Implicit Models and Probability - are degrees of belief/truth/existence a complete free-for-all?

Or, to put it another way, as long as you model your statements using the grammatical framework of our modern logical idioms, is it appropriate practice to assign a probability to any utterance at all, even in the absence of any logically interpretable semantic models of what you mean?

Context:

Users @Thinkingman and @Kristian-berry have been asking some interesting (if somewhat tangential to each other) questions on this site around modal existence claims in theology and set theory. A line that I see both independently exploring through their questions addressing the concept of a "possibly existing thing" has been to take an Actualist, Realist, Degrees of Being/Probability approach. (See: Bayesianism and Pseudoscience, Existence tropes and Degrees of existence)

Intuitively, anything that forms a boolean set space can be understood with a probability measure. So, there is nothing that says that a sensible Kolmogorov-axiomatic weighting cannot be given for any statement in a first-order formal language being modelled. In fact, a lot of modern Set theory has put its stock behind the wholesale exploitation of this idea in Forcing - the creation/discovery of models consistent with the basic underlying theory but with variations accounting for the different boolean possibilities given some semantic information.

(I was going to say "consistent first order formal model" instead of language, but let's just assume that the demonstrably inconsistent theories get a trivial weighting of "all zeroes", given that classically, trivialism follows from contradiction. This is not a logical pluralism question.)

What appears to follow from this is a licence for separation of the semantics of the language in favour of a syntactic consideration of probability weights applied to sentences. As long as probabilities are assigned in line with the Kolmogorov axioms, an Actualist and Realist doesn't have to really refer to alternative possible situations or non-actual beings, but simply to assign a non-terminal value as a degree of belief/truth/existence, to the sentence that one is maintaining a position for, and to complement this value with an enabling actualist, realist commitment to the mathematical model that one takes to be operative in accounting for the Uncertainty/Unsettledness of reality.

Or, to simplify, there is only one reality, and it's super mathematically complex, such that it contains all of its own potential for variability, and we are free to use that mathematical complexity to account for assigning degrees of belief and existence to sentential language more or less arbitarily (as long as we respect the rules of probabilistic coherence).

Making the concern more concrete

Something about this position deeply unnerves me, and it might just be the pragmatics of what it means to utter a statement, but I find it bizarre that one can maintain a propositional attitude about a sentence without actually knowing anything about how to map the sentence to a body of propositional sense.

Let's say someone asks me about my degree of belief in the statement "Grues exist". Properly speaking, I don't know what a grue is. I should like to say that I have no degree of belief at all about the statement - neither that Grues do exist nor that they do not.

However, if someone were to say to me "Since you do not have either positive evidence either for or against Grues, you should maintain an ambivalent position, and to do so is to give it a probabilistic weight of 0.5", I should find it hard to explain to them why the sentence "Grues exist" in itself should not be the sort of thing that cannot get a probabilistic value. After all, "Grue" is, seemingly, an established word in our common language, and following Carnap, the syntax of the language is perfectly amenable to this kind of evaluation.

What it does seem open to me to say, perhaps, is that I can only make sense of what it means to assign such a value in the context of a purely analytic boolean-valued probability model of the world. That is, that I'm only giving it a value in a probability measure as a kind of academic mathematical exercise - it doesn't really mean anything. It can't - I have no idea what Grues are.

There is an implicit model of what it means for "Grues exist" to be true - that there is some set or property that we take to be the semantic category of things that are grues, and that this set has at least one member or that at least one thing in the world has the property of being a grue. By contrast, it's less clear what it would canonically mean for "Grues exist" to be false, except perhaps in that (as per Wittgenstein) one is actually speaking about the totality of facts and that this totality does not include Grues, subject to some clear boolean logic interpretation of the logical composite "Grue" in negative terms of the other things that do exist.

But without my knowing what "Grue" means, it seems as though my probabilistic evaluation of it must only be on the basis of the compositional possibility of using the word "Grue" in syntactic modelling. This is (to me) a very different kind of assertion to the implicit model of asserting that "Grues exist" is true.

