7.3.3. Any and every

We will conclude with some issues concerning the word any. It was noted in 3.1.3 that this word should be replaced (usually by some) when a sentence is analyzed as truth functional compound. Thus Tom didn’t see anything becomes ¬ Tom saw something and If anyone backs out the trip will be canceled becomes Someone will back out → the trip will be canceled. But sentences containing any can—in most cases—also be understood to state direct affirmative generalizations and can be analyzed using a universal quantifier as the main logical operation. When they are seen in this way, the truth-functional structure that appears to give the overall form of the sentence will be confined to the quantified predicate. Thus the examples above could be analyzed as follows:

These analyses are, for the time being at least, preferable to analyses as truth-functional compounds since we do not yet have a perspicuous way of analyzing quantifier phrases containing some.

The indefinite article a is interchangeable with any in many cases like these—e.g., Tom didn’t see a thing—so they constitute another sort of case (on top of those noted in 7.3.1) in which a may be used to state a generalization. (It’s also true that any is interchangeable with a in many cases like those noted in 7.3.1—e.g., Any dog likes bones.) But a cannot be used very successfully in place of any in the second example above. Something like If even one person backs out, the trip will be canceled does work, but that is comparable to replacing anyone by someone.

It would be grammatical to put every in place of any in the examples above; but the meaning would be quite different, and the new meaning could be captured only by an analysis as truth-functional compounds:

These two sets of examples can be generalized to a rule of thumb: in contexts where any and every convey a different meaning, the significance of any can be captured by a generalization having a scope wider than some other operator while the significance of every will be captured by generalization having a scope narrower than this operator. The contexts in the examples above, negations and the antecedents of conditionals, are the most common ones where any and every convey different meanings; but we will encounter another such context in the next section. Contexts like these (along with some others where the operators are not ones we will study) are the chief contexts in which any can be used grammatically. Thus any can serve to avoid a potential ambiguity in the relative scope of generalization and other operations.

When operators of the relevant sorts are stacked up, any tends to mark wider scope than only the one of them with narrowest scope. For example, on its most natural interpretation,

amounts to

so the generalization has a scope wider than the negation but narrower than the conditional. The statement made when the generalization has widest scope can be expressed using any, but it has a different form:

We will look at the phrase there is in 8.1. For now, it is enough to note that it permits us to use the relative clause that Tom didn’t find. This clause serves grammatically to give any wider scope than the negation; and, as a result, the ability of any to assume a scope wider than some operation is held in reserve for the conditional.

There are other cases where we cannot analyze a sentence containing any as a truth functional compound even if we replace any by some. For example, If Alex hears anything, he’ll tell us about it cannot be analyzed as a conditional because replacing the pronoun it by its antecedent would change the meaning; while it is not clear what claim is being made by If Alex hears anything, he’ll tell us about anything, it is clear that it differs in meaning from the original sentence—as does If Alex hears something, he’ll tell us about something. This means that we cannot get around the following analysis:

Notice that this form is the restatement using an unrestricted universal of the restricted universal quantification (∀x: Hax) Tasx. The latter symbolic form could turn up as the analysis of the sentence Alex will tell us about anything he hears, and this is a case where the word any cannot be replaced by some without changing the meaning (try it). In our original example, this replacement is possible (at least in colloquial speech), but it employs an exceptional use of some. The sentence we get—namely, If Alex hears something, he’ll tell us about it—is used to state a generalization, not to claim the existence of an example.