Difference between biconditional and logical equivalence

Solution 1:

Basically, the distinction is between talking about a specific situation versus all possible situations.

Suppose I have two different propositional variables $p$ and $q$. Then:

• $p$ and $p\wedge p$ are logically equivalent. (Remember that "$\wedge$" means "and.")

• $p$ and $q$ are not logically equivalent.

• However, $p\iff q$ might be true (e.g. if both $p$ and $q$ happen to be true).

This is ultimately a distinction between talking about general necessities versus specific situations. The keyword here is "model." In the setting of propositional logic (there are other logics), a model is just a specific assignment of truth values to the propositional variables in the language. E.g. suppose our language has propositional atoms $p, q, r$. Then "$p$ and $q$ are true, $r$ is false" (or rather, the function $\nu: \{p, q, r\}\rightarrow\{true, false\}$ sending $p$ and $q$ to $true$ and $r$ to $false$) is a model. Note that given a model, we can also talk about the truth values of more complicated sentences in that model: e.g. "$p\wedge r$" is false according to the model above.

(Indeed, we can prove by "structural induction" that an assignment of truth values to propositional variables uniquely extends to an assignment of truth values to all propositions, which respects the obvious rules - e.g. if $\varphi$ and $\psi$ are both assigned "true," then $\varphi\wedge\psi$ must be assigned "true," and so forth. Sometimes a model is defined as a truth assignment to all propositions, which satisfies these reasonable rules; the proof described in the previous sentence means that we can get away with the simpler definition above.)

When we say that two sentences are logically equivalent, we mean that there is no model in which they have different truth values. The expression "$p\iff q$," however, is a (compound) sentence which is (in general) true in some models and false in others. This is the distinction:

• The notion of "logical equivalence" is talking about what things are possible in general.

• When we say that a sentence is true/false, we are talking about its truth/falsity in a specific model.

For example, in the model $\nu$ defined above the sentence "$p\iff q$" is true, even though $p$ and $q$ are not logically equivalent (exercise).

At this point, it's useful to introduce a bit of terminology: a sentence which is true in every model is called a tautology. When we say "$\varphi$ and $\psi$ are logically equivalent," we're just saying "$\varphi\iff \psi$ is a tautology."

A bit of more advanced material

There is also a "relative" version of this. Suppose $\varphi$ is some proposition which is true in every model in which the proposition $\psi$ is also true. (Note that this just means that the proposition "$\psi\implies\varphi$" is a tautology.) Then we write "$\psi\models\varphi$."

The value of this new symbol is that it lets us generalize considerably: if $\Gamma$ is a set of propositions, we write "$\Gamma\models\varphi$" iff $\varphi$ is true in every model where every proposition in $\Gamma$ is true. If $\Gamma$ is infinite, this is meaningfully different from just talking about tautologies (since "$\Gamma\implies\varphi$" isn't actually a proposition).

However, one of the most important theorems in logic - the compactness theorem - states that if $\Gamma\models\varphi$ then there is some finite subset $\{\gamma_1, \gamma_2,...\gamma_n\}\subseteq\Gamma$ such that $\{\gamma_1, \gamma_2,...,\gamma_n\}\models\varphi$. And this just means that the proposition "$(\gamma_0\wedge\gamma_1\wedge...\wedge\gamma_n)\implies\varphi$" is a tautology. So via the compactness theorem we can reduce questions about the relation "$\models$" to questions about tautologies, but that's far from obvious.

(And there are important logics which don't have this property, so actually they are meaningfully different in general.)

Solution 2:

Think of the material biconditional $\LARGE\leftrightarrow$ as a logical operation on two sentences rather than as equivalence. Unlike the latter, it is not making an assertion nor in a metalangauge.

• The statement $\Large P\equiv Q$ asserts the logical equivalence of $P$ and $Q.$ It means that $\large P\leftrightarrow Q$ is a (logical) validity, i.e., is true regardless of interpretation.

  What's the difference between $↔$ and $≡$ ?

• The statement $\Large P {\iff} Q$ asserts the (not necessarily logical) equivalence of $P$ and $Q.$ This means that that $\large P \leftrightarrow Q$ is either analytically or synthetically true (perhaps in an axiom system), i.e., that $P$ and $Q$ have the same truth value in the given interpretation.

  $\Large P {\iff} Q$ can be read as ‘$P$ is true if and only if $Q$ is true’ or ‘$P$ being true is a necessary and sufficient condition for $Q$ to be true’.

P.S. Symbolic logic is an area rife with conflicting notation, terminology and even notions; my understanding is eclectically evolving.