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Harry Gensler. Introduction to Logic (2017 3 ed). p 169.

“Unless” is also equivalent to “if not”; so we also could use “(∼B ⊃ 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 $\equiv$ if not? OED Third Edition, June 2017. Screenshot.

A. adv. Only in conjunctional phrases followed by than or that.

  1. 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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    $\begingroup$ The OED link does not work; no one can see what's there unless someone fixes it... $\endgroup$ – Barry Cipra Sep 5 at 22:59
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    $\begingroup$ Why should an answer about modern usage reference only an obsolete usage which requires a "than" or "that" not often found in modern usage? Or are you looking for a history of its changing uses over time? $\endgroup$ – aschepler Sep 5 at 23:05
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    $\begingroup$ Indeed, @aschepler , the Oxford definition seems singularly unhelpful. $\endgroup$ – Lubin Sep 5 at 23:10
  • $\begingroup$ @BarryCipra Sorry. Can you access it now? $\endgroup$ – NNOX Apps Sep 6 at 0:48
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    $\begingroup$ 1. OED links are available to individual or institutional subscribers only. Please do not use links on Math.SE that are behind paywalls. 2. By OED convention, I believe any sense preceded by a dagger (the † symbol) is obsolete. It may therefore not be a suitable basis for determining a technical meaning. In this particular case, I think (I have OED2 at home, no subscription to OED3) the cited sense was no longer extant after maybe 1600 or so. 3. The usual meaning (both everyday and technical) of "A unless B" is that B is a necessary but possibly not sufficient condition for not A. $\endgroup$ – Brian Tung Sep 6 at 16:39
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This is probably too long for a comment.

First, “$p$ unless $q$” is explained in your first link as “if not $q$, then $p$”, which seems exactly right to me.

As for dictionary definitions, my American Heritage Dictionary says, “except on the condition that”, in other words, that $p$ will be the case at all times, the only possible exceptions occurring when $q$ holds.

To my mind, the two say the same thing.

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  • $\begingroup$ Thanks, but I'd like to work and start solely from Definition 1 above. I'm uncertain if your answer does? $\endgroup$ – NNOX Apps Sep 6 at 0:49
  • $\begingroup$ It absolutely does not, @Edge . That’s why I would toss the edition of the Oxford Dictionary that you’re using into the waste-bin. $\endgroup$ – Lubin Sep 6 at 3:38
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    $\begingroup$ No. That entry specifically denotes an obsolete use. The OED is not worthless, but this application is. $\endgroup$ – Brian Tung Sep 6 at 16:54

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