Abstract
The analyses of unless and until lie at the intersection of logic and linguistics. They crop up in papers about tense connectives [1], quantification [15], anaphora [7], polarity and duality [17, 18] and in classical theorems of tense logic [10]. Unless and until are morphologically similar, and in some contexts, they even appear to be ‘interchangeable’. In this paper we give compositional analyses showing the interrelatedness of these two connectives. In addition, we use this case study to draw some broader methodological points. The locus classicus on the logic of unless is Quine’s Elementary Logic [20] where he sets forth three methodological dogmas. We dub these Quine’s Three Dogmas of Linguistic Negativism and argue that these three dogmas not only give a misleading account of the interplay between logic and linguistics but that rejecting them leads to discovering a unified compositional analysis.
Until and unless you discover that money is the root of all good, you ask for your own destruction.
—Ayn Rand, Atlas Shrugged
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Notes
- 1.
Here and throughout the paper we use colored fonts to make our analyses more perspicuous. Our convention is that occurrences of unless and if are blue, occurrences of quantifiers and later occurrences of until and temporal particles or operators are green, negative elements are red, and other colors such as orange and purple highlight other features.
- 2.
Although they appear interchangeable, there are differences. Unless supports inferences of uncertainty whereas until need not. If I say, “I’ll stay unless you have a replacement”, this often implies that I’m uncertain whether you have a replacement or not. However, if I say, “I won’t leave until you have a replacement” it might be the case that I know you will have a replacement in the near future and that I’m postponing my leaving until that happens.
- 3.
Similar linguistic phenomena occur in other languages (e.g., Russian):
The translation for unless in Russian is literally if not, drawing a suggestive cross-linguistic parallel.
- 4.
The terminology of ‘negativism’ is adapted from Hao Wang’s [22] characterization of Quine’s philosophical views as logical negativism. Not only are Quine’s theses largely negative—the rejection of analyticity, the indeterminacy of translation, the view that modal logic is ‘conceived in sin’—but also they have been negative in their influence. “... [Both] Carnap and Quine have inadequate conceptions of logic and apply logic in philosophy in misleading manners, which do not do justice to logic in its more developed state and, through the conspicuity of their work, give to philosophers the wrong ideas about logic.” [22]
- 5.
We are grateful for a reviewer for calling our attention to the myth of vel and aut at http://plato.stanford.edu/entries/disjunction/#MytVelAut. Commutativity holds for the logician’s or in either the inclusive or exclusive senses.
- 6.
Section 3.7 expands on the notion that unless is not simply replaceable by if not.
- 7.
This restriction to “simple chains of natural equivalences” is to avoid unwanted implications, for instance, that “all tautologies say that same thing.”
- 8.
It turns out that this observation of ours was made independently and, more insightfully, by von Fintel [4].
- 9.
Within the realm of mathematics the relevant cases are intuitively ‘universal’ and ‘timeless’, and so the logician’s classical transformations preserve equivalence of all of the following: (i) If a natural number is evenly divisible by 2, then it is even; (ii) If a natural number is not even, then it is not evenly divisible by 2; (iii) Unless a natural number is even, it is not evenly divisible by 2; (iv) A natural number is evenly divisible by 2 only if it is even.
- 10.
Our analysis may even be useful to explain some occurrences of a non-commuting or. Consider the famous declaration by the American patriot Patrick Henry: “Give me liberty or give me death” (we are indebted to Gillon [7] for this example). Whether or not this declaration was intended as inclusive or exclusive (perhaps Henry invites the liberty of death if he can’t have the liberty in life), Henry’s defiant disjunction is not intuitively equivalent to: “Give me death or give me liberty”. Our analysis suggests that non-commuting occurrences of or can be paraphrased by a fronted unless:
Then the exceptional cases of the non-commutativity of or can be explained by the non-commutativity of the fronted unless which, in turn, is explained by the failure of contraposition: the cases in which Patrick Henry is not granted his liberty are different from the cases in which he is not given death.
- 11.
PPIs, in contrast to NPIs, appear in affirmative or positive contexts; they do not require negation in order to be licensed.
- 12.
Important differences remain. Löbner [16] notes that propositions connected by unless are typically complete or perfective states of affairs, whereas until typically connects incomplete or imperfective processes or phases of states of affairs. Furthermore, he notes that the complement clause that accompanies unless, unlike that of until, typically has a negative polarity [16]. These observations, in our view, should not deter, but be invitations for deeper, compositional analyses.
- 13.
There is nothing grammatically wrong with the following conjoined and disjoined unless clauses: (i) Rosemary can’t sleep unless she has her pillow and unless it is quiet; (ii) Rosemary can’t sleep unless she has her pillow or unless it is quiet; (iii) Rosemary can’t sleep if she doesn’t have her pillow and (if) it isn’t completely quiet; (iv) Rosemary can’t sleep if she doesn’t have her pillow or (if) it isn’t completely quiet.
Note the optionality of if in the second clause in examples (iii) and (iv). Here, the equivalences among (i) and (iii) and (ii) and (iv) can be explained by our analysis and the following pair of propositional theorems:
T50 (P \(\rightarrow \) Q) \(\wedge \) (P \(\rightarrow \) R) \(\leftrightarrow \) (P \(\vee \) Q \(\rightarrow \) R); T60 (P \(\rightarrow \) Q) \(\vee \) (P \(\rightarrow \) R) \(\leftrightarrow \) (P \(\wedge \) Q \(\rightarrow \) R).
- 14.
- 15.
According to our analysis of the punctual sense of until, we do not need to stipulate ad hoc that after Prince Charming kissed her, Sleeping Beauty woke up because this will be a logical consequence of analyzing until in terms of an exclusive unless:
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Acknowledgements
We wish to thank Sebastian Löbner, Hans Kamp, Revantha Ramanayake, Richard Larson, Thomas Graf, Jiwon Yun, Chris Collins, Paola Cepeda and John Foulks for helpful discussion. We also thank Alan Bale and Luis Alonso-Ovalle for pointing out some important resources, and we especially thank the reviewers of the TbiLLC proceedings for constructive critique. We express our gratitude for the hospitality of Rusiko Asatiani, Matthias Baaz, Nick Bezhanishvili, David Gabelaia and the organizers of the Tenth International Tbilisi Symposium on Language, Logic and Computation and the Georgian Academy of Sciences and Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam. Any shortcomings or errors are our own responsibility.
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Mar, G., Manyakina, Y., Caffary, A. (2015). Unless and Until: A Compositional Analysis. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds) Logic, Language, and Computation. TbiLLC 2013. Lecture Notes in Computer Science(), vol 8984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46906-4_12
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