able to draw a simple or singular conclusion (because there is no simple claim among the premises). — Leontiskos

In other words, this is an invalid form: <If A then B; If C then D; Therefore, E>. When something of that form claims to be valid we have a meta-contradiction, where either the supposedly invalid form is not invalid, or else the inference in question is not permissible. More precisely, we have a metabasis eis allo genos, which is at best quasi-permissible.

(Note that these sorts of exceptions created by (b∧¬b) are popping up in many places. For example, "(A∧C)↔C is invalid for any (non-A) substitution of C," and yet (b∧¬b) creates an exception.)

One could of course be very analytical and simply say that we must choose between, say, the principle that all conditionals are conditional, and the permissibility of introducing standing contradictions into formulae. Still, I think it is fairly obvious that classical propositional logic is built for the former rather than the latter. There are special moves built-in to the system—in this case RAA—that provide a way to navigate such meta-difficulties, but unlike standard inferences RAA is "a mode of argumentation rather than a specific thesis of propositional logic" (IEP). RAA pertains to the boundary of the system, not the interior.

[Using (b∧¬b) within formulas] is a bit like putting ethanol fuel in your gasoline engine and hoping that it still runs. — Leontiskos

In my opinion the creators of classical propositional logic never intended for the system to accommodate standing contradictions. ...Or in the language I used earlier, "internal contradictions," or contradictions contained within the "interior logical flow of argumentation."

I don't think that's quite right, depending on what you meant by "generally". — Srap Tasmaner

Thanks for bringing it back to the beginning, which is what I was also trying to do in my last post.

That proposes another link, and I would suggest that in everyday reasoning the truth of (3) requires the falsity of (1), even though P→~Q does not entail ~(P→Q), which indeed does seem to be a problem for material implication. — Srap Tasmaner

This is what we were just talking about on page 8 of the related thread, "What can we say about logical formulas/propositions?"

I think people do recognize the difference even in everyday reasoning, and would accept that (2) is the simple contradiction of (1), and that (3), while also denying (1) a fortiori, is a much stronger claim. — Srap Tasmaner

Agreed, but I stand by my original claim that (1) contradicts (3).

If no Englishmen are honorable, then it stands to reason that not all of them are, but that's a much stronger claim than simply denying that being English entails being honorable. — Srap Tasmaner

Yes, this is the square of opposition. "Some S are P" and "No S are P" are contradictories, whereas "All S are P" and "No S are P" are "contraries." But "contraries," when used in this sense, are also contradictories.

But your point is well made. The two parties would be speaking "contraries" and not merely contradicting.

(For some reason I did not receive a notification of your post.)

I understand that you'd think that B∧¬B should be able to be replaced by any proposition P, but that is not the case.

Example:
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid. — Lionino

To bring this full circle, consider your post which started us off on this long trek:

((p→q)∧(p→¬q)) and (p→(q∧¬q)) are the same formula — Lionino

Or, "((A→B)∧(A→¬B)) and (A→(B∧¬B)) are the same formula."

Now it is surely true that < ((A→B)∧(A→C)) and (A→(B∧C)) are the same formula >.

But are these the same sort of equivalence?

1. ((A→B)∧(A→¬B))↔(A→(B∧¬B))
2. ((A→B)∧(A→C))↔(A→(B∧C))

I want to say that (B∧¬B) and (B∧C) are not meta-logically equivalent, and because of this the truth tables are misleading.

-

Or going back to this:

I understand that you'd think that B∧¬B should be able to be replaced by any proposition P, but that is not the case.

Example:
(A∧(B∧¬B))↔(B∧¬B) is valid
But (A∧C)↔C is invalid. — Lionino

Logically speaking, (A∧C)↔C is invalid for any (non-A) substitution of C. That's just what it means for a formula to be invalid. Yet when we substitute (B∧¬B) for C it magically becomes valid. There is a kind of meta-contradiction here insofar as (A∧C)↔C is both invalid and not invalid. It is invalid according to the truth table, and it is not invalid given the fact that we can substitute some C which makes the formula valid.

