Mutual exclusion

[Paul Howard]

I’m confused about the following paragraph from “The Construction of Semantic Space.” Since I was a child, the definition in my mind for mutual exclusion is what you’re saying is non-contradiction. Did I learn it wrong? In my thinking, the heads-or-tails feature of flipping a coin has two mutually exclusive possibilities of heads or tails, because one result disqualifies the other. If we add the remote possibility of the coin balancing on the edge, the heads and tails remain mutually exclusive possibilities. That’s also different from what you’re saying in the paragraph below.

“Assume that there are two contradictory propositions P and not-P. Aristotelian logic says that only one of P or not-P can be true, implying that {both P and not-P} cannot be true. This is called the principles of Non-Contradiction (NE). Aristotelian logic also says that at least one of P or not-P must be true, implying that {neither P nor not-P} cannot be true. This is called the principle of Mutual Exclusion (ME). If we allow “both” (NC) and “neither” (ME) we are breaking the fundamental principles of Aristotelian logic.”

https://blog.shabda.co/2017/03/14/construction-semantic-space/

Thanks! Hare Kṛṣṇa

[Ashish Dalela]

Heads or tails is not the only possibility.

1. A coin toss could lead to a coin standing sideways, which would be neither heads nor tails. Logic eliminates that possibility: A coin toss must be either head or tails.

2. A coin could be a small box, such that when you toss it, the box opens and the outer side of the bottom and top parts turn up. Logic eliminates that too: A coin toss cannot show both heads and tails (the outer sides of the box being seen).

3. A coin could be a small box, such that when you toss it, the box opens and the inner side of the bottom and top parts turn up. Logic eliminates that too: A coin toss cannot be neither heads nor tails (the inner sides of the box being seen).

4. After a toss, the claim that “I got a coin” is a logically invalid answer. The answer must be either “I got a head” or “I got a tail”. Hence, “both” is eliminated.

5. The coin cannot disappear into thin air upon being tossed. That is also a logically invalid answer. The answer cannot be “I got neither head nor tail”.

So what is logic? It is the elimination of conceivable scenarios. We restrict the universe of conceivability and we call it impossible, when it is possible.

So, heads OR tails depends on two things — (a) it is not both heads and tails, and (b) it is not neither heads nor tails. Yes it is true that this is commonly called mutual exclusion. But it combines two claims: (a) a law of non-contradiction, and (b) a law of excluded middle. Law of non-contradiction says: Both A and not-A are not true. The law of excluded middle says: Neither A nor not-A are not true.

I shortened the law of excluded middle into “mutual exclusion”, because it seems to me that most people haven’t heard of the “law of excluded middle” but they have heard of mutual exclusion. I always clarify that this means the law of excluded middle, by defining “mutual exclusion” as the “law of excluded middle”.

The easier formulation is — the categories “neither” and “both” are eliminated. This means many things. A coin cannot disappear into thin air if tossed. A coin cannot stand sideways if tossed. A coin cannot open up like a box such that both outer faces turn up. A coin cannot open up like a box such that both inner faces turn up. After the coin is tossed you cannot say “I got a coin” instead of “head” or “tail”.

Logic is a limitation on what you can say and how the world must be conceived.

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