I can't speak to the history, but if you think about the truth table, A implies B is the same as (not A) or B. en.wikipedia.org/wiki/Truth_table#Applications

There is no combination of statements A,B where "A implies B" is true but "Not A or B" is not, or vice-versa. The difference, as I see it, between the mathematical definition of "implies" and the colloquial one is that the mathematical one does not necessitate the existence of a clear chain of reasoning leading from A to B. However, the existence of a clear chain of reasoning is an aesthetic, not a logical, condition. (How would you logically define "clear" for example?)

@Speakpigeon can you think of an exmaple, yourself, where A -> B is (at least broadly) true, but B -> A is not true? If you can, you can transform that example into (not A) or B and see for yourself how it works out.

I also don't believe that there is any logical distinction to be made between language and metalanguage. there in your question is your answer.

Already asked here.

For a discussion about the conditional in mathematical reasoning, see your previous post.

Regarding the material conditional, see your previous post.

I’ve been touring through this book lately - academic.oup.com/edited-volume/42053 It’s really good. I think you’d find a lot in there to compare and contrast your thinking with.

For any two propositions P and Q, it is a tautology that: P → Q or Q → P, when → is meant as a binary operation on true or false values. But we wouldn't say that either P implies Q or Q implies P. So there is a nuance that the mathematician has been trying to point to.

I mean this in all sincerity, it might be better for you not to use any implication symbol when you’re dealing with but rather to use the logical implication ‘⊨’ to talk about statements in Classical Logic. Otherwise it won’t make sense to you. It would also be helpful for you to prove the Deduction Theorem for the horseshoe, and really reason about why A⊨B is saying the same as ⊨(~A v B) in Classical Logic.

@MichalRyszardWojcik "So there is a nuance that the mathematician has been trying to point to" A nuance? It is not true that P → Q is true if P ⊃ Q is true. I don't think we can call that a "nuance". I think it is more like a failure of the horseshoe P ⊃ Q as a model of P → Q.

@PW_246 "it might be better for you not to use any implication symbol . . . to talk about statements in Classical Logic" I'm not talking about what you call "Classical Logic". I'm talking about human logic and whether the horseshoe P ⊃ Q implies the implication P → Q. It does not. This is all we need to understand. - 2. "the logical implication ‘⊨’" The logical implication?! I don't know that there is also an illogical implication. You are probably confusing with something else.

Logical implication ‘⊨’ is also called semantic entailment, while a material conditional/implication is an object-language conditional. You’ll be surprised to know that I agree that P⊃Q⊨P→Q is not generally true. However, I do hold that that if ⊨P⊃Q, then ⊨P→Q. In English, if a material conditional from P to Q is valid/logically necessary then the implication from P to Q is also valid/logically necessary.

@PW_246 "Logical implication ‘⊨’ is also called semantic entailment" So what difference do you think there is between "P implies Q, P → Q and P ⊨ Q?! - 2. "I agree that P⊃Q⊨P→Q is not generally true" Why do you think it is not, though? - 3. "In English, if a material conditional from P to Q is valid/logically necessary then the implication from P to Q is also valid/logically necessary." I don't see the difference with P⊃Q⊨P→Q but you're presumably going to bring in the notion of tautology?

@PW_246 "Logical implication ‘⊨’" You are already speaking nonsense for there is only one sort of implication. - 2. "I agree that P⊃Q⊨P→Q is not generally true" This, too, is off. It is just not true, rather than "not generally true". - 3. "the implication from P to Q is also valid/logically necessary" This is also academic nonsense. An implication is just true or false, not "valid". And implication on its own is also not "necessary" or not even "necessarily true". Rather it is just either true or false. So I look forward to your possible clarifications . . .

@Speakpigeon the difference is that there are logics that allow for multiple conditionals, but the entailment relation for a logic must be at the meta-level in order for us to be able to talk about it coherently. It isn’t just awkward to nest semantic entailment turnstiles—you can’t do it. You can come up with a conditional that models a notion of entailment, but the relation at the meta-level is between sequences of formulas as opposed to between formulas.

@speakpigeon ‘P⊃Q⊨P→Q’ is literally saying that for any formulas P,Q it holds that any model of P⊃Q is a model of P→Q . If you don’t differentiate between the meta-level and the object level, you can’t talk about models. Also, if you think formulas take exactly one of exactly two truth values, then you’re talking about Classical Logic. Maybe you can skate by if you don’t adopt substitution, but there would have to be a reason and a way. Either way, I hope you think P⊃P⊨P→P

@PW_246 "the difference is that blah-blah-blah" You are not making sense. There is just one logic, namely human logic, so talking of "logics" as you do is just absurd. It is obvious therefore that you are not even talking about logic. You're talking about mathematical theories, which are about . . . nothing. - 2. "logics that allow for multiple conditionals" There is just one conditional. All conditional statements follow the same logic. So any theory which "allow for multiple conditionals" are just not about the conditional.

