# Logical Interpretations of Probability

According to Wikipedia's page on probability interpretations...

Logical probabilities are conceived (for example in Keynes' Treatise on Probability) to be objective, logical relations between propositions (or sentences), and hence not to depend in any way upon belief. They are degrees of (partial) entailment, or degrees of logical consequence, not degrees of belief.

Why, then, would the logical interpretation of probability (especially Keynes' version) be considered epistemic rather than objective? It seems to be based on logical relations which hold in a mind-independent sense.

The article continues...

Frank P. Ramsey, on the other hand, was skeptical about the existence of such objective logical relations and argued that (evidential) probability is "the logic of partial belief". (p 157) In other words, Ramsey held that epistemic probabilities simply are degrees of rational belief, rather than being logical relations that merely constrain degrees of rational belief.

This does seem to be genuinely epistemic, but it's not clear how it differs from the Bayesian interpretation of probability, aside from not explicitly referencing Bayes' Theorem as the decider of rational belief.

How does the logical interpretation of probability stand as its own epistemic interpretation, rather than being partly objective and partly subsumed into the Bayesian interpretation?

• Logical probabilities are epistemic because they are properties of propositions or of information, rather than being properties of the way the world is. Bayesianism is one form of Ramsey's logic of partial belief. There is some more information in my answer to this question: philosophy.stackexchange.com/questions/30856/… – Bumble Aug 23 '20 at 4:50
• There are two different senses of "objective", as in relating to the reality itself and as in relating to the state of evidence. There are objective claims about the state of available evidence given one's knowledge, objective but epistemic. – Conifold Aug 23 '20 at 5:12

A useful paradigm for this is to think about a deck of cards. Shuffle it up, and draw one. What is the probability that the top card is black?

A logical interpretation of this would be to say "Okay, what is the state space of the possible state of the cards, and in what proportion of that space do we say that the top card is black?". You look at the 52 cards, you spot that the space splits neatly into 26 of each colour, and in the understanding that the deck has been properly randomised you conclude that "The probability is 0.5, because that is the proportion of the state space that is black"

Ahh, says Ramsey, hold on a second. This idea of being 'properly randomised' begs the very question at work here. We constructed a model of the deck of cards on the basis of the evidence observed. The 'relation' of probability at work in any given card draw isn't just pure mathematics, but also depends on features outside the model, such as whether the deck is stacked, whether any cards are duplicates, whether the dealer is using sleight of hand and so on.

Logical models of probability give us a very useful framework for how to distribute our estimates effectively, but they're not the whole or even a strictly necessary part of the story. It can even result in inappropriate attributions of confidence, in that most people using probability estimates do not generally give good evidence for the models being used to assess the probabilities of individual events they predict.

Ramsey, as a subjectivist, would argue that we make our judgements of probability on the basis of confidence, not on a mechanical statement of known facts.

However, his opponent ought not, strictly speaking, be said to be presenting an account of the "objective metaphysical chance" of the top card in our example being black. Why not? Well, having shuffled the deck of cards, a mechanical process which puts the sequence of cards in some order, the top card of the deck is now fixed.

If you freeze time at the point at which the shuffle is finished, and consider various branching futures from this point which vary only in accordance with the laws of physical possibility, you are not now going to find some possible futures where the card is black and some where the card is red. That is, the objective metaphysical chance that the card on top is black can have exactly one of two values - 0 or 1.

In fact, this is the same objective metaphysical chance as that of the top card being exactly the 9 of diamonds - it either is, or it isn't. We aren't currently in a position of any kind of metaphysical flux - the shuffle has concluded, the deck is in some sequential order, and all that remains is for us to find out what that order is.

This more metaphysical concept of chance does have some relevance in Physics, in that some of our Quantum Physical models potentially have an element of indeterminacy written into their known principles. But this isn't generally what people talk about when they refer to the logical model of probability as objective - what they mean, rather, is a more epistemic point, that the parameters of a model of assigning probabilities to events can be determined independently of the beliefs of any observers involved.

It's important to notice that the section being referenced is talking about the interpretation of logical probabilities, not the logical interpretation of probabilities. The second suggests a discussion of the logic that lies behind probabilistic assessments, which is based in the mathematics of distributions. Logical (or epistemic, or inductive) probabilities are 'weight of evidence' contexts that do not apply or leverage the mathematics typical of probability and statistics. To use the example from the page, when someone says "the extinction of the dinosaurs was probably caused by a large meteorite" they mean that there is a weight of evidence that conforms to the theory that an impact event led to the extinction of the dinosaurs.

Really, the language of probability should not be used here at all; we ought to say something like: "We assess the theory of meteoric extinction to be true based on the accumulation of evidence consistent with it". But language is imprecise and conventional, so all we can do is note that 'probably' has a non-probabilistic sense in this context.

This gets to the heart of the two different interpretations. In Keynes' view we have a number of different propositions — propositions about, say, the presence of impact crystals, or odd chemical isotopes, or fracture patterns in the Earth's crust — that lead to a conclusion about a meteoric impact. The word 'probably', in this view, points to a recognition that new propositions might be made reflecting new observations, or that the current propositions might be organized in different ways, either of which might ultimately change the conclusion. The logic is sound as it stands, but is not immutable, and so we have to allow for the fact that it might change. However, Ramsey prefers to view the issue as a belief justified by the given weight of evidence, opposed to other beliefs that exist but which do not enjoy the benefit of evidentiary support. 'Probably' in that sense means that this is the belief we ought to hold in the juridical sense: the belief least subject to doubt. But in truth the distinction between a 'theoretical proposition' and a 'justified belief' in this case is razor thin; more a matter of impression management than any more substantive concern.