The Law of the Excluded Middle

In the Metaphysics, Book IV, Section 4, Aristotle presents the law of the excluded middle. In modern terms, the law of the excluded middle can be expressed by the truth table for (p and not-p), which is false in all cases (a contradiction).

The following quotations are taken from "The Philosophy of Aristotle", edited by Renford Bambrough, translated by J L Creed and A E Wardman. (Perhaps you would like to read through the full passage from which I will be citing first. Note: the translations are different.)

From page 58:

There are some people who, as we have said, both maintain that the same thing can be and not be and say that it is possible to hold this view. This is the view of many who study nature. We have assumed here that it is impossible for the same thing to be and not to be at the same time, and on the basis of this we have shown that of all principles this is the least open to question. Some people, through their lack of education, expect this principle, too, to be proved; for it does show a lack of education not to know of what things we ought to seek proof and of what we ought not. For it is altogether impossible for there to be proofs of everything; if there were, one would go on to infinity, so that even so one would end up without a proof; and if there are some things of which one should not seek proof, these people cannot name any first principle which has that characteristic more than this.
So, the law of the excluded middle is an axiom. There is no proof for it. Aristotle tries to support the axiom with an ad hominem attack on the lack of education of those who dispute it, and by stating categorically that it is the most deserving axiom, because no one can name any other that is more deserving. This last argument seems circular to me. We don't have this axiom in natural language. For example, "this statement is false" can occur in natural language. Also modern physical theory provides us with some phenomena that appear to both "be" and "not be" at the same time: the wave-particle duality of photons, for example.

From page 59:

Firstly, then, this much is clearly true: that the phrase "to be" or "not to be" means something definite; so that not everything can be both in a given state and not in a given state.
"Not everything" - but my contention is that some things can (like Schrodinger's cat). So I don't want to include the law of the excluded middle as an axiom in my natural language processing system. My hypothesis is that my system contains Aristotle's. I will endeavor to show that my logic bot can handle most of Aristotle's examples in the following passages. ("> " precedes interaction with the bot. Blockquoted passages are Aristotle, the non-indented passages are my comments.)
Then, we would allow that "man" means one thing only - let it be "two-footed animal."
> man is a two-footed animal
What I mean when I say that something "means one thing" is this: if man is such and such, then for anything that is a man that is what being a man will be.
> Socrates is a man.
> a man is man
> is Socrates a two-footed animal?
Nor does it make any difference if someone says that a word means more than one thing, provided the meanings are limited in number; for each different account could be given a different name.
"Could be" - but we don't have to. I want to explore what happens if I don't. Natural language permits polysemy, relying on context to disambiguate between meanings. Inevitably, some ambiguity remains; my hypothesis is that it is that ambiguity that allows natural language to change and adapt to new conditions (for example, allowing multiple meanings permitted natural language to adapt the words "web" and "click" to the different environment created by computers and the internet).
I refer to cases where, for instance, one might say that "man" meant not one but many things, and that "two-footed animal" was the account of one of them, but that there were many others, though they were limited in number;
> man is "not one but many things"
> what is man?
man is a two-footed animal, not one but many things
then one could apply a particular name to each different account.
Again, one could, but I choose to follow natural language in allowing a single name to have multiple "accounts".
If, however, this were not the case, and one were to say that a word had an infinite number of meanings, then plainly there could be no account of anything; for to mean no single thing is to mean nothing;
I don't claim that a word has to be able to have infinite meanings, but I do think that the meanings of a word are expandable - words can add meanings.
I also think Aristotle is making use of a linguistic ambiguity to attempt a proof that infinite meanings means a word means nothing: "no single thing" can mean "nothing", it is true, but "no single thing" can also mean "many things".
and if words mean nothing, there is an end to discussion between people and, indeed, really to reflection with oneself. For it is not possible to think without thinking of some single thing; and if it is possible to think of this single thing, it must be given a single name.
I don't agree with any of this: I think it is possible to think of two things at once (puns for instance), and I don't think there is any compelling necessity that requires each single thing to have its own single name.
Let us assume then, as we said at the outset, that a name means something, and means just one thing.
I'm not going to make this assumption, and I'm going to submit the examples Aristotle gives in the following discussion to my logic bot, to show that it can handle the examples he gives without crashing or going into an infinite loop.
It is then not possible for "being a man" to mean the same as "not being a man",
> "being a man" means the same as "not being a man"
if "man" means not only something that can be attributed to only one thing, but means also just one thing itself.
I think what Aristotle is saying is: if "being a man" means "being a two-footed animal", then "being a man" cannot mean "not being a man" because its one meaning allotment has already been taken up. If that is what he's saying, since my bot is not subject to the "one meaning per word" rule, "being a man" can mean both "two-footed animal" and "not being a man":
> "being a man" means "being a two-footed animal"
> what does "being a man" mean?
not being a man, being a two-footed animal
(For we do not expect everything that can be attributed to one thing also to mean just one thing; if that were so, "musical," "white," and "man" would all mean one thing, so that all things would be one, since they would share the same definition.)
I'm not sure what he's getting at here. Something like if "man is musical" and "man is white", musical must mean white, and if man means two-footed animal, they all must mean two-footed animal. Again, my bot (and natural language) is under no such constraints.
It will not be possible, either, for a thing to be and not to be the same thing, unless one is talking ambiguously - as, for instance, if what we call "man" were to be called by others "not-man."
I want the bot to be able to deal with humans talking ambiguously.
> a two-footed animal is not-man
But our problem is not whether it is possible for the same thing to be and not to be man in name, but whether it is possible in fact.
I'm interested in the names; and modern physical theory seems to indicate that photons for example can be both a wave and a not-wave.
If "man" and "not-man" do not mean different things, plainly "not being a man" will also be the same as "being a man," so that "to be man" will be "to be not-man," since they will be one.
Again, I don't see any problem with allowing this:
> "being a man" is man
> "not being a man" is not-man
> not-man is "not being a man"
> what is man?
..., not being a man, ..., being a man
> what is not-man?
..., not being a man, ..., being a man
> is man not-man?
(That is what "being one" means, being like "cloak" and "garment," with one and the same account being given of both.)
Aristotle appears to be saying, if cloak and garment mean the same thing, they are the same thing. This is true in my system if the relation between cloak (or garment) and its definition is symmetric:
> cloak means an article of clothing
> garment means an article of clothing
> is cloak garment?
> an article of clothing is garment
> is cloak garment?
So that if these two are one, "being a man" and "being not-man" will mean one and the same thing. But in fact it has been shown that they mean different things.
> "not being a man" means "being not-man"
> is "being a man" "being not-man"?

If I want to make "being a man" and "being not-man" different things:
> why is "being a man" "being not-man"?
being a man is being not-man because: being a man is not being a man, and not being a man is being not-man
> forget "being a man" is "not being a man"
> is "being a man" "being not-man"?

In conclusion, I think Aristotle's axiom forbidding p and not-p is arbitrary and creates significant problems for any system attempting to deal with natural language. Modern physics provides evidence that reality may not conform to the law of the excluded middle, either.