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Professor Vern R. Walker

Hofstra University School of Law
Hempstead, New York 11549   USA

The following is adapted from Vern R. Walker,
Discovering the Logic of Legal Reasoning, 35 Hofstra Law Review 1687, 1693-96 (2007)
Copy of full text available for download in .pdf format


A primary strategy for deciding similar cases similarly is to develop and apply substantive legal rules, which prescribe particular outcomes for particular types of cases. The substantive rules of law state the conditions under which particular types of governmental action are justified. Formal logic represents such rules as “conditional propositions.” A “proposition” is the descriptive content of an assertion or statement. It is capable of being either true or false, and is usually expressed in ordinary language by a sentence or a clause. A “conditional” proposition has the logical form “if p, then q,” where p and q stand for two constituent propositions. In the terms of this conditional schema, a legal rule states that if proposition p (the condition) is true, then this fact warrants that proposition q (the conclusion) is also true. A warranted conclusion can then warrant additional inferences, based on additional rules, and can ultimately help justify action or inaction.

While traditional logic has focused on propositions as having one of two values (“true” and “false”), the dynamic process of rule-based legal reasoning is better understood as assigning to propositions one of three values (“true” / “undecided” / “false”). When a legal proceeding begins, all propositions that form the conditions of the applicable legal rules are “undecided.” Participants in the legal process produce evidence and arguments to persuade the decision-maker (whether judge, regulator, or factfinder) to change the values of those propositions to either “true” or “false.” Put another way, the legal rules identify the propositions that are relevant within the type of proceeding, but the particular proceeding begins with the decision-maker being neutral on whether the conditions for applying those rules are satisfied or not.

A major feature of rule-based legal reasoning is the distinction between prima facie case and affirmative defense. Some rules state the conditions under which governmental action is justified (the prima facie case for the proponent of the action), while other rules state exceptions or affirmative defenses—that is, conditions under which the prima facie line of reasoning is defeated. These latter rules, which logicians call “defeaters,” function as a kind of negation. If the defeater condition is determined to be true, then the conclusion is false. Legal rules exhibit defeater logic when they state an exception to a normal rule or an affirmative defense to a prima facie case, and it is common in such circumstances to place the burden of proof for the defeater proposition on the party invoking the exception or raising the affirmative defense. For example, the law of battery has rules governing when the defendant has a privilege to act that constitutes an exception or defense to the prima facie case (for example, when the defendant was acting reasonably in making a lawful arrest or in defending herself from intentionally inflicted bodily harm).

We can visualize systems of substantive rules of law (such as the law of battery) as inverted “implication trees” — trees that map the conditions of rules, or the implications of proving issues of fact. Figure 1 shows a partial implication tree for the law of battery. The nodes of an implication tree are propositions to be proved or disproved, and the top node of a tree is the ultimate issue to be proved before some governmental action is justified (for example, entering a court judgment for the plaintiff). Each level of each branch extending downward from the top node states the logical conditions for proving the immediately higher proposition. Figure 1 illustrates both a conjunctive level (stating a conjunction of conditions connected by “AND”) and a disjunctive level (stating a disjunction of conditions connected by “OR”). A conjunction is true if, but only if, all of the conjuncts are true, while a disjunction is true if, but only if, at least one disjunct is true. A branch can also state a defeating condition (connected by “UNLESS”), the truth of which determines the conclusion to be false, even if the prima facie branch is true. Because rules tend to have multiple conditions for making an inference to a single conclusion, successively lower levels of a rule tree tend to expand horizontally as they expand downward. The shape of the inverted implication tree therefore tends to be triangular, with the single ultimate issue as the apex at the top, dependent for its truth or falsehood upon combinations of factual issues that terminate the branches along the triangle’s base at the bottom.

Traditional logicians are used to the deductive logic of mathematics, in which proof begins with axioms (at the top) and deduces conclusions (downward). They may therefore overlook this fundamentally different orientation of legal reasoning. The rule-based deductions of legal reasoning do not rest upon self-evident axioms, but rather upon the truth or falsehood of the relevant issues of fact, which are defined by rules adopted by legal authorities. The soundness of the reasoning depends upon the appropriateness of the rules governing the reasoning, and (at bottom) upon the plausibility of the relevant evidence. Moreover, conclusions that rest upon uncertain and incomplete evidence are at best plausible and conditionally true. The rule-based reasoning of law can be only as sound as the factfinder’s evaluation of the evidence.

Another feature of rule-based legal reasoning that challenges traditional deductive logic is the possibility of changing the rules themselves as a result of the reasoning. For example, within common law systems, courts have inherent authority to elaborate new legal rules that apply to the very case being decided, as well as to future cases. Even when the authority is legislative and the legal rules are derived from statutes or regulations, a court or administrative agency has considerable discretion to elaborate new rules of application in pending cases. Whenever a court or agency explicates a new definition for a legal term, or interprets a legal phrase, or carves out an exception to an existing rule, it creates a new rule. Such new rules may create new conditions that extend the branches of the rules tree, or create exceptions that add defeaters to the tree. Courts sometimes also overrule prior cases, hold statutes unconstitutional, or vacate administrative regulations, thus removing branches from the tree. Under the rule of law, however, the action of changing a substantive rule is itself governed by legal rules and must be justified in each particular case. Such reasoning about the rules themselves can be considered a second-order aspect of legal reasoning.

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