The simple if-conditional is not the only conditional in English. The phrase only if is used to mark a compound which limits the possibilities for the truth of its main clause. It does this by asserting a denial of the main clause that is conditional on the failure of the subordinate clause, so it can be thought of as a hedged denial. As this suggests, the only-if-conditional can be paraphrased using the if-conditional and negation, with ψ only if φ expressed symbolically as ¬ ψ ← ¬ φ.
Like the if-conditional the only-if-conditional has implicatures. It suggests that the truth of its subordinate clause is a necessary condition for the truth of its main clause (while the if-conditional suggests that the truth of the subordinate clause is a sufficient condition). There is a secondary implicature of each conditional in which it suggests the truth of the other conditional, and this can make each seem to say that same thing as a conjunction of the two, a compound known as a biconditional. However, these secondary implicatures are easily canceled. The biconditional ψ if and only if φ can be expressed symbolically as (ψ ← φ) ∧ (¬ ψ ← ¬ φ), or (φ → ψ) ∧ (¬ φ → ¬ ψ) when arrows are reversed.
A third sort of conditional is marked by the English word unless. It hedges the main clause by asserting a limitation on the possibility of its failure, saying this can happen only when the subordinate clause is true. The effect is to assert the main clause conditional on the denial of the subordinate clause, and the unless-conditional can be stated using the if-conditional and negation, with ψ unless φ expressed as ψ ← ¬ φ. Like the other two conditionals, the unless-conditional carries implicatures, both core implicatures and easily canceled secondary ones.
The symbolic analyses of the conditionals can be captured by the rough formulas: only if = not unless (i.e., ψ only if φ = not ψ unless φ) and unless = if not. In these terms, only if = not if not.