Abstract
Higginbotham (1986) argues that conditionals embedded under quantifiers (as in ‘no student will succeed if they goof off') constitute a counterexample to the thesis that natural language is semantically compositional. More recently, Higginbotham (2003) and von Fintel and Iatridou (2002) have suggested that compositionality can be upheld, but only if we assume the validity of the principle of Conditional Excluded Middle. I argue that these authors' proposals deliver unsatisfactory results for conditionals that, at least intuitively, do not appear to obey Conditional Excluded Middle. Further, there is no natural way to extend their accounts to conditionals containing ‘unless'. I propose instead an account that takes both ‘if' and ‘unless' statements to restrict the quantifiers in whose scope they occur, while also contributing a covert modal element to the semantics. In providing this account, I also offer a semantics for unquantified statements containing ‘unless'.
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Notes
- 1.
No student will succeed if they goof off is equivalent to: for every student, it’s false that he will succeed if they goof off, which in turn is equivalent to: for every student, he will goof off and he won’t succeed. Similarly, No student will succeed unless he works hard is equivalent to: for every student, it’s false that he will succeed unless he works hard, which, on the assumption that “unless” means or, is equivalent to: for every student, he will not succeed and he will not work hard. Here and for the rest of the paper I will make occasional reference to the truth functional equivalence of ‘no x (A)’ and ‘every x (not A)’ when both quantifiers have wide scope over the sentence, as do Higginbotham (2003) and von Fintel and Iatridou (2002). This is not intended as a claim about the semantics of ‘no’, nor as a claim that the two constructions are everywhere intersubstitutable, but merely as the observation that they are truth functionally equivalent when they have wide scope over the sentence in which they occur.
- 2.
Higginbotham (2003) and von Fintel and Iatridou (2002) discuss counterexamples of this nature, though they use them to object to ‘restrictive analyses’, which I will consider below. I am indebted to them for the structure of the counterexamples presented in this section of the paper.
- 3.
One might deny that (7) and (8) really are false, and claim instead, for example, that they are simply indeterminate, or lack a truth value. Certainly the defender of CEM as a general principle should argue for some such claim. I will not discuss such a possible defense here, but rather the discussion will proceed on the highly intuitive assumption that this is a genuine counterexample to CEM. It is worth noting, though, that it is far easier to convince oneself that (7) and (8) are indeterminate, than it is to convince oneself that that their quantified counterparts (9) and (10) are:
-
(9)
No fair coin will come up heads if flipped.
-
(10)
Every fair coin will come heads if flipped.
(9) and (10) strike most people as quite clearly false. Thus even if one is inclined to reject (7) and (8) as counterexamples to CEM on the grounds that they are indeterminate rather than false, one still needs an explanation of why (9) and (10) seem quite clearly false and not at all indeterminate. Any natural extension of the Simple Solution to cases of indeterminacy would predict that the quantified statements should be indeterminate if their embedded conditionals are indeterminate.
-
(9)
- 4.
I am indebted to Jim Higginbotham and David Chalmers for pointing this out to me.
- 5.
A quantificational adverb is an adverb such as “always”, “sometimes”, “often”, “never”, and so on. Lewis (1975) argued that these adverbs quantify over cases or situations. Thus, for example, the sentence “John always wins” is to be analyzed to mean that all relevant situations involving John are ones in which he wins.
- 6.
I.e. parts of possible worlds; see Kratzer (1989). In our discussion, nothing will hang on the use of situations rather than worlds. (An account that uses situations rather than worlds is useful in dealing with so-called ‘donkey’ sentences, such as “if a farmer owns a donkey, he beats it” (Heim, 1990). We will not be concerned with such sentences here.)
- 7.
There is a fair amount of contextual variability associated with the restricting nominal “student” here. I have been eliding the details of this restriction, other than including a parenthetical ‘relevant’ in my representation of the logical form of these statements. There is far more that needs to be said here. In particular, it seems that some contextual restrictions allow the extension of the restricted nominal to change across the possible situations, while others do not. For example, if I say “every student will succeed if they work hard” with my introductory logic class in mind, there is a reading of the sentence on which it applies to any students who might possibly take my class. The utterance would then be a commentary on how I run my course. On this reading, the statement is false if the likes of Bill is even a possible member of my class. There is another reading of the sentence, though, on which it only applies to the students that have actually enrolled in my class, and thus understood is a commentary on the intellectual abilities of these students. On this reading, it does not matter whether Bill might have enrolled – that he has not in fact enrolled is enough to discount him from the evaluation of the statement. We should, I think, understand this variability as part of the general phenomenon of contextual variability in nominals – the property picked out by “is a student” might be such that its extension does not vary across the relevant possible situations, or it might be less rigid. (We could also locate difference between the readings in the set of relevant possible worlds we are considering. The proposal presented here is neutral between the two implementations, however, I am inclined to locate the restriction in the restricted nominal.) It should be noted, though, that it is less clear how these two readings would be generated, if we understood the statement to be quantifying over actual individual students, and attributing conditional properties to them, as we would under the Simple Solution. Unless we take the quantificational NP to range over possible individuals, it may be hard to avoid the consequence that the only available readings of the statement should be ones that pertain to the students that are, in fact, members of my class.
