# What is the difference between "logical equivalence" and "material equivalence"?

Wikipedia offers this as the difference between "logical equivalence" and "material equivalence":

Logical equivalence is different from material equivalence. Formulas p and q are logically equivalent if and only if the statement of their material equivalence (P ⟺ Q) is a tautology.

Material equivalence is associated with the biconditional. However, I am still unclear what the difference is between the two.

I want to make sure I am using the terms correctly. Recently to avoid confusion I dropped the adjective "logically" in front of the following use of "equivalent":

Using De Morgan's laws, ¬(A ∧ ¬B) is equivalent to ¬A ∨ B.

If there is any difference between the two terms, what is it? Perhaps an example of the correct use of each would help clarify the difference.

If there isn't any difference I probably shouldn't use either one.

Wikipedia contributors. (2019, February 13). Logical equivalence. In Wikipedia, The Free Encyclopedia. Retrieved 11:22, August 3, 2019, from https://en.wikipedia.org/w/index.php?title=Logical_equivalence&oldid=883191333

The answer is suggested by the quote that you provided:

Logical equivalence is different from material equivalence. Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology.

There is a difference between being true and being a tautology. Once you see this you can see the difference between material and logical equivalence. So, for instance, if P = 'today is Saturday' and Q = 'the year is 2019' then (P ↔ Q) is true, because both P and Q are true (at the time of writing), but (P ↔ Q) is not a tautology, because tomorrow P will be false and Q will remain true.

A similar difference exists between material and logical implication, which is also important to remember. With the same example, (P → Q) is true but not a tautology, so P does not logically imply Q. This similarity is to be expected as material equivalence is simply bi-directional material implication, and logical equivalence is bi-directional logical implication.

This is why it can be misleading to write 'P → Q' without specifying if one means logical implication, material implication, or some other kind of conditional relation.

• Could we say that material implication (⇒) is a binary operator — that is to say a ternary relation — about truth, while logical implication (⊢ or ⊨) is a binary relation about syntactic or semantic causality? Jul 14, 2020 at 23:12
• @Maggyero You could say that, though I would not use 'causality'. Jul 15, 2020 at 3:27
• What is the difference with causality? Jul 15, 2020 at 8:30

It may be enriching to view the distinction in a broader setting:

P ↔ Q is plainly a compound proposition: Any propositions P and Q are joined by the material equivalence (biconditional) connective in the language of propositional logic.

However, logical equivalence of P and Q, P ⟛ Q, is a theorem and requires to be shown that both P ⊢ Q and Q ⊢ P hold, what, alternatively for propositional logic, amounts to showing that P ↔ Q is a tautology.