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Colin Fine answered on

Unless" does not equal "or" 'directly and intuitively'.

This contradicts the textbooks beneath. Who is correct? Let me ask this here, as I prefer answers from logicians.

Usually, "P unless Q" is "symbolized as P ∨ Q. See :

Lande N.P. Classical logic and its rabbit-holes: A first course (2013), pages 55-7.

The wedge symbol is used to translate “or” and “unless.” A previous chapter explained that “unless” is equivalent in meaning to “if not.” This equivalence holds in propositional logic as well, but in propositional logic it is usually simpler to equate “unless” with “or.” For example, the statement “You won’t graduate unless you pass freshman English” is equivalent to “Either you pass freshman English or you won’t graduate” and also to “If you don’t pass freshman English, then you won’t graduate.” As the next section demonstrates, the wedge symbol has the meaning of “and/or”—that is, “or” in the inclusive sense. Although “or” and “unless” are sometimes used in an exclusive sense, the wedge is usually used to translate them as well.

Hurley P. A Concise Introduction to Logic (13 edn, 2018), page 319.

Translate “unless” as “or.”

Gensler H. Introduction to Logic (3 edn 2017), 132.

In addition, the word “unless” sometimes functions like the word “or.” For example, the statement “You can’t go to the party unless you clean your room,” can be rewritten as “Either you clean your room or you can’t go to the party.”

Baronett S. Logic (5 edn 2022), 318

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  • The first text by Lande uses the right arrow notation which is usually referred to as 'implies', ie, "if I do not have the surgery implies that I will die", and in reverse, "if I die implies that I did not have the surgery". I'm not sure if we can entirely link this to being the same as the logical OR operation, which is represented by the v symbol in classical logic notation.
    – DJ_Davey
    Jan 31 at 20:25
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    No. The more direct symbolization of "P unless Q" is ¬ Q → P, which is logically equivalent to P ∨ Q. This is because the material conditional is defined to satisfy A → B ≡ ¬ A ∨ B. Colloquial connectives are loaded with linguistic functions aside from logical ones, so they rarely match each other 'directly and intuitively'. Material conditional and inclusive or especially. "But", "but for", "unless", "except" also convey contrastive emphasis that elementary logic ignores by design.
    – Conifold
    Jan 31 at 21:21
  • @Conifold good answer, you should post it as one. Jan 31 at 23:36
  • Colin's answer was downvoted, it seems odd to take him as an equivalent authority to logic textbooks. Feb 1 at 20:03

4 Answers 4

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In introductory logic textbooks, it is customary to understand logical connectives purely in terms of their truth conditions. When you do this, 'unless' turns out to have the same truth conditions as 'inclusive or' in classical logic. This approach has the virtue of simplicity, but it only works up to a point. It is misleading to say that 'unless' means the same thing as 'or'. Meanings are much more fine-grained than truth conditions.

To see why the truth conditions of 'unless' and 'or' agree, consider some examples:

  1. Unless you hurry, you'll miss the train. If you don't hurry, you'll miss the train. You hurry or you miss the train or both. What is ruled out in each case is you not hurrying and you not missing.

  2. Swimming is forbidden unless a lifeguard is present. If there is no lifeguard present, swimming is forbidden. There is a lifeguard or no swimming or both. What is ruled out is swimming and no lifeguard.

  3. Unless you hand over your wallet, I'll shoot you. If you don't hand over your wallet, I'll shoot you. Hand over your wallet, or I'll shoot you, or both. What is ruled out is you not handing over your wallet and you not getting shot.

To amplify the point that meanings go beyond mere truth conditions, consider an important difference between 'unless' and 'or'. 'Or' is commutative, but 'unless' typically is not.

  1. Unless it rains, I'll go for a walk tomorrow. This is not the same as: Unless I go for a walk, it rains tomorrow.

  2. Unless you hand over your wallet, I'll shoot you. This is not the same as: Unless I shoot you, you'll hand over your wallet.

'Unless' is a kind of conditional, and like other conditionals, it is not straightforwardly reducible to a truth function, except in simple cases. Usually with a conditional, the antecedent is logically, epistemologically or temporally prior to the consequent. This is why 'unless' is not commutative in ordinary usage.

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There are two translations, linked by classical logic. Of your citations, Hurley is least incorrect; "unless" can be translated as a disjunction ("or"), or as an implication ("if" & "then"). For more details, see this Maths SE discussion. This would also apply to linear logic.

Intuitionistically, we have to pick one translation, because there is no longer an equivalence between disjunction and implication; usually, "unless" is treated as implication, and would be translated as such.

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Nope.

'Unless' as 'or' and 'but' as 'and' are two cases that show that natural language is far richer in meaning than logical relationships. Meaning is a complicated topic, but suffice it to say, that truth-conditional semantics is only one theory of meaning. Here are some other theories of semantics. What's also important to understand is that moving from natural language to formalized semantics means one often picks and chooses what sort of meaning that the natural language conveys gets carried over to the model of meaning expressed as a formal system.

Consider that natural language can be modeled by both propositional logic and predicate logic. Is one model more correct than the other? It all depends on the requirements of the formal system being built. In instances, prefer to use propositional logic and others predicate logic, and in others still higher-order logic. In each case, we leave some meaning behind.

Thus 'unless' and 'or' have the same truth-conditional meaning, but unless obviously carries some additional meaning. Consider this YT video that relates the difference between 'if' and 'unless'. 'If' is used to convey the notion of conditionality. Unless is an exclusion from a generality. And the use of 'or' simply conveys options or choice. To build on her example:

If I meet you on the streets of Chicago, either you and I can or cannot talk in English. This conveys a condition of ability and possibility. I speak English, but if you only speak Spanish, then we cannot speak English together. This is because it is a condition or requirement we both speak English confers ability. We might not be able to speak at all with each other unless we can find a translator to speak through. Now, with 'unless', this conveys a disjunct that is an exception to the truth set out in the former claim. Notice too the copula is used for predication. We can construct a formal model of this natural language using propositional, predicate, or modal logic as we wish.

The moral of the story is that formal systems of logic are impoverished compared to far richer formalisms that have a broader semantic scope. Consider Montague grammar. There are even richer grammars. One example is HPSG. And formal semantic systems that combine mathematical logic and phrase grammars might even include formalisms for discourse structures, as in discourse representation theory (SEP), which is an older example of dynamic semantics.

Therefore, in translating English into a formal logic, we might leave out various layers of meaning that the natural language contains. Propositional logic deals in implication of entire propositions. First-order logic deals with predicates. Modal logic deals with possibility and certainty. Thus, sometimes we simplify the meaning of a word to fit the framework of a logic by leaving out some of the meaning.

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I would say the theory is still sound in a literal sense. I am educated in computer science and would say that logic of unless is equivalent to the computer science XOR (eXclusive OR) which says that a logical state is either one or the other, with the additional exclusion of it being both at the same time. So we could implicitly deduce that 'unless' = 'or'. Additionally, any if..then rules are simply vehicles for arriving at our result/calculation. Any if..not rule we could compare to the logical NAND function, and A XOR B = (A NAND (A NAND B)) NAND (B NAND (A NAND B)), which is enough to fry my brain! This is just my view on it and there is just cause for debate here.

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