- 8.
Treating ‘unless’ as meaning ‘if … not’ is the most obvious way to fill out the claim that Meadow really does satisfy the relevant ‘unless’-statement: It’s true that Meadow will succeed if she doesn’t work hard. Higginbotham (2003) proposes that we handle ‘unless’ in this manner, and claims that a compositional treatment of quantified ‘unless’-statements is possible so long as ‘unless’ is assimilated to ‘if … not”. (Higginbotham provides few details, so it is not clear whether he proposes this to deal with situations such as Meadow’s, or for some other reason.) Besides a general desire not to simply dismiss as pragmatic any phenomenon that threatens semantic simplicity, there are other considerations that weigh against treating ‘unless’ as ‘if … not’. Geis (1973) produces a battery of reasons not to equate ‘unless’ with ‘if …not’, and I refer my reader to his excellent article for more detailed discussion than I can provide here.
Geis notes that ‘unless’ and ‘if … not’ behave differently with respect to the possibility of coordinate structures. There is no obstacle to conjoining clauses containing ‘if … not’, but we cannot do the same with clauses containing ‘unless’. Compare, for example:
John will succeed if he doesn’t goof off and if he doesn’t sleep through the final.
*John will succeed unless they goof off and unless he sleeps through the final.
‘Unless’ and ‘if … not’ also interact differently with negative polarity items. Naturally, negative polarity items can occur in the scope of ‘if … not’. They cannot, however, occur in the scope of ‘unless’:
John won’t succeed if he doesn’t ever attend class.
*John won’t succeed unless he ever attends class.
As a final point against the identification of ‘if … not’ and ‘unless’, we should note that clauses containing ‘if … not’ can be modified by ‘only’, ‘even’, ‘except’, while clauses containing ‘unless cannot:
John will succeed only if he doesn’t goof off.
*John will succeed only unless they goof off.
John will succeed even if he doesn’t work hard.
*John will succeed even unless he works hard.
John will succeed except if he doesn’t work hard.
*John will succeed except unless he works hard.
I will take these considerations and others in Geis (1973) to tell strongly against the identification of ‘unless’ with ‘if … not’ that Higginbotham (2003) suggests, and so this particular means of deriving the falsity of “Meadow will succeed unless she works hard” is untenable. Perhaps other means might be proposed, but I do not know of any other such proposals.
- 9.
I am assuming here that ‘usually’ can be understood as ‘most’, and so am setting aside any additional normative or otherwise modal import ‘usually’ may possess; nothing will hang on this simplifying assumption.
- 10.
The formulation of von Fintel’s uniqueness clause needs to be amended in order for these to be strictly equivalent, but it is a minor adjustment, and is independently motivated. As it stands, von Fintel has the following as his uniqueness clause:
∀S (Q [C – S] [M] → R ⊆ S)
However, the clause, as it stands, is violated if there are ‘irrelevant’ R-situations (i.e. situations that are in R, but not in C). That is, statements such as “John will succeed unless he doesn’t work hard” would be predicted to be false if there are possible situations outside of the contextually relevant ones in which John doesn’t work hard – situations in which, e.g., John dies in a freak accident. To see that the uniqueness clause is violated, take S to be the contextually relevant situations in which John doesn’t work hard (i.e. S = C∩R). This difficulty is easily remedied by rendering the uniqueness clause as:
∀S (Q [C – S] [M] → C∩R ⊆ S)
The adjustment is minor, and surely reflects von Fintel’s original intentions. Once we have made this adjustment, the two clauses are provably equivalent when the quantifier in question is a universal.
- 11.
I am indebted to John Hawthorne for bringing this phenomenon to my attention.
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Acknowledgments
Many thanks to John Hawthorne, Richard Larson and Paul Pietroski, Christopher Viger, and an anonymous reviewer for extremely helpful discussion and comments on an earlier draft of this paper. A much earlier version of the paper was presented in 2003 to the Australian National University Philosophical Society, and at the Carleton University Conference on Recent Research in Semantics, under the title of “Compositional Conditionals”. I am indebted to both audiences for their questions and comments.
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Leslie, SJ. (2009). “If”, “Unless”, and Quantification. In: Stainton, R.J., Viger, C. (eds) Compositionality, Context and Semantic Values. Studies in Linguistics and Philosophy, vol 85. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8310-5_1
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