Bayesianism: Logical Propositions

[This post is part of a series on Bayesian epistemology; see Index]

Is it true that the Earth is round? Does Newtonian mechanics accurately describe the celestial mechanics? Before we answer such questions it seems that we first need to assume that they are meaningful. We need to assume that such propositions are true or false, in the sense that they accurately (or not) reflect and describe reality*. This is the first assumption of Baysianism.

Assumption 0: Domain of Discourse: We are concerned with the truth value of a set of basic propositions (A,B,C….) about the world, each of which can be either True or False, and with the truth of all the complex propositions that can be constructed from them.

Most explanations of Bayesianism don’t even bother listing this as an assumption, but it actually isn’t quite as trivial as it appears to be**. The most important objection to it is that thinking of descriptions of the world – propositions about it – as either “true” or “false” is generally simplistic and naive. The Earth isn’t really round – it’s shape is somewhat oblong, it vibrates, and if you insist on perfect accuracy you will find no physical object has a well-defined shape due to quantum effects. But clearly, saying that the Earth is round is more correct than saying that it is flat. The “truth” of a statement is therefore not a simple matter of “true” or “false”. There are grades of truth, grades of accuracy. The same description can even be less or more true in different ways (e.g. more accurate, but less complete). And there is good cause to think that we would never have a complete and fully accurate description of reality, Truth with a capital “T”, as that would require an infinitely detailed description and our minds (and records) are finite.

While all of this is true, it is nevertheless clearly useful to think in such dichotomies. Indeed, even when we aren’t we’re really only thinking about the truth of more elaborate propositions, like “It is more accurate to say that the Earth is round than that it is flat”. Even in computer applications and mathematical derivations attempting to make use of more “fuzzy” thinking*** the programs and theorems are written down as they “truly” are, not just fuzzily.

I therefore believe that this objection ultimately fails in practice. Bayesian thinking indeed refers to rough dichotomies, but in many cases and in many ways such thinking is perfectly valid and acceptable. The limited nature of this discourse, however, should be borne in mind, and ideally we should find ways to adapt and change our discourse as greater accuracy and precision are needed.

Another objection along these lines is that meaning is holistic. This line of thought objects to the entire attempt to determine the truth of some set of individual propositions. The proposition “the Earth is round”, for example, is meaningless on its own, without an understanding of what “Earth”, “round”, and so on mean. Meaning is only conferred by a network of intertwined concepts, and judging the truth of individual propositions in isolation is therefore impossible. Instead of trying to ascertain the truth of propositions, we should be considering the mental structures and representations underlying them.

I find this objection to be even weaker than the first. While meaning is indeed holistic, there is nothing wrong with approaching the problem of knowledge by considering specific propositions. They still represent truth claims about the world, and as such can be judged to be true or false. Understanding how to consider logically- and conceptually-related propositions should certainly be a concern, but this does not undermine the Bayesian approach.

The division into basic and complex propositions is not problematic, as far as I can see. The moment you admit to a domain of discourse containing true and false claims, you automatically can generate complex claims from it and it is only reasonable to want to consider their truth as well. As we shall see, it is precisely the construction of this complex structure that allows Bayesianism to proceed.

* This is the “Correspondence Theory of Truth”, and implicitly presumes “Realism”. Formally at least, Baysianism can be advanced without this metaphysical burden – merely having two truth values is enough, regardless of your theory of truth. In practice, however, I (and every Baysian I’ve read about) uses Truth in the correspondence sense and believes in Realism, so I’ll leave it at that.

** The only one I know that discusses it, however, is Mark Colyvan, in “The philosophical significance of Cox’s theorem”.

*** Allowing truth values to be continuous between 0 and 1, rather than 0 or 1, leads to “fuzzy logic”.

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