This is what I was warning about in my first posts on metabasis eis allo genos:

[Using (b∧¬b) within formulas] is a bit like putting ethanol fuel in your gasoline engine and hoping that it still runs. — Leontiskos

(Note that to say, "It works fine, just look at the truth table!," is exactly like saying, "The ethanol fuel works fine, just look at my gas gauge!")

So then why is it that the logic cannot capture the English, "A does not imply B"? Is it because the English represents a denial without any corresponding affirmation?

If so, it seems that I was wrong when I said that to deny a conditional in English is necessarily also to affirm an opposed conditional:

In natural language when we deny a conditional we at the same time assert an opposed conditional — Leontiskos

In English we can deny in a manner that does not affirm the negation of any proposition, and this violates the way that propositional logic conceives of the LEM. In fact, going back to flannel’s thread, this shows that a contradiction in English need not take the form (A ^ ~A). In English one can contradict or deny A without affirming ~A.

...but then again maybe to say “Not A and not ~A” is only open to Buddhist-type logic or Buddhist-type English. Even if that is so, perhaps what is available more broadly is the denial of a consequence without any attendant affirmation, such as, “That does not follow from this, and I make no claim about what does follow,” as I claimed <here>. In this way one undercuts an inference and deprives the conclusion (or consequence) of its validity without falsifying the conclusion. Thus one can say, “A does not imply B,” without making any positive assertion, conditional or otherwise. Apparently the relation between a negation and an affirmation differs in English and in logic.

Edit: This may actually be key to understanding A→(B∧¬B), for the contradiction is nonsensical or unstable when taken in a particular sense, and this is why the standard logical operations cannot be applied to it in the same way. A reductio ad absurdum may be parallel to the English move of denying a conditional without affirming anything in the same move. If a reductio affirmed something in the same move then there would be no and-elimination step, and if that were so then a reductio would be identical to a modus tollens, which it is not. The affirmation involved as the final step of a reductio only takes place "after" the and-elimination step. The contradiction is repugnant regardless of which conjunct is preferred (or of which supposition was originally made), and this makes sense because what is proximately aimed at in a reductio is contradiction per se—a universal concept—rather than the application of any truth value to a variable. The application of the truth value to the variable is what is remotely aimed at, and will only take place after the contradiction and the and-elimination have already occurred.

Here's the thing: nearly every one of your posts in this thread contains factual errors. — Banno

Assertions, assertions, and more assertions. And when asked to provide substantiation, you fly like a little bird, even when one goes ahead and does the setup work for you:

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. . . — Leontiskos

- So many of your posts evidence a strong desire to avoid serious philosophical engagement at all costs, but then every so often you make a real contribution and keep people guessing. Eventually, though, one stops playing the onerous guessing game.

Something like this. We see ourselves. Self-interest is somehow shared interest in these cases.. or something approximating that squared circle of care. — AmadeusD

Right. We see that the thing that we are interested in is the same thing they are engaged in, namely rational deliberation and a search for something better. Even if our conclusions diverge, our pursuit is akin.

IN practical terms, it probably solves it. But the arguments remain unchanged :P — AmadeusD

Of course, but there are also more robust versions of this, where instead of speaking about understanding and love one speaks about truth and goodness, which are said to be "convertible" with one another, and which both bridge the gap between subjective and objective. If seeking and knowing truth is good/enjoyable/lovable, then we are self-motivated towards it.

But I suspect there are exceptions to these rules: cases where understanding or truth does not lead to love or an attribution of goodness.

- Deflection yet again.

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. — Banno

See the response I gave:

This is perhaps my favorite proof for the modus tollens thus far. The question is whether that second step justifies the modus tollens. Does the "law of non contradiction" in step two allow us to think of the contradiction as a simple kind of falsity, which requires no truth-assignment? And if so, does that thing (whatever it is), allow us to draw the modus tollens? These are the questions I have been asking for 12 pages.

