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This is a basic question about implication. In constructive logic, we have $\Rightarrow$ for implication and $\Leftrightarrow$ for bi-implication. If we want to express only one-way implication, we can perhaps have the following: $$(A\Rightarrow B)\wedge(B\not\Rightarrow A).$$ That is, $A$ implies $B$ but not vice versa. But as far as I know, $\not\Rightarrow$ is not a connective in the object language but is used at the metal-level. Is there any way to express $B\not\Rightarrow A$ in the object language? Thanks in advance!

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    $\begingroup$ $B\wedge\neg A$? $\endgroup$ Nov 9, 2020 at 21:00
  • $\begingroup$ Since you mention constructive logic, let me point out that, unless I've mad a mistake, $\neg(B\to A)$ is constructively equivalent to $\neg((\neg B)\lor A)$ (even though $B\to A$ is weaker than $(\neg B)\lor A)$) and also to $(\neg\neg B)\land(\neg A)$. But these are weaker than $B\land(\neg A)$. $\endgroup$ Nov 10, 2020 at 1:14
  • $\begingroup$ At the meta-level, $==>$ is not used to express material implication, but logical implication : " A logically implies B iff there is no possible case in which A is true and B is false". Make sure you understand the difference between material implication and logical implication. See Seymour Lipschutz, Outline of set theory (additional chapter at the end of the book). $\endgroup$ Jun 29, 2023 at 7:56

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$$\lnot (B \implies A)$$ equivalently, $$\lnot (\lnot B \lor A)$$ equivalently, $$B \land \lnot A$$

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You can use material implication as

$$(P\implies Q)\iff (\lnot P \vee Q)$$

So, you can replace $ B$ does not imply $ A $ by $$\lnot( \lnot B \vee A) $$ or $$B \wedge \lnot A$$

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If we want to express only one-way implication, we can perhaps have the following: $$(A\Rightarrow B)\wedge(B\not\Rightarrow A).$$ Is there any way to express $B\not\Rightarrow A$ in the object language?

It sounds like you want $$∀x(Ax→Bx)\;∧\;∃x(Bx∧¬Ax)$$ or, equivalently, $$∃y∀x\,\Big((Ax→Bx)∧By∧¬Ay\Big).$$

Notice that $$(A\to B)\wedge\color{red}{B \land\lnot A},$$ is tautologically equivalent to the simpler $$\color{red}{B \land\lnot A}.$$

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