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Logic, Meaning, and ConversationSemantical Underdeterminacy, Implicature, and their Interface$

Jay David Atlas

Print publication date: 2005

Print ISBN-13: 9780195133004

Published to Oxford Scholarship Online: October 2011

DOI: 10.1093/acprof:oso/9780195133004.001.0001

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(p.231) Appendix 2 On Hitzeman (1992) on ‘Almost’

(p.231) Appendix 2 On Hitzeman (1992) on ‘Almost’

Logic, Meaning, and Conversation
Oxford University Press

I offer two arguments reductio ad absurdum to show that x almost F’d does not entail x did not F. Suppose for adjectives, determiners, or verbs F, the statement schema A(almost F) entails the schema A(not F). (a) Then Almost all swans are almost white entails Almost all swans are not white, which entails Not all swans are not white, which entails Some swans are white. But the proposition Some swans are white is just what a speaker who asserts Almost all swans are almost white chooses the word almost to avoid conveying, (b) It is intuitively evident that if there were no white swans, it could still be true that almost all swans were almost white. However, if A(almost F) entailed A(not F), and there were no white swans, it would be false that almost all swans were almost white, as we have just seen in (a). Conclusion: A(almost F) does not entail A(not F). The simple entailment explanation of the ordering of almost and not quite is the wrong explanation. What is the relationship between almost and not quite?

Hitzeman (1992) believes that the transition in (a) from Almost all swans are almost white to the entailed Almost all swans are not white requires that I assume (incorrectly in her view) that almost all swans is an upward-entailing (i.e., upward monotonic in the sense of J. Barwise and R. Cooper 1981) generalized quantifier noun phrase. Actually, what I imputed to the entailment theorist was the claim that x is almost white analytically entails x is not white, for free x, or, equivalently (in Classical but not Intuitionistic logic), that ∀x(x is almost white) analytically entails ∀x(x is not white). On what I take to be Hitzeman’s assumption—viz., that {x: x is almost white} ⊆ {x: x is not white}—her claim, that the hypothesis that almost all swans is upward-entailing is a necessary condition for the correctness of my (a), is false (even (p.232) though, on her, admittedly plausible, assumption about the subset relation between the extensions of the predicates ‘x is almost white’ and ‘x is not white’, the hypothesis is a sufficient condition for my claim).

For example, for the non-monotonic generalized quantifier Only Tom (Atlas 1996b), the entailment theorist about almost would certainly be committed to the claim that Only Tom was almost drunk entails Only Tom was not drunk. The status of this claim as an entailment claim is unaffected by the fact that only Tom is not an upward-entailing generalized quantifier. In fact, on the traditional, and incorrect, view that only Tom is a downward monotonic generalized quantifier, the traditional entailment theorist would be caught in a logical cleft stick: he would want to claim that ‘x is almost drunk’ analytically entails ‘x is not drunk’, and so necessarily {x: x is almost drunk} ⊆ {x: x is not drunk}, and he would want to claim that the substitution of the superset Not Drunk for its set Almost Drunk in an allegedly downward-entailing quantifier only Tom necessarily preserves truth—which is an absurd conjunction of claims. So the theorist cannot both be traditional about the downward monotonicity of only Tom and simultaneously hold that almost F entails not F. (This result, alone, is worth the price of admission, since both views are standardly held, and they are logically inconsistent. No semantic theorist can consistently hold that only Tom is downward monotonic and that almost F entails not F.) The solution to the traditional difficulty is that neither part of the traditional position is correct: only Tom is non-monotonic, and almost F does not entail not F. Hitzeman’s argument, which confuses a sufficient with a necessary condition, does not undermine my (a).

Since Hitzeman (1992) thinks that my reductio argument requires the assumption that almost all N is upward monotonic, she attempts to defang my argument by arguing that almost all N is not upward monotonic. She offers an alleged counterexample to the plausible, upward monotonicity of almost all N, plausible even to Hitzeman because the following argument (A) seems valid to her: (A) Almost all dogs run, Every individual that runs movesAlmost all dogs move. Her alleged counterexample is the alleged invalidity of the argument: (B) Almost all men are fathers, Every individual that is a (biological) father is maleAlmost all men are male. But how one could think that the premises of this argument (B) might be true while simultaneously its conclusion not true is quite beyond me. If it is true that almost all men are fathers, and that every individual that is a (biological) father is male, then it is surely true that almost all men are male, even though it is also true that all (biological, non-trans sexual) men are male. So what is Hitzeman’s objection to argument (B)—that is, to the upward monontonicity of almost all N? She asserts that the concluding sentence Almost all men are male is “strange,” but not “strange” in the way she thinks that the analytical truth All men are male is “strange.” She thinks the former is “strange” because it (allegedly) entails Some man is not male!

But for her to argue in this way against the validity of argument (B) fails for two different reasons: first, because the issue is not strangeness of the concluding sentence of (B) but its truth in any model in which the premises are true and, second, because even if “strange” meant ‘false’, her argument overtly begs the question against my view that almost does not entail not. She argues: almost entails not; so Almost all men are male is “strange,” thus necessarily false; so the (alleged) invalidity of argument (B) shows that Atlas cannot assume that Almost all N is an upward monotonic (p.233) quantifier. This argument just blatantly begs the question against my position that almost does not entail not. And, as I argued above, she (falsely) thinks that upward monotonicity is necessary (by contrast with sufficient) for the first claim of my reductio argument against the entailment of not by almost to be sound.

Hitzeman (1992: 236) has been kind enough to say that she “believe[s] that the most interesting argument against the entailment hypothesis is…due to Atlas (1984a)”—that is, the argument in question here. Since the semantically most sophisticated attack on the pragmatic position of Sadock (1981) and me (Atlas 1984a) that I am defending here is Hitzeman’s (1992), involving as it does an attempt to defeat my argument by arguing (incorrectly) for the non-monotonicity of almost all N, I find it a highly instructive argument.

According to criteria for upward monotonic generalized quantifier noun phrases analogous to the ones discussed in Zwarts (1996, 1998) and Atlas (1996b: 282) for downward monotonic generalized noun phrases, we have the following obvious criteria for upward monotonicity of the NP:

  1. a NP(VP 1 and VP 2) ⊩ NPVP 1 & NPVP 2

  2. b NP(VP 1 and VP 2) ⊩ NPVP 2 v NPVP 2

  3. c NPVP 1 v NPVP 2NP(VP 1 or VP 2)

  4. d NP(VP 1 and VP 2)⊩NPVP 1

  5. e NP(VP 1)⊩NP(VP 1 or VP 2)

where and entails the Boolean intersection interpretation of the extensions of the Verb Phrases, and or means nothing more restrictive than the Boolean union of the extensions of the verb phrases. These criteria (a) to (e) have obviously correct instantiations for almost all N, for example, (a) Almost all college boys smoke and drinkAlmost all college boys smoke & Almost all college boys drink. Thus the evidence is powerful that almost all N is upward monotonic, contrary to Hitzeman’s (1992) claim and consistent with my argument against almost F entailing not F.