Why does “unless” mean “if not”?
Harry Gensler. Introduction to Logic (2017 3 ed). p 169.
“Unless” is also equivalent to “if not”; so we also could use “$({\sim}B \supset D)$ (“If you don’t breathe, then you’ll die”).”
Nicholas JJ Smith, Logic: The Laws of Truth (2012). p 115.
The statement “P unless Q” means that if Q is not true, P is true—so we translate it as $¬ , Q→P$.
Using solely the original meaning of "unless" below, please expound why? How does definition 1 below ≡ if not? I know that definition 1 is obsolete, but I'm interested in the etymology. OED Third Edition, June 2017. Screenshot.
†A. adv. Only in conjunctional phrases followed by than or that.
- Forming a conjunctional phrase introducing a case in which an exception to a preceding negative statement (expressed or implied) will or may exist: (not) on a less or lower condition, requirement, etc., than (what is specified). Obsolete.
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The quotes from the books are exactly the kind of ridiculously naive and over-simplified linguistics in introductions to logic that I criticize in this blog article. Particularly for everyday natural language expressions, you will find no shortage of ways where this "translation" provides only a clumsy interpretation, is inadequate, is outright wrong, or is just nonsensical. This also has nothing to do with how mathematicians/logicians use formal logic.
Even then, your question is strange. There's no reason to expect every definition of "unless" to be even clumsily equivalent to this "translation". There's also no reason to expect the etymology of "unless" or this obsolete form have anything at all to do with this logical interpretation. This is almost certainly not the form of "unless" the authors of these books were intending, especially as the given example doesn't use "than" or "that".
Either you're asking how the natural language meaning of "if not" relates to this obsolete form of "unless", in which case this is not a mathematics question; or you're unreasonably asking how this "translation" corresponds to a meaning of "unless" that is clearly not the meaning the authors intended. It would be like asking why "not" is "translated" to logical negation but under the condition that "not" is defined as "nothing" because "not" (allegedly) derives from "nought" which means (among other definitions) "nothing".
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According to the definition you quote, “unless” gives an exception to a preceding negative statement. An exception to a statement is a condition in which the statement does not apply. Therefore the statement does apply only if the condition is not fulfilled (because otherwise the exception applies, and therefore the statement does not). Therefore, as far as logic is concerned, “unless” translates into “if not”.
In describing an exception, the “unless” phrase also suggests that the condition is generally not fulfilled, but that part is irrelevant from a logical point of view (the logic only cares about the truth/falsehood of statements, not about the probability that they will be found to be true).
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