In ordinary discourse, when we say that $A$ implies $B$, we shall formalize it by writing the following: $$A\rightarrow B$$ But when we say that $A$ does not imply $B$, we cannot formalize it as the following: $$\neg(A\rightarrow B)$$ Because by material implication, we have the following equivalences for the second formula: $$\neg(A\rightarrow B)\Longleftrightarrow\neg(\neg A\vee B)\Longleftrightarrow A\wedge\neg B$$ But the ordinary meaning of the sentence "$A$ does not imply $B$" is that $B$ does not follow from $A$: if $A$ is true, $B$ is either false or undecidable. I wonder how "$A$ does not imply $B$" is formally expressed in the object language (not at the meta-level). Thanks!
from Hot Weekly Questions - Mathematics Stack Exchange
Fred
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