So, what could one say about ¬(A→B) in English? — Lionino
On the flip side, can the English meaning of "A does not imply B" be converted to logical formulas? — Lionino
A→B is somewhat straightforward, A implies B. And logic here agrees with our intuition. — Lionino
(2+2=4) implies (Kamala Harris is a presidential nominee). — flannel jesus
It actually does.But it doesn't really match our intuition at all.
let me rephrase: it doesn't match MY intuition, and many other people. — flannel jesus
here is the trouble: if ¬(A→B) is true and A is false, B is true. — Lionino
So, what could one say about ¬(A→B) in English? — Lionino
And what about the following formulas:
A→(B∧¬B);
A→¬(B∧¬B);
¬(A→(B∧¬B))? — Lionino
The key is that in English we prescind from many things that material implication does not prescind from — Leontiskos
In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. — Leontiskos
To be fair, if ¬(A→B) is true and A is false, anything is true. — bongo fury
¬(A→B) means A without B. — bongo fury
Thus for ~(p -> q) I'd say, "It is not the case that p implies q." — tim wood
Or second, relying on the equivalence of (p -> q) <=> (~p v q), I'd say, "Either it is not the case that p, or (it is the case that) q." — tim wood
In any case, going back and forth between "logical formulas" and natural language" is always going to be problematic. — tim wood
yet ¬(A→B) and ¬B entail A. — Lionino
If A does not imply B and [regardless of whether] B is false, can we really infer that A is true? — Lionino
¬(A→B) means A without B
B is true
Therefore A is true
Does that make intuitive sense to you? — Lionino
What about the following example?
Rain without wetness
Wetness
Therefore rain. — Lionino
So what logic does in this case is to set out explicitly two ways of using "or" of which we were probably unaware. After understanding this we are able to say clearly whether we are using an exclusive or an inclusive "or". Prior to that logical analysis, we were probably unaware of the distinction, let alone which we were using.in English, there is no lexical distinction between inclusive-or and exclusive-or, but A∨B is inclusive-or, meaning the result is also True if both are True. — Lionino
Yep. Even if you add the irrelevant and contradictory P2, which is going to make everything true anyway. — bongo fury
Rain without wetness
Wetness
Therefore rain. — bongo fury
¬(A→B) and ¬B entail A. I[f] A does not imply B and B is false, can we really infer that A is true? — Lionino
P1: ¬(A→B)
P2: B is true
Concl.: A is true — Lionino
Rain without wetness
Wetness
Therefore rain. — Lionino
Ah, well, hmm. No doubt in some circumstances it has to be done; writing laws comes to mind. But the caveat being that natural language is about communication while logic is about demonstration. Two very different animals - two different languages - though sometimes they're on the same path and drink from the same stream.So I think it is very much worthwhile to look into how we can bring language into logic. — Lionino
A -> B is false in a given interpretation if and only if (A is true in the interpretation and B is false in the interpretation).
A |= B is true if and only if every interpretation in which A is true is an interpretation in which B is true.
A |- B iff and only if there is a derivation of B from A.
Example:
"If Grant was a Union general, then Grant was under Lincoln." True in the world of Civil War facts. But false in some other worlds in which Grant was a Union general but, for example, Lincoln was not president.
"Grant was a Union general" entails "Grant was under Lincoln". Not true, since there are worlds in which Grant was a Union general but, for example, Lincoln was not the president.
"Grant was a Union general" proves "Grant was under Lincoln". Not true, since there are not other premises along with "Grant was a Union general" to prove "Grant was under Lincoln". — TonesInDeepFreeze
In classic symbolic logic, a -> b is true, according to its truth table, if, for example, a is true and b is true.
(2+2=4) implies (Kamala Harris is a presidential nominee). These is true in classical logic. But it doesn't really match our intuition at all. — flannel jesus
If you are asking what is the most accurate English translation of the intended meanings in ordinary symbolic logic, just put in:
"it is not the case that" where '~" occurs
"if ____ then ____" where '____ -> ____' occurs
"and" where '&' occurs
"or" where 'v' occurs — TonesInDeepFreeze
[1] There are instances in which A is true but B is false.
[2] It is not the case that A entails B (same as above).
[3] It is not the case that A implies B (where 'implies' means the material conditional).
[4] It is not the case that A implies B (where 'implies' means a connective other than the material conditional). — TonesInDeepFreeze
'rain without wetness', 'wetness', 'rain' are not sentences. — TonesInDeepFreeze
Here is the incongruence:I don't get it, but I'm confident I could get it using natural English. Is there a substantial difference? — javi2541997
if ¬(A→B) is true and B is false, A is true. If we read it as such, we would have it "If A does not imply B, and B is false, A is true". Surely that can't be the case, otherwise obviously false sentences such as "An equation being quadratic does not imply it has real solutions, the equation does not have real solutions, therefore it is quadratic." would follow. So we can't read ¬(A→B) as "A does not imply B". — Lionino
The English phrase "A does not imply B" typically means "There are instances in which A is true but B is false". By your list, that does not mean the same as the material conditional. — Lionino
If 'it is not the case that if A then B' is to be understood as the third option, we are simply circling back. — Lionino
What is a phrase in English that unambiguously corresponds in meaning to ¬(A→B)? — Lionino
"There is rain without there is wetness". — Lionino
First, that is not idiomatic — TonesInDeepFreeze
I don't think that "It is not the case that" is usually ambiguous. — TonesInDeepFreeze
"If A then B" is understood differently by different people in different contexts.
So any ambiguity in "It is not the case that if A then B" stems from "If A then B".
So specify what you mean by "If A then B", then you will have specified what you mean by "It is not the case that if A then B". — TonesInDeepFreeze
(1) "If A then B" if and only if "Every instance in which A is true is an instance in which B is true".(material conditional) — TonesInDeepFreeze
Saying A→B is "if A then B" does not provide a solution to the matter of unambiguously converting A→B to English."If A then B" is understood differently by different people in different contexts. — TonesInDeepFreeze
A→B is also true whenever A is false. — Lionino
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