@PW_246 "in order for us to be able to talk about it" So you add a notion of "semantic entailment" or "logical implication", but distinct apparently from the notion of just implication, just so you can understand each other when talking about . . . the implication? Whoa. You guys have a serious problem. I repeat. This is really blind logicians each describing the same logical elephant as if it was different things. Logic you think is in the eye of the (blind) beholder? Go and say that to Aristotle. He will have a good laugh.

@PW_246 "You can come up with a conditional that models a notion of entailment" You are funny. The conditional is what it is. It not a model, of anything. We use the conditional when we want to convey the idea that some implication is true. - 2. "but the relation at the meta-level is between sequences of formulas as opposed to between formulas" You're not making sense. How sequences of anything, formulas or not, could possibly be true or false?!

@PW_246 "* ‘P⊃Q⊨P→Q’ is literally saying that for any formulas P,Q it holds that any model of P⊃Q is a model of P→Q .*" So, not real logic. The one question in logic is whether some sentence implies some other sentence. The implication P ⊃ Q ⊢ P → Q is not true, and this is all we need to know. 2. "If you don’t differentiate between the meta-level and the object level, you can’t talk about models." Good, because we don't need to do that. All we need is to understand is how the implication P → Q applies to the real world.

@PW_246 "if you think formulas take exactly one of exactly two truth values, then you’re talking about Classical Logic" I do and no, I certainly don't talk about Horseshoe logic. 2. "I hope you think P⊃P⊨P→P" As you explained it, P⊃P⊨P→P just doesn't make any logical sense. I can only say that the implication P ⊃ P ⊢ P → P is of course true although only logicians could possibly be interested in the fact. We sometimes insist on some particular self-evident truths, but this one I think doesn't even qualify.

In what way could a simple explanation of semantic entailment works w.r.t. models not make logical sense? Also, how are you supposed to construct models for your view of logic without having a clear difference between the meta-language and the object language? Do you not believe in generalization over properties?

@PW_246 "Also, how are you supposed to construct models for your view of logic without having a clear difference between the meta-language and the object language?" Occam's razor comes to mind. Why would I bother with a notion for which there is no clear indication that it is necessary?

@PW_246 What think you understand of logic is not about logic at all. It is only what you understand of academic theories purporting to be about logic. You don't seem to have a clue. To give you a lead, distinguish between logic and application thereof. Logic is really fundamental to human thought (and presumably animal thought if any). Everything else (precisely) follows. Whereas academics started from an inevitably ill-conceived and ultimately false mathematical theory, and then had to add purely theoretic adjuncts to fill the many gaps between theory and reality whenever they appear.

You’re conflating reasoning with logic. Maybe logos is closer to what you’re talking about, but it doesn’t matter since logic is normative as well as positive. Some modes of human reasoning are invalid, and some things that must be true seem like they can’t be. We’re not gods; logic is for us to formalize notions of reasoning so we know exactly what and why we’re saying what we’re saying. What does Occam’s razor have to do with checking for validity? The whole problem with you is that you refuse to substantiate your claims with a clear semantics. You can’t just shoot from the hip.

Humans are perfectly capable of being happy and well without ever thinking about Logic proper; understanding consequences of a situation with which you have real-world experience is not the same as having a conceptual understanding of the abstract notion of logical consequence. You should expect kindergartners to fail to prove that sqrt(2) is irrational because that’s just how far our society has gotten us given our brain power. There are almost always gaps between our perception of reality and reality itself. Why then would our ability to reason not have gaps and inconsistencies?

@PW_246 "You’re conflating reasoning with logic. *" Please don't make stuff up. To say that logic is a mental capacity doesn't imply that reasoning is the same as logic. 2. "*The whole problem with you is that you refuse to substantiate your claims with a clear semantics." Yeah, I'm no logic teacher. - 3. "You can’t just shoot from the hip." Why not?! As long as I speak the truth!

@PW_246 "Humans are perfectly capable of being happy and well without ever thinking about Logic proper" Ok, I guess we have to stop here because you clearly don't understand much of what I say. I said that we have a logical capacity, not that we "think about logic". This is just a ridiculous misrepresentation of what I said.

You shoot from the hip only as long as you speak the truth, which means having a more open mind and more nuanced views.

@PW_246 "understanding consequences of a situation with which you have real-world experience is not the same as having a conceptual understanding of the abstract notion of logical consequence." Of course not (yet another platitude) but you need a logical capacity to understand "consequences of a situation with which you have real-world experience".

You said logic is a mental capacity because humans have a logical capacity. You said that your view of logic is informed by that humans are logical. I’m saying you have to actually think about the fine details of pretty much anything to understand it. I’m saying that an understanding of logical capacity is not the same thing as a theory in the field of logic.

@PW_246 "Why then would our ability to reason not have gaps and inconsistencies?" Reasoning by analogy is just really bad logic. - 2. Our logic is what it is. Prove to me that it doesn't work. - 3. Our logical capacity is obviously limited, but our ability to reason logically is even more limited and subject to errors. You're the one confusing reasoning and logic!

@PW_246 "I’m saying that an understanding of logical capacity is not the same thing as a theory in the field of logic." Oh, good, that definitely clears things up, thanks! Have a good night.