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Can anybody please explain to me how "if $p$, then $q$" is logically equivalent to "$q$ unless ~$p$" ? My problem is with all truth possibilities of $p$ and $q$ except for when both are true. It seems in those situations they would not have the same logical meaning. For e.g: "if $p$, then $q$" is true if $p$ is false and $q$ is true. However, shouldn't this possible combination render "$q$ unless ~$p$" false? Feel free to use truth tables or whatever method that would help in explaining.

Kalernor
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2 Answers2

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Logicians tend to translate "$p$ unless $q$" as

$$ p \vee q $$

which is equivalent to both

$$ \sim p \to q $$

and

$$ \sim q \to p $$

You can get a reading of the English sentence "$p$, unless $q$" on which it's making a kind of conditional assertion. It's asserting that $p$, but only so long as $q$ isn't true. If $q$ turns out to be true, then an assertion of "$p$, unless $q$" simply falls silent on the question of whether or not $p$. If you're hearing this reading, then the English sentence "$p$ unless $q$" would just be ruling out the possibility that both $p$ and $q$ are false---and that's just the possibility that "$p \vee q$" rules out.

$$ \begin{array}{c c | c} p & q & p \vee q \\\hline T & T & T\\ T & F & T\\ F & T & T\\ F & F & F \end{array} $$

But, you can get another reading of the sentence "$p$, unless $q$", on which it's not merely saying that $p$ is true, given that $q$ is false. You can get a reading on which it's additionally saying that $p$ will be false if $q$ is true.

You're probably hearing that second reading of "$p$, unless $q$". In that case, you'd want to translate that into propositional logic with "$\sim p \equiv q$"; or, equivalently, "$p \equiv \sim q$"; or, equivalently, "$\sim(p \equiv q)$".

$$ \begin{array}{c c | c | c | c} p & q & \sim p \equiv q & p \equiv \sim q & \sim(p \equiv q) \\\hline T & T & F & F & F\\ T & F & T & T & T\\ F & T & T & T & T\\ F & F & F & F & F \end{array} $$

Which of these translations is correct isn't properly a question for logic to settle. It's a question for linguistics to settle. I generally explain this to my students and tell them that they should just get used to the fact that logicians tend to use "$p$, unless $q$" in the first way.

JDG22
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  • Pick up just about any introductory logic text which includes a section on translation and talks about "unless" and you'll see it translated with inclusive disjunction (or something else which is equivalent to inclusive disjunction).

    Moreover, the person asking the question was asking about the claim that "$p \to q$" is logically equivalent to "$q$ unless $\sim p$". But "$p \to q$" is equivalent to "$q \vee \sim p$", so the person asking the question was obviously dealing with a text or instructor who told them that "unless" is to be translated as inclusive disjunction.

     – JDG22 Feb 24 '16 at 02:53
  • here's what I found at the top of a google search (but you can consult any introductory text to find the same):

    https://en.wikibooks.org/wiki/Formal_Logic/Sentential_Logic/Translations#Unless

     – JDG22 Feb 24 '16 at 02:55
  • Sorry, I overstated things. "Unless" is rather ambiguous. Sometimes it means "if not", sometimes if means "if not and only if not". Because $p\lor q\equiv \neg q\to p$, "if not" can be rendered with $\lor$, which will be equivalent to $\oplus$ when $p,q$ are mutually exclusive. But sometimes the meaning of "p unless q" is $p\leftrightarrow \neg q$. – BrianO Feb 24 '16 at 03:37
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Natural languages are ambiguous, so if you want to check a logical equivalence you should "translate" your propositions to the language of some form of logic (propositional logic, predicate logic, ...). The title of your question is mistaken but then in the description you write it right. "If $P$ then $Q$" is equivalent to "$Q$ unless $\lnot P$" because to make the implication $P \implies Q$ be true when $Q$ is false it must be the case that $P$ is also false (which meand $\lnot P$ is true), otherwise the implication doesn't hold. $Q$ unless $\lnot P$ should be "translated" as $Q \lor \lnot P$

la flaca
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