zaterdag 20 maart 2021

A semi-formal question on theories and facts

In what follows I pose a semi-formal question on theories and facts. A theory of the world is a non-empty class of explanatory propositions. Let T be the class of all theories of the world. What a theory needs to explain are certain facts of the world. Let F be the class of facts of the world that a theory of the world would have to explain. I abstract from specific issues concerning temporality, so facts are to be understood timelessly. The members of F are true propositions that report facts. Loosely I refer to these propositions themselves as facts.

Now, for each t in T, let C(t) be the class of all logical entailments of t. Thus t is a subset of C(t). Class C(t) is not necessarily a subclass of F. For F might only contain “atomic” facts while C(t) might contain "composite" facts (e.g., conjunctions or even disjunctions of atomic facts). Moreover, even in case F contains both atomic and composite facts, it might still be the case that t contains "merely instrumental" propositions. Merely instrumental propositions are not about the world. They are non-factual and lack a truth value.

Although a theory t doesn't have to explain facts that are not in F, C(t) must be logically compatible with those facts. A theory t in T satisfies fact f in F just in case f is a member of C(t). I take it that to entail is to explain. So, if theory t entails fact f, then t both satisfies and explains f. Theories have to do explanatory work. Each theory thus satisfies at least one member of F. Again, the empty class is not a theory. Ideally a theory satisfies all f in F, but this is not required for it to be a theory.

A theory is not a subset of F. For explanation is not a mere "repetition" of facts. The class F itself is thus not a theory. Further, a theory isn't obtained by applying the logical operations of 'and', 'or', or 'not' - or equivalents thereof - to a subset of F. For explanation isn't a matter of merely "reorganizing" facts either.

A theory also needs to be such that its subset of (reorganized) F-facts "vanishes" compared to F. For, again, the task of theories is to explain and not to significantly simply repeat or reorganize facts. It's important to exclude "lazy" theories as well. A lazy theory is a theory that includes both some (reorganized) facts from F and a collection of non-factual propositions that are completely unrelated to the facts and thus trivially logically compatible with the facts.

For all theories t in T and all facts f in F we write t(f) if and only if [t satisfies f] or [f is logically compatible with C(t)]. Moreover, for all t in T, let s(t) in [0,1] be the explanatory scope of theory t. If s(t)=0 the class of facts in F that are satisfied by theory t vanishes compared to F. If s(t)=1 the class of facts in F that are not satisfied by t vanishes compared to F. Suppose that there is an explanatory scope value s such that in the class of all theories having an explanatory scope greater than s, there is exactly one t* in T such that t*(f) for all f in F.

Would it then be reasonable to assert that all propositions in C(t*) are true and that t* satisfies all members of F? After all, what else would be a “natural” explanation of such an exceptional state of affairs? If the answer to the former question is 'yes', all propositions of t* are true factual statements about the world and t* explains all the facts that need to be explained. Theory t* would thus be both a "realistic" and "ultimate" theory of reality.

The above suggests a hybrid heuristic. Given classes T and F, choose a sufficiently large threshold for explanatory scope and select in the class of remaining theories the unique theory that satisfies or is at least compatible with all the facts in F. This theory would then be the real final theory of the world. Said heuristic is hybrid since it combines the concepts of satisfiability and compatibility. The former is more familiar to realistic or correspondentistic accounts of the nature of theories, while the latter is more familiar to anti-realists or coherentists.

Once T and F are given the hybrid heuristic seems open to computation in the sense that automated theorem provers may be used to execute the process of raising the explanatory scope threshold until in the remaining set of theories the desired unique theory is found.

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