# Conditionals and Implications

Until recently, I wasn't aware of the subtle difference in $$\to$$ and $$\implies$$. I've read the previous answers, and want to put it all together to corroborate my understanding.

$$P \to Q$$ is a propositional statement formed by the binary logical connective $$\to$$ and two statements $$P,Q.$$ I see $$P \to Q$$ as a promise, thereby motivating its truth table where it's only false when $$P$$ is true and $$Q$$ is false.

Say, we are proving the theorem $$n\text{ is even}\implies n=2m\text{ for some integer }m.$$ Now, when we write at the end of our proof that $$P \implies Q$$, what we are saying is that if $$P$$ holds, then $$Q$$ also holds, if truth value of $$P$$ is 1, then truth value of $$Q$$ is also 1. That is, through our proof, we are denying there being any case of $$P$$ true and $$Q$$ false. So if I am to relate $$P \implies Q$$ to $$P \to Q$$, I'd say that $$P \implies Q$$ asserts that it cannot be the case that $$P$$ holds and $$Q$$ doesn't hold, thereby ultimately saying that $$P \to Q$$ is always true (including trivial vacuous cases), i.e. $$P \to Q$$ is a tautology.

$$P \iff Q$$ asserts if $$P$$ holds, then $$Q$$ also holds; and if $$Q$$ holds, then $$P$$ also holds. In relation to $$P \leftrightarrow Q$$, it says that $$P \leftrightarrow Q$$ is a tautology, i.e. $$P \leftrightarrow Q$$ is always true, thereby saying that the truth values of $$P$$ and $$Q$$ always go together as the same; and that it cannot be the case that truth values of $$P$$ and $$Q$$ are different. $$P \iff Q$$ is also called logical equivalence.

I've read that $$\implies$$ and $$\iff$$ are not connectives, but that they are making meta statements (statements about about propositions $$P$$ and $$Q$$). Are $$\implies$$ and $$\iff$$ just shorthand then for saying, "if you start with $$P$$, you can deduce $$Q$$" and "if you start with $$Q$$, you can deduce $$P$$" etc. ?

• Mostly. It is much preferred to use $\to$ as the logical connective within a conditional statement, to distinguish it from the use of $\!\implies\!$ as logical entailment between statements, but many texts do still use $\!\implies\!$ for both purposes. Jul 28, 2022 at 4:25
• It is possible to derive $(A\implies B)\iff \neg (A\land \neg B)$ from "first principles." Here is my formal proof using a form of natural deduction: dcproof.com/DeriveImplies.html (requires some knowledge of the basic methods of proof). I make no distinction between the different kinds of implication that you describe. Aug 1, 2022 at 2:24

when we write at the end of our proof that $$P{\implies}Q,$$ we are asserting that it cannot be the case that $$P$$ holds and $$Q$$ doesn't hold,

Yes.

thereby ultimately saying that $$P \to Q$$ is always true, i.e. $$P \to Q$$ is a tautology.

No; based on your described context, $$(P{\implies}Q)$$ likely means that the conditional $$(P \to Q)$$ is universally true under the assumption of mathematical axioms/definitions, that is, that something like $$\forall x \;\forall y \;\Big(P(x,y) \to Q(x,y)\Big)\tag1$$ is mathematically true. Here, $$(P \to Q)$$ is not generally a tautology, and, typically, statement $$(1)$$ is not a logical validity.

In the context of your above sentence, the word always (equivalently: necessarily) is not well-scoped (and actually redundant), and tripping you up: in the current theory, $$(P \to Q)$$ is (always) true and $$Q$$ is (always) true whenever $$P$$ is true.

In short: mathematical truths are generally not tautologies.

$$P{\iff}Q$$ asserts that $$P \leftrightarrow Q$$ is a tautology, i.e. $$P \leftrightarrow Q$$ is always true. $$P{\iff}Q$$ is also called logical equivalence.

Similarly, based on your described context, $$(P{\iff}Q)$$ asserts that $$P$$ and $$Q$$ are mathematically equivalent, but asserts neither that they are logically nor that they are tautologically equivalent.

Are $$\implies$$ and $$\iff$$ just shorthand for saying, "if you start with $$P$$, you can deduce $$Q$$" and "if you start with $$Q$$, you can deduce $$P$$" etc. ?

Based on your described context, yes (assuming mathematical definitions and results).

I've read that $$\implies$$ and $$\iff$$ are not connectives, but that they are making meta statements.

Based on your described context, these two symbols are connectives but not material connectives.

I wasn't aware of the subtle difference in $$\to$$ and $$\implies$$.