What I am saying is that knowledge of (2) does not give us knowledge of (1) — Leontiskos
"Not A without B" translates A→B into English. — A claim I attribute to 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 BHowever, 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
If ¬A is true, not A without B is true — Lionino
Everything else is allowed. That everything else includes ¬A. — Lionino
The question is not whether ¬A is allowed, but whether ¬A ⊢ A→B. — Leontiskos
Forms relating to ¬¬(A→B):
"Not(A without B)"
"Not A without B"
"No A without B" — Leontiskos
I think this is simply incorrect. — Leontiskos
(2) is false. — Leontiskos
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. — Leontiskos
Forms relating to ¬¬(A→B):
"Not(A without B)"
"Not A without B"
"No A without B" — Leontiskos
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
Only if you think A→B does not stand for Not A without B. — Lionino
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
By double negation ¬¬(A→B) is simply not A without B. — Lionino
I think they are there implicitly in "not A without B" as spoken. — bongo fury
That second premise(¬B) is superfluous to the conclusion (A). — Relativist
But the logic conclusion says otherwise. — Relativist
¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck") — Relativist
Daniel's answer seems to falsify «not A without B» without falsifying A→B. — Leontiskos
I was thinking of ¬¬(A→B)↔¬(A∧¬B). This is not the same as your interpretation of "Not A without B." — Leontiskos
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
¬(A∧¬B) is also no A without B. It says that A=1, B=0 is false. — Lionino
¬(A→B) = It is not the case that ("all bluebirds fly" implies "Fred is a duck")
— Relativist
is not true. — Lionino
Would anyone interpret — Leontiskos
The formula (A→B) cannot be used in all semantic instances of "if A then B". — Relativist
But the mapping to semantics is critical. — Relativist
Would anyone interpret A→B as A implies B if they weren't taught about symbolic logic, like 99% of the world? — Lionino
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
Sorry, this whole Benjamin thing is too confusing for me to keep up. — Lionino
at least some 1 in every 50 people would interpret it as no A without B — Lionino
Actually, yes, I think they would. People tend to understand that arrows signify directionality, in the sense of starting point → destination. — Leontiskos
Sure: 2% of people might interpret it as, "No A without B," but that doesn't make for a very good translation. — Leontiskos
"No A without B in the domain of A-B pairs." — Leontiskos
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"? — Leontiskos
However, what about ¬(A→B)? What can we say about this in English? — Lionino
Fine, your opinion against mine. — Lionino
"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
"No A without B in the domain of A-B pairs."
— Leontiskos
That is already implied by the phrase. — Lionino
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
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
Yes, because it doesn't lead to absurds in English. — Lionino
This misplaces the negations, acting as if the second negates the first when the opposite is true. — Leontiskos
Are you not equivocating between language speakers and non-language speakers? — Leontiskos
To think that the English entails whatever the logic entails is to beg the question and assume that the English perfectly maps the logic. — Leontiskos
(A→B),¬B |= ¬A
¬A |= (A→B) — Leontiskos
What absurdities does it lead to? — Leontiskos
C↔¬A, C |= (A→B) — Leontiskos
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