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I am a bit puzzled about the semantics of the $\iff$ notation. I understand clearly its meaning in a logical prepositions context. However, $\iff$ can also be understood as a boolean algebra operator with its own truth table: $ T \iff T = T, T \iff F = F, etc$

This is somewhat unusual as one is typically accustomed to think of operators like $\land$ , (boolean AND), $\lor$ (boolean OR), $\oplus$ (boolean XOR) etc. when talking about boolean operations. But, like I said $\iff$ can also viewed as an binary boolean operator with its own truth table.

Therein lies the conundrum: how do I express notationally, e.g. that $A \iff B$ is equivalent to $A \implies B \land B \implies A$ ?

If I were to write:

$ (A \iff B) \iff [(A \implies B) \land (B \implies A)]$

Then I think the notation is problematic as it uses the same symbol ($\iff$) with two different semantics / nuances:

  • the first instance of $\iff$ uses it as a boolean algebra operation
  • the second instance of $\iff$ uses it to convey the idea of logical equivalence

It also seems somewhat cyclical.

Any thoughts?

Marcus Junius Brutus
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    Yes, there is the need to have two symbols: one for the propositional connective called bi-conditional: it is part of the formal language, like the negation operator $\lnot$, and it is used to produce formulas, and the other one for the relation of (logical) equivalence between formulas, that is "outside" the language (the meta- context). Unfortunately, there is no common agreed convention about the use of the available symbols: $\leftrightarrow, \Leftrightarrow, \equiv$. – Mauro ALLEGRANZA Sep 30 '21 at 08:22
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    The simplest approach is: to explicitly define a "convention" when you start your "text" and use it consistently. – Mauro ALLEGRANZA Sep 30 '21 at 08:24
  • @MauroALLEGRANZA I am glad I wasn't just over-analyzing this. – Marcus Junius Brutus Sep 30 '21 at 08:59
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    Some years ago, when I used a implication operation (in the context of Boolean or Heyting algebras, and more generally residuated lattices, and perhaps even more general settings, I don't remember right...) I used $\to$ for the operation symbol, and $\Rightarrow$ for the implication between formulas, and I always felt confused about the use of $\Rightarrow$ as an operation symbol, but it happens; as Mauro Allegranza wrote, there is no agreement about this notation... – amrsa Sep 30 '21 at 10:26
  • Here is my answer to your question, and its counterpart for → ⇒ ⊨ ⊢. – ryang Nov 19 '21 at 17:49

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