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Source: p 139, Introduction to Logic (2 ed, 2010) by Harry J. Gensler.

[1.] Translate “unless” as “or.” ...     [eg: A unless B =  B unless A = Either A or B].

[2.] “Unless” is also equivalent to “if not”...    [eg: A unless B =  A, if not B].

Though phrased differently, the above is asserted too in p 319, A Concise Introduction to Logic (12 Ed, 2014) by P J Hurley. However, both books don't dig deeper or explain 1 and 2.

So would someone please explain why unless can be substitued as in 1 and 2?

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  • 1
    What is your overall goal with questions like this? Are you interested in an explanation of the development of the meaning of words for its own sake, or do you have the impression that it will help you learn these words? If it's the later, this is not the most effective way to learn. You should instead look at the words in context, and practice hearing them, using them in your own speech, reading them, and using them in your writing. May 1, 2015 at 3:35
  • @sumelic I apologise if my question offends. In response to Are you interested in an explanation of the development of the meaning of words for its own sake, or do you have the impression that it will help you learn these words?, my answer is the former, though the latter sometimes helps. Does this clarify?
    – user5306
    May 1, 2015 at 3:38
  • Yes. I don't find your questions offensive, but they can be frustrating because it's hard to know what you will find "reasonable and intuitive". If the main reason for these questions is simply interest in how the meanings of words develop, I would recommend studying historical linguistics, in particular the topic of semantic drift, rather than going one-by-one through words. I feel that you would benefit from learning more about general ways and patterns in which a word's meaning can change. May 1, 2015 at 3:43
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    It's certainly not true that in general A unless B = B unless A. For instance, We leave Monday unless it rains. does not equal It rains unless we leave Monday. There are situations where that's true, provided there's a binary-valued predicate: He's alive unless he's dead = He's dead unless he's alive, though either sentence seems a fairly pointless thing to say.
    – jlawler
    May 1, 2015 at 4:31
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    This is about formalised logic, not English, and definitely not etymology!
    – curiousdannii
    May 1, 2015 at 5:17

3 Answers 3

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Start with 2.: A unless B = A, if not B
A, if not B = if not B then A
In general, if x then y = not x or y (for the "material implication" of logic)
So, if not B then A = not not B or A (from the above, with x as not B and y as A)
In general, not not x = x
So, not not B or A = B or A (substituting B for not not B)
In general, x or y = y or x
So, B or A = A or B

Thus, A unless B = A or B

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  • Logically, it's provable. To the extent that if p, then q represents the functor , Reduction of Implication states that (p ⊃ q) is equivalent to (¬p ⋁ q), so (¬q ⊃ p) is equivalent to (¬¬q ⋁ p), which reduces to (q ⋁ p) by Reduction of Double Negation, and that to (p ⋁ q) by Commutativity.
    – jlawler
    May 1, 2015 at 15:26
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    @jlawler, yes, John. That's sort of what I said, isn't it?
    – Greg Lee
    May 1, 2015 at 15:43
  • Yes, but you were speaking English. BTW, for those who speak logic, link at umich.edu/~jlawler/logicguide.pdf; I'd intended to put it in the comment above, but forgot.
    – jlawler
    May 1, 2015 at 15:50
  • +1. Thanks. But to clarify, you assumed that unless = if not* in your first step, right? Your answer proves only: A if not B = A or B.
    – user5306
    May 1, 2015 at 16:11
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    I said above that it's up to the judgement of English speakers to say. I'm an English speaker, and that's my intuitive judgement about these English expressions. It's as simple as that. It's an opinion. You're welcome to have a different opinion. It is this sort of fact that grammarians try to describe and explain. But the facts come first, before the theories.
    – Greg Lee
    May 4, 2015 at 4:02
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When glossing a logical proposition with English words, we are not really using the English language system; it's just a way to represent the unambiguous mathematical concepts in a mnemonic and easy-to-pronounce manner. For this reason, mathematicians and logicians like to present specialized definitions of the words they use. That is what is being done in this passage. You'll only confuse yourself if you try to use the ordinary meaning of English words in this context.

I find this a rather poor definition for an introductory text; I'm not a mathematician, so it may in fact not be unambiguous, but I find myself unsure if the author means for this to represent exclusive disjunction or inclusive disjunction. It seems to me to technically be a definition of inclusive disjunction. Usually in logic, the phrase "Either A or B" is taken to mean that exactly one of the following 3 propositions is true:

  1. A is true and B is true
  2. A is true and B is false
  3. A is false and B is true

However, in ordinary English, "A, unless B" seems to additionally imply "if B is true, A is not true". Clearly the author means instead for "A, unless B" to represent "if B is true, A may or may not be true", but that is an extremely unfortunate definition of the word "unless" in my opinion.

I'm afraid I can't say much about the linguistic aspect of this question.

Evidence from the 2002 edition: In the 2002 edition, page 56, accessed through Google Books, Gensler uses the notation "D ∨ B" as equivalent to "D unless B". The ∨ sign is used for inclusive disjunction, so that does indeed appear to be what he means with this supposed translation of logic into ordinary English words. In other words, by "A unless B" he really means "Either A, or B, or both".

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You need to bear in mind that logic textbooks do not purport to provide translations of natural language into logical symbolism. Instead, they provide conventialized translations of standard logical symbols and expressions of English. It is pretty standard that in Philosophy 150, about half the time students screw up the exercise of matching natural language sentences to logical formalism, because they don't heed the strict conventions in the discipline of logic regarding the meaning of "and" (for example).

Philosophy.stackexchange.com is a good venue for asking questions about the conduct of logic as a discipline. IMO, questions about the conventions of philosophers and logicians is off topic here, and you would be better served focusing, here, on questions about natural language use. Do you want to know about the meaning of "unless"? Then skip the distractor of quirky usage in logic textbooks.