Logic and the Law
Lawmakers, lawyers, and judges are all amateur logicians. When lawyers and judges try cases or render opinions, much of their reasoning is analogical, probabilistic or even rhetorical, which puts it out of the reach of deductive logic and its proof procedures. This more informal reasoning is constrained by deductive logic — one is not allowed to reach contradictory conclusions — but the (consistent) conclusions that are arrived at rarely have the force of proof.
When lawmakers draft rules or lawyers draft contracts, on the other hand, deductive logic comes into play more substantially. The aim of written rules or contract terms is to define a set of consequences as precisely as possible so as to box in later conclusions as much as possible. Part of this boxing-in will be an informal appeal to the formal meanings of terms like 'and', 'or', 'if' and 'not'. In combination with the non-logical terms that represent (actual or potential) statements of fact, they bring into play the formal notions of model enumeration and mechanical decision procedures. The models of a rule are all of the factual scenarios that make the conclusion of the rule true. Decision procedures allow us prove whether any proposed factual scenario is covered by the rules.
So, although may legal documents such as court briefs or case law decisions, which assert informal arguments, can only be adjudicated informally, a substantial percentage of legal documents assert pedantically precise rules which “want to be” decided deductively. They implicitly depend on correct logic. No other profession (science included) has such a large collection of natural language documents that already approximate deductive logic.
The Laws of Thought
In two monographs published in 1847 and 1854, (the second of which titled An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities) the English mathematician George Boole introduced the distinction between the parts of sentences that denote individual statements of fact — the parts that are directly true or false — and the logical parts (such as 'if' and 'or') that combine the alleged facts into larger, compound statements of fact. The distinction was not new, but his contribution was to reduce all of the many expressions for these logical connectives to just three primitives ('and', 'or' and 'not). You could derive every other logical connective as some combination of these three. This allowed him to express his laws of thought as an algebra -- a system of equations over variables whose values are either true or false, instead of the more familiar arithmetic algebra where the values of variables are numbers. This is somewhat counterintuitive to our informal concept of rules, where we think of 'if' as the primary operator because it introduces the notion of conditionality. Certain facts are true, the rules say, only if other facts are true. But 'if' can be reduced to a combination of 'not' and 'or', and the conditional semantics remain intact.
This distinction between the basic factual assertions and the logical connectives among them became the foundation for present day propositional logic, where the two parts are called, respectively, propositions and truth functions. There are typically five truth functions in propositional logic ( 'and', 'or', 'not', 'only if', 'if and only if') but they can still be reduced to Boole's three primitives.
The intended boxing-in of logical consequences when we write rules is entirely a matter of the truth functions. They give rise to the deductive derivations. From a logical point of view, it doesn't really matter what the basic propositions say. They are just variables representing truth values. So the mental health rule from the Understanding section above:
Psychiatric (mental health) service to an outpatient is a covered service only if it constitutes an active preventative, diagnostic, therapeutic, or rehabilitative service with respect to emotional or mental disorders, and only if (a) the service is furnished by a group practice organization, by a hospital, or by a community mental health center or other mental health clinic which furnishes comprehensive mental health services, or if (b) the service is furnished to a patient of a day care service with which the Board has an agreement under section 49(a)(3), or to the extent of twenty consultations during a benefit period (as defined in regulations), if the service is furnished otherwise than in accordance with clause (a) or (b).
looks like this to the propositional logician:
------------------------------------------------------------------------------------ only if ------------------------------- , --------------------- , -------------------- , or ---------------------------------------------------------------------------- ------------- and only if (a) -------------------------------------------- , -------------- , or ------------------------------------ or other ---------------------------------------------------------------------------, or if (b) ------------------- ---------------------------------- , or ----------------------, if ------------------------- otherwise than in accordance with clause (a) or (b).
As you can see, parsing the rule into propositions and truth functions is not just a matter of locating 'and', 'or' and 'not.' A lot of Boolean logic is packed into English adverbs such as: also, besides, hence, even, except, however, moreover, nevertheless, notwithstanding, so, despite, unless, whether, otherwise, as well as punctuation such as ',' and ';'. English legalese adds its own set of idiomatic phrases for expressing conditionality such as: 'subject to', 'unless the contrary is shown', and 'pursuant to the following'.
By marshaling the phrases in natural language rules into their propositional and logical parts, the formal logician puts the rules into a form that can be used to deduce the logical part of the boxing-in intension. The amalgamation of Boolean operations produces one big Boolean expression over the basic propositions. This expression tells you what the total propositional consequences of your logic are. A very efficient decision procedure over this logic, known as the binary decision diagram, becomes the LogicMap.
The nodes in the LogicMap are the basic propositions. They mean whatever the rule authors intended them to mean. So deciding the truth value of each when you walk the map on your way to the final conclusion is independent of the logic. The boxing-in constraints for limiting the interpretation of these — such as avoiding vague or ambiguous predicates, and providing clear definitions — we cannot help you with. We provide the paths that you must follow as you make these decisions. This is the implied logic.