And yet this seems to be the level on which probability talk takes place (particularly in the realm of Bayesian learning and Large language models) - it's a level that makes very little reference to the implicit model of the first order commitments of my statements in order to determine how to update one's degree of beliefs or generic information about the statements in question.

On this level, it's hard to see why there should be any ground for factivity. This is the old Quine/Carnap point revisited - meaning posits have to get their semantic sense through contact with the world in order for the analytic/synthetic distinction to hold up, but our probabilities seem to just float at the realm of syntax.

If we were saying that what we mean to say when we give a probability of 0.5 to the statement "Grues exist" is that there is a division of the state space of reality, carved appropriately at the joints of evidence or methodologically rigorous practice, into two essentially equiprobable halves, such that in one half of those states Grues exist and in the other half Grues do not exist, then I think that would seem to be a perfectly sensible statement. However, this does not appear to be guaranteed by the assignment of a probabilistic measure to the sentence "Grues exist", and given what I've said so far, that appears to be normal in practice of the syntactic treatment of probability.

So! Have I missed something crucial methodologically about what the concept of a "degree" of belief or truth facilitates, perhaps as part of a wider scientific, mathematical or theological practice? Is this suggesting that something needs to be done methodologically before the theological/mathematical interpretation of the modality of the unverified/indeterminate possibility? Or, perhaps, are the Actualist Realists happy with a kind of pragmatist approach, that much of what we're saying with probability-talk doesn't actually mean much beyond its utility in communication - and if so, is this just as true of the theology or mathematics as it is of everything else?

• A complicating factor is the question of accepting the Kolmogorov axioms in full, as they are. If unitarity is waived, then something about the truth-value semantics, how they bridge to the probability semantics, will be altered, and perhaps by this alteration we could clarify how some of the syntax falls out of the structure of truth in the system, where the structure is the referent of the truth-value terms in their algebraized network. Then, with or without accepting unitarity, we'd have a slightly clearer picture? Commented Sep 2, 2023 at 18:33
• ""Since you do not have either positive evidence either for or against Grues, you should maintain an ambivalent position, and to do so is to give it a probabilistic weight of 0.5"," no, that would be deeply naive. A better approach would be to have a Beta(1,1) hyper-prior on the probability that Grues exist.
– user6527
Commented Sep 2, 2023 at 18:42
• @DikranMarsupial in the case of Grues (green turns to blue at an arbitrary but fixed distant time as the famous new riddle offered by Goodman), and seems OP centers their degree of belief about its existence around 0.5, then your Bayesian priori about such degree would be better chosen as Beta(0.5, 0.5) than uniform arguably... Commented Sep 2, 2023 at 19:40
• @DikranMarsupial, thanks for the clarification; and you’re quite right, my operative concept of a “non-terminal” intermediate value is more simplistic than it has to be. The relevant object isn’t the truth value 0.5 (/some real in the [0,1] interval) but the result of revising evidential updates applied to an initial prior distribution, and the mathematics isn’t committed to taking that arbitrary sample before embarking on updating it through Bayes’ law etc. That’s a really useful clarification, and it’s quite a sophisticated form of realism that emerges. Commented Sep 3, 2023 at 16:08
• @DikranMarsupial my premise is a conjunction and the reason for Jeffreys like prior lies in the former, both my and your symmetric priors are uninformative since their same flatness around 0.5, yet for Grues even 1 sample after that time is ok for Goodman to reveal the inductive entrenchment of either green/blue or grue. The habit forming Bayesian-Laplace like succession process is besides the point here and was not Goodman's central interest (maybe Hume's interest in the original problem of induction, which Goodman criticized). Thus in fact no likelihood needed here, only falsification... Commented Sep 4, 2023 at 0:36

3 Answers

It makes sense to distinguish between what is true/false about the world and what we as reasoning agents believe about the world. Our beliefs are based on partial information. This does not mean that our beliefs float freely or have a semantic disconnect with the way the world is. Having true beliefs has a considerable effect on how successful we are. Believing things that are false can have disastrous consequences, even leading to the death of the believing agent. Probabilities have a utility in decision making, not just communication.

That we can use probabilities as an approximate calculus of partial belief is just a fact of psychology. For one thing, many cognitive psychologists explicitly use Bayesian models to understand how people reason and revise their beliefs and this approach seems to hold up pretty well. For another, Bruno de Finetti showed how to use Dutch book arguments to derive a probabilistic calculus for partial belief, based on a few simple assumptions about what constitutes rational betting behaviour.

More generally, it just seems completely natural to think of probabilities as degrees of belief. In my experience, everybody is born a Bayesian and they do not imagine that probabilities are anything other than degrees of belief until they have their understanding of probability corrupted by going on a statistics course. Incidentally, I wouldn't put too much emphasis on Kolmogorov. Many Bayesians prefer Richard Cox's axiomatisation, and there are several others, including some in which all probabilities are conditional.

In the case of propositions where I have no relevant information at all, or where I don't even know the meaning of the words involved, it makes sense to suspend assigning a degree of belief. Unless I am compelled to make a decision, I don't see any reason to assign a probability to the existence of grues, if I don't know what a grue is.

There are also other ways to cope with a lack of information. We could deploy a second level of probability and speak of the probability that a given first level probability statement is correct. This is what Captain Hocken does in the The Naked Gun, when he says, "Doctors say he has a 50-50 chance of living. Though there's only a 10% chance of that". Although meant as a joke, this does make sense. There's a 10% chance the doctors have made a correct assessment of the probability. Another option is to use upper and lower bounds on probabilities, e.g. as in Dempster-Shafer theory.

As to whether it is appropriate to assign a probability of 0.5 to some unknown on the basis of the principle of indifference, this is quite problematic. It really only works in simple cases. I wrote more about this is my answer to this question: Is the principle of indifference invalid?

• You could infinitely play that game of "X% chance that the assessment of the assessment of the ... of the assessment of the claim is true". Where to stop? Commented Sep 3, 2023 at 9:24
• Secondly, the issue with these Bayesian probability assignments is that they are entirely subjective - they are a matter of feeling more than anything, which doesn't allow for objective analysis and poses as a red herring to the actual logical proposition at hand. Commented Sep 3, 2023 at 9:25
• @csstudent1418 Hmmm. Have you ever actually used Bayesian statistics? Maybe you should pick up a good textbook. Bayesianism isn't some weird hippy theory under which everything is subjective and anything goes. It is a finely honed set of techniques that are widely used in science and commerce. It stands as an alternative to frequentist statistics, and just because people are usually taught frequentism first in introductory stats classes, this does not make Bayesianism any less useful or less important. Commented Sep 3, 2023 at 11:20
• They would want to but the number will be irredeemably subjective. You are correct that the updating process is not subjective but that is only because likelihoods given hypotheses are measurable. The inverse in an objective sense doesn’t exist. The prior is crucial and based on plausibility which is entirely subjective. Subjectivity in, subjectivity out.
– user62907
Commented Sep 3, 2023 at 21:17
• @thinkingman Exploration companies can and do this. Exploration companies that don't go bust. It is not irredeemably subjective. You are just displaying a complete ignorance of how important, how useful and how common the use of Bayesian statistics is. Buy some textbooks and learn about it. Get some real world experience. You are indulging in speculation based on no knowledge whatsoever and this is a pointless thing to do. Commented Sep 4, 2023 at 2:09

Having trouble assigning a probability to a proposition isn’t just a matter of the sentence not making sense or the sentence containing a term you don’t know the meaning of.

The trouble is deeper with the notion of probability itself and can be seen with statements that do mean something to you. What does it mean to assign a probability to the statement “fairies exist” or “god exists” or even that “humans exist”?

The only coherent notions of meaning when it comes to these statements is the subjective degree of belief concept that many Bayesians use by defining it as how much you would bet on a proposition. So for example, if your degree of belief is 0.9, you would bet 0.9 units on a bet that returns 1 unit.

But this just begs the question: why do these subjective bets matter? It is not hard to imagine how the whole world may be psychologically certain or feel close to certain about the earth being flat, as it once was, and having it turn out that the earth was a sphere. And since when did we care about how a person behaves, feels, or what he experiences when trying to determine truth about the world?

The point is that no attachment of a probability to a proposition makes sense in that it doesn’t actually tell you anything about the world. The best way to say you don’t know something is to simply say…you don’t know. The best way to say that you feel more confident in X than Y is to simply say…you feel more confident in X than Y.

Attaching a probability to a proposition seems to imply that you can know something more or less than another. But that to me is to pretend to have knowledge that what one lacks.

• "why do these subjective bets matter?" - because in the real world we actually do get different payoffs depending on what we choose to do. We literally make bets, or we take actions that have payoffs we can't fully predict in advance, which amounts to a bet. If you limit yourself to binary "believe/disbelieve," with no room for degrees of uncertainty, then you won't be able to make good decisions in a variety of real-life situations. Commented Sep 3, 2023 at 0:08
• Of course bets are useful. No one denied that. The answer is about the ontological nature of probabilities when it comes to beliefs. With that being said, we make bets every single day without assigning probabilities to propositions. Therefore, as an empirical matter, they are unneeded. More importantly, something being useful doesn’t imply its truth. Living life assuming the flat earth model won’t bring you much of a consequence even though it’s demonstrably false
– user62907
Commented Sep 3, 2023 at 0:16
• So if option 1 has a payoff of 12 if it works out and 0 if it doesn't, and option 2 has a payoff of 15 if it works out and 0 if it doesn't, and option 1 is more likely to work out than option 2, which option should you choose? You can't decide; it's not enough information just to know the rank order of the probabilities. You need the actual probabilities, or at least a gut sense telling you roughly how likely each is. There's a reason financial analysts and hedge fund managers and gamblers and physicists etc. use probabilities. They do work. Commented Sep 3, 2023 at 4:02
• @thinkingman "The only coherent notions of meaning when it comes to these statements is the subjective degree of belief " that is factually incorrect. There is also objective Bayesianism, which is an extension of propositional logic to accommodate reasoning under uncertainty.
– user6527
Commented Sep 3, 2023 at 11:54
• @csstudent1418 it shows no such thing. If X is "a sequence of 10 flips of a fair coin that come up heads each time" and Y is "[standard binomial probability calculation]" then you have not shown that X does not exist even though it is unlikely.
– user6527
Commented Sep 3, 2023 at 11:57

Well, actually assigning consistent probabilities to everything in a way that conforms to the Kolmogorov axioms is, in practice, impossible. Humans cannot do it. Computers can't do it. It's too hard.

Bayesian inference is in general an NP-hard problem. This means that when the problem size increases beyond a handful of variables, exact Bayesian inference cannot be done.

Think about it. If you believe in some axioms, such as the ZF axioms, then to assign consistent probabilities to every statement, you also have to believe in all theorems that follow from the ZF axioms, and disbelieve all statements that can be disproved from the ZF axioms. That means, for example, that you hold the correct belief in the Riemann hypothesis; if the Riemann hypothesis is provable, you should give it probability 1, and if it is not provable, you should give it probability 0. But no one has yet been able to solve the Riemann hypothesis. And the same is true for any number of other unsolved conjectures in mathematics, too. So you cannot assign consistent probabilities!

The best we can hope for is to try to approximately hold consistent probabilities. We can approximate Bayesian inference by various heuristics and algorithms.

This is why when we speak of human degrees of belief, it's technically more accurate to speak of "credences" rather than probabilities. Human credences do not necessarily sum to 1. If a person holds a high credence in a set of statements that together imply a conclusion C, it does not mean the person holds a high credence in C. Human credences are inconsistent with themselves. They do not satisfy the Kolmogorov axioms.

But this does not mean human credences are totally arbitrary. If a person investigates an issue, gathering evidence and studying the arguments on all sides of the question, the person is likely to change their credences over time. They would perform approximations of Bayesian updates, and they would also notice some of the inconsistencies in their own credences, and try to update their credences to fix these inconsistencies.

The "truth" could be understood as the credences a person would hold, after sufficient iterations of investigation.