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So is this correct to say that 2 + 2 = 4 ≡ 3 + 2 = 5, since both are true statements? It's a simple question but usually when logical equivalence is mentioned it's mostly seen between two propositions with variables, hence I just want to make sure.

To further elaborate this with another example, would $\forall x(P(x) <-> Q(x))$ mean $P(x_{1}) ≡ Q(x_{1}), \dots , (x_{n}) ≡ Q(x_{n})$, where domain is set ${x_{1}, \dots, x_{n} } $

Another example would be from Rosen's Discrete textbook, where they make compound proposition logically equivalent/≡ to T (true). Hence it seems reasonable to assume that one side (and hence both sides) of a statement that claims logical equivalence to be just statements that are just plain-out true (T), unless I'm misinterpreting something here!:

Discrete Math textbook

Kindly please let me know and clarify :)

Bob Marley
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  • i would be inclined to put $\iff$ rather than $\equiv.$ Or, if using $\equiv,$ put parentheses around the equalities: $$(2+2=4)\equiv (2+3=5)$$ But that's only because we often use $\equiv$ for numbers rather than for logical equivalence. – Thomas Andrews May 21 '24 at 22:45
  • Thank you for the answer! I just edited question to give another better example. What are your thoughts for that? – Bob Marley May 21 '24 at 22:52
  • Rosen uses $↔$ for the bi-conditional (see page 9) and $≡$ for logical equivalence (page 25). The relation between the two is clearly stated in DEFINITION 2 page 25: "The compound propositions p and q are called logically equivalent if p ↔ q is a tautology." – Mauro ALLEGRANZA May 22 '24 at 06:02

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Those are both basic arithmetic statements and are true, however they are not considered logically equivalent under formal logic. In formal logic, logical equivalence is when two statements/propositions have the same truth value under all possible conditions. In this case, the two arithmetic statements are true independently, but they are not logically equivalent because they do not have the same truth value under all conditions.

Rufus Gordon-Heywood
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