Difference between Logical and Tautological equivalences. Examples of the questions I was given on this matter: 1)(pVq)->r is the tautological equivalence of (p->r)∧(q->r) 2)~p->(qVr) is the tautological equivalence of ~q->(pVr) I have to demonstrate which one, if any of those two are true statements.

niouzesto

niouzesto

Answered question

2022-09-07

I've googled already for an explanation and examples that show the difference between logical and tautological equivalences. I understand that a tautological equivalence is first a logical one, but not necessary vice versa. Besides that as far as I've seen they are the same. Are the truth tables the same? What could be a good example that shows the differences between both?
Examples of the questions I was given on this matter: 1) ( p V q ) > r is the tautological equivalence of ( p > r ) ( q > r ) 2 )   p > ( q V r ) is the tautological equivalence of   q > ( p V r ) I have to demonstrate which one, if any of those two are true statements.
On this basis I have to understand the difference between tautological and logical equivalences, why one and not the other, both or none.

Answer & Explanation

Kristopher Beard

Kristopher Beard

Beginner2022-09-08Added 18 answers

Step 1
My understanding is that a tautological equivalence is an equivalence due to truth-functional considerations alone. So, for example, a truth-functional analysis tells us that A B is equivalent to B A. But note that that same truth-functional analysis also tells us that x P ( x ) x Q ( x ) is equivalent to x Q ( x ) x P ( x ), so even though these expressions are first-order logic expression, it already is an equivalence on the basis of truth-functional considerations alone.
Step 2
On the other hand, x P ( x ) x Q ( x ) is logically equivalent to x ( P ( x ) Q ( x ) ), but this equivalence cannot be established on the basis of a purely truth-functional analysis: you also need to understand the meaning of the quantifiers and how they work in order to see that these are equivalent. So, this would be an example of a logical equivalence, but that is not a tautological equivalence.

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