# Is assigning a probability to uncertainty a contradiction?

By contradiction here, I don’t necessarily mean a logical contradiction, although perhaps that also does apply here.

My question is along the lines of this: Is the very notion of assigning a probability to uncertainty a contradiction, or if not, meaningless?

Assigning a probability to a proposition implies that different propositions can have different probabilities. But if different propositions have different probabilities, then you can be sure of certain propositions better than others.

But relative “suredness” implies knowledge. However, if one is uncertain of each proposition, then you are trying to know something that you don’t know. It is akin to knowing uncertainty. Isn’t this contradictory?

The only exception to this I can think of is you have an event X and Y where X includes Y. For example, betting on a dice landing on 2 instead of an even number seems ridiculous, but that is only because the former is a part of the latter.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Sep 5, 2023 at 15:02
• my apologies in advance, but i really do not understand what you might mean. i am, like everyone else is, uncertain who will win the next england international, but i am offered odds anyway, and perhaps i will take the bet. if we were certain of everything that we could guess the likelihood of, then...
– user67675
Sep 6, 2023 at 1:32

I can suggest examples that contradict your line of thinking. Suppose I hold a complete pack of cards and hand one to you. You might assign a probability of 1:52 that the card will be the six of spades, for instance. Now suppose I sort all of the spades out of the pack and discard the rest, so I am holding only 13 cards. If I now pass you a card, you will assign a probability of 1:13 that it is the six of spades.

In the two scenarios above, you do not know for certain which card you will be given, but you do have some information about the range of possible outcomes, so the nature of your uncertainty varies from one scenario to the other, and that justifies the allocation of different probabilities.

• Ati sundar, ati sundar. Sep 5, 2023 at 6:50
• No, but that's not my point. The point I'm failing to make is that you can, in certain circumstances, bound your degree of uncertainty and quantify it in a way that allows you to assign probabilities with a sensible rationale, whereas your headline question suggested that was a contradiction. Sep 5, 2023 at 9:38
• "When you say that the probability is 1/52, all you are really saying is that there are 52 cards and one of them is a six of spades." you are confusing what a probability is with the way that a probability is calculated. "These aren’t knowledge statements about the future." they are when you add the model, which is that when you draw a card, any individual card is equiprobable. Then the probability of drawing a spade is the sum of the probability of drawing each of the spades according to that model of how cards are drawn. Sep 5, 2023 at 12:37
• @AgentSmith Could you translate what that expression means into English? Sep 5, 2023 at 17:44
• @Galen, "ati" = most and "sundar" = beautiful, in hindi (language of India), use to express joy/wonder/awe. Close English words would be "splendid", "remarkable", "awesome" "superb". Sep 6, 2023 at 0:30

No, there is no contradiction, the axioms of probability are derived from very reasonable requirements.

I'm sure we can agree that there are things we can be certain of, e.g. logical tautologies, e.g. 2 + 2 = 4.

I'm sure we can agree that there are things we can be sure are impossible, e.g. logical impossibilities, such as intersecting parallel lines in Euclidean geometry.

There will be pairs of propositions where we can say that one proposition/statement/events that are more plausible than others, such as "drawing a royal flush in poker" and "rolling a 6 on a fair die". In other words, we can rank events by their plausibility.

If we assign impossibilities a probability of 0 and a certainty a probability of 1, then any statement must have a plausibility between 0 and 1 (as it can't be "ultra certain"). The probabilities are assigned to observe the rankings of plausibility, e.g.

p(2+2=4) > p(roll 6) > p(royal flush) > p(intersecting parallel lines).

This provides us with a basis for using a number (a probability) to represent the relative plausibilities of statements/events.

There are a few other considerations, such as consistency, so the plausibilities of compound events must be lower than the plausibilities of their components (as it is a conjunction), etc.

If you work out the consequences of this you end up with the axioms of probability, as worked out by several mathematicians, e.g. R. T. Cox

There, I have saved you a few minutes of basic research.

Suppose you're playing poker. You don't know the other players' hands, but the probabilities of different types of hands can be calculated easily. If you hold a straight or a flush, the probability that you hold the winning hand is high. But if you only hold a pair, the chance that it will win the pot is much lower. So you should bet more when you hold a strong hand like a straight or flush than a weak one like a pair.

You can't be sure of successful outcomes (this is the "uncertainty"), but probabilities allow you to compare different propositions ("a pair will win the pot" versus "a flush will win the pot") and make decisions about how to proceed in the game. While you may not get the expected result in any particular hand, if you follow the probabilities for the long run you will be as successful as possible.

Even when we can't calculate probabilities exactly, as in dice or card games, approximations are useful. For instance, it's not possible to calculate the odds in physical sports. But it's reasonable to believe that an established champion at the sport will do better than a newcomer. Upsets happen, but they're rare. These are not totally random events, skill biases the probabilities. But there's still uncertainty because people are not perfect.

An intriguing question. Here's what I brought to the party. A contradiction is a feature of certainty and not uncertainty. If I say God exists (100% certain) then it contradicts God doesn't exist (100% certain). However if I say God may exist (not 100% certain), it doesn't contradict any other statement; even its own negation viz. God may not exist (not 100% certain) doesn't contradict it. Amazing! Hmmmm.