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Suppose that P, Q and R are the statements that satisfies:

(1) $P\iff Q$ ( i.e. P $\vdash$ Q and Q $\vdash$ P)

(2) $P\iff R$ ( i.e. P $\vdash$ R and R $\vdash$ P)

(3) Q and R are the statements that never mean the same thing.

(For example, Q: $\forall$ $n$ $\in$ $\mathbb{N}$ , $\psi$($n$) > $n$ , R: $\forall$ $n$ $\in$ $\mathbb{N}$ , $\psi$($n$) > $n^2$. Clearly, Q and R mean different things.)

(4) P iff Q and P iff R satisfy the validity of a logical argument.

(For example, let the statement P is "Blue is the color" and Q is "Eyes are part of the body". Since P and Q are tautologies, P iff Q . But it does not satisfy the validity of a logical argument.

I think Q and R are equivalent statements because of the condition (1), (2).

But I can't understand how the statements P, Q are equivalent satisfied the condition (3), (4).

What does this mean? Is there something I'm missing?

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  • $\begingroup$ Equivalence is transitive. $\endgroup$ – Invisible Feb 22 at 11:52
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Logical equivalence of two statements does not suggest that they mean the same thing.

It simply indicates that they are equally truthful - that whatever interpretation makes one statement true will also make the other true.


Consider that all tautologies are true, and therefore equivalent, but they do not need to express the same idea.

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  • $\begingroup$ Thank you for your answer. But I want the following case; (4) P iff Q and P iff R satisfy " the validity of a logical argument ". For example, let the statement P is "Blue is the color" and Q is "The eyes are part of the body". Since P and Q are tautologies, P iff Q . But it dose not satisfy the validity of a logical argument. Sorry for the inexperienced question. $\endgroup$ – user81787 Feb 22 at 10:55
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(For example, let the statement P is "Blue is the color" and Q is "Eyes are part of the body". Since P and Q are tautologies, P iff Q . But it does not satisfy the validity of a logical argument.

This all seems wrong to me.

First, these P and Q are not tautologies. They are true, but not necessarily true; one can think of logically possible worlds where blue is not 'the color' (I'm actually not even sure what you mean by this ...), and where eyes not part of the body.

And second, if you have some P and Q that are tautologies, then they do validly imply each other.

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There are lots of standards of equivalence! In this case, we can distinguish between logical equivalence and equivalence of meaning. For an example of logical equivalence, consider:

(a) "It's raining or it's not raining." (b) "If 2 + 2 = 5, the moon is made of cheese".

Both of these are tautologies, so they're both always true. This means they're logically equivalent. But facts about the weather certainly don't mean the same thing as facts about mathematics and cheese.

Can you try to come up with some statements $P, Q, R$ that make the conditions (1) and (2) down hold? That might help clear things up too.

Of course, one difficulty here is that, as sketched, logical equivalence is a technical notion: whether two statements are logically equivalent is something you can figure out by writing up some truth tables. Equivalence of meaning, though, as I've put it, is an intuitive, loose notion. Logicians, linguists, and philosophers have put forward a few technical proposals, but all that matters here is that it's clearly something different from logical equivalence. Logical equivalence is coarser than meaning equivalence: if two statements are meaning equivalent, they're logically equivalent, but not vice versa.

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  • $\begingroup$ Thank you for your kind reply. There was not enough explanation for my question, so I added the condition (4). $\endgroup$ – user81787 Feb 22 at 11:04

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