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The idea of fact is complex in its epistemic, metaphysical and linguistic connotations, but in the end it has compelling force, and the power to shock when the predicted practical effects are not actualized.

In the Stanford Encyclopedia of Philosophy we find this application of set theory to facts mapping to 'worlds'.

The concept would rely on a power set (P) or set of all possible sets of facts (F), containing individual or elementary facts, x, mapped to the worlds, W, where these facts exist.

In probability theory, the existence of different set cardinality for the natural and real numbers prevents assigning probabilities to each possible set of outcomes, and the idea of sigma-algebra is introduced as a compromise to limit the measurement of probabilities to only those sets in the power set that are measurable.

This idea might not have been introduced in the philosophical approach to facts. It makes no sense in a way since facts are not always quantifiable.

However, facts are indeed mapped (measured) to at least an ordinal or scale of relative importance, and some are discarded, while other are used.

In this regard, some facts are 'not F-measurable' (they have no probability - read, 'significance' or 'qualitative value') while others are highly weighted for importance or in terms of utils in economics, for instance.

I wonder if including the concept of 'measurable' or 'assignable' sets through sigma algebras (sets of events we are subjectively interested in) within the power set of facts could enrich this approach to facts as mappings between propositions and worlds, perhaps introducing injectivity in a more consistent manner.

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  • I guess you can identify a fact with the set of possible worlds where it holds, and its probability with the measure of that set. Not all authors agree to that, or to set operations making sense for facts. But more to the point, why would ranking by "importance" have anything to do with probability? Even if "significance" or "utility" is quantified somehow there is little reason to expect that it would obey probability rules. This identification is more suitable for quantifying which facts are more or less "likely", whatever that means. – Conifold Feb 11 at 23:46
  • You want probabilities (and measures in general) to be well-behaved under at least countable intersections and unions. You'd like uncountable unions, but then the measure of the unit interval would be 1 as its length, on the one hand, or zero on the other, as the union of its singletons. So we settle for countable additivity, hence sigma algebras. I don't think this is a particularly deep or metaphysically significant point, it's just mathematical convenience. A sigma algebra is the biggest class of sets on which measure makes sense. – user4894 Feb 11 at 23:55
  • @user4894 Beautiful summary of the motivation of a sigma algebra! – Math tourist Feb 12 at 1:59
  • @Conifold I am not proposing at all a probability measure on facts. Not at all. I don't know exactly where I am going with this, but I see there are many hierarchies of facts, and any mapping to the $W$ is bound to be non-reproducible, or challengeable, unless there is more structure imposed on this mapping. – Math tourist Feb 12 at 2:02
  • You can restrict what is considered fact by choosing a smaller collection of subsets, and if you also believe that facts are subject to the standard set operations then this collection will be an algebra. I am not sure why you would want a sigma algebra unless you do want to put a probability measure on it. Facts not in the algebra will then be excluded from the distinction of factness, perhaps because people do not believe that there is a way to verify/falsify them. – Conifold Feb 12 at 20:48

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