Is the Law of Excluded Middle an allowed argument in court?

Question

Is the Law of Excluded Middle a valid deduction rule in court? If not, is it reasonable to say that all arguments in court must be "constructive in nature"?

As an example, consider this idealized and simplified scenario with the following facts:

1. If Alice's phone is not dead, she will see the news of Bob's death.
2. If Alice's phone is dead, she will ask Clare for a charger.
3. If Alice meets up with Clare, Clare will tell her about Bob's death.

Suppose I want to argue that Alice knows about Bob's death. Note that without knowing whether Alice's phone is dead or not, we can not constructively derive this fact. However if we assume the law of excluded middle, we can argue:

• If Alice's phone is not dead, by 1, our statement is true.
• If Alice's phone is dead, by 2 and 3, our statement is true.

My question is whether this kind of argument can be used in court, or that we must know whether Alice's phone is dead or not first concretely. Does the burden of proof change anything (i.e. whether this argument is made by the prosecutor or not)?

Answer 1

Is the Law of Excluded Middle an allowed argument in court?

Yes, but admissibility does not imply conclusiveness. The former is a procedural matter whereas the latter is an issue of epistemic nature.

The scenario you outline seems inconclusive because the three premises fail to bridge the gap between the state of the phone and Alice's cognitive state. Two examples are:

• The conditional character of premise #3 renders it unclear whether Alice managed to reach Clare.
• Alice's state of mind (such as being sleepy, drugged, distracted, overwhelmed, etc.) could prevent her from grasping, or even paying attention to, the news.

Other obvious gaps are left out because you intended the scenario to be a simplified one.

Accordingly, in this scenario the law of excluded middle is inapposite on epistemic --rather than procedural-- grounds.

Does the burden of proof change anything (i.e. whether this argument is made by the prosecutor or not)?

Generally speaking, being plaintiff or prosecutor makes no difference in that regard. Furthermore, the burden of proof would switch to the defendant if the defendant raises an affirmative defense that centers on that issue.

Answer 2

I think that no legal argument needs the principle of excluded middle. Your argument only needs

4. Either Alice's phone is dead or Alice's phone is not dead.

That's much weaker than the principle of excluded middle, which says P ∨ ¬P for all P, not just one particular P. Many specific sentences of the form P ∨ ¬P are constructively provable, so there's no reason for the judge/jury to reject premise 4 on form alone, even if they reject classical logic.

You might be able to prove premise 4 constructively in an axiomatic theory of cell phones, but even if you can't, you can't prove the other three premises either. The argument's success depends on the judge/jury's acceptance of the premises, and it seems far more likely that they'll accept the fourth one than the other three.

Answer 3

While you're better off asking lawyers, this question is fully within in the scope of the forum.

That being said, any lawyer can make any argument they want. The question is whether it will persuade the judge, who are generally sophisticated thinkers. There are times when LEM makes sense. Consider a PD trying to argue his client was both in two places at once. That violates LEM so the prosecutor argues LEM.

But there are times when LEM is inapplicable to reasoning such as motive. Do people have motives that never conflict? Of course not. There's a complex and overlapping motivation. Morality and law come in shades of gray.

So, the answer to your question is that LEM applied to reasoning is often acceptable in court, particularly because it's the simpler way to reason, occurs frequently in legal contexts, and applying graded membership, for instance, is more complex and tougher to understand.

What this demonstrates is that logic is abstraction which models experience. Consider the relation between the transitivity of space and models of space. One needn't have mutually exclusive mathematical structures. The simplest case is a pair of overlapping Venn diagrams. In topology, neighborhoods of points overlap. Whether or not one uses LEM to describe a Venn diagram depends on the diagram it models. Some models are disjoint, and some have a union of extension. The logic that applies must cohere with the geometry. This why the Laws of Thought are abandoned for non-classical logic which has superior descriptive and predictive power as a thinking tool.

In argumentation particularly using natural language and informal logic, persuasiveness is the metric by which logic and fallacy occur in rhetoric and polemics, not obedience to logical formalisms.

Answer 4

In principle, of course it is an allowed argument. Why shouldn't it be? But whether it carries any weight depends on whether you apply it properly. Your start with the statement that Alice's phone must either have been dead or not dead, and then conclude that either way she must have known about Bob's death. But there are other options. Alice might have mislaid her phone. Perhaps she was out of range of a signal. Perhaps she had it switched off. Perhaps it was broken. Perhaps she had fallen out with Clare. Perhaps she couldn't find Clare. Perhaps she had just borrowed Clare's charger without asking. Perhaps... There could be any number of reasons why your conclusion could be false.

So yes, you can appeal to the LEM to say that it is unnecessary to prove that Alice's phone was alive or dead because a particular conclusion follows regardless of which was true, but your appeal will success only if the conclusion really does follow.

Answer 5

No. The issue is that your clause "in court" is ambiguous. One can certainly imagine a situation where the use of excluded middle will be disallowed in court. Say, mathematician A publishes a paper claiming that in constructive mathematics, the Extreme Value Theorem (EVT) of the Calculus is false. Mathematician B proceeds to ridicule mathematician A's article is absurd. Mathematician A sues mathematician B for defamation. When the case reaches the court, Mathematician B wishes to present a proof (by contradiction) of the EVT. If the judge were to allow him to use the law of excluded middle, B would win the case. Otherwise, A would win the case.

As far as the usual courtroom situations where there only finitely many possibilities, the law of excluded middle typically holds both in classical and in intuitionistic logic, and certainly in a court of law.