The MIN Connective
MIN is one of the four major logical connectives that we use in modeling the evidence assessment of a factfinder. It is a generalized form of the logical connective AND. But we have to use it very carefully.
MIN determines how the plausibility of the conclusion is calculated. Within our default-logic framework, the assertion that is the conclusion of the inference (such as a finding of fact by the factfinder) is placed at the top of the tree, and the “children” assertions are placed on the immediately lower level. The children provide evidentiary support for the conclusion, and state the conditions under which the conclusion is plausible. If there is more than one child, then the siblings are connected together by an appropriate logical connective, such as MIN (see the diagram below). The children assertions in combination constitute proof of the parent conclusion, and the logical connective determines how the combination works in determining the plausibility of the conclusion.
MIN has a precise logical meaning. The MIN connective assigns to the conclusion the lowest plausibility-value possessed by any of its (children) conditions. It generalizes the “AND” connective (conjunction) for use in many-valued logics. In the logic framework used by the Lab, any particular evidentiary assertion has a plausibility-value drawn from an ordinal, seven-valued scale with the following possible values: “highly plausible” / “very plausible” / “slightly plausible” / “undecided” / “slightly implausible” / “very implausible” / “highly implausible.” The MIN connective assigns to the conclusion the lowest value possessed by any of the conditions (“highly plausible” being the highest value). The plausibility of the conclusion of the inference is only as strong as its weakest premise.
MIN means much more than merely “relevant.” The conditions should all be relevant to reaching the conclusion (i.e., determining that it is plausible), but the MIN connective requires that the weakest single condition or premise determines the plausibility of the conclusion. We therefore use the MIN connective only when all of the conditions as a set provide a single, complex line of support for the conclusion, and each condition is necessary for reaching the conclusion. We use MIN only when any one of these individual conditions necessarily decreases the plausibility of the conclusion.
In the image below taken from the logic model for the Casey decision, the special master was making a finding under Prong 3 of the Althen test of causation in fact (namely, whether a “proximate temporal relationship” exists between the vaccination and the onset of symptoms). This proposition (a requirement that the petitioner must prove under the Althen test) appears in bold font at the top of this segment of reasoning. The special master in fact concluded that “[P]etitioner’s onset of symptoms occurred within an appropriate time period after vaccination” (the assertion in regular font directly below the top proposition). In the Lab’s modeling, the bold font indicates a three-valued proposition established by a legal rule as an issue of fact in all appropriate cases (as indicated by the citation to “Althen, 418 F.3d at 1278”), and the regular text indicates a seven-valued assertion made within the particular case (here, the Casey case).
In the Lab’s Legal Apprentice™ model for the Casey decision (above), this finding by the special master rests on two assertions (both of which the special master found plausible, as indicated by the green circular icon before each of the assertions):
MIN [1 of 2]: “[I]f the varicella vaccination was the cause of petitioner’s injuries,” then the “petitioner’s onset of symptoms occurred within an appropriate time period after vaccination.” (The annotations in brackets indicate that this assertion is found on the cited page of the decision, and that three experts in the case agreed to this assertion, including the petitioner’s expert and the respondent’s expert.)
MIN [2 of 2]: The varicella vaccination caused her encephalomyeloneuritis. (The support for this assertion occurs elsewhere in the decision.)
These two assertions together form the basis for the finding, and they are appropriately connected together and to that conclusion by MIN – to the extent that either assertion is implausible, the conclusion is similarly implausible on this reasoning. If the conditional assertion in MIN [1 of 2] is implausible, then MIN [2 of 2] alone is insufficient basis to draw the conclusion. Also, if MIN [2 of 2] is implausible, then the conditional in MIN [1 of 2] alone is insufficient to reach the finding. The finding is only as strong as the weaker of these two premises.
MIN models the conjunctive part of the meaning expressed by many ordinary-language words. These include, in English, words such as “and,” “also,” “moreover,” “however,” “but,” “yet,”“nevertheless,” and “although.” I say “the conjunctive part of the meaning,” because many of these words express conjunction plus some other meaning, such as surprise that the second assertion should be plausible (“yet”) or that it asserts something contrary to expectation (“nevertheless”). Modeling correctly with MIN or any other logical element, however, is not a mechanical translation of any English words, but an interpretation of the underlying logical meaning.
One advantage of the MIN connective that makes it useful in everyday reasoning is that it does not require knowledge about the degree of independence among the premises or children conditions. Because the plausibility of the conclusion is set equal to the plausibility of the least-plausible condition, any entailments, implications, or statistical associations among those conditions have no independent effect on the plausibility of the inference. This permits plausible default reasoning on the basis of limited information, while allowing more precise analysis to occur in the future, if additional information becomes available that is helpful in solving a particular problem.