The Material Conditional
Boole’s original algebra of thought did not have a truth-functional operator for ‘if’ as in ‘A if B’. This was added later by logicians around the turn of the 20th century to round out propositional logic. It became known as the material conditional, but it is not a primitive operator. ‘A if B’ was defined in terms of ‘not’ and ‘or’, as ‘not B or A.’ When undergraduates take an introductory course in logic, they often find this reduction very unintuitive. It captures the intuition that the conditional ‘B implies A’ is false if B is true but A is not, but it also means that ‘B implies A’ is true in all other cases — the expected one where both A and B are true, and the unexpected ones where both A and B are false, and when B is false and A is true. Logic was formalized around the turn of the century primarily to prove theorems about mathematics, and for these purposes, this reduction works well.
The material conditional also works surprising well for analyzing the logic of legal rules expressed in natural language, human intuitions aside. It works well, that is, as long as the conditionals are asserted positively. When rules assert negative conditionals on the other hand, as in ‘B does not imply A’, problems arise if we treat ‘imply’ as the material conditional. The negation of ‘not B or A’ is ‘B and not A’. So from ‘The existence of God does not imply that all prisoners go free’ we could deduce that ‘God exits and all prisoners do not go free.’ The violated intuition here is that a negated conditional is not asserting that the premise is true and the conclusion is false, but that it is possible that the premise is true and the conclusion false. This possibility means that the premise does not guarantee the conclusion. The text is asserting a non-rule, if you will, agnostic about the premise or the conclusion.
Legal rules often contain these negative conditionals, particularly when articulating exceptions to prior rules. These exception cases can usually be handled without resorting to negative conditionals in the logic by adding the negation of the exception conditions to the premises of the overriden rule. So the original material conditional remains positive, but the absence of the exception conditions become part of it.
There are, however, occasions where an erstwhile conditional is unconditionally denied in a legal text. From a logical point of view, these non-rule assertions are superfluous. For any given set of sufficient conditions that imply a conclusion in a text, there are infinitely many (unmentioned) insufficient conditions that don’t imply the conclusion. Adding one or more of them to the logic won’t change the deductive consequences. Rule makers add these unconditional non-rules to texts to influence future adjudicators of the rules, letting them know of possible interpretations that they have already, explicitly ruled out. Don’t even think of it! Because of this, LogicMaps will preserve these non-rule assertions (when they are unconditional) by resorting to a little bit of alethic modal logic (the logic of possibility and necessity) to represent the negated conditional as other than material.
Here is an example:
The negated conditional occurs toward the end of the text in part (b). Clicking on the Insufficient rule roundtangle takes you to its LogicMap:
Insufficient Conditions for:
An alien is lawfully present in the United States (for the purposes of 8 U.S.C. 1611(b)(2))
Notice how the right terminal node disclaims any relevance of this rule to the conclusion. The left terminal, with a negative color, is also logically irrelevant to the conclusion, but as we have said, it is there to stake a claim against extra-logical reasoning on the part of others.
Decidability
Decidability is an important feature for systems of formal logic. A system is decidable if there exists an effective procedure (a decision procedure) for determining, in a finite number of steps, whether an arbitrary expression in the logic is valid. Many of the more expressive systems of logic are provably undecidable. There is no such procedure. The propositional logic that we use here is decidable. One of its most efficient decision procedures, the binary decision diagram, can quickly decide whether any arbitrary logical expression over a set of propositions is true or false, given a truth value assignment for each of its propositions. This decision procedure is regularly used to optimize the performance of digital switching circuits. It always yields a final true or false decision in such applications because the circuits amount to a functional definition of the logical expression they correspond to.
In the case of legal rules, we are not so much interested in the truth or falsity of the rule itself (we assume the rule is true because it is stipulated), but about the relation between its conditional premises and its conclusion. So the LogicMap, which emulates the binary decision diagram, applies the decision procedure to all of the ways the collection of premises can be collectively true or false. When these conditions are asserted by the rule to be both necessary and sufficient for the conclusion, the LogicMap is a complete decision procedure for the truth or falsity of the conclusion. But if the conditions are only necessary, or only sufficient, the LogicMap constitutes only a partial decision procedure for just falsity, or just truth. The other terminal node in these maps means ‘don’t know’. This is important to bear in mind when you navigate a LogicMap, because any rule category other than necessary and sufficient limits what you can conclude about the rule’s conclusion. The positive and negative terminal nodes always correspond to the truth or falsity of the conditions. One or the other side may not always give you a definitive answer about the rule’s conclusion.
This is especially true with LogicMaps of unconditional non-rules like the one above. From a logical point of view, they decide nothing at all, on either side. Their LogicMaps are not really decision procedures at all. We render them as maps to show the extra-logical intent of the rule drafters. The paths leading to the negative side are paths you are explicitly forbidden to consider as relevant to the conclusion.