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If $A \implies C$ is true, then does this imply that either of the following two sentences are true?

$$A \land B \equiv C \land B$$

$$A \lor B \equiv C \lor B$$

Intuitively, I would guess that the first sentence is true since if the first half is true, then $A$ and $B$ is true, and if $A$ is true, $C$ must also be true so $A$ and $C$ must be true. I would guess that the second sentence is false since $B$ and $C$ could be false but $A$ could be true, making the LHS true and the RHS false.

If my intuitions are right, would that mean that you can "substitute" the consequence of a material conditional into a sentence sometimes, but not always. If so, what are the rules for doing this?

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  • $\begingroup$ For the first, you explain why $\implies$ holds but you ignore the other implication. What you say about the second sentence, "B and C could be false but A could be true" is impossible since $A\implies C$ is true. $\endgroup$ Sep 18, 2021 at 21:37
  • $\begingroup$ We have "$0 \implies 1$" true. With "B true" is the equivalence false. Similarly, the same choice of A,C but "B false" gives non-equivalence. $\endgroup$
    – user376343
    Sep 18, 2021 at 21:39
  • $\begingroup$ I think you would do better to draw up some truth tables rather than rely on your intuitions. Just considering the cases where $A$ is false will reveal some problems with your thinking. $\endgroup$
    – Rob Arthan
    Sep 18, 2021 at 22:01
  • $\begingroup$ next question in probability: can we just plug in the conditional like $E[XY|X=5] = E[5Y|X=5]$ ? $\endgroup$
    – BCLC
    Sep 19, 2021 at 17:57

2 Answers 2

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Assume $A$ is false and $C$ is true.

Then $A \implies C$ is true.

If B is true then $A \land B \equiv C \land B$ is false because LHS is false and RHS is true.

If B is false then$ A \lor B \equiv C \lor B$ is false because LHS is false but RHS is true. Thus none of the above equivalents is true.

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  • $$A → C \;\not\vDash\; (A ∧ B) \leftrightarrow (C ∧ B)$$ counter-interpretation: $(A,B,C)=(0,1,1)$
  • $$A → C \;\not\vDash\; (A ∨ B) \leftrightarrow (C ∨ B)$$ counter-interpretation: $(A,B,C)=(0,0,1)$
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