[Source:] pp 114-115, Symbolic Logic: A First Course (NOT 4 ed, 2011), by Gary Hardegree.
For want of brevity, I rewrite 'is on duty' as 'remains'.

1. the pool may NOT be used unless a lifeguard remains

2. Following the dictionary definition, this is equivalent to:

3. the pool may NOT be used except when a lifeguard remains

... = 4. the pool may NOT be used if a lifeguard is NOT on duty,
and the pool may be used if a lifeguard remains.

5. which, as noted earlier, is equivalent to the following biconditional,

6 . the pool may be used if and only if a lifeguard remains

By comparing 6 with the original statement 1, we can discern ...

7. the strong sense of ‘unless’   is equivalent to   ‘if and only if not’

How does comparing 6 with ... 1 mean 7? Please explain and show all steps and thought processes? This book appeared to have omitted some key steps.

  • 1
    "The pool may be used if and only if a lifeguard remains" = "The pool may not be used if and only if not a lifeguard remains". Now compare directly to 1 and note we can substitute "unless" for "if and only if not". QED.
    – commando
    May 4 '15 at 0:11

"Unless" specifies a necessary condition. The presence of a lifeguard is necessary to allow using the pool. It is not a sufficient condition: For example, if the lifeguard is present but tells everyone to stay out of the pool, then the use of the pool is not allowed. This is logically the same as "except", but very different from "if and only if".

  • 1
    I concur. The author of the example in the text book has made the common error of confusing "unless" with "unless and only unless". He appears to be calling this "the strong sense of unless", but it is better to think of this as an example of what linguists call conditional perfection.
    – Bumble
    May 19 '16 at 21:56

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