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
it's intuitive that
A→B means not(A without B).
So it's intuitive that
¬(A→B) means A without B. — bongo fury
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
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
"Even if lizards were purple, they would not be smarter." — Leontiskos
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. — Leontiskos
My conclusion thus far is that «A does not imply B» can't be translated to logical language. — Lionino
Saying «A implies B» is A→B, but «A does not imply B» doesn't take the ¬ operator anywhere. — Lionino
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. — Leontiskos
In natural language when we deny a conditional we at the same time assert an opposed conditional — Leontiskos
So then why is it that the logic cannot capture the English, "A does not imply B"? — Leontiskos
In English one can contradict or deny A without affirming ~A. — Leontiskos
My conclusion thus far is that «A does not imply B» can't be translated to logical language. I attempted several different ways in flannel jesus' thread but none worked. — Lionino
So it's intuitive that
¬(A→B) means A without B. — bongo fury
it's intuitive that
A→B means not(A without B). — bongo fury
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
and use instead "not A without B", which is exactly understood in English as coexistence. — Lionino
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
You mean that saying "He is not beautiful" is not necessarily the same as saying "He is ¬beautiful"? — Lionino
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
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
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
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
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
if A is false then we can say A→B, and yet your English does not capture this move. — Leontiskos
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
¬A ⊢ A→B — Leontiskos
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