See my posts <here> and <here> for some of the curious differences between (φ^~φ) and ¬(φ^~φ). — Leontiskos

There is nothing "putative" about the use of MP — Banno

You've pulled a 180! Earlier you literally rejected my characterization of the argument as a modus tollens and said:

Modus Tollens tells us that "Given ψ→ω, together with ~ω, we can infer ~ψ". In the first example you do not have ~ω. It might as well be a Reductio, although even there it is incomplete. — Banno

After you decried that you wanted nothing to do with the modus tollens, I replied:

What is at stake is meaning, not notation. To draw the modus tollens without ¬(B∧¬B) requires us to mean FALSE. You say that you are not using a modus tollens in the first argument. Fair enough: then you don't necessarily mean FALSE. — Leontiskos

Again, this has all been addressed at some length in these earlier posts, including the problem for the modus tollens where (B∧¬B) and ¬(B∧¬B) are equally capable of furnishing the modus tollens with its second premise. At this point I am bored of revisiting old material as you continue to contradict yourself.

Such derivations have been presented here by several folk, including the one from the IEP given above. — Banno

IEP proves my point, "Whitehead and Russell in Principia Mathematica characterize the principle of “reductio ad absurdum” as tantamount to the formula (~p →p) →p of propositional logic. 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."

Look at proof 10.5. It is a proof of (~A→A)⊢A in an axiomatic system. — Banno

Which is of course not RAA, unless you want to follow Whitehead and Russell in their idiosyncratic view. You're just Googling at random to try to support a claim you made, a claim which you said was elementary.

- Perhaps. I am thinking of the example that Janus gave elsewhere.

P: Lizards are purple
S: Lizards are smarter

1. (P→¬S)
2. "P does not imply S"

I think the English sense is never falsified by the logical sense, and in that way it would seem to hold. The problem is that the logical sense can be falsified by alternative English senses, given that English has a more robust notion of implication than material implication. So you can't go in the other direction. Ergo, you cannot translate (P→¬S) as, "If lizards are purple, then they are (necessarily) not smarter," even though you can draw the conclusion, "P does not imply S" ("Lizards' purpleness does not imply lizards' smartness").

Edit: So we might say that (1) guarantees (2) but (2) does not guarantee (1). Thus I admit that it doesn't count as a real translation.

Edit2: I think Janus' argument is special insofar as it makes use of a Cambridge property, and in that case (1) and (2) seem to be the same.

- How is the logical statement different from the English statement «A does not imply B»?

If B is always false whenever A is true, then surely «A does not imply B». The logic covers the English but the English is not captured by the logic.

My conclusion thus far is that «A does not imply B» can't be translated to logical language. — Lionino

Is there something wrong with: (A→¬B)?

(This is why I added a parenthetical edit to my last post, which is about the OP of the other thread.)

Saying «A implies B» is A→B, but «A does not imply B» doesn't take the ¬ operator anywhere. — Lionino

Yes, for it is not possible to capture the negation of the idiosyncrasies of material implication while simultaneously capturing the negation of the notion of implication or conditionality. One or the other must be lost. English abandons the first and propositional logic the second.

Ok, so your "A without B" is not that "it is possible to have A without B", but that "there is A without B". I guess that can make sense as ¬(A→B) ↔ (A∧¬B). — Lionino

The oddity is that there is not parity between a conditional and its negation:

it's intuitive that

A→B means not(A without B).

So it's intuitive that

¬(A→B) means A without B. — bongo fury

1. A→B
2. ¬(A→B)

A conditional, by its very name, signifies that which is not necessary. (1) is therefore conditional in that it neither commits us to A, ¬A, B, or ¬B. It retains something of the hypothetical nature of natural-language conditionals.

(2) is not a conditional in this sense, for it commits us to both A and ¬B.

In natural language when we deny a conditional we at the same time assert an opposed conditional; we do not make non-conditional assertions. In natural language the denial of a conditional is itself a conditional. But in propositional logic the denial of a conditional is a non-conditional.

See:

As noted in my original post, your interpretation will involve Sue in the implausible claims that attend the material logic of ~(A → B), such as the claim that A is true and B is false. Sue is obviously not claiming that (e.g. that lizards are purple). The negation (and contradictory) of Bob's assertion is not ~(A → B), it is, "Supposing A, B would not follow." — Leontiskos

Given material implication there is no way to deny a conditional without affirming the antecedent, just as there is no way to deny the antecedent without affirming the conditional. — Leontiskos

And:

You are thinking of negation in terms of symbolic logic, in which case the contradictory proposition equates to, "Lizards are purple and they are not smarter." Yet in natural language when we contradict or negate such a claim, we are in fact saying, "If lizards were purple, they would not be smarter." We say, "No, they would not (be smarter in that case)." The negation must depend on the sense of the proposition, and in actuality the sense of real life propositions is never the sense given by material implication. — Leontiskos

"If lizards were purple then they would be smarter."

