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I am trouble with the following question:

"The connective “unless” can be ambiguous, and this exercise will pinpoint the ambiguity. We awake at dawn, and we are told We will have a picnic today unless it is raining at 10 A.M. Let $P$ be "We will have a picnic today" and Q be "it is raining at 10 A.M." $P$u $Q$ denote “P unless Q.” (This is not a standard notation.) Complete as much of a truth table as possible for PuQ, and discuss any ambiguous lines.

My attempt $$\begin{array} {|c|} \hline P & Q & PuQ& \\ \hline F & T & T& \\ \hline T & F & T \\ \hline T & T & F \\ \hline \end{array}$$

Mathman
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  • I can't even understand what it means for a connective to be ambiguous. – Git Gud May 29 '14 at 15:55
  • @GitGud I think you can think of a connective as ambiguous if you can't know what truth value it will take on. – Doug Spoonwood May 29 '14 at 15:56
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    @GitGud - the natural language is ambiguous. Thus, we have some "fuzzyness" in translating "unless" with a truth-functional connective which, "by construction", cannot be ambiguous. – Mauro ALLEGRANZA May 29 '14 at 16:15
  • @MauroALLEGRANZA I see, they meant the natural language connective, not the logical one. – Git Gud May 29 '14 at 16:24

4 Answers4

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See the discussion in this post and the extract from Stephen Cole Kleene, Mathematical logic (1967 - Dover ed 2002).

According to his proposal [page 64] :

$A \lor B$ is $A$ unless $B$ [usually] and is $A$ except when $B$ [usually].

An "informal" argument in support of this "translation" is the following.

We can rewrite our "P unless Q" :

"We will have a picnic today" unless "it is raining at 10 A.M."

as :

if "it is not raining at 10 A.M.", then "we will have a picnic today".

But this is "if not Q, then P", which in the truth-functional model is equivalent to :

$Q \lor P$.

But there is an ambiguity in "translating" from the natural language into the truth-functional model, as referred in the question.

If we decide for XOR, we have to leave the "equivalence" of "P unless Q" with "if not Q, then P".

I agree with StumpyLeg's answer; three case are "clear-cut". The doubt is with the $T-T$ case: here lies the choice between OR and XOR.

Community
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Mauro ALLEGRANZA
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  • Yes, I agree. Indeed, for some reason, "if not Q then P" sounds more intuitive than does $Q\lor P$ but as you note, they are truth-functionally equivalent. – amWhy May 29 '14 at 16:14
  • @amWhy - YES. I think that "we logician" (sorry for this ...) are accustomed to "blame" $\rightarrow$, but in fact we have some problem also with $\lor$ (see intuitionsitic logic). Of course the problem is "trivial": it is the problem of mapping a "fuzzy" world (natural language with nuances) into a "bi-valent" model. But in many cases it works ! – Mauro ALLEGRANZA May 29 '14 at 16:17
  • It is rather interesting that we use it as XOR in most cases. So it is not that obvious. – Mathman May 29 '14 at 16:45
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$P$, $Q$ both false is clear-cut: $P u Q$ is false. (If there's no rain and no picnic, the announcement was erroneous.) If there's any ambiguity, it's the T/T case (last line in the OP's table). It comes down to whether "unless" should be read as OR (then the statement is true) or as XOR (then it's false). That seems to be the intent of the example, since rain would normally nix a picnic but the announcement doesn't explicitly commit to cancellation in case of rain.

StumpyLeg
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  • So how would one know what is explicit "enough"? – Mathman May 29 '14 at 16:33
  • One can't always expect a hard-and-fast rule for how to interpret natural language and translate it into a formalism like truth-functional logic. (You can always make up hard-and-fast rules, but that's not what you're asking.) However, if someone were to say, "We'll have a picnic unless it's raining at 10 A.M.--in which case we may or may not, depending on how bad it is," then that's OR. Whereas "We'll have a picnic unless it's raining at 10 A.M., in which case we'll stay inside" means XOR. – StumpyLeg May 29 '14 at 16:51
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One might be tempted to say that "$P$ unless $Q$" is the same as "if not $Q$, then $P$". In fact this is also the semantic used e.g. in PERL, which seems to be the only programming language that allows if $condition then functioncall(); to be written as functioncall() unless !$condition;.

But in natural language the use may vary. For example, "We'll go to the movies unless it rains" may be interpreted differenetly than "If it doesn't rain, we'll go to the movies". The former seems to suggest much more strongly that also "If it rains then we do not go to the movies" holds.

Hagen von Eitzen
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unless = if not

P unless Q = P if not Q

I think?

BCLC
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