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Lets suppose we have two propositions p and q:

  1. p = a ^ b
  2. q = c ^ d

Are these propositions logically equivalent?

Both propositions define the same boolean function F = {((0, 0), 0), ((0, 1), 0), ((1, 0), 0), ((1, 1), 1)}, so they seem to be logically equivalent.

But I know that proposition (p ↔ q) is always true if and only if p and q are logically equivalent. From this perspective p and q aren't logically equivalent because proposition ((a ^ b) ↔ (c ^ d)) isn't always true.

As a result, I am a little confused about the meaning of the term logical equivalence. The purpose of the question is to gain a deeper understanding of the meaning of the term logical equivalence.

user341
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1 Answers1

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You are right, in all you say. But if you look very closely at the boolean function which they define, the answer becomes more clear.

They define the same boolean function amd they are equivalent only if we make an intermediate step which identifies/exchanges the letter $a,b$ with $c,d$. If we consider them as defining boolean functions with four inputs the functions which they define are not equal, so the formulae are not equivalent. This is the convention.

So by convention, these formulae are not logically equivalent, for the reasons you pointed out.

8bc3 457f
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