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Relations of ideas are ideas that are absolutely certain through either demonstration or purely through intuition. They are ''a priori'', in that they are discoverable independent of experience. This categorization does not necessitate ideas to carry information dependent on the world and thus ideas falling into this category are independent of any existing thing. They are universal constants in that they hold true in all worlds. It should be noted, relations of ideas cannot provide any new information about the world. These types of propositions are simply a means used to help understand more complex ideas. They can be thought of as symbols or a series of simpler ideas describing a larger more complex idea. Common examples usually include geometry or math as formal sciences fall within this categorization. Examples of such statements include 'a square’s sides add up to 360 degrees' or '1 + 1 = 2'. Alternatively, a worded proposition may look something like 'when you run, you move your body,' or, 'all bachelors are unmarried'. Relations of ideas can never be denied as their denial would imply a contradiction in the very definition of the terms within the proposition.[[CiteRef::Hume (1975)]]
Matters of fact are the complete opposite of relations of ideas. Matters of fact are ''a posteriori'' statements and thus based on experience. They are, for this reason, not independent of worldly experience. Unlike relations of ideas, matters of fact do not hold true in all possible worlds. The contrary of matters of fact imply no contradiction and such statements cannot be established by demonstration. Matters of fact can show new information about the world but rely on the experience of the world. Examples of such statements include 'the sky is blue', or 'water is odourless', or 'all guitars have 6 frets.' It should be noted that false statements, such as the last example, can still be matters of fact. The level of coherence within false statements or contrary statements remains the same as within true statements despite being incorrect. In this sense, contrasting statements are, too, matters of facts.[[CiteRef::Hume (1975)]]
The reason behind this distinction was simple; it was to provide criteria by which to organize scientific statements. Through this distinction, all statements were categorized into either matters of fact or relations of ideas. This also ultimately meant that there was no type of idea which was certain and provided information about the world. In the case of matters of fact, propositions are reliant on senses and due to the fallibility of the senses, have no certainty. In the case of relations of ideas, propositions can be proven with absolute certainty through the use of other relations of ideas. Unfortunately, however, these statements cannot give any new information about the world. This distinction was often taken by the scientific community as a strike at Newton’s theory of motion. [[CiteRef::Kant (2007)]] Such a distinction has large consequences in the fields of science, religion, and even philosophy due to its prevention of certain real world statements. As an example this distinction would make useless the attempt to try to prove non physical entities as matters of fact.