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Suppose I want to say that the conditional statement $P\implies Q$ is also equivalent to $(\sim P)\lor Q$.

Writing it like this is confusing: $P\implies Q \equiv (\sim P)\lor Q$

I have a few solutions.

  1. $``P\implies Q" \equiv ``(\sim P)\lor Q"$
  2. $(P\implies Q) \equiv ((\sim P)\lor Q)$
  3. Let A denote the statement $P\implies Q$ and let B denote the statement $(\sim P)\lor Q$. Then, $A \equiv B$

Which ones are correct, and which ones are recommended?

bluesky
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2 Answers2

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The conventional order is: not, and, or, implies, iff. This means that most readers would understand the first solution correctly even without the quotes (which are a clumsy substitute for additional brackets).

The second and third are obviously correct as well, but the real question behind your question is readability and then you should consider context and audience; for example, people like me have a subjective preference for English; and sparsely used additional brackets may avoid confusion among readers who do not know the conventions by heart, such as yourself.

Lieven
  • 2,462
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Suppose I want to say that the conditional statement $P\implies Q$ is equivalent to $(\sim P)\lor Q.$ Writing it like this is confusing: $$P\implies Q \equiv (\sim P)\lor Q$$

Since your goal is to be systematic and clear, just replace the overloaded symbol (material conditional / logical implication / mathematical implication / etc.) with the material conditional : $$P\to Q \quad\equiv\quad (\sim P)\lor Q.\tag A$$ This isn't ambiguous, nothwithstanding that I've dropped the outermost parentheses from the left and right formulae. If the usual precedence convention is in force, then even just $$P\to Q \quad\equiv\quad \sim P\lor Q.\tag B$$ This asserts that the sentence $$(P\to Q) \quad\leftrightarrow\quad \sim P\lor Q,$$ that is, $$(P\to Q) \quad\leftrightarrow\quad ((\sim P)\lor Q),$$ is logically true.

ryang
  • 44,428
  • Thank you for your answer. I previously was not aware of the confusion arising from the different symbols. If you don't mind, could you take a look at a new question I posted? https://math.stackexchange.com/questions/4931518/justifying-a-swap-from-mathematical-implication-to-material-conditional-and-wh – bluesky Jun 12 '24 at 16:30