The denial is, "Even if lizards were purple, they would not be smarter." It is not, "Lizards are purple and they are not smarter." The logical negation is the English counterexample.

---

The deeper issue here is that there is no uncontroversial way to translate between English and formal logic, because English has inherent meaning where logic has none. Logical meaning is derived from English meaning, and not vice versa. Because of this the "meaning" of a logical sentence is merely what can be done with it, or what it can be transformed into, and no one transformation is more central to its "meaning" than any other. This is what I was trying to get at on the first page.

Bongo did a good job of using English to capture the range of the logical possibilities, but at least one problem arises in that the English negation and the logical negation are substantially different. As you pointed out in the other thread, a central aspect of an English negation of a conditional is that the consequent and only the consequent is negated (e.g. If <lizards were purple> then NOT<lizards would be smarter>).

(What this then means is that to unequivocally claim that ' scenario does not represent a contradiction is to rely exclusively on a "bug" of material implication, and only those who are able to contextualize material implication within a larger whole will be able to consider the question more fully.)

It's not rocket science.

We use a word to mention a thing.

We use a word in quote marks to mention the word. — bongo fury

Yes, but for some there is a great deal to be gained by a misunderstanding.

- That's fair. I was trying to elaborate the general principle of my first quote. "Letting someone off" requires some variety of involuntariness, and forgiveness is one species of letting someone off.

The point is that for Aristotle forgiveness requires not only culpability but also some measure of involuntariness. When you focus on blame to the exclusion of all else you overlook this, and you also overlook the fact which many have pointed out, namely that blame is a transitional state, not an end in itself. Your excessive aversion to blame of any kind reifies it in a way that makes it an end in itself. You thus fall into ignoratio elenchus, and your case depends on a necessary imputation of bad motives to your interlocutor. Your argument presupposes that they must be interested in blame as an end in itself.

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

In a primarily inferential system like classical propositional logic some can be derived and some cannot (i.e. some rules of inference are technically superfluous). For example, if one has access to the disjunctive syllogism then they also have effective access to the classical modus tollens (over the material implication). So there will be unique and irreducible inference-axioms in any inferential system, but my claim is that RAA is uniquely unique.

The text that we used for philosophical logic was the second edition of Harry Gensler's Introduction to Logic, which is very wide ranging. The chapter that precedes, "Chapter 6: Basic Propositional Logic," is a chapter on induction. In the last two pages of that chapter he points out the irreducibility of modus ponens and brings in Aristotle (link). Note that the beginning of the first sentence is, "Some suggest that we approach justification in inductive logic the same. . ."

Now justifying the inference-axiom of modus ponens is not overly problematic (although it does require induction and/or intellection), but justifying the inference-axiom of RAA is rather more problematic. This difficulty has shown up often in this thread in the way that folks wish to continue treating RAA glibly even when more fundamental questions are being asked. This goes to my point that it is easier to reject a RAA than a MP. To accept the premises of a MP is to already have implicitly accepted the conclusion. Not so for an RAA, as Tones' claim demonstrates.

I don’t forgive someone for mistakes they make due to understandable limitations of knowledge. Only a particular sort of imperfection is a prerequisite for forgiveness, and that is blame. — Joshs

On the contrary, see:

Along the same lines, in the Nicomachean Ethics Aristotle says that the belief that someone acted at least partially involuntarily is what makes forgiveness possible. Even the simple admission, "My bad: I regret how that turned out. It wasn't what I wanted," is a variety of involuntariness that can go a long way to predisposing the aggrieved party towards forgiveness. — Leontiskos

And:

If there is a car crash, again one needs to identify the fault; sometimes it might be the brakes, and sometimes it might be the driver. There was one recently in which a child was killed - the fault was in the driver, but it was not alcohol, but epilepsy. The driver was unaware of their epilepsy because they had not been diagnosed. They were found not guilty of causing death by dangerous driving. — unenlightened