In research articles in logic one can sometimes find claims such as the following: the proposition X is (logically) stronger than the proposition Y. For instance, P & Q can be said to be stronger than P because P & Q entails P, but not the other way around. Here is my question: of course, in a propositional language, if two propositions are equivalent they are also equally strong (or weak); but does the converse hold, i.e., is it sufficient for two propositions in a propositional language to be equally strong (weak) for them to be equivalent? I'd say yes, but I'm not sure.
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2$(a\implies b\land b\implies a)\equiv(a\iff b)$. – Feb 01 '21 at 16:59
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If you think $(a\implies b\lor b\implies a)$ is a tautology equivalent to the law of excluded middle then perhaps yes since you have a total order of propositions – Henry Feb 01 '21 at 17:31
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" if two propositions are equivalent they are also equally strong " I wouldn't agree with this. All tautologies are equivalent. But, they are not equally strong in their ability to serve as axioms for a propositional calculus. Some tautologies can serve as single axioms for a system while many others cannot under the same rule(s) of inference. – Doug Spoonwood Feb 01 '21 at 17:46