A theory is a causal chain, or statements of a causal chain, which have been proven logically/mathematically to be true. In other words, a statement is made about a causal relationship ("X because Y" or "X therefore Y") and that statement is tested. Evidence is gathered and presented regarding the statement, and evaluated; this might include the results of observation or experiment, but in all cases evidence must be grounded in the objective, external world.

A hypothesis is an unproven theory - e.g. it is a statement of a causal chain or relationship, perhaps (one hopes) including an explanation of why it is likely to be true. It has not, however, been subject to definitive or rigorous testing! If I stated "The sky is blue because blue dye evaporates the fastest and humans have made lots of dye in their history so that the sky is permeated with predominantly blue dye," that's a hypothesis.

Although no-one is under any obligation or onus to get these right, if you're serious about your discussions, it is very difficult to have a conversation about science, reality, or even philosophy unless this convention is adhered to by all parties. Otherwise, the conversation degenerates into metasyntactic semantics.

Webster 1913 himself has a quote from Sir W. Hamilton regarding this very conundrum:

This word [theory] is employed by English writers in a very loose and improper sense. It is with them usually convertible into hypothesis, and hypothesis is commonly used as another term for conjecture. The terms theory and theoretical are properly used in opposition to the terms practice and practical. In this sense, they were exclusively employed by the ancients; and in this sense, they are almost exclusively employed by the Continental philosophers.

In more modern English, this means the word theory should be used like this: "In theory, X, but in practice, Y." In a scientific context, the word hypothesis is preferred, as in "Z is just a working hypothesis."

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