#### Propositions - the carriers of probability

A probability is a number between 0 and 1 which expresses someone's degree of confidence in the truth of some proposition. A proposition is simply a statement of fact like "This page is over 1000 words long". In reality, every such statement can be either true or false. You would only talk about a "probability" if you were unsure which of the both (true or false) was the case. However, what you always *do* know up-front, is that the proposition is either false or true, but not both, and not something in between either.

What about statements like "This page is entertaining and informative"? How can it *really* be either true or false? Doesn't it just depend on who is judging it? Well, it does, until you define some way of measuring "entertaining and informative" which does not involve a single person's tastes. But let's say that we agreed on some voting scheme in which all potential evaluators would participate in. Then the "entertaining and informative" would no longer be up to your or my opinion only - it would become more of an objective property of this page. And yes, without having seen the actual ratings, you could be unsure about this property (how everyone has rated it). So you could assign different probabilities to all the possible "entertaining and informative" ratings it might have. In other words, you would then have propositions like "The entertainment rating is 0/10" or "The informativeness rating is 9/10", and of course each of them could be true or false, but not at the same time. You might feel more confident that this page has good ratings than bad ratings and express this by numbers using your probability assignment when asked about it.

The thing I'd like you to consider is that when we are discussing probabilities, we are talking about our degree of uncertainty about some concrete propositions. If the propositions appear fuzzy and their truth seems undecidable in principle, then we have to become more specific first and clarify what we mean before we can even start talking about and asking questions about probabilities. Obviously, if we don't even know what our questions are about, we cannot expect any definite and useful answers.

Incidentally, propositions like "a die throw result is 4" or "a coin throw outcome is heads" are very clear. Pretty much everyone agrees on what they mean and could check their truth just like anyone else. Now you see one reason why these sorts of propositions are so eagerly used in classroom introductions to probability. Still, there are many other propositions that just as concrete and a lot more fun to think about than these trivial examples.

Finally, note that the very reason why we talk about probabilities of propositions is that, although they are verifiable in principle (their truth could be checked - and we know how), they may be quite hard to verify in practice. Maybe the proposition is about something that has not happened yet; it could also just as well be about some past event. If we were able to directly find out whether it's true or false, we would of course just do it and we wouldn't waste time talking about its "probability". Probability is for situations where we have to *infer* the truth of a proposition from whatever indirect clues we *can* collect without doing miracles or spending a fortune.

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