Exhaustive and exclusive alternatives

Definitions

Here we are taking different alternatives to be described by statements.

• A set of statements is exhaustive when at least one of them is true in any logically possible situation. ("exhaustive" because they do not leave out any situation)
• A set of statements is exclusive when there is no logically possible situation in which more than one of them is true. ("exclusive" because the truth of one excludes the truth of the others)
• In other words, if a set of statements is both exhaustive and exclusive, then in any logically possible situation, exactly one of them is true.

Examples

• Neither exhaustive nor exclusive - a=5, a>4
• Exhaustive but not exclusive - a>4, a<10
• Exclusive but not exhaustive - a>4, a=1
• Exclusive and exhaustive - a>0, a=0, a<0

Exercises

• "The market will go up. The market will go down." Why are they not exhaustive alternatives?
• "P and Q", "Neither P nor Q" - Are they exclusive? exhaustive?
• "John is happy", "John is sad" - Are they exclusive? exhaustive?