Immanuel Kant distinguishes between analytic and synthetic propositions and further introduces the concept of synthetic it{a priori} judgements in his epistemology. Here's the analysis of each option:
1. (A) ‘All bodies are extended' is an analytic judgement:
This is correct. In an analytic judgement, the predicate concept is already contained within the subject concept. In this case, the concept of "bodies" inherently includes the property of "extension" (taking up space). Hence, it is an analytic judgement.
2. (B) ‘All bodies are heavy' is a synthetic judgement:
This is correct. In a synthetic judgement, the predicate concept adds new information that is not contained in the subject concept. The property of "heaviness" is not inherently part of the concept of "bodies" and requires empirical observation. Thus, it is a synthetic judgement.
3. (C) 'A straight line between two points is the shortest one' is an analytic proposition:
This is incorrect. While it might seem analytic, Kant would consider this a synthetic it{a priori} proposition because the relationship between "straight line" and "shortest" involves spatial intuition, not merely logical analysis.
4. (D) 'The law of conservation of matter' is a synthetic it{a priori principle:}
This is correct. Synthetic it{a priori} principles add new knowledge about the world that is universal and necessary, but not derived from experience alone. The law of conservation of matter is an example of such a principle because it is foundational to natural science and not purely empirical.
Thus, the correct answers are (A), (B), and (D).