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My question is simple and I guess that it might be false even though I do not explain anything.

Suppose that the problem is to show that two compound propositions are "logically equivalent". The generic approach for the problem is to simplify those propositions using well known relations that are logically equivalent.

Question: Can we use the rule of inferences to show that two propositions are logically equivelent?

My answer is No. This is because the rule of inferences to verify a validity of arguments only considers "True" premises. However, to show logically equivalent, we need to consider all possible cases.

I tried to find counter examples, but it failed.

Could you give some counter examples? or Is it possible to use the rule of inferences to show the logically equivalent?

filter hash
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    "Can we use the rule of inferences to show that...?" YES. Use the rules to prove $\varphi \vdash \psi$ and $\psi \vdash \varphi$. – Mauro ALLEGRANZA Apr 05 '22 at 08:11
  • @MauroALLEGRANZA Thanks. You mean that if both sides are valid arguments, then they are logically equivalent right? – filter hash Apr 05 '22 at 08:43

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$φ→ψ$ and $ψ→φ$ both being valid means precisely that $φ↔ψ$ is valid.

In other words, knowing that $$φ→ψ\tag1$$ and $$ψ→φ\tag2$$ are valid arguments is knowing that $φ$ and $ψ$ are logically-equivalent sentences.

To be clear, this requires neither $φ$ nor $ψ$ to be true in the current interpretation; the inference rules don't care about this. If $φ$ and $ψ$ are indeed also true, then we specifically call $(1)$ and $(2)$ sound arguments.

ryang
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  • Thank you for your answer! – filter hash Apr 05 '22 at 14:21
  • No problem! Just to point out that you can upvote/downvote multiple answers on top of check-marking one answer; these actions influence the site's search results, cleanup activities, and other behind-the-scenes processes. – ryang Apr 05 '22 at 14:36
  • The first sentence should be "φ→ψ and ψ→φ both being valid means precisely that φ↔ψ is valid." As written, your first sentence is ambiguous, if not outright stating "Both (φ→ψ being valid means precisely that φ↔ψ is valid) and (ψ→φ being valid means precisely that φ↔ψ is valid)." – Acccumulation Apr 05 '22 at 18:45
  • @accumulation Do notice that the singular verb “means” signals that the two facts are being collectively referred to as a unit rather than separately; that is, “means” being a singular verb precludes the very misinterpretation (reading ‘both’ in the sense of ‘each’) that you suggest. In any case, I have edited the Answer. -) – ryang Apr 05 '22 at 19:55
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I think the question deals with two distinct problems

(1) Does validity require true premises?

(2) What is the relation between semantics and syntax?

As to question (1)

Validty is sometimes explained as truth-preservation.

But that does not mean that a valid reasoning rquires true premises.

Metaphorically, consider logic as an inference machine.

What the constructor of the machine guarantees is that, in case you use true premises as input, the machine will give you back a true conclusion as output: the machine is " truth-preserving".

Of course, you can use false premises as input; and the machine will give you as output a conclusion that logically follows from the premises. But the constructor gives no guarantee as to the truth of this conclusion.

As to question (2)

If the logical system you are using is complete, then, any logical truth ( semantic anspect) is also a theorem ( syntactic aspect).

An equivalence is the conjunction of two logical truths, namaley, of two logical implications.

Vince Vickler
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  • Thank you for your answer. But, I did not toally understand what you want to say. In this context, what is the complete? If the premises are false, then the rule of inference yields the false statement, right? Did you want to explain to me this fact? – filter hash Apr 05 '22 at 09:08
  • No, in case the premises are false, the rules give you either a false statement or a true one. Logic has nothing to say as to what happens when the premises are false. – Vince Vickler Apr 05 '22 at 09:10
  • (1) All cows are birds. (2) All birds are mammals. (3) All cows are mammals. False premises, true conclusion ( which logically follows from the premises). – Vince Vickler Apr 05 '22 at 09:11
  • (1) Einstein was a physicist & Einstein was an astronaut. Therefore Einstein was a physicist. False premise, true conclusion ( by & elimination). – Vince Vickler Apr 05 '22 at 09:13
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    Thanks!!!! I fully understood. Thanks again. – filter hash Apr 05 '22 at 09:48
  • You're welcome. – Vince Vickler Apr 05 '22 at 09:51
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I'd like to make a comment on your original answer. You say:

My answer is No. This is because the rule of inferences to verify a validity of arguments only considers "True" premises. However, to show logically equivalent, we need to consider all possible cases.

You answer doesn't work, because rules of inference do cover all possible cases, and not just those cases where the premises are true. Indeed, rules of inference, and indeed logic in general, doesn't care about truth at all: all it cares about is whether something validly follows from something else. And yes, we often phrase this in terms of truth: "If the premises are true, then the conclusion is true". But note the if: we are not saying that the premises are true. Rather, we are saying: for all logically possible cases, is it the case that if the premises hold, the conclusion holds as well?

For equivalence this is even more clear. Here, we are asking: For all possible cases or worlds, do we have that the one statement holds if and only if the other statement holds? And note that that would be satisfied not only when both statements are true, but also when both statements are false. So yet again, we are not saying that anything specific claim is True.

So, your argument does not work.

In fact, as others have pointed out, you can demonstrate logical equivalence between $\phi$ and $\psi$ using rules of inference (i.e. using formal proofs or derivations). And there are two basic ways to do this:

First, you can show, as Mauro says, that $\phi \vdash \psi$ as well as $\psi \vdash \phi$, i.e. do 2 individual proofs that respectively show that they imply each other. Two statements are logically equivalent if and only if they ligically imply each other.

Second, you can show that $\vdash \phi \leftrightarrow \psi$, i.e. do one proof that shows that the biconditional between them is a logically valid statement. Any biconditional $\phi \leftrightarrow \psi$ is a logical truth (a necessarily tru statement) if and only if $\phi$ and $\psi$ are logically equivalent.

Bram28
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