External negation of conditionals occurs in sentences beginning with ‘It is not true that if’ or similar phrases, and it is not rare in natural language. A conditional may also be denied by another with the same antecedent and opposite consequent. Most often, when the denied conditional is implicative, the denying one is concessive, and vice versa. Here I argue that, in natural language pragmatics, ‘If $A$, $\sim B$’ entails ‘$\sim$(if $A, B$)’, but ‘$\sim$(if $A, B$)’ does not entail ‘If $A$, $\sim B$’. ‘If $A, B$’ and ‘If $A$, $\sim B$’ deny each other, but are contraries, not contradictories. Truth conditions that are relevant in human reasoning and discourse often depend not only on semantic but also on pragmatic factors. Examples are provided showing that sentences having the forms ‘$\sim$(if $A, B$)’ and ‘If $A$, $\sim B$’ may have different pragmatic truth conditions. The principle of Conditional Excluded Middle, therefore, does not apply to natural language use of conditionals. Three squares of opposition provide a representation the aforementioned relations.
Negation of Conditionals in Natural Language and Thought