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It is argued that there is a difference between these probabilities. When a dice lands on 6, it is argued that because it could have landed on 1-5 by the nature of physical laws, the probability is 1/6. Logically though, the dice could have exploded midair. It could have turned into a bird and flew. It could have decided to never land again. In a way, these break physical laws compared to the other more physical possibilities.

But the dice already landed on 6. It couldn’t have been the case that it landed on 5 any more than it being the case that it could have turned into a bird. All of those possibilities are now impossible.

So why should we differentiate them?

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  • Because probabilities are assigned to possibilities, not to actualities, and possibilities determine the sample space. But it is not about logic vs physics. It is physically possible that the dye explodes or, less dramatically, lands on edge or corner, but discounting rarely occurring events makes for more tractable models that are still useful for making decisions in all too common conditions of uncertainty. See Hintikka for criticism of the "one-world assumption", that we can come to know how things actually are without considering any alternatives.
    – Conifold
    Commented Jan 8, 2023 at 8:40
  • Bumble's answer seems thorough enough. Consider reading Interpretations of Probability (SEP). If the probability of probability being impacted by a outlier among physical events is negligible, we simply ignore it.
    – J D
    Commented Feb 7, 2023 at 21:21

3 Answers 3

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Your question contains some confusion. There are several different accounts of how to understand or use 'probability'. The most common are:

  1. Classical - that probabilities are understandable in terms of some real or hypothetical symmetry.
  2. Frequentist - that probabilities represent some long-run frequency of a series of events.
  3. Logical - that probabilities are a partial degree of entailment of a proposition by other propositions.
  4. Subjective - that probabilities are a degree of belief or degree of uncertainty in a proposition on the part of a real or ideal reasoning agent.
  5. Propensity - that probabilities arise from physical dispositional properties.

What makes them different is that they represent fundamentally different quantities.

In the case of rolling a real die, there is no guarantee that the probability of it landing on a six is 1/6. The die is only symmetrical within some specific engineering tolerance when it was made. If it has been rolled a lot before, its edges may become worn unevenly. The action of rolling it may not yield a perfectly uniform distribution of outcomes. The die may come to rest on an edge or a corner. (Coins when tossed come to rest on their edge more often than you might think.) The probability of the die exploding is very low, since we are aware that such events are extremely rare.

Commonly when we speak of the probability that a die will land six when it is next rolled, we are speaking of the appropriate degree of uncertainty that attaches to this proposition, given all the available evidence. In the absence of any evidence of bias, an assessment of approximately 1/6 is reasonable. On the other hand, if the die has already been rolled and landed six, and we can see that it is six, then updating our belief with this information yields a probability of 1, since we are now certain.

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Why are physical and logical probabilities considered separate?

The two are separate because logical or mathematical probability can deal in certainties, and so avoid the noise of the real world.

In the example in the question, it might well be that a bad batch of plastic causes the die to explode or fall apart in mid-air. But that possibility occurs only in the real world. In mathematics or logic, such eccentric outcomes are ignored, and, to a certainty, 1/6 is the probability.

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It seems apropos, given the issue raised in the OP, to divide the possibility space into the following two categories:

  1. Imagined Possibility Space (theoretical probability): A die can land on 1 through 6 OR turn into a bird and fly away OR disappear OR you get the idea.
  2. Experienced Possibility Space (experimental probability): There are only 6 possibilities when you roll a die.

We never see everything possible in the imagined possibility space being actualized in the experienced possibility space as when we conduct an experiment (trials) except perhaps in the hands of a skilled magician ( that's telling, oui?). Thus, possibilities like a die transforming into a bird and flapping away or vanishing into thin air aren't included when calculating theoretical probabilities of die rolls.

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