Question I have trouble understanding what people mean when they use a phrase with the word exception. To me it sounds like a contradiction. So my question has two parts: A) Is using the term exception ever legitimate? B) Does the term "except" usually contradict the general rule that comes before it? For example, All ice cream should be taxed, except vanilla. This seems that the quantifier "all" is false if a member is excluded. For example, All students passed the final exam except Roy. Seems to me this means only Roy failed the final exam and the quantifier "all" makes the sentence false. Please help me make sense of the term exception. Thanks for your help.

Accepted:
July 22, 2010

Comments

I see what you're thinking: that in sentences such as:

(1) All teams lost except Spain

we give in one hand what we take with the other. We are affirming that all teams lost and also that Spain did not lose. You're right that this would indeed be a contradiction.

But I don't think the logical structure of such sentences is as you propose. The issue depends on what logicians call the relative scope of the terms "all" and "except". You understand (1) to mean:

(2) (all teams lost) and (Spain did not lose)

which is indeed a contradiction. Logicians would actually make a few changes to bring out more clearly the logical structure of (2):

(2') (each team is such that it lost) and (it is not the case that Spain lost)

Again, this is a contradiction.

But a more accurate analysis of how the sentence (1) is usually meant is this:

(3) all teams except Spain lost

where a more perspicuous representation of the logical structure of this is really:

(3') each team is such that if it's not Spain then it lost.

This is not a contradiction: it is false only if there is at least one team other than Spain that did not lose.

(2') is a conjunction, one of whose components is a universally quantified claim, "each team is such that it lost". By contrast, (3') is a universally quantified claim. (2') claims in part that each team has the property that it lost; by contrast, (3') claims that each team has the (more logically complex) property that if it's not identical to Spain then it lost.

Logicians say that in (2') the "each" has narrow scope, while it has wide scope in (3').

Hope this helps.