Intuitionistic logic is what we get when we take ordinary everyday classical logic and drop the Law of the Excluded Middle (LEM), in other words, we drop the law that says that for any proposition, either it or its negation is true. This is such an ingrained notion that it's hard to imagine giving it up. For example, it seems obvious that either it's raining or it's not raining. But there are good reasons for not taking it for granted in mathematics. The main issue is that sometimes when we use LEM we find that we can prove the existence of something, but have no way of constructing it. Mathematics is full of such proofs. The best known example is probably the proof that an irrational number raised to the power of an irrational number can be rational. Some mathematicians (and many computer scientists) only like constructive proofs, ie. proofs that actually exhibit the thing whose existence is being proved. Using intuitionistic logic is a good way to always force this to be true. So, for example, in classical logic, you might find you have proved "A or B", but not have a proof of A or a proof of B. But in intuitionistic logic, if you prove "A or B", then you must have proved A or you must have proved B.

Now (British) criminal law is more lax than mathematical logic when it comes to proof. You only need to prove something beyond reasonable doubt, rather than providing a rigorous derivation from axioms. (Though admittedly mathematicians rarely do this in practice.) But criminal law does have one place where it has higher standards of proof than classical logic: you can't necessarily convict someone of "A or B" unless you have a proof of A or a proof of B. A proof of "A or B" will not do.

It seems that there is enough evidence to show that Kate Moss recently used a controlled substance of class A or class B. Unfortunately, the law requires either a proof that she had been using a class A drug, or a proof that she had been using a class B drug. A proof that she had been using one or the other will not do. And hence Kate Moss cannot be prosecuted. In this regard, the law is Intuitionistic. If you're going to have a legal system that doesn't recognise LEM then you really need to carve up crime-space as a semilattice so you can charge people with the

*join*of two crimes :-)

## 15 comments:

It looks like you're wrong in last 2 sentences of the 2nd paragraph. In CLASSIC logic you MUST have proved A or MUST have proved B because of LEM. And if the law was of classic logic then Kate should be prosecuted, but that's not true.

(Sorry if my English is not very correct)

Not that it matters, since it is symmetric, but wouldn't they charge her with the join rather than the meet?

Interesting post. It's pretty clear that the legal system doesn't support the law of excluded middle at all:

If we are taking beyond reasonable doubt to be the equivalent of proof, then the opposite of guilty would be innocent. But there is a middle ground, when you are neither certain someone is guilty or innocent. Some legal systems even have a separate verdict for this, but all systems are careful to say "not guilty" instead of "innocent" when acquiting.

... or admit that

logicallyit doesn't make any sense to "punish" people for using recreational drugs anyway...Luke,

Yes, I guess the join would be more conventional. Fixed.

@Qrilka—The following is a proof (sketch) for A-or-B, but does not contain a proof of A and does not contain a proof of B. Suppose the negation of A-or-B holds. Then it must be the case that not-A and not-B (for if A holds, then A or B holds, and if B holds, then A or B holds). Then, we proceed to find some contradiction in having not-A and not-B (this, of course, would depend on the particular A and B that under consideration). Then we have derived a contradiction from assuming the negation of A-or-B, and we conclude A-or-B.

An example: As premises: we know that given any two distinct individuals, at least one of them is happy, and that Anita and Bill are distinct individuals. We can now prove the proposition that Anita is happy or Bill is happy. We will not, however, be able to prove that Anita is happy, or be able to prove that Bill is happy.

Adam: all systems are careful to say "not guilty" instead of "innocent" when acquiting (sic).

All systems are also careful to say "guilty" instead of "not innocent" when convicting.

Your point being?

@Adam, Scots law indeed allows "guilty", "not guilty" or "not proven". In English courts (and maybe others) the jury is allowed to backtrack, the judge may ask them to determine if the defendant is guilty of A or iff they are not guilty of A then "in the alternative" also consider if they guilty of B, where B is a lesser offense than A.

What does this have to do with intuitionistic vs classical logic? The D.A.'s office needs to be confident of the precise section of the U.S. code actually violated before deciding to take up a case. Supposing you could indeed prosecute and convict for either first-degree homicide or jaywalking, what does it mean to sentence a cat to either a $10 fine or death by lethal injection? (A way out: fix wavefunctions so those darn things don't collapse.)

Whether "proof of" distributes over "or" is incidental to the real crux of intuitionistic vs classical logic. It's that LEM gives you a free proof in one but not the other. It's a free proof we use all the time, criminal law notwithstanding. If you rule out all the impossible, whatever remains, however improbable, must be the truth, cites a whodunnit novelist. Applying the LEM is easy: just carry out the burden of the universal quantifier.

Returning to Kate Moss, it looks like there are gradations involved: was it a "hard-core" war-on-drugs drug like cocaine or pep-pills like ecstasy. If the crown prosecutor can't be sure it's one and not the other, why shouldn't I believe the powder's actually neither, merely body talc?

> The D.A.'s office needs to be confident of the precise section of the U.S. code actually violated before deciding to take up a case.

And an intuitionist considering "A or B" often needs to know precisely which proposition, A or B, they are going to prove before they take it up. If you don't see why one is similar to the other then you're using a different similarity metric on the space of things to write blog posts about. Fair enough, I've noticed these things vary between people.

This post is just the opportunity to mention that Kate Moss has given a new semantics to the proverb "A rolling stone gathers no moss". You see, when Mick Jagger decided to marry a model, he chose Jeri Hall, and not Kate.

-- Paul

Hello,

I'm curious, is the use of the word "unfortunately" there merely a figure of speech or do you actually believe that goodness would be better served by making her suffer for having the insolence of deciding the chemistry of her own mind?

I know that the drug issue is besides the point of your posting; my reason for writing this is merely the feeling that a pro-drug-war stance held by a mathematician would be most economically explained by not having considered the issue in detail yet :)

Anonymous,

I'm with Bertrand Russell on this. Logic and rationality can't tell you what goals to try to achieve, but they can provide good tools for achieving those goals. I'm only considering the law as a means to someone else's ends, and judging goodness with respect to those ends.

I don't see where's there's enough evidence to prove it was "A or B". I see evidence that she snorted SOMETHING, but it could have been baking soda, no-doz, or flour.

This case would not have been issued in the U.S. either, according to my wife, who used to be a prosecutor.

fred,

I'm responding to what the director of public prosecutions said:

"The issue was not whether the white powder that Kate Moss was snorting was cocaine or talcum powder.

"The law required us to prove that it was either a class A drug or a class B drug. We could only base our case on one of these options."

He seems to be explicitly claiming that the issue was not over whether or not this was baking soda but purely over which drug it was.

I have heard of another case in which it was likely a crime had been committed but where it was impossible to determine which precisely one: something along the lines of the Hans Reiser case where the body was missing so it was impossible to distinguish between a homicide or an abduction. But I don't seem to be able to find references to it - in fact, I wouldn't know what to search on.

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