How did logical positivists distinguish analytic statements from synthetic statements?

Logical positivists distinguished between analytic and synthetic statements based on their source of truth and their relationship to empirical verification.1

• Analytic statements: These are considered true or false solely by virtue of the meanings of the terms they contain or the logical conventions underlying the use of symbols. They are a priori and do not describe the world; rather, they serve to elucidate meanings already implicit in the subject. Examples include the propositions of logic and mathematics, such as "all bachelors are unmarried". Because their validity is established by the "relation of ideas," they cannot be refuted by empirical observation.2341

• Synthetic statements: These are statements that purport to express matters of material fact. Their truth or falsity is contingent upon empirical evidence, and they require verification through observation and experiment. According to the principle of verification, a synthetic statement is meaningful only if there is a method of testing its truth against the world. If a statement cannot be verified through such empirical methods, logical positivists regarded it as meaningless.142