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.")

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- 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

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I'm late to the game, and I'm sorry if this has already been brought up. But just in case it hasn't, here's my response to the Op:

However, what about ¬(A→B)? What can we say about this in English? The first thought is "A does not imply B". But here is the trouble:

if ¬(A→B) is true
and B is false,
A is true. — Lionino

No, your conclusion (A is true) is not valid. You seem to be interpreting “¬(A→B)” as: “¬A->¬B”, and that’s invalid. “¬(A→B)” just means that the truth value of A does not give us a clue as to the truth value of B. A better English translation of ¬(A→B) is : it is not the case that A implies B

Consider these substitutions:
A=All bluebirds fly
B=Fred is a duck
This is consistent with ¬(A→B) being true. If we discover Fred is a pigeon then B is false, but it tells us nothing about whether or not all bluebirds fly.

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

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You can infer A from ¬(A→B) by De Morgan.
¬(A→B)
¬(¬A∨B) (definition of material implication)
¬¬A∧¬B (de Morgan)
A∧¬B (double negation) — Lionino

I concede your point, but what you have proven is that:
¬(A→B)
Implies A
(Which I confess seems counterintuitive - see below*).

You had said: If A does not imply B, and B is false, A is true

That second premise(¬B) is superfluous to the conclusion (A).
--------------------------------------

*Now suppose we apply the logic to these statements:
A=All bluebirds fly
B=Fred is a duck

¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")... which is certainly true because the antecedent has no bearing at all on the consequent
(¬A∨B) = "not all bluebirds fly" or "Fred is a duck"
...
A∧¬B: All bluebirds fly and Fred is not a duck

Problems:
Despite the fact that ¬(A→B) is a true statement...
1) it is NOT true that all birdirds fly (hatchlings don't fly),
2) My pet duck is actually named Fred.
But the logic conclusion says otherwise.
Something ain't right. I had to dig out my 1973 Logic textbook to understand the problem, but I'd like to see if anyone can identify the problem on their own.

Only if you think A→B does not stand for Not A without B. — Lionino

The "without" reading of A→B does need brackets when written:

Not (A without B)

i.e. ¬(A & ¬B)

I think they are there implicitly in "not A without B" as spoken. So the spoken phrase does clarify the logic of →.

But perhaps they are needed explicitly when the phrase is written. I mean,

(not A) without B

seems a willful misunderstanding. And gives (¬A) & (¬B).

But brackets will prevent that particular misunderstanding.

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.

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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».

¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")
— Relativist

is not true. — Lionino

The statement "It is not the case that ("all bluebirds fly" implies "Fred is a duck") IS true. But you're right that it's not equivalent to :
¬(A→B)

But why isn't it? It's because there is no material implication. The formula (A→B) cannot be used in all semantic instances of "if A then B".

I don't think I ever realized this before. When I took sophomore logic (50 years ago!), we concentrated on formulaic proofs. But the mapping to semantics is critical.

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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.

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.

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However, what about ¬(A→B)? What can we say about this in English? — Lionino

The solution you have arrived at is the idea that ¬(A→B) means, "A without B," and therefore (A→B) means "Not(A without B)." This misplaces the negations, acting as if the second negates the first when the opposite is true. What you are really saying is that ¬¬(A→B) means "Not(A without B)," and that (A→B) and ¬¬(A→B) are linguistically interchangeable.

What is really happening?

• A∧¬B means "A without B"
• ¬(A∧¬B) means "Not A-without-B"

Then:

• (A∧¬B) ↔ ¬(A→B)
• ¬(A∧¬B) ↔ (A→B)

And then you assume that the '↔' is applicable not only for logic, but also for English, thus:

• ¬(A→B) means "A without B"
• (A→B) means "Not A-without-B"

This is almost identical to the problems in "Do (A implies B) and (A implies notB) contradict each other?" In both cases formal logical equivalence is being conflated with semantic equivalence.

The problem was isolated in <this post>. A→B and ¬(A∧¬B) (or ¬A∨B) are not the same sentence. A→B directly supports relations like causality, whereas the other two do not. Further, the only way to prove A→B from ¬(A∧¬B) is via an indirect proof such as RAA, which is an equivalence and not a derivation. "If P then Q," and, "Not A-and-not-B" are two different claims, both in logic and in English.

-

We can see this with an example.

A: I stop eating
B: I lose weight

The implication form is A→B ("If I stop eating, then I will lose weight"). This describes a relation between eating and weight. It means that to stop eating leads to losing weight, and that if one is not losing weight then they have not stopped eating (modus tollens).

The conjunction form is ¬(A∧¬B) ("It is not the case that, it is true that I stop eating and it is false that I lose weight"). This says that A and ¬B cannot coexist. There is no relation posited between A and B.

The relation can be inferred from the conjunction, but it is not the same as the conjunction:

1. ¬(A∧¬B)
2. __Suppose A
3. __∴ B
4. ∴ A→B

(4) follows from (1) and (2), but it is not equivalent to (1), despite the fact that the truth tables are the same. Put differently:

1. ¬(A∧¬B)
2. A
3. ∴ B
4. [Meta-step: ¬(A∧¬B), A ⊢ B. Therefore, A→B given ¬(A∧¬B)]

(One could also show this with RAA)

Fine, your opinion against mine. — Lionino

In your opinion arrows do not connote directionality? Do you think there is a reason logicians introduced the inference A→B over and above the conjunction ¬(A∧¬B)?

"2% of the population might interpret 龍 as 'dragon', but that doesn't make for a very good translation". You see how that doesn't work? — Lionino

Are you not equivocating between language speakers and non-language speakers? If only 2% of native speakers interpret 龍 as what we mean by 'dragon' in English then yes, it is a bad translation.

"No A without B in the domain of A-B pairs."
— Leontiskos

That is already implied by the phrase. — Lionino

You think the English phrase, "No A without B," implies that we must be thinking about the entire domain of speech in terms of A-B pairs? This seems clearly incorrect. In English when we say, "No pizza without heartburn" we in fact order a salad ("C"), and this satisfies the condition just like Gregory's answer does.

It is saying there is no A, if there is no B. From A→B, ¬B, we infer ¬A — (A→B),¬B|=¬A. From A→B, C, we infer nothing about A because the value of B hasn't been declared. From A→B, C, ¬B, we infer ¬A, because C doesn't interfere — (A→B),¬B, C|=¬A. — Lionino

You are again conflating the logic with the English. To think that the English entails whatever the logic entails is to beg the question and assume that the English perfectly maps the logic. That is what we are considering, not what we are assuming.

Regarding the modus tollens, the English does support it but, again, this is not the same as whether ¬A entails the truth of the conditional. These are not the same thing:

• (A→B),¬B |= ¬A
• ¬A |= (A→B)

Specifically:

• (A→B),¬B |= ¬A
• ¬A |= (A→B)
• «No A without B»,¬B |= ¬A
• ¬A |= «No A without B»

Does (4) hold? It is questionable, but if it doesn't then the translation limps, and if it does then this also holds: < C |= «No A without B» >, in which case the translation also limps since C does not semantically entail (A→B). Either way the translation limps.

I think you took my "everything else is allowed" to mean literally everything else (C), but I meant "every other values of A and B". — Lionino

But that's not what the English means. It is an arbitrary restriction of the English meaning. After all, if it's not being interpreted in favor of its literal meaning, then what is it being interpreted in favor of?

Part of the puzzle here is that in reality negations always obtain within a scope. For example, if C=salad, then C=¬A (not pizza). When we are within the same scope, C must always be either A or ¬A, and since C=¬A, C |= (A→B).

(Propositional logic seems to assume, prima facie, not only the commonsensical idea that C is neither A nor B, but also the deeply counterintuitive idea that C is neither ¬A nor ¬B. Usually if C is neither A nor B then it must be both ¬A and ¬B.)

Yes, because it doesn't lead to absurds in English. — Lionino

What absurdities does it lead to?

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One of them was the matter of putting logical formulas into natural language (English in our case) — that matter was essential for the purpose of correctly interpreting some statements. — Lionino

A paper that shows how Medieval Aristotelian logic was in some ways more robust than current logic is Gyula Klima's, "Existence and Reference in Medieval Logic." He uses Russell's King of France example rather than conditionals.