How did logical positivists argue that a priori statements are linguistic conventions?

Logical positivists, such as A.J. Ayer and Rudolf Carnap, argued that a priori statements are linguistic conventions rather than insights into the nature of reality. They characterized these propositions as having three primary features:

• Linguistic: They are statements about how we propose to use language. For example, asserting that "a straight line is the shortest distance between two points" is simply a report of how one defines "straight" and a declaration of the intent to use that term exclusively for the shortest path.1
• Conventional: Because these statements are definitions, they express conventions for which alternative definitions may exist. One could, for instance, choose to define terms differently based on the paths of light rays.1
• Analytic: These statements are analytic because the predicate merely repeats part or all of the subject term. Consequently, they provide no new information about the world; they are strictly about meanings and possess no empirical content.1

By framing a priori knowledge this way, logical positivists sought to demonstrate that the importance of rationalistic systems was illusory, as such systems rely on statements that do not describe nature.1