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I think that the difference has something to do with object language and meta language, but I'm not sure.

I've heard that $\leftrightarrow$ is a connective on a proposition level, whereas $\iff$ is a statement about propositions. But where does $\equiv$ come in?

I've seen texts that use either ($\leftrightarrow$ and $\iff$) or ($\iff$ and $\equiv$). But how do those three relate if they are all used at the same time? Would such use imply that there can be a meta language of a meta language? If not, which pairing makes more sense to use?

user1147844
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NilsK
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    There is no agreed convention about this. If your source is a textbook, then it ought to define the notation it uses. In papers, you often just have to infer what the author means from the context. If the subject area is mathematical logic, then $\equiv$ is often used for syntactic identity, $\leftrightarrow$ for object language bi-implication and $\iff$ for metalanguage "iff", but many authors have different conventions. – Rob Arthan Nov 28 '20 at 15:52
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    They mean whatever the author intends them to mean. They all are used for effectively the same purposes in general, however certain authors may apply some sort of nuance to them if they see fit like you describe. – JMoravitz Nov 28 '20 at 15:53
  • Thank you @RobArthan. What exactly do you mean by syntactic identity? – NilsK Nov 28 '20 at 15:55
  • Formulas $s$ and $t$ are syntactically identical if they are the same when viewed just as sequences of symbols. – Rob Arthan Nov 28 '20 at 15:57
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    I'll just repeat what all the others have said: there is no standard convention for these. But yes, if a text uses two of these, it's probably to differentiate between material equivalence and logical equivalence. I have seen no text that uses al three. – Bram28 Nov 28 '20 at 16:11
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    My posts about the difference between $\equiv$ and $\iff$ (here) and the difference between $\implies$ and $\rightarrow$ (here) are apropos. Echoing the other commenters here with my usual caveat: "symbolic logic is an area rife with conflicting notation, terminology and even notions; my understanding is eclectically evolving." – ryang Nov 28 '20 at 16:19
  • @NilsK syntactic identity means equal as strings – Poscat Mar 18 '24 at 07:43

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↔ short left right arrow - has not default meaning, reference If_and_only_if.

⟺ double short left right arrow - has not default meaning, but can be used in books about List of logic symbols.

≡ congruent - has not default meaning. Look up congruent in the context of modulo division classes. ⟺ double short left right arrow might indicate that too.

For reference purposes look at Operators Without Built-In Meanings. These are symbols available both in LATEX and MathML. In these they have just the names I gave in first place and are entered in a short form. These names are known to google for example, ↔ short left right arrow LATEX or w3.org operator dictionary for MatML as appendix.

Steffen Jaeschke
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