The MAX Connective

MAX 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 OR.

Like MIN, MAX determines how the plausibility of the conclusion is calculated. Within the Lab’s 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 MAX (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.

Image of a Logic Drawing Illustrating the MAX Connective

MAX has a precise logical meaning. The MAX connective assigns to the conclusion the highest plausibility-value possessed by any of its (children) conditions. It generalizes the “OR” connective (disjunction) 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 MAX connective assigns to the conclusion the highest value possessed by any of the conditions (“highly plausible” being the highest value on the scale). The plausibility of the conclusion of the inference is as strong as its strongest single premise. This is why it is a generalized version of “OR,” because the conditions are equally acceptable alternatives, any one of which provides adequate support for the conclusion.

MAX means much more than merely “relevant.” Ideally the conditions would not contain any assertion that is irrelevant to inferring that the conclusion is plausible, but the MAX connective only requires that the single strongest condition or premise determines the plausibility of the conclusion. We therefore use the MAX connective only when each of the assertions in the conditions set provides sufficient support for the conclusion. We use MAX only when any one of these individual conditions can establish 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 1 of the Althen test of causation in fact (namely, whether a “medical theory causally connect[s]” the vaccination and the alleged adverse effect). 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 “There is an adequate medical theory of causation” (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).

Image of a Legal Apprentice Model Illustrating the MAX Connective

In the Lab’s Legal Apprentice™ model for the Casey decision (above), this finding by the special master rests on two alternative lines of reasoning argued by the petitioner (in this case, the special master happened to find each of the alternatives plausible, as indicated by the green circular icon before each of the assertions):
MAX [1 of 2]: The varicella vaccine can cause “a direct viral infection,” which could “negatively affect the nervous system.” (The annotations in brackets indicate that this assertion is found on the cited page of the decision, that the source is the special master, and that the basis is expert testimony.)
MAX [2 of 2]: The varicella vaccine can cause “an immune-mediated inflammatory response,” which could “negatively affect the nervous system.”
These two assertions provide alternative bases for the finding, and they are appropriately connected together and to that conclusion by MAX. The petitioner argued for two possible causal scenarios mediating the vaccination and the alleged injury: a direct viral infection and an immune-mediated inflammatory response. To the extent that either causal path is plausible, the required finding under Althen is similarly plausible. In fact, the special master found both of the paths to be plausible, and ultimately found that “[t]he attenuated virus in the varicella vaccine both directly attacked petitioner’s nervous system and caused an immune-mediated inflammatory response in her nervous system” (Casey decision, at p. 27). MAX is therefore a generalized version of “inclusive OR”: since either alternative can establish the conclusion, the plausibility of both alternatives also does so.

MAX models the logical disjunction expressed by such English words as “or” and “either … or … ,” as well as such phrases as “in the alternative.”

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