Let be the set of propositions considered by some rational logician (call her Sue). Further, suppose that is closed under the propositional connectives , , . Here are two related but different preorders on :
- if logically entails .
- if Sue considers at least as likely to be true as is.
Let be the equivalence relation defined by iff and let similarly be defined by iff .
Then we know what type of structure is: since we’re assuming classical logic in this article, it’s a Boolean algebra. What type of structure is ?
We can at least come up with a couple of examples. Since Sue is a perfect logician, it must be that if , then . If Sue is extremely conservative, she may decline to offer opinions about whether one proposition is more likely to be true than another except when she’s forced to by logic. In this case, is equal to and therefore again a Boolean algebra.
In the other extreme, Sue may have opinions about every pair of propositions, making a total ordering. A principal example of this is where is isomorphic to a subset of and Sue’s opinions about the propositions were generated by her assigning a probability to every proposition .
What’s in between on the spectrum from logic to probability? Are there totally ordered structures not isomorphic to or a subset? More ambitiously: every Boolean algebra has operations , , , while has operations , , which play similar roles in the computation of probabilities (note that is partial on ). How are these related and does every structure on the spectrum from logic to probability have analogous operations?
These structures (i.e., structures of the form for some acceptable in a sense to be defined below) were called scales and defined and explored in a very nice paper by Michael Hardy.
The Definition of a Scale
Modding out by the equivalence relations once and for all, the general setup is that we have a map (induced by the identity function on in the above setup) from a Boolean algebra to a poset . What should be true of ?
Since if a proposition logically entails a proposition , Sue will consider at least as likely to be true as , we should have that implies ( will now be the ordering in either or , depending on context). In fact, we should have that implies .
Actually we should have more: For example, it should be the case that if , then . In general, if is a propositional formula where appears negatively (that is, all occurrences of are negated in a normal form of ), then should imply and the reverse is true if appears positively in . Furthermore, if we can require that the inequality be strict.
Finally, we should require that not just if , but even if it only holds that . That is, even if doesn’t logically entail , if you consider more likely to be true than , you should consider more likely to be true than . A similar generalization to holds as above.
These considerations are equivalent to Hardy’s definition:
Let be a Boolean algebra, be a poset, and . Then is called a basic scaling if:
- is strictly increasing, so that implies .
- preserves relative complementation, so that if and , then , where is the relative complement .
Hardy proves that the relative complement operation is well-defined on , that is, that depends only on , , and . Note however, that it is a partial operation: even if in , there is no guarantee that there such that , , .
A scale is then defined as a poset together with a partial relative complement operation which is the range of a basic scaling.
Hardy’s paper gives many examples of scales, including a few pretty wild ones. Here’s one: Let be the boolean algebra of subsets of . Let iff or . Let iff . This defines a basic scaling to a scale .
What does it look like? Every element except for has an immediate predecessor, and every element except for has an immediate successor. Therefore, it is partitioned into “galaxies” together with in initial galaxy and a final galaxy . Between any two galaxies that are comparable, there are uncountably many galaxies and infinite antichains of galaxies.
Analogues of , ,
We already know that there are appropriate analogues of in all scales, since we know that relative complementation carries over in a well-defined way from the domain Boolean algebra.
What about ? Hardy proves the following:
- If in , then depends only on and . In this case we define to be .
- For , if exists then .
It turns out that, for any , the operation is a partial injective map. Let be its inverse.
Hardy calls a scale divided if the necessary condition for existing given by (2) above is also sufficient. He proves:
For any divided scale, and , in Boolean algebra ,
In other words, all divided scales do have a operation, which satisfies the appropriate law from probability theory.
Finding an analogue of or is trickier, and, when he wrote the paper, Hardy only knew how to do it in the case that the scale is linearly ordered and Archimedean, defined as follows:
Let . Then is called infinitesimal if there is an infinite subset such that for such that for all .
A scale is called Archimedean if it is divided and has no nonzero infinitesimals.
The idea behind the definition of infinitesimal is that, assigning the Boolean algebra a total measure of 1, the measures of the elements of must approach 0.
In that case, you can define a division as follows: Let be the maximum number of times can be subtracted from , let be the maximum number of times that result can be subtracted from , and so on. The quotient is then defined as the continued fraction:
Then the map maps the scale injective to a subscale of (in particular, preserving ). Thus, can be pulled back